Download Nonmonotonic Reasoning

Nonmonotonic Reasoning
Alexander Bochman
Computer Science Department,
Holon Institute of Technology (H.I.T.)
November 24, 2011
1
What is nonmonotonic reasoning
The field of nonmonotonic reasoning is now an essential part of the logical approach to Artificial Intelligence (AI). There exists a vast literature on the topic,
including a number of books [Antoniou, 1997; Besnard, 1989; Bochman, 2001;
2005; Brewka, 1991; Lukaszewicz, 1990; Makinson, 2005; Marek and Truszczyński,
1993; Schlechta, 1997; 2004]. Two collections are especially useful: [Ginsberg, 1987] is a primary source for the early history of the subject, while the
handbook [Gabbay et al., 1994] provides overviews of important topics and
approaches. [Minker, 2000] is a most recent collection of survey papers and
original contributions to logic-based AI. This chapter has also benefited from
a number of overviews of the field, especially [Reiter, 1987a; Minker, 1993;
Brewka et al., 1997; Thomason, 2003].
The relationship between nonmonotonic reasoning and logic is part of a
larger story of the relations between AI and logic (see [Thomason, 2003]). John
McCarthy, one of the founders of AI, has suggested in [McCarthy, 1959] and
consistently developed a research methodology that used logic to formalize the
reasoning problems in AI1 . McCarthy’s objective was to formalize common sense
reasoning used in dealing with everyday problems.
In a sense, nonmonotonic reasoning is an outgrowth of McCarthy’s program.
But though commonsense reasoning has always appeared to be an attractive
standard, the study of ‘artificial reasoning’ need not and actually has not been
committed to the latter. The basic formalisms of nonmonotonic reasoning could
hardly be called formalizations of commonsense reasoning. Still, in trying to
cope with principal commonsense reasoning tasks, the suggested formalisms
have succeeded in capturing important features of the latter and thereby have
broken new territory for logic.
Artificial Intelligence has practical purposes, which give rise to problems
and solutions of a new kind, apparently different from the questions relevant for
philosophers. The authors of first nonmonotonic theories have tried, of course, to
express their formalisms using available logical means, ranging from the classical
1 See
[Lifschitz, 1991a] for an overview of McCarthy’s research program.
1
first order language to modal logics. McCarthy himself has always believed
that anything that can be expressed, can be expressed in first order logic (he
considered this a kind of Turing thesis for logic). Still, from its very beginning,
logical AI has created formalisms and approaches that had no counterpart in
existing logical theories, and refined them to sophisticated logical systems. This
was achieved mostly by people that were not logicians as their primary speciality.
It is even advantageous to see nonmonotonic reasoning as a brand new approach
to logical reasoning; this would save us from hasty attempts to subsume such a
reasoning in existing logical formalisms at the price of losing the precious new
content. Already at this stage of its development, nonmonotonic reasoning is
not yet another application of logic, but a relatively independent field of logical
research that has a great potential in informing, in turn, future logical theory
as well as many areas of philosophical inquiry.
The origins of nonmonotonic reasoning within the broad area of logical AI
lied in dissatisfaction with the traditional logical methods in representing and
handling the problems posed by AI. Basically, the problem was that reasoning
necessary for an intelligent behavior and decision making in realistic situations
has turned out to be difficult, even impossible, to represent as deductive inferences in some logical system.
In commonsense reasoning, we usually have just partial information about
a given situation, and we make a lot of assumptions about how things normally
are in order to carry out further reasoning. For example, if we learn that Tweety
is a bird, we usually assume that it can fly. Without such presumptions, it would
be almost impossible to carry out the simplest commonsense reasoning tasks.
Speaking generally, human reasoning is not reducible to collecting facts and deriving their consequences; it embodies an active epistemic attitude that involves
making assumptions and wholesale theories about the world and acting in accordance with them. We do not only perceive the world, we also give it structure
in order to make it intelligible and controllable. Commonsense reasoning in this
sense is just a rudimentary form of a general scientific methodology.
The way of thinking in partially known circumstances suggested by nonmonotonic reasoning consists in using justified beliefs and reasonable assumptions that can guide us in our decisions. Accordingly, nonmonotonic reasoning
can be described as a theory of making and revising assumptions in a reasoned
or principled way ([Doyle, 1994]). Of course, the latter are only beliefs and
assumptions, so they should be abandoned when we learn new facts about the
circumstances that contradict them.
The sentence “Birds (normally) fly” is weaker than “All birds fly”; there is
a seemingly open-ended list of exceptions – ostriches, penguins, Peking ducks,
etc. etc. So, if we would try to use classical logic for representing “Birds fly”,
the first problem would be that it is practically impossible to enumerate all
exceptions to flight with an axiom of the form
(∀x).Bird(x)&¬P enguin(x)&¬Emu(x)&¬Dead(x)&... ⊃ F ly(x)
This fact indicates that our commonsense assumptions are often global in
character, saying something like “the world is as normal as possible, given the
2
known facts”. The second crucial problem is that, even if we could enumerate
all such exceptions, we still could not derive F ly(T weety) from Bird(T weety)
alone. This is so since we are not given that Tweety is not a penguin, or dead,
etc. The antecedent of the above implication cannot be derived, in which case
there is no way of deriving the consequent. Nevertheless, if told only about a
particular bird, say Tweety, without being told anything else about it, we would
be justified in assuming that Tweety can fly, without knowing that it is not one
of the exceptional birds. So the problem is how we can actually make such
assumptions in the absence of information to the contrary.
This suppositional character of commonsense reasoning conflicts with the
monotonic character of logical derivations. Monotonicity is just a characteristic
property of deductive inferences arising from the very notion of a proof being
a sequence of steps starting with accepted axioms and proceeding by inference
rules that remain valid in any context of its use. Consequently, if a set a of
formulas implies a consequence C, then a larger set a ∪ {A} will also imply C.
Commonsense reasoning is non-monotonic in this sense, because adding new
facts may invalidate some of the assumptions made earlier.
In his influential “frames paper” [Minsky, 1974], Marvin Minsky proposed
the notion of a frame, a complex data structure meant to represent a stereotyped and default information. While Minsky’s description of a frame was
informal, central to his notion were prototypes, default assumptions, and the
unsuitability of classical definitions for commonsense concepts. In the appendix
entitled “Criticism of the Logistic Approach”, Minsky explained why he thinks
that logical approaches will not work. To begin with, he directly questioned
the suitability of representing commonsense knowledge in a form of a deductive
system:
There have been serious attempts, from as far back as Aristotle, to
represent common sense reasoning by a ”logistic” system .... No
one has been able successfully to confront such a system with a
realistically large set of propositions. I think such attempts will
continue to fail, because of the character of logistic in general rather
than from defects of particular formalisms.
Minsky doubted the feasibility of representing ordinary knowledge effectively
in the form of many small, independently true propositions. On his opinion, such
a “logical” reasoning is not flexible enough to serve as a basis for thinking.
The strategy of complete separation of specific knowledge from general rules of inference is much too radical. We need more direct ways
for linking fragments of knowledge to advice about how they are to
be used.
As a result of this deficiency, traditional formal logic cannot discuss what
ought to be deduced under ordinary circumstances. Minsky was also one of the
first who mentioned monotonicity as a source of the problem:
MONOTONICITY: ... In any logistic system, all the axioms are
necessarily “permissive” - they all help to permit new inferences
3
to be drawn. Each added axiom means more theorems, none can
disappear. There simply is no direct way to add information to
tell such the system about kinds of conclusions that should not be
drawn! To put it simply: if we adopt enough axioms to deduce what
we need, we deduce far too many other things.
As yet another problematic feature, Minsky mentioned the requirement of
consistency demanded by Logic that makes the corresponding systems too weak:
I cannot state strongly enough my conviction that the preoccupation with Consistency, so valuable for Mathematical Logic, has been
incredibly destructive to those working on models of mind. At the
popular level it has produced a weird conception of the potential capabilities of machines in general. At the “logical” level it has blocked
efforts to represent ordinary knowledge, by presenting an unreachable image of a corpus of context-free “truths” that can stand separately by themselves.
First theories of nonmonotonic reasoning could be viewed as providing logical
answers to Minsky’s challenge.
2
Pre-history: problems and first solutions
Long before the emergence of first nonmonotonic systems, there have been a
number of problems and applications in AI that required and used some forms of
nonmonotonic reasoning. In fact, it is these problems and solutions, rather than
the strategic considerations of McCarthy or Minsky, that influenced the actual
shape of subsequent nonmonotonic formalisms. Initial solutions to commonsense
reasoning tasks worked (though in restricted applications), and this was an
incentive for trying to provide them with a more systematic logical basis.
On a most general level, nonmonotonic or default reasoning is intimately
connected to the notion of prototypes in psychology and natural kinds in philosophy. Just as the latter, default assumptions cannot be defined via necessary
and sufficient conditions, but involve a description of “typical” members. The
problem of representing and reasoning with such concepts has reappeared in
AI as a practical problem of building taxonomic hierarchies for large knowledge
bases. The basic reasoning principle in such hierarchies is that subclasses inherit
properties from their super-classes.
Much more complex logical issues have arisen when the organization of
a domain into hierarchies has been allowed to have exceptions. The theory
of reasoning in such taxonomies has been called nonmonotonic inheritance
(see [Horty, 1994] for an overview). The guiding principle in resolving potential conflicts in such hierarchies was a specificity principle ([Poole, 1985;
Touretzky, 1986]): more specific information should override more generic information in cases of conflict. Though obviously related to nonmonotonic reasoning, nonmonotonic inheritance relied more heavily on graph-based represen-
4
tations than on traditional logical tools (see, e.g., [Touretzky, 1986]). Nevertheless, it has managed to provide a plausible analysis of reasoning in this restricted
context. The relations of nonmonotonic inheritance to general nonmonotonic
formalisms, and especially the role of the specificity principle have been an active
area of research.
Default assumptions of a different kind have been ‘discovered’ in the framework of already existing systems, such as reasoning in databases and planning
domains, and even in formulation of ordinary problems, or puzzles. A common
assumption or, better, convention in such systems and problem formulations has
been that positive assertions that are not explicitly stated should be considered
false. Attempts to formalize this assumption have led to Reiter’s Closed World
Assumption principle and McCarthy’s circumscription (see below).
But first and foremost, the problem of default assumptions has shown itself
in attempts to represent reasoning about actions and change.
2.1
The frame problem
This, then, are the three problems in formalizing
action: [the qualification, frame, and ramification
problem]. Other than that, no worries. [Ginsberg, 1993]
It is difficult to overestimate the importance of the frame problem for AI. It
is central to virtually every interesting area of AI, such as planning, explanation,
and diagnosis. As such, solving the frame problem is necessary for the whole
logical approach to AI.
As was stated in [McCarthy and Hayes, 1969], a computer program capable
of acting intelligently in the world must have a general representation of the
world in terms of which its inputs are interpreted. It should decide what to do
by inferring in a formal language that a certain strategy will achieve its assigned
goal. This paper introduced also the main problem that prevented an adequate
formalization of this task - the Frame Problem. Basically, the problem was how
efficiently determine which things remain the same in a changing world.
As was rightly noted in [Thomason, 2003], the frame problem arises in the
context of predictive reasoning, a type of reasoning that has been neglected in the
traditional tense-logical literature, but is essential for planning and formalizing
intelligent behavior. Prediction involves the inference of later states from earlier
ones. Changes in this setting do not merely occur, but occur for a reason.
Furthermore, we usually assume that most things will be unchanged by the
performance of an action. It is this inertia assumption that connects reasoning
about action and change with nonmonotonic reasoning. In this reformulation,
the frame problem is to determine what stays the same about the world as time
passes and actions are performed – without having to explicitly state each time
all the things that stay the same ([Morgenstern, 1996]).
The frame problem has a number of formulations and levels of generalization.
The initial description of the problem in [McCarthy and Hayes, 1969] was made
in the specific context of a situation calculus, an instance of first-order logic
5
especially formulated for reasoning about action. In this model, changes are
produced by actions, so the basic relation is Result between an action, an initial
situation, and a situation resulting from the performance of the action. In order
to specify how propositions (fluents) do not change as actions occur (e.g., a red
block remains red after we have put it on top of another block), the authors
suggested to write down special axioms they called frame axioms. Of course, a
huge number of things stay the same after a particular action, so we would have
to add a very large number of frame axioms to the theory. This is precisely the
frame problem. More generally, it is the persistence problem [Shoham, 1988]:
the general problem of predicting the properties that remain the same as actions
are performed, within any reasonable formalism for reasoning about time.
On a more general level, the frame problem has been understood as a temporal projection problem encompassing the persistence, ramification and qualification problems, as well as the problem of backward temporal projection, or
retrodiction.
The ramification problem was formulated first in [Finger, 1987] and concerns
the necessity of taking into account numerous derived effects (ramifications) of
actions, effects created by logical and causal properties of a situation. Taking
an example from [Lin, 1995], suppose that a certain suitcase has two locks, and
is open if both locks are open. Then the action of opening one lock produces
an indirect effect of opening the suitcase if and only if the other lock is open.
Derived effects should be taken into account when combined with the above
mentioned inertia assumption, since they override the latter. Thus, if an action
of opening one lock is performed, its derived effect overrides the default inertia
assumption that the suitcase remains closed.
The ramification problem has raised general questions on the nature of causation and its role in temporal reasoning. It was an incentive for the causal
approach to the frame problem, described later.
The qualification problem is the problem of specifying what conditions must
be true in the world for a given action to have its intended effect. It was
introduced in [McCarthy, 1980] and described as follows:
It seemed that in order to fully represent the conditions for the
successful performance of an action, an impractical and implausible
number of qualifications would have to be included in the sentences
expressing them. Common sense reasoning is ordinarily ready to
jump to the conclusion that a tool can be used for its intended purpose unless something prevents its use. Considered purely extensionally, such a statement conveys no information; it seems merely
to assert that a tool can be used for its intended purpose unless it
can’t. Heuristically, the statement is not just a tautologous disjunction; it suggests forming a plan to use the tool.
If I turn the ignition key in my car, I expect the car to start. However, many
conditions have to be true in order for this statement to be true: The battery
must be alive, the starter must work, there must be gas in the tank, there is no
potato in the tailpipe, etc. - an open-ended list of qualifications. Still, without
6
knowing for certain about most of these facts, I normally assume that turning
the key will start the car.
The Qualification Problem has turned out to be one of the most stubborn
problems for the representation of action and change. The majority of subsequent nonmonotonic formalisms, such as default logic, have failed to deliver the
intended conclusions due to the fact that they do not provide natural ways of
formalizing the specificity principle, mentioned earlier in discussing nonmonotonic inheritance, according to which more specific defaults should override more
general defaults.
The idea behind first nonmonotonic solutions to the frame problem has been
to treat inertia as a default: changes are assumed to occur only if there is some
reason for them to occur. In action-based formalisms, the absence of change is
inferred when an action is performed unless a reason for the change can be found
in axioms for the action. Attempts to formalize such a reasoning obviously required a nonmonotonic reasoning system. In one of the earliest attempts to formalize such a reasoning, [Sandewall, 1972] used a modal operator U N LESS A
meaning “A cannot be proved” for dealing with the frame problem, namely
for expressing the inertia claim that every action leaves any fluent unaffected,
unless it is possible to deduce otherwise. One of the principal motivations of
McCarthy and Reiter for the study of nonmonotonic reasoning was the belief
that it would provide a solution to the frame problem.
The frame problem has ‘survived’, however, the first generation of nonmonotonic formalisms. [Hanks and McDermott, 1987] have suggested the Yale Shooting Anomaly and demonstrated that the apparently plausible nonmonotonic approaches to the frame problem fail. Actually, it was a major factor in conversion
of McDermott, one of the founders of nonmonotonic reasoning, to an antilogicist
(see [McDermott, 1987]).
The Yale Shooting Anomaly has shown, in effect, that a simple-minded
combination of default assumptions measured only by the number of defaults
that are violated can lead to spurious solutions. Still, the development of the
nonmonotonic approach to the frame problem has continued and led to new,
more adequate solutions (see, e.g., [Morgenstern and Stein, 1994; Sandewall,
1994; Shanahan, 1997]). In addition, monotonic solutions to the frame problem
has been suggested in [Schubert, 1990; Reiter, 1991] and successfully applied to
quite complex formalization problems. These solutions were based, however, on
writing explicit frame axioms stating what does not change when an action is
performed.
2.2
Procedural nonmonotonicity
Procedural solutions to the frame problem have been popular in AI since its earliest days. Perhaps the best known is the planning program STRIPS suggested
by Fikes and Nilsson in 1971. Given an initial state, a goal state, and a list of
actions, STRIPS finds a sequence of actions that achieves the goal state. In the
course of planning, however, it must reason about what changes and what stays
the same after an action is performed. STRIPS avoids (and thereby solves)
7
the frame problem by assuming that if an action is not known to change some
feature, it does not. In a setting devoid of deductive inferences, this principle
is easy to represent procedurally by associating with each action a list of preconditions that must be satisfied in order for the action to be performed, along
with an add list and a delete list. The add list is the set of statements that gets
added to the current state after an action is performed, and the delete list is
the set of statements that gets deleted from the current state after the action is
performed. By its very nature, STRIPS cannot handle conditional actions, and
worked only for limited ontologies devoid of causation and other constraints on
combination of propositional atoms.
Also in other areas of AI researchers have routinely been implementing procedural nonmonotonic reasoning systems, usually without reflecting on the underlying reasoning patterns on which their programs rely. Typically these patterns
were implemented using the so-called negation-as-failure, which occurs as an
explicit operator in programming languages like PROLOG.
In the database theory there is an explicit convention about the representation of negative information that provided a specific instance of nonmonotonic reasoning. For example, the database for an airline flight schedule does
not include the city pairs that are not connected, which clearly would be an
overwhelming amount of information. Instead of explicitly representing such
negative information, databases implicitly do so by appealing to the so-called
closed word assumption (CWA) [Reiter, 1978], which states that if a positive
fact is not explicitly present in the database, its negation is assumed to hold.
For simple databases consisting of atomic facts only, e.g. relational databases,
this approach to negative information is straightforward. In the case of deductive databases, however, it is no longer sufficient that a fact not be explicitly
present in order to conjecture its negation; the fact may be derivable. For this
case, Reiter defined the closure of a database as follows:
CW A(DB) = DB ∪ {¬P (t) | DB 2 P (t)}
where P(t) is a ground predicate instance. That is, if a ground atom cannot
be inferred from the database, its negation is added to the closure. Under the
CWA, queries are evaluated with respect to CWA(DB), rather than DB itself.
Logically speaking, the CWA singles out the least model of a database.
Consequently, it works only when the database possesses such a least model,
e.g., for Horn databases. Otherwise it becomes inconsistent. For example, for
a database containing just the disjunctive statement A ∨ B, neither A nor B
is deduced, so both ¬A and ¬B are in the closure, which is then inconsistent
with the original database. A suitable generalization of CWA for arbitrary
databases, the Generalized Closed World Assumption, has been suggested in
[Minker, 1982].
The Prolog programming language developed by Colmerauer and his students [Colmerauer et al., 1973] and the PLANNER language developed by Hewitt ([Hewitt, 1969]) were the first languages to have a nonmonotonic component. The not operator in Prolog, and the THNOT capability in PLANNER
8
provided default rules for answering questions about data where the facts did
not appear explicitly in the program.
In Prolog, the goal not G succeeds if the attempt to find a proof of G using
the Prolog program as axioms fails. Thus, Prologs negation is a nonmonotonic
operator: if G is nonprovable from some axioms, it needn’t remain nonprovable
from an enlarged axiom set. The way this procedural negation is actually used
in AI programs amounts to invoking the rule of inference “From failure of G,
infer ¬G.” This is really the closed world assumption. This procedural negation
can also be used to implement other forms of default reasoning; this has led to
developing a modern logic programming as a general representation formalism
for nonmonotonic reasoning.
A different formalization of the CWA was proposed in [Clark, 1978] in an
attempt to give a formal semantics for negation in Prolog. Clark’s idea was
that Prolog clauses provide sufficient but not necessary conditions on the predicates in their heads, while the CWA is the assumption that these sufficient
conditions are also necessary. Accordingly, for a propositional logic program Π
consisting of rules of the form p ← a, queries should be evaluated with respect
to its completion, which is a classical logical theory consisting of the following
equivalences:
_
p ↔ {∧ai | p ← ai ∈ Π},
for any ground atom p. The completion formulas embody two kinds of information. As implications from right to left, they contain the material implications
corresponding to the program rules. In addition, left-to-right implications state
that an atom belongs to the model only if one of its justifications is also in the
model. The latter justification requirement has become a central part of the
truth maintenance system, suggested by Jon Doyle.
2.3
Justification-based truth maintenance
Doyle’s truth maintenance system (TMS) can be viewed as one of the first
rigorous solutions to the problem of representing nonmonotonic reasoning (see
[Doyle, 1979]). In particular, it introduced the now familiar notion of nonmonotonic justification, subsequently used in default and modal nonmonotonic
logics.
The idea of the TMS was to keep track of the support of beliefs, and to use
the record of these support dependencies when it is necessary to revise beliefs.
In a TMS, part of the support for a belief can consist in the absence of other
beliefs. This introduced nonmonotonicity.
The TMS represented belief states by structures called nodes that the TMS
labeled as either in or out (of the current state). The TMS also recorded sets
of justifications or reasons for each node in the form of rules A//B c read
as “A without B gives c”, meaning that the node c should be in if each node
in the set A is in and each node in the set B is out. The TMS then seek to
construct labelings for the nodes from these justifications, labelings that satisfy
two principles:
9
• stability - a node is labeled in iff one of its reasons is valid in the labeling
(i.e., expresses hypotheses “A without B” that match the labeling);
• groundedness - labelings provide each node labeled in with a noncircular
argument in terms of valid reasons.
The TMS algorithm and its refinements had a significant impact on AI
applications, and called for a logical analysis. It provided a natural and highly
specific challenge for those seeking to develop a nonmonotonic logic. In fact,
both the modal nonmonotonic logic of [McDermott and Doyle, 1980] and the
default logic of [Reiter, 1980] can be seen as logical formalizations of the above
principles. Namely, each of these theories formalized nonmonotonic reasoning
by encoding groundedness and the presence and absence of knowledge in terms
of logical provability and unprovability (consistency).
It is interesting to mention, however, that Jon Doyle himself has always felt
a discrepancy between his original formulation and subsequent logical formalizations:
In the first place, the logical formalizations convert what in many
systems is a fast and computationally trivial check for presence and
absence of attitudes into a computationally difficult or impossible
check for provability, unprovability, consistency or inconsistency.
This inaptness seems especially galling in light of the initial problemsolving motivations for nonmonotonic assumptions, for which assumptions served to speed inference, not to slow it. ([Doyle, 1994])
One of the most important features of Doyle’s system was its emphasis on
justifications and the role of argumentation in constructing proper labelings.
This theme has been developed in subsequent theories.
3
Coming of age
Nonmonotonic reasoning obtained its impetus in 1980 with the publication of
a seminal issue of the Artificial Intelligence Journal, devoted to nonmonotonic
reasoning. The issue included papers representing three basic approaches to
nonmonotonic reasoning: circumscription [McCarthy, 1980], default logic [Reiter, 1980], and modal nonmonotonic logic [McDermott and Doyle, 1980]. These
theories suggested three different ways of meeting Minsky’s challenge by developing formalisms that do not have the monotonicity property.
On the face of it, the three approaches were indeed different, beginning with
the fact that they were based on three altogether different languages – the classical first order language in the case of circumscription, a set of inference rules
in default logic, and modal language in modal nonmonotonic logic. Still, behind these differences there was a common idea. The idea was that default
conditionals or inference rules used in commonsense derivations can be represented, respectively, as ordinary conditionals or inference rules by augmenting
their premises with additional assumptions, assumptions that could readily be
10
accepted in the absence of contrary information. In this respect, the differences
between the three theories amounted to different mechanisms of making such
default assumptions.
3.1
Circumscription
McCarthy’s circumscription was based on classical logic, and focused in a large
part on representation techniques. He stressed that circumscription is not a
“nonmonotonic logic”, but a form of nonmonotonic reasoning augmenting ordinary first order logic.
The first paper [McCarthy, 1980] connected the strategic ideas of [McCarthy
and Hayes, 1969] with the need for nonmonotonic reasoning, and described a
simplest kind of circumscription, namely domain circumscription. The second
paper [McCarthy, 1986] provided more thorough logical foundations, and introduced the more general and powerful predicate circumscription approach.
As a first description in [McCarthy, 1980], McCarthy characterized circumscription as a kind of conjectural reasoning by which humans and intelligent
computer programs jump to the conclusion that the objects they can determine
to have certain properties are the only objects that do.
The result of applying circumscription to a collection A of facts
is a sentence schema that asserts that the only tuples satisfying a
predicate are those whose doing so follows from the sentences of
A. Since adding more sentences to A might make P applicable to
more tuples, circumscription is not monotonic. Conclusions derived
from circumscription are conjectures that A includes all the relevant
facts and that the objects whose existence follows from A are all the
relevant objects.
Thus, circumscription is a tool of making conjectures. Conjectures may be
regarded as expressions of probabilistic notions such as “most birds can fly” or
they may be expressions of standard, or normal, cases. Such conjectures sometimes conflict, but there is nothing wrong with having incompatible conjectures
on hand. Besides the possibility of deciding that one is correct and the other
wrong, it is possible to use one for generating possible exceptions to the other.
Although circumscription was originally presented as a schema for adding
more formulas to a theory, just as Reiter’s CWA or Clark’s completion, it can
also be described semantically in terms of restricting the models of the theory
to those that have minimal extensions of (some of) the predicates and functions. Let P be a set of predicate symbols that we are interested in minimizing
and Z another set of predicate symbols that are allowed to vary across compared models. Predicates other than P and Z are called the fixed symbols. Let
A(P ; Z) be a first-order sentence containing the symbols P and Z. A (parallel
predicate) circumscription chooses models of A(P ; Z) that are minimal in the
extension of predicates P, assuming that these models have the same interpretation for all symbols not in P or Z. This characterization can be concisely written
11
as a second-order formula. More detailed description of circumscription in its
different forms can be found in [Lifschitz, 1994b].
The importance of predicates and functions that vary across compared interpretations has been recognized in [Etherington et al., 1985]. Without these,
it would be impossible to infer new positive instances of any predicates from
the preferred models. Accordingly, the general problem with using circumscription in applications amounted to specifying the circumscription policy: which
predicates should be varied, which should be minimized, and with what priority.
Widely varying results occur depending on these choices. In fact, as was noted
already in [McCarthy, 1980], the results of circumscription depend on the very
set of predicates used to express the facts, so the choice of representation has
epistemological consequences in making conjectures by circumscription.
The above models of circumscription can also be described as preferred models with respect to an appropriate preorder on all models. This view has later
led to the generalization of circumscription to a general preferential approach
to nonmonotonic reasoning.
3.1.1
Abnormality theories
It is somewhat misleading to reduce the essence of circumscription to minimization. Viewed as a formalism for nonmonotonic reasoning, the central concept of
McCarthy’s circumscriptive method is an abnormality theory - a set of conditionals containing the abnormality predicate ab that provides a representation
for default information. [McCarthy, 1980] justified the need in introducing such
auxiliary predicates as follows:
When we circumscribe the first order logic statement of the [missionaries and cannibals] problem together with the common sense facts
about boats etc., we will be able to conclude that there is no bridge
or helicopter. “Aha”, you say, “but there won’t be any oars either”.
No, we get out of that as follows: It is a part of common knowledge
that a boat can be used to cross a river unless there is something
wrong with it or something else prevents using it, and if our facts
don’t require that there be something that prevents crossing the
river, circumscription will generate the conjecture that there isn’t.
The price is introducing as entities in our language the “somethings”
that may prevent the use of the boat...
Using circumscription requires that common sense knowledge be expressed in a form that says a boat can be used to cross rivers unless there is something that prevents its use. In particular, it looks
like we must introduce into our ontology (the things that exist) a
category that includes something wrong with a boat or a category
that includes something that may prevent its use. Incidentally, once
we have decided to admit something wrong with the boat, we are
inclined to admit a lack of oars as such a something and to ask
questions like, “Is a lack of oars all that is wrong with the boat?”.
12
Some philosophers and scientists may be reluctant to introduce such
things; but since ordinary language allows “something wrong with
the boat” we shouldn’t be hasty in excluding it. Making a suitable
formalism is likely to be technically difficult as well as philosophically
problematical, but we must try. We challenge anyone who thinks he
can avoid such entities to express in his favorite formalism, “Besides
leakiness, there is something else wrong with the boat”.
In [McCarthy, 1986], McCarthy proposed a uniform principle for representing
default claims in circumscription.
It turns out that many common sense facts can be formalized in a
uniform way. A single predicate ab, standing for “abnormal” is circumscribed with certain other predicates and functions considered
as variables that can be constrained to achieve the circumscription
subject to the axioms. This also seems to cover the use of circumscription to represent default rules.
Many people have proposed representing facts about what is “normally” the case. One problem is that every object is abnormal in
some way, and we want to allow some aspects of the object to be
abnormal and still assume the normality of the rest. We do this
with a predicate ab standing for “abnormal”. We circumscribe ab z.
The argument of ab will be some aspect of the entities involved.
Some aspects can be abnormal without affecting others. The aspects themselves are abstract entities, and their unintuitiveness is
somewhat a blemish on the theory.
For example, to say that normally birds fly, we can use
∀x : Bird(x) ∧ ¬ab aspect1(x) ⊃ F ly(x).
Here the meaning of ab aspect1(x) is something like “x is abnormal with respect to flying birds”. There can be many different aspects of abnormality, and
they are indexed according to kind. The circumscription would then minimize
abnormalities, allowing relevant predicates to vary (e.g., F ly).
An important advantage of representation with abnormality predicates is
that, by asserting that a certain object is abnormal in some respect, we can
block, or defeat, an associated default rule without asserting that its consequent
is false. For example, by asserting ab aspect1(T weety), we block the application
of the default “Birds fly” to Tweety without asserting that it cannot fly. This
feature has turned out to be useful in many applications.
Abnormality theories have been widely used both in applications of circumscription, and in other theories. Some major examples are inheritance theories
[Etherington and Reiter, 1983], logic-based diagnosis [Reiter, 1987b], naming
defaults in the abductive approach of [Poole, 1988a], general representation of
defaults in [Konolige and Myers, 1989] and reasoning about time and action.
Abnormality theories have brought out, however, several problems in the
application of circumscription to commonsense reasoning. One of the most
13
pressing was the already mentioned specificity problem arising when there are
conflicting defaults. In combining two defaults, “Birds fly” and “Penguins can’t
fly”, the specificity principle naturally suggests that the second, more specific,
default should be preferred. A general approach to handle this problem in circumscription, suggested in [Lifschitz, 1985] and endorsed in [McCarthy, 1986],
was to impose priorities among minimized predicates and abnormalities. The
corresponding variant of circumscription has been called prioritized circumscription. [Grosof, 1991] has generalized prioritized circumscription to a partial order
of priorities.
3.1.2
Further developments and applications
[Lifschitz, 1985] described the concept of parallel circumscription and also treated
prioritized circumscription. He addressed also the problem of computing circumscription and showed that, in some cases, circumscription can be replaced by
an equivalent first-order formula. Further results in this direction have been
obtained in [Doherty et al., 1995].
Independently of McCarthy, [Bossu and Siegel, 1985] have provided semantic
account of nonmonotonic reasoning for a special class of minimal models of a
first-order theory. [Lifschitz, 1987a] has proposed a more expressive variation of
circumscription, called pointwise circumscription which, instead of minimizing
the extension of a predicate P (x) as a whole, minimizes the truth-value of P (a)
at each element a.
[Perlis, 1986] and [Poole, 1989b] showed the inadequacies of circumscription
to deal with counterexamples like the lottery paradox of Kyburg. In a lottery,
it is known that some person will win, yet, for any individual x, the default
should be that x does not win. If these facts are translated in a straightforward
way into circumscription, it is impossible to arrive at the default conclusion,
for any given individual x, that x will not win the lottery. To remedy this and
similar anomalies, [Etherington et al., 1991] proposed a scoped circumscription
in which the individuals over whom minimization proceeds are limited by a
scoping predicate.
We mentioned earlier that Reiter’s Closed World Assumption works only
for theories that have a unique least model. For this case, it has been shown
in [Lifschitz, 1985] that CWA is equivalent to circumscription (modulo unique
names and domain closure assumptions). Similarly, it has been shown in [Reiter, 1982] that an appropriate circumscription always implies Clark’s predicate
completion, but not vice versa. In other words, the minimal models determined
by circumscription form in general only a subset of models sanctioned by completion. Recently, it has been shown in [Lee and Lin, 2004] that circumscription
can be obtained from completion by augmenting the latter with so-called ‘loop’
formulas.
In the earliest circumscriptive solutions to the frame problem, the inertia
rule was stated using an abnormality predicate. This formalization has succumbed, however, to the Yale Shooting Problem, mentioned earlier. [Baker,
1989] presented another solution to the Yale Shooting problem in the situation
14
calculus, using a circumscriptive inertial axiom. A circumscriptive approach
to the qualification problem was presented in [Lifschitz, 1987b]; it used an explicit relation between an action and its preconditions, and circumscriptively
minimized preconditions, eliminating thereby unknown conditions that might
render an action inefficacious. Further details and references can be found in
[Shanahan, 1997].
3.2
Default Logic
A detailed description of default logic, its properties and variations can be found
in [Antoniou and Wang, 2006]. Accordingly, we will sketch below only general
features of the formalism, sufficient for determining its place in the nonmonotonic reasoning field.
In both circumscription and modal nonmonotonic logic (see below), default
statements are treated as logical formulas, while in default logic [Reiter, 1980]
they are represented as inference rules. In this respect, Reiter’s default logic
has been largely inspired by the need to provide logical foundations for the
procedural approach to nonmonotonicity found in deductive databases, logic
programming and Doyle’s truth maintenance. In particular, default logic begins
with interpreting “In the absence of any information to the contrary, assume
A” as “If A can be consistently assumed, then assume it.”
A default is a rule of the form A : b/C, intended to state something like: ‘if
A is believed, and each B ∈ b can be consistently believed, then C should be
believed’. A is called a prerequisite of a default rule, b a set of its justifications.
The flying birds default is represented by the rule Bird(x) : F ly(x)/F ly(x).
Reiter defined a default theory to be a pair (D, W ), where D is a set of default
rules and W is a set of closed first-order sentences. The set W represents what
is known to be true of the world. This knowledge is usually incomplete, and
default rules act as mappings from this incomplete theory to a more complete
extension of the theory. They partly fill in the gaps with plausible beliefs.
Extensions are defined by a fixed point construction. For any set S of firstorder sentences, Γ(S) is defined as the smallest set satisfying the following three
properties:
1. W ⊆ Γ(S);
2. Γ(S) is closed under first-order logical consequence;
3. If A : b/C belongs to D and A ∈ Γ(S) and ¬B ∈
/ S, for any B ∈ b, then
C ∈ Γ(S).
Then a set E is an extension of the default theory iff Γ(E) = E, that is, E
is a fixed point of the operator Γ.
The above definition embodies an idea that an extension must not contain
“ungrounded” beliefs, i.e., every formula in it must be derivable from W and
the consequents of applied defaults in a non-circular way. It is this property
that distinguishes default logic from circumscription. Though a standard way
15
of excluding unwanted elements employs a minimality requirement, minimality
alone is insufficient to exclude ungrounded beliefs. Reiter’s idea can also be
expressed as follows. At first stage we assume a conjectured extension and use
it to determine the set of applicable inference rules, namely default rules such
that their justifications are consistent with the assumption set. Then we take
the logical closure of these applicable inference rules, and if it coincides with the
candidate extension, the latter is vindicated. In this sense, an extension is a set
of beliefs which are in some sense “justified” or “reasonable” in light of what
is known about the world (cf. [Etherington, 1987]). This interpretation can
be seen as a paradigmatic form of general explanatory nonmonotonic reasoning
that we will discuss later.
Extensions are deductively closed sets that are closed also with respect to
the rules of the default theory. Moreover, they are minimal such sets, and hence
different extensions are incomparable with respect to inclusion. Still, not every
minimal theory closed with respect to default rules is an extension.
Multiple extensions of a default theory are possible. The perspective adopted
on these in [Reiter, 1980] was that any such extension is a reasonable belief set
for an agent. A typical example involves Richard Nixon who is quaker and
republican. Quakers (typically) are pacifists. Republicans (typically) are not
pacifists. One might conclude that Nixon is a pacifist using the first default, but
also that Nixon is not a pacifist because he is a republican. In this situation the
default logic generates two extensions, one containing the belief that Nixon is a
pacifist, the other one containing the belief that he is not. Speaking generally,
there are many applications where each of the extensions is of interest by itself.
For instance, diagnostic reasoning (see below) is usually modeled in such a way
that each diagnosis corresponds to a particular extension.
A default theory does not always have extensions, a simplest example being
the default theory (∅, {true : ¬A/A}). However, an important subclass consisting of normal default rules of the form A : B/B always has an extension.
On Reiter’s intended interpretation, normal defaults provided a representation
of commonsense claims “If A, then normally B”. Reiter developed a complete
proof theory for normal defaults and showed how it interfaces with a top-down
resolution theorem prover. He has considered this proof theory as one of the
advantages of default logic. Moreover, extensions of normal default theories are
semi-monotonic: if E is an extension of a normal default theory (W, D) then
the normal default theory (W, D ∪ D0 ) has an extension E0 such that E ⊆ E0 .
In other words, additional defaults may augment existing extensions or produce
new ones, but they never destroy the extensions obtained before.
Reiter has mentioned in [Reiter, 1980] that he knows of no naturally occurring default which cannot be represented in this form. However, the later paper
[Reiter and Criscuolo, 1981] showed that in order to deal with default interactions, we need at least semi-normal defaults of the form A : B ∧C/C. Moreover,
though some authors2 have questioned the usefulness of defaults which are not
semi-normal, Paul Morris has shown in [Morris, 1988] that such default rules
2 E.g.,
[Brewka et al., 1997].
16
can provide a solution for the Yale Shooting Anomaly.
There has been a number of attempts of providing a direct semantic interpretation of default logic. It was suggested already in [Reiter, 1980] that default
rules could be viewed as operators that restrict the models of the set of known
facts W . Developing this idea, [Etherington, 1987] has suggested a semantics
based on combining preference relations ≥δ on sets of models determined by
individual defaults δ. For normal default theories, extensions corresponded to
preferred sets of models. For full generality, however, an additional condition
of Stability was required which stated, roughly, that the corresponding model
set is ‘accessible’ from M OD(W ) via a sequence of ≥δ -links corresponding to
defaults δ with unrefuted justifications. This operational, or quasi-inductive
description of extensions has been developed later in a number of works – see
[Makinson, 2003; Antoniou and Wang, 2006].
Default logic is a more general and more expressive formalism than circumscription. In one direction, [Imielinski, 1987] has shown that even normal
default theories cannot be translated in a modular way to circumscription. In
the other direction, [Etherington, 1987] has shown that minimizing the predicate P , with all other predicates varying, corresponds to the use of the default
: ¬P (x)/¬P (x). Speaking generally, circumscription corresponds to default
theories involving only simplest normal defaults without prerequisites; skeptical reasoning in such theories is equivalent to minimization with respect to
the ordering on interpretations determined by the sets of violated defaults (cf.
Theorem 4.7 in [Poole, 1994a]).
[Etherington and Reiter, 1983] used default logic to formalize inheritance hierarchies, while [Reinfrank et al., 1989] have shown that justifications of Doyle’s
truth maintenance system can be directly translated into default rules, and then
extensions of the resulting default theory will exactly correspond to admissible
labelings of TMS.
A large part of the expressive power of default logic is due to the representation of defaults as inference rules. This representation avoids some problems arising with formula-based interpretations of defaults (such as contraposition) and provides a natural framework for logic programming and reasoning in
databases. Unfortunately, this representation also has its problems, problems
considered by many researchers as more or less serious drawbacks.
One kind of problems concerned an apparent discrepancy between commonsense claims “If A then normally B” and their representation in terms of normal
defaults A : B/B. For example, default logic does not allow for reasoning by
cases: From “Italians (normally) like wine” and “French (normally) like wine”
we cannot conclude that the person at hand that is Italian or French likes wine.
This is because, in default logic, a default can only be applied if its prerequisite
has already been derived. A more profound problem has been pointed out in
[Makinson, 1989]. A general class of preferential nonmonotonic inference relations (see below) satisfies a natural Cumulativity postulate stating that, if A
entails both B and C, then A ∧ B should entail C. Makinson has shown, however, that default logic does not satisfy Cumulativity (see [Antoniou and Wang,
2006]). This discrepancy was a first rigorous indication that default logic can17
not be directly subsumed by the preference-based approach to nonmonotonic
reasoning.
A problem of a different kind has been noticed in [Poole, 1989b] that concerned joint consistency of justifications of default rules. Namely, the definition
of extensions only enforces that each single justification of an applied default
is consistent with the generated extension, but nothing guarantees the joint
consistency of the justifications of all applied defaults (see again [Antoniou and
Wang, 2006] for examples and discussion).
These and other problems have led to a number of proposed modifications of
default logic. Most of them are described in [Antoniou and Wang, 2006]. Also,
driven mainly by the analogy with logic programming, a number of authors have
suggested to generalize the notion of extension to that of a partial extension
– see [Baral and Subrahmanian, 1991; Przymusinska and Przymusinski, 1994;
Brewka and Gottlob, 1997] (cf. also [Antonelli, 1999; Denecker et al., 2003]).
Unfortunately, these modifications still have not gained widespread acceptance.
A generalization of a different kind has been proposed in [Gelfond et al.,
1991], guided by the need to provide a logical basis for disjunctive logic programming, as well as more perspicuous ways of handling disjunctive information. A disjunctive default theory is a set of disjunctive defaults, rules of the
form a : b/c, where a, b, c are finite sets of formulas. The authors described a
semantics for such default theories that constituted a proper generalization of
Reiter’s default logic, as well as the semantics for disjunctive databases from
[Gelfond and Lifschitz, 1991]. They also have shown how the above mentioned
Poole’s problem of joint justifications can be avoided by using disjunctive default
rules.
[Marek and Truszczyński, 1989] introduced the notion of a weak extension
as a default counterpart of stable expansions in autoepistemic logic and models
of Clark’s completion in logic programming (see below). Weak extensions can
be defined3 as fixed points of a modified operator Γw , obtained by weakening
the third condition in the above definition of Γ to
(3’) If A : b/C ∈ D, A ∈ S and ¬B ∈
/ S, for any B ∈ b, then C ∈ Γ(S).
A number of studies have dealt with elaboration of default logic as a logical
formalism. Thus, [Marek and Truszczyński, 1989] suggested a more ‘logical’
description of default logic using the notion of a context-depended proof as a
way of formalizing Reiter’s operator Γ. [Marek et al., 1990] described a general
theory of nonmonotonic rule systems based on abstract rules a:b/A having an informal interpretation similar to Doyle’s justifications: ‘If all a’s are established,
and none of b’s is established now or ever, conclude A’. The authors studied
abstract counterparts of extensions and weak extensions in this framework. An
even more general default system in the framework of the domain theory has
been developed in [Zhang and Rounds, 1997]. A reconstruction of default derivability in terms of restricted Hilbert-type proofs has been described in [Amati
et al., 1994]. Finally, a sophisticated representation of default logic as a sequent
3 Cf.
[Halpern, 1997].
18
calculus based on both provability and unprovability sequents has been given in
[Bonatti and Olivetti, 2002].
Reiter has mentioned in [Reiter, 1987a] that, because the defaults are represented as inference rules rather than object language formulas, they cannot
be reasoned about within the logic. Thus, from “Normally canaries are yellow”
and “Yellow things are never green” we cannot conclude in default logic that
“Normally canaries are never green”. As far as we know, a first default system
with such meta-rules allowing to infer derived default rules from given ones has
been suggested in [Thiele, 1990] (see also [Brewka, 1992]). [Bochman, 1994]
introduced default consequence relations as a generalization of both default and
modal formalizations of nonmonotonic reasoning. Default consequence relation
is a logical (monotonic) inference system based on default rules of the form
a:b C similar to that in [Marek et al., 1990]. The basic system was required
to satisfy the following postulates:
Monotonicity
Cut
If a : b A and a ⊆ a0 , b ⊆ b0 , then a0 : b0 A.
If a : b A and a, A : b B, then a : b B.
Consistency
A:Af
together with postulates securing that default rules respect classical entailment
both in premises and conclusions. A binary derivability operator associated
with a default consequence relation was defined as Cn(u, v) = {A | u : v A}.
This operator simplified a characterization of extensions and weak extensions,
called expansions in [Bochman, 1994], due to their correspondence with stable
expansions of autoepistemic logic. Namely, a set u of propositions is an extension
of a default consequence relation if u = Cn(∅, u), and it is an expansion if
u = Cn(u, u) (where u denotes the complement of u).
Reiter’s default rules A:b/C were translated as the rules A:¬b C. Then
it was shown that extensions of the resulting default consequence relation correspond precisely to extensions of Reiter’s default logic. Moreover, it has been
shown that the following additional postulates also preserve extensions:
Reflexivity
A: A.
Negative Factoring If a, B : b f and a : b, B A, then a : b A.
The resulting class of default consequence relations can be viewed as an underlying monotonic logic of Reiter’s default logic. In contrast, the logic adequate
for reasoning with expansions can be obtained by adopting instead a powerful
alternative postulate:
Factoring
If a, B : b A and a : b, B A, then a : b A.
Such default consequence relations have been called autoepistemic. In autoepistemic consequence relations, extensions collapse to expansions (weak extensions). It was shown that this logic provides an exact non-modal counterpart
of Moore’s autoepistemic logic (see below).
19
3.3
Modal nonmonotonic logics
The third seminal paper in the 1980 issue of Artificial Intelligence, [McDermott
and Doyle, 1980], is the beginning of a modal approach to nonmonotonic reasoning. McDermott and Doyle provided first a broad picture of instances of
nonmonotonic reasoning in different parts of AI, and stressed the need for a formal logical analysis of these phenomena. The theory was also clearly influenced
by the need to provide a formal account of truth maintenance. As a general suggestion, the authors proposed to expand the notation in which logical inference
rules are stated by adding premises like ‘unless proven otherwise’.
The modal nonmonotonic logic was formulated in a modal language containing a modal operator M p with the intended meaning “p is consistent with
everything believed”. In accordance with this understanding, a direct way of
making assumptions in this setting might consist in accepting an inference rule
of the form “If 0 ¬A, then ` M A”. Such a rule, however, would be circular
relative to the underlying notion of derivability. And the way suggested by the
authors was a fixed-point construction much similar to that of Reiter’s default
logic, but formulated this time entirely in the modal object language.
For a set u of propositions, let us denote by M u the set {M A | A ∈ u},
and similarly for ¬u, etc. In addition, u will denote the complement of u with
respect to a given modal language. Using this notation, a set s is a fixed point
of a modal theory u, if it satisfies the equality
s = Th(u ∪ M ¬s)
The set of nonmonotonic conclusions of a modal theory was defined in [McDermott and Doyle, 1980] as an intersection of all its fixed points.
The initial formulation of modal nonmonotonic logic has turned out to be
unsatisfactory, mainly due to the fact that it secured no connection between
a modal formula M C and its objective counterpart C, so that even the nonmonotonic theory {M C, ¬C} was consistent. In response to this latter difficulty,
[McDermott, 1982] developed a stronger version of the formalism based on the
entailment relation of standard modal logics instead of first-order logic. In fact,
the decisive modification was an adoption of the Necessitation rule A ` LA.
The corresponding fixed points were defined now as sets satisfying the equality
s = CnS (u ∪ M ¬s),
where CnS is a provability operator of some modal logic S containing the necessitation rule.
In what follows, we will call such fixed points S-extensions (following [Konolige, 1988] and [Schwarz, 1990], but in contrast with the influential subsequent
terminology of Marek and Truszczyński who have called them S-expansions).
As has been shown by McDermott, the stronger is the underlying modal
logic, the smaller is the set of extensions, and hence the larger is the set of
nonmonotonic consequences of a modal theory. It has turned out, however, that
the modal nonmonotonic logic based on the strongest modal logic S5 collapses
20
to a monotonic system. So the resulting suggestion was somewhat indecisive,
namely a range of possible modal nonmonotonic logics without clear criteria
for evaluating the merits of the alternatives. However, this indecisiveness has
turned out to be advantageous in the subsequent development of the modal
approach.
Neither Doyle or McDermott pursued the modal approach much beyond
these initial stages. Actually, both have expressed later criticism about the
whole logical approach to AI.
3.3.1
Autoepistemic logic
As an alternative response to the problems with initial formulations of modal
nonmonotonic logics, Robert C. Moore proposed his autoepistemic logic (AEL)
in [Moore, 1985]. Autoepistemic logic has had a great impact on the development of nonmonotonic reasoning; for a certain period of time it has even been
considered as a correct replacement of the theory of McDermott and Doyle.
Moore has suggested that modal nonmonotonic logic should be interpreted
not as a theory of default (defeasible) reasoning, but as a model of (undefeasible)
autoepistemic reasoning of an ideally rational agent. Following a suggestion
made by Robert Stalnaker in 1980 (published much later as [Stalnaker, 1993]),
Moore reconstructed nonmonotonic logic as a model of an ideally rational agent’s
reasoning about its own beliefs. He argued that purely autoepistemic reasoning
is not defeasible; it is nonmonotonic because it is context-sensitive. He defined a
semantics for which he showed that autoepistemic logic is sound and complete.
Instead of the modal consistency operator M , he used the dual belief operator L. Moore stressed the epistemic interpretation of a default rule as one
that licenses a conclusion unless something that the agent knows blocks it.
Nonmonotonicity can be achieved by endowing an agent with the introspective
ability, namely an ability to reflect on its own beliefs in order to infer sentences
expressing what it doesn’t believe. Introspective here means that the agent is
completely aware about his beliefs: if a formula p belongs to the set of beliefs
B of the agent, then also Lp has to belong to B, and if p does not belong to B,
then ¬Lp must be in B. Sets of formulas satisfying these properties were first
discussed by Stalnaker, who called them stable sets.
Stable sets are essentially sets of formulas globally valid in S5-models (see
[Konolige, 1988]). As has been shown in [Moore, 1985] (and implicitly already
in [McDermott, 1982]) a stable set is uniquely determined by its objective (nonmodal) propositions. This fact has provided later a basic link between modal
and non-modal approaches to nonmonotonic reasoning.
A stable expansion of a set u of premises was defined as a stable set that is
grounded in u. Again, the formal definition was based on a fixed point equation:
s = Th(u ∪ Ls ∪ ¬Ls)
The groundedness condition ensured that every member of an expansion has
some reason tracing back to s. As in default logic, conflicting AEL rules can
lead to alternative stable sets of beliefs a reasoner may adopt.
21
Moore has argued that the original nonmonotonic logic of McDermott and
Doyle was simply too weak to capture the notions they wanted, while [McDermott, 1982] strengthened it in a wrong way. He observed that, in the modal
nonmonotonic logic, the component Ls is missing from the ‘base’ of the fixed
points, which means that McDermott and Doyle’s agents are omniscient as to
what they do not believe, but they may know nothing as to what they do believe.
A difference between the nonmonotonic modal and autoepistemic logics can
be seen on the theory {LP →P }, which has a single fixed point that does not
include P , though it also has a stable expansion containing P . As argued by
Moore, this makes the interpretation of L in nonmonotonic modal logic more like
“justified belief” than simple belief. On the other hand, already [Konolige, 1988]
argued that the second expansion is intuitively unacceptable. It corresponds to
an agent arbitrarily entering P, hence also LP, into her belief set.
Since nonmonotonic S5 collapses to monotonic S5, Moore (following Stalnaker) has suggested to retreat not to S4, as was suggested in [McDermott,
1982], but to K45 obtained from S5 by dropping reflexivity LP → P . Moore
has also shown that the axioms of K45 do not change stable expansions.
[Levesque, 1990] generalized Moore’s notion of a stable expansion to the full
first-order case. He provided a semantic account of stable expansions in terms
of a second modal operator O, where O(A) is read as “A is all that is believed.”
In this approach, nonmonotonicity was pushed entirely into the scope of the O
operator. Levesque’s ideas have been systematically presented and applied to
the theory of knowledge bases in [Levesque and Lakemeyer, 2000].
It has been shown in [Konolige, 1989] that autoepistemic logic is a more expressive formalism than circumscription (see also [Niemelä, 1992]). The relation
between autoepistemic and default logic, however, has turned out to be more
complex, and it has created an important incentive for further development of
a theory of nonmonotonic reasoning.
3.3.2
Unified modal theories
[Konolige, 1988] has attempted to translate default logic into autoepistemic
logic, and vice versa. To this end, he suggested to rephrase a default rule
A : B1 , . . . Bn /C as a modal formula (LA ∧ ¬L¬B1 ∧ · · · ∧ ¬L¬Bn ) ⊃ C.
Despite Konolige’s intentions, however, this translation has turned out to
be inappropriate for representing default rules. Furthermore, [Gottlob, 1994]
has shown that there can be no modular translation of default logic into autoepistemic logic, even if we restrict ourselves to normal defaults, or just to
inference rules without justifications. Still, [Marek and Truszczyński, 1989] has
shown that Konolige’s translation works for prerequisite-free defaults. As to the
general case, Marek and Truszczyński have shown that the translation provides
instead an exact correspondence between stable expansions and weak extensions.
A more organized picture started to emerge with the result of [Schwarz,
1990] according to which autoepistemic logic is just one of the nonmonotonic
logics in the general approach of [McDermott, 1982]. Namely, it is precisely the
nonmonotonic logic based on K45, or even KD45, if we ignore inconsistent ex22
pansions4 . This result, as well as difficulties encountered in interpreting default
logic, revived interest in the whole range of modal nonmonotonic logics based
on ‘negative introspection’ in the works of Marek, Truszczyński and Schwarz.
A minimal modal nonmonotonic logic in this range is based on a modal logic
N that does not contain modal axioms at all (see [Fitting et al., 1992]). Corresponding N -extensions have been introduced in [Marek and Truszczyński, 1990]
and called iterative expansions. A systematic study of modal nonmonotonic logics based on different underlying modal logics can be found in [Marek et al., 1993;
Marek and Truszczyński, 1993]. This study has shown the importance of many
otherwise esoteric modal logics for nonmonotonic reasoning, such as S4F, SW5,
and KD45. All these logics have, however, a common semantic feature: in the
terminology of [Segerberg, 1971], each of them is characterized by certain Kripke
models of depth two having a unique final cluster (see [Schwarz, 1992a]).
An adequate modal interpretation of defaults logic in modal nonmonotonic
logics of [McDermott, 1982] has been suggested in [Truszczyński, 1991]. The
interpretation was based on the following more complex translation:
A : B1 , . . . Bn /C ⇒ (LA ∧ LM B1 ∧ · · · ∧ LM Bn ) ⊃ LC
Under this interpretation, a whole range of modal logics between T − and
S4F can be used as a “host” logic. In addition, the translation can be naturally
extended to disjunctive default rules of [Gelfond et al., 1991]. In this sense,
modal nonmonotonic logics also subsumed disjunctive default logic. A different
translation of default logic into the modal logic KD4Z has been suggested in
[Amati et al., 1997].
[Bochman, 1994; 1995b] introduced the notion of a modal default consequence relation (see also [Bochman, 1998c]). These consequence relations were
defined in a language with a modal operator, but otherwise involved the same
rules as general default consequence relations, described earlier. A default consequence relation in a modal language has been called modal if it satisfied the
following two modal axioms:
A : LA and
: A ¬LA.
Basically, the transition from default to modal nonmonotonic logics amounts
to adding these default rules to a default theory. A similar translation has been
used in [Janhunen, 1996] and rediscovered a number of times in later publications. Modal default consequence relations have turned out to be a convenient
tool for studying modal nonmonotonic reasoning. Thus, both autoepistemic
reasoning and reasoning with negative introspection acquired a natural characterization in this framework. In particular, S-extensions of a modal theory u
have been shown to coincide with extensions of the least modal default consequence relation containing u and the modal axioms of the modal logic S.
Under certain reasonable conditions, modal consequence relations have turned
out to be reducible to their nonmodal, default sub-relations in a way that
4 KD45 is obtained from S5 by replacing the T axiom LA⊃A with a weaker D axiom
LA⊃¬L¬A.
23
preserved the associated nonmonotonic semantics. These results were used in
[Bochman, 1998c] for establishing a two-way correspondence between modal and
default formalizations. In particular, it has been shown that modal autoepistemic logic is equivalent to a nonmodal autoepistemic consequence relation,
described earlier.
Lin and Shoham have suggested in [Lin and Shoham, 1992] a bimodal system
purported to combine preferential and fixed-point approaches to nonmonotonic
reasoning. The bridge was provided by a preference relation on models that
minimized knowledge for fixed sets of assumptions. Semantic models of a bimodal system are triples (Mk , Mb , V ) such that Mk and Mb are sets of possible
worlds, Mb ⊆ Mk , and V is a valuation function assigning each world a propositional interpretation. Mk and Mb serve as ranges of two modal operators K
and B representing, respectively, the knowledge and assumptions (beliefs) of an
introspective agent. A model M1 is preferred over a model M2 , if they have
the same B-worlds, but M1 has a larger set of K-worlds. Finally, a preferred
model is a model M = (Mk , Mb , V ) such that Mk = Mb , and there is no model
M1 that is preferred over M.
Lin and Shoham were able to show that, by using suitable translations, default and autoepistemic logic, as well as the minimal belief logic of [Halpern
and Moses, 1985] can be embedded into their system. [Lifschitz, 1991b; 1994a]
described a simplified version of such a bimodal logic, called MBNF (Minimal
Belief with Negation-as-Failure), extended it to the language with quantifiers,
and considered a representation of default logic, circumscription and logic programming in this framework. Yet another bimodal theory, Autoepistemic Logic
of Minimal Beliefs (AELB) has been suggested in [Przymusinski, 1994] as an extension of Moore’s autoepistemic logic and has been shown to subsume circumscription, CWA and its generalizations, epistemic specifications and a number
of semantics for logic programs.
It has been shown in [Lifschitz and Schwarz, 1993] that a significant part of
MBNF, namely theories with protected literals, can be embedded into autoepistemic logic (see also [Chen, 1994]). Furthermore, it has been shown in [Schwarz
and Truszczyński, 1994] that in most cases bimodal nonmonotonic logics can be
systematically reduced to ordinary unimodal nonmonotonic logics by translating the belief (assumption) operator B as ¬K¬K (see also [Bochman, 1995c]
for a similar reduction of MBNF).
Finally, a partial (four-valued) generalization of modal and default nonmonotonic logics has been studied in [Denecker et al., 2003].
3.4
Logic Programming
Logic programming has been based on the idea that program rules should have
both a procedural and declarative (namely, logical) meaning. This double interpretation was intended to elevate the programming process by basing it on
a transparent and systematic logical representation of real world information.
Fortunately, it has been discovered quite early that logic programming with
negation as failure allows us to express significant forms of nonmonotonic rea24
soning. Moreover, general nonmonotonic formalisms, described earlier, inspired
the development of new semantics for logic programs. These developments have
made logic programming an integral part of nonmonotonic reasoning research
functioning as a general computational mechanism for knowledge representation
and nonmonotonic reasoning (see [Baral, 2003]).
A normal logic program is a set of program rules of the form A ← a, not b,
where A is a propositional atom (the head of the rule), a and b are finite sets
of atoms (forming the body of the rule), and not is negation as failure.
The semantics for logic programs have gradually evolved from definite rules
without not to more general program rules involving negation as failure. For
definite logic programs, such a semantics has been identified with the unique
minimal model of the corresponding logical theory [van Emden and Kowalski,
1976]. Both the Closed World Assumption (CWA) [Reiter, 1978] and completion
theory [Clark, 1978] have provided its syntactic characterization. This semantics has been conservatively extended in [Apt et al., 1988] to stratified normal
programs. It has been called the perfect semantics in [Przymusinski, 1988]. See
[Shepherdson, 1988] for a detailed description of these initial semantics.
Unlike the CWA, Clark’s completion is naturally definable for any normal
program. It has turned out, however, that the corresponding supported semantics for logic programs produces inadequate results when applied to arbitrary
(i.e., non-stratified) programs. In response to this problem, two competing semantics have been suggested for arbitrary normal programs: the stable model
semantics [Gelfond and Lifschitz, 1988], and the well-founded semantics [van
Gelder et al., 1991]. A large literature has been devoted to studying the relationships between these three semantics, as well as between the latter and the
nonmonotonic formalisms.
Under the stable model semantics, a normal program rule A ← a, not b
corresponds to a rule a : ¬b/A of default logic (see [Marek and Truszczyński,
1989]). This translation establishes a one-to-one correspondence between stable
models of a normal logic program and extensions of the associated default theory.
In this sense, logic programs under the stable semantics capture the main idea, as
well as primary representation capabilities, of default logic. Still, this embedding
of logic programs into default logic is unidirectional, since not every default
theory corresponds in this sense to a logic program.
The stable model semantics is sufficient, in particular, for an adequate representation of Doyle’s truth-maintenance system [Doyle, 1979]. If we translate
each nonmonotonic justification as a program rule then there is a 1-1 correspondence between stable models of the resulting logic program and admissible
labelings of the original justification network [Elkan, 1990].
Guided partly by the correspondence with default logic, [Gelfond and Lifschitz, 1991] have suggested to include classical negation ¬ into logic programs,
in addition to negation-as-failure not. They argued that some facts of commonsense reasoning can be represented more easily when classical negation is
available. [Alferes and Pereira, 1992] defined a parameterizable schema to encompass and characterize a range of proposed semantics for such extended logic
programs. By adjusting the parameters they have been able to specify several
25
semantics using two kinds of negation. [Gelfond, 1994] expanded the syntax
and semantics of logic programs even further to an epistemic formalism with
modal operators allowing for the representation of incomplete information in
the presence of multiple extensions.
[Gelfond and Lifschitz, 1988] showed that the stable model semantics is also
equivalent to some translation of logic programs into autoepistemic theories.
[Pearce, 1997] has suggested that a certain three-valued logic called here-andthere (HT) can serve as an underlying logic of the stable model semantics.
Continuing this line of research, [Lifschitz et al., 2001] have shown that HT
provides a characterization of strong equivalence for logic programs under the
stable semantics5 .
A stable model of a logic program is also a model of its completion, while
the converse does not hold. [Marek and Subrahmanian, 1992] showed the relationship between supported models of normal programs and expansions of autoepistemic theories. [Fages, 1994] has shown, however, that a natural syntactic
acyclicity condition on the rules of a program, subsequently called “tightness”,
is sufficient for coincidence of the stable and supported semantics. As a practical consequence, this has reduced the task of computing the stable models for
tight programs to classical satisfiability [Babovich et al., 2000]. Finally, as a
most important recent development, [Lin and Zhao, 2002] showed that a classical logical description of the stable semantics for an arbitrary normal program
can be obtained by augmenting its Clark’s completion with what they called
“loop formulas”. This formulation has opened the way of computing the stable
semantics of logic programs using SAT solvers.
Unlike the stable semantics, the well-founded semantics suggested in [van
Gelder et al., 1991] does not correspond to any of the nonmonotonic formalisms,
described earlier. It determines, however, a unique model for every normal
logic program. This semantics has been elaborated upon by Przymusinski to a
theory of partial stable models [Przymusinski, 1991b; 1990]; the well-founded
model constituted the least model in this hierarchy. Przymusinski has shown,
in effect, that the well-founded semantics can be seen as the three-valued stable
semantics (see also [Dung, 1992]). [Fitting, 1991] has extended the well-founded
semantics to a family of lattice-based logic programming languages. Denecker
has characterized the WFS as based on the principle of inductive definitions
- see [Denecker et al., 2001]. Finally, [Cabalar, 2001] has shown that a twodimensional variant of the logic HT, called HT 2 , can serve as the underlying
logic of WFS.
An argumentation-theoretic analysis of the different semantics for normal
logic programs has been suggested in [Dung, 1995a]. He has shown, in particular, that the stable, partial stable and well-founded semantics are based on
different argumentation principles of rejecting negation-as-failure assumptions
(see below). An alternative representation of normal logic programs and their
semantics has been given in [Bochman, 1995a; 1996b] in the framework of de5 Two programs are strongly equivalent if they have the same stable model semantics for
any extension of these programs with additional rules.
26
fault consequence relations from [Bochman, 1994]. The framework has provided
a relatively simple representation of various semantics for such programs, as well
as allowed to single out different kinds of logical reasoning that are appropriate
with respect to these semantics. It has been shown, in particular, that stable and supported models are precise structural counterparts of, respectively,
extensions and expansions of general default consequence relations.
An important line of research in logic programming has concentrated on
extending the expressive capabilities of program rules by allowing disjunction
in their heads, namely to rules of the form c ← a, not b, where c is a set of
atoms (treated disjunctively). For such disjunctive logic programs, Reiter’s
Closed World Assumption has been generalized to the Generalized Closed World
Assumption (GCWA) [Minker, 1982] and its derivatives. The model theoretic
definition of the GCWA states that one can conclude the negation of a ground
atom if it is false in all minimal Herbrand models. This makes disjunctive
programming quite close in spirit to McCarthy’s circumscription - see [Oikarinen
and Janhunen, 2005] for a detailed comparison. Initial semantics for disjunctive
programs containing negation in premises of the rules have been summarized in
[Lobo et al., 1992].
[Przymusinski, 1991a] introduced the stable model semantics for disjunctive logic programs that generalized corresponding semantics for normal logic
programs. Depending upon whether only total (2-valued) or partial (3-valued)
models are used, one obtains the disjunctive stable semantics or the partial
disjunctive stable semantics, respectively. He showed that for locally stratified disjunctive programs, both disjunctive semantics coincide with the perfect
model semantics.
Unfortunately, absence of clear grounds for adjudicating semantics for logic
programs has resulted in a rapid proliferation of suggested semantics, especially
for disjunctive programs. This proliferation has obviously created a severe problem. [Dix, 1991; 1992] has provided a systematic analysis of various semantics
for normal and disjunctive programs based on such properties as associated nonmonotonic inference relations, modularity, relevance and the principle of partial
evaluation.
Even the two-valued stable semantics for disjunctive logic programs is already not subsumed by default logic, but requires its extension to disjunctive
default logic [Gelfond et al., 1991]. It can be naturally embedded, however, into
more general modal nonmonotonic formalisms such as Lifschitz’ MBNF [Lifschitz, 1994a] that we mentioned at the end of the preceding section. These formalisms have suggested, in turn, a generalization of disjunctive program rules to
rules involving negation as failure in heads, namely rules of the form not d, c ←
a, not b, where c and d are sets of atoms - see [Lifschitz and Woo, 1992; Inoue
and Sakama, 1994]. The stable model semantics is naturally extendable to such
generalized programs in a way that preserves the correspondence with these nonmonotonic formalisms. Moreover, the alternative well-founded and partial stable
semantics are representable in a many-valued extension of such a modal nonmonotonic system, suggested in [Denecker et al., 2003]. Yet more powerful generalization has been introduced in [Lifschitz et al., 1999]) that made use of pro27
gram rules with arbitrary logical formulas in their heads and bodies. Such results
as the correspondence between the stable semantics and completion with loop
formulas have been extended to such programs (see [Erdem and Lifschitz, 2003;
Lee and Lifschitz, 2003]).
[Bochman, 1996a] has suggested a general scheme for constructing semantics
of logic programs of a most general kind in the logical framework of biconsequence relations that we will describe later in this study. A detailed description
was given in [Bochman, 1998a; 1998b]. Briefly, the formalism involved rules
(bisequents) of the form a : b c : d that directly represented general program
rules not d, c ← a, not b. The notion of circumscription of a biconsequence
relation was defined, and then a nonmonotonic completion was constructed as
a closure of the circumscription with respect to certain coherence rules. The
strength of these rules depended on the language L in which they were formulated. Finally the nonmonotonic semantics of a biconsequence relation was
defined as a pair of sets of propositions in the language L that are, respectively,
provable and refutable in the nonmonotonic completion. A uniform representation of major existing semantics for logic programs was provided simply by
varying the language L of the representation. A related, though more ‘logical’, representation of general logic programs and their semantics has been
suggested in [Bochman, 2004b] in the framework of causal inference relations
from [Bochman, 2004a] that will be described later.
3.5
Argumentation theory
Since Doyle’s seminal work on truth maintenance [Doyle, 1979], the importance
of argumentation in choosing default assumptions has been a recurrent theme
in the literature on nonmonotonic reasoning (see, e.g., [Lin and Shoham, 1989;
Geffner, 1992]). In a parallel development, a number of researchers in traditional
argumentation theory have shown that, despite influential convictions and prejudices, the ordinary, human argumentation is within the reach of formal logical
methods (see [Chesñevar et al., 2000] for a survey of this development).
A powerful version of abstract argumentation theory, especially suitable for
nonmonotonic reasoning, has been suggested in [Dung, 1995b]. We will give
below a brief description of this theory.
Definition. An abstract argumentation theory is a pair hA, ,→i, where A is a
set of arguments, while ,→ a binary relation of an attack on A.
If α ,→ β holds, then the argument α attacks, or undermines, the argument β.
A general task of argumentation theory consists in determining sets of arguments
that are safe (justified) in some sense with respect to the attack relation. To
this end, we should extend first the attack relation to sets of arguments: if Γ, ∆
are sets of arguments, then Γ ,→ ∆ is defined to hold if α ,→ β, for some α ∈ Γ
and β ∈ ∆.
Let us say that an argument α is allowable for the set of arguments Γ, if Γ
does not attack α. For any set of arguments Γ, we will denote by [Γ] the set of
28
all arguments allowable by Γ, that is
[Γ] = {α | Γ 6,→ α}
An argument α will be said to be acceptable for the set of arguments Γ, if Γ
attacks any argument against α. As can be easily checked, the set of arguments
that are acceptable for Γ coincides with [[Γ]].
Using the above notions, we can give a quite simple characterization of the
basic objects of an abstract argumentation theory.
Definition. A set of arguments Γ will be called
• conflict-free if Γ ⊆ [Γ];
• admissible if it is conflict-free and Γ ⊆ [[Γ]];
• a complete extension if it is conflict-free and Γ = [[Γ]];
• a preferred extension if it is a maximal complete extension;
• a stable extension if Γ = [Γ].
A set of arguments Γ is conflict-free if it does not attack itself. A conflictfree set Γ is admissible if and only if any argument from Γ is also acceptable for
Γ, and it is a complete extension if it coincides with the set of arguments that
are acceptable with respect to it. Finally, a stable extension is a conflict-free
set of arguments that attacks any argument outside it. Any stable extension
is also a preferred extension, any preferred extension is a complete extension,
and any complete extension is an admissible set. Moreover, as has been shown
in [Dung, 1995b], any admissible set is included in some complete extension.
Consequently, preferred extensions coincide with maximal admissible sets. In
addition, the set of complete extensions forms a complete lower semi-lattice: for
any set of complete extensions, there exists a unique greatest complete extension
that is included in all of them. In particular, there always exists a least complete
extension of an argumentation theory.
As has been shown in [Dung, 1995a] the above objects exactly correspond
to the semantics suggested for normal logic programs. Thus, stable extensions
correspond to stable models, complete extensions correspond to partial stable
models, preferred extensions correspond to regular models, while the least complete extension corresponds in this sense to the well-founded semantics. These
results have shown, in effect, that the abstract argumentation theory successfully captures the essence of logical reasoning behind normal logic programs (see
also [Kakas and Toni, 1999]).
The notion of an argument is often taken as primitive in argumentation theory, which even allows for a possibility of considering arguments that are not
propositional in character (e.g., arguments as inference rules, or derivations). As
has been shown in [Bondarenko et al., 1997], however, a powerful instantiation
of argumentation theory can be obtained by identifying arguments with propositions of a special kind called assumptions (see also [Kowalski and Toni, 1996]).
29
Thus, for logic programs assumptions can be identified with negation-as-failure
literals of the form not A, while consistency-based justifications of default rules
(see above) can serve as assumptions in default logic.
Slightly simplifying definitions from [Bondarenko et al., 1997], an assumptionbased argumentation framework can be defined as a triple consisting of an underlying deductive system, a distinguished subset of propositions Ab called assumptions, and a mapping from Ab to the set of all propositions of the language
that determines the contrary α of any assumption α. For instance, the contrary of the negation-as-failure literal not A in logic programming is the atom
A itself, while ¬A is the contrary of the justification A in default logic. A set
of assumptions Γ attacks an assumption α, if it implies its contrary, α. It has
been shown that the main nonmonotonic formalisms such as default and modal
nonmonotonic logics, as well as semantics of normal logic programs are representable in this framework. A dialectical proof procedure for finding admissible
sets of assumptions has recently been described in [Dung et al., 2006].
In [Bochman, 2003b] an extension of an abstract argumentation framework
was introduced in which the attack relation is defined directly among sets of
arguments. The extension, called collective argumentation, has turned out to
be suitable for representing semantics of disjunctive logic programs. In collective argumentation ,→ is an attack relation on sets of arguments satisfying the
following monotonicity condition:
(Monotonicity)
If Γ ,→ ∆, then Γ ∪ Γ0 ,→ ∆ ∪ ∆0 .
Dung’s argumentation theory can be identified with a normal collective argumentation theory in which no set of arguments attacks the empty set ∅ and
the following condition is satisfied:
(Locality)
If Γ ,→ ∆, ∆0 , then either Γ ,→ ∆ or Γ ,→ ∆0 .
In a normal collective argumentation theory the attack relation is reducible
to the relation Γ ,→ α between sets of arguments and single arguments, and the
resulting theory coincides with that given in [Dung, 1995a].
The attack relation of an argumentation theory can be given a natural fourvalued semantics based on independent evaluations of acceptance and rejection
of arguments. On this interpretation the attack Γ ,→ ∆ means that at least one
of the arguments in ∆ should be rejected whenever all the arguments from Γ
are accepted. The expressive capabilities of the argumentation theory depend,
however, on the absence of usual ‘classical’ constraints on the acceptance and
rejection of arguments, so it permits situations in which an argument is both
accepted and rejected, or, alternatively, neither accepted, nor rejected. Such
an understanding can be captured formally by assigning to any argument an
arbitrary subset of the set {t, f }, where t denotes acceptance (truth), while f
denotes rejection (falsity) (cf. [Jakobovits and Vermeir, 1999]). This interpretation is nothing other than the well-known Belnap’s interpretation of four-valued
logic (see [Belnap, 1977]).
30
The language of arguments can be extended with a global negation connective
∼ having the following semantic interpretation:
∼A is accepted iff A is rejected
∼A is rejected iff A is accepted.
An axiomatization of this negation in argumentation theory can be obtained
by imposing the following rules on the attack relation (see [Bochman, 2003b]):
A ,→ ∼A
∼A ,→ A
If a ,→ A, b and a, ∼A ,→ b, then a ,→ b
(AN)
If a, A ,→ b and a ,→ b, ∼A, then a ,→ b
It turns out that the resulting N-attack relations are interdefinable with
certain consequence relations.
A Belnap consequence relation in a propositional language with a global
negation ∼ is a Scott (multiple-conclusion) consequence relation satisfying
the postulates
(Reflexivity)
A A;
(Monotonicity) If a b and a ⊆ a0 , b ⊆ b0 , then a0 b0 ;
(Cut) If a b, A and a, A b, then a b,
as well as the following two Double Negation rules for ∼:
A ∼∼A
∼∼A A.
For a set u of propositions, ∼u will denote the set {∼A | A ∈ u}. Now, for
a given N-attack relation, we can define the following consequence relation:
a b ≡ a ,→ ∼b
(CA)
Similarly, for any Belnap consequence relation we can define the corresponding attack relation as follows:
a ,→ b ≡ a ∼b
(AC)
As has been shown in [Bochman, 2003b], the above definitions establish an
exact equivalence between N-attack relations and Belnap consequence relations.
In this setting, the global negation ∼ serves as a faithful logical formalization of
the operation of taking the contrary from [Bondarenko et al., 1997]. Moreover,
given an arbitrary language L that does not contain ∼, we can define assumptions as propositions of the form ∼A, where A ∈ L. Then, since ∼ satisfies
double negation, a negation of an assumption will be a proposition from L.
Such a ‘negative’ representation of assumptions will agree with the applications
of the argumentation theory to other nonmonotonic formalisms described in
[Bondarenko et al., 1997].
31
3.6
Abduction and causal reasoning
It does not seem necessary to argue that abduction and causation are essential
for the human, commonsense reasoning about the world. It has been gradually
realized, however, that these kinds of reasoning are essential also for efficient
‘artificial’ reasoning. Moreover, a new outlook on these kinds of reasoning can
be achieved by viewing them as special, though important, instances of nonmonotonic reasoning.
3.6.1
Abductive reasoning
Abduction is the process of finding explanations for observations. The importance of abduction for AI can be seen already in the Minsky’s frame paper
[Minsky, 1974]. A frame that contains default and stereotypical information
should be imposed on a particular situation:
Once a frame is proposed to represent a situation, a matching process
tries to assign values to each frame’s terminals, consistent with the
markers at each place. The matching process is partly controlled by
information associated with the frame (which includes information
about how to deal with surprises) and partly by knowledge about
the system’s current goals.
[Israel, 1980] criticized first nonmonotonic formalisms by objecting to the
centrality of deductive logic in these formalisms as a mechanism for justification.
He argued that what we need is not a new logic, but a good scientific methodology. Abductive reasoning to a best explanation requires rational epistemic
policies that lie, on Israel’s view, outside nonmonotonic logics. [McDermott,
1987] levied a similar criticism about the absence of a firm theoretical basis
behind diagnostic and other programs dealing with abduction:
This state of affairs does not stop us from writing medical diagnosis
programs. But it does keep us from understanding them. There is
no independent theory to appeal to that can justify the inferences a
program makes .... these programs embody tacit theories of abduction; these theories would be the first nontrivial formal theories of
abduction, if only one could make them explicit.
Despite this criticism, however, formal properties of abductive inference
methods and relations to other formalisms have been actively explored. [de
Kleer, 1986] has developed the Assumption-based Truth Maintenance Systems
(ATMS), a method for finding explanations in the context of a propositional
Horn-clause theories, when the hypotheses and observations are positive atoms.
Similarly to Doyle’s JTMS [Doyle, 1979], described earlier, the ATMS was developed by de Kleer as a subcomponent of a more general problem-solving system.
The main inference mechanism in the ATMS is the computation of a label at
each node. A label for an atom c is a set of environments, that is, sets of hypotheses that explain c. The ATMS differed from Doyle’s TMS, however, in
32
keeping track of multiple explanations or contexts, and especially in using only
monotonic propositional inference.
More general systems of logical abduction used more general theories. The
most prominent of these is the Clause Maintenance System (CMS) of [Reiter and
de Kleer, 1987]. To generalize the ATMS definition of explanation, the authors
considered a propositional domain theory consisting of general clauses and drew
on the concept of prime implicates to extend the inference technique of the
ATMS. Prime implicates can be used to find all the parsimonious explanations
of unit clauses. It was shown in [Reiter and de Kleer, 1987] that the label in
ATMS is exactly the set of such explanations, so the ATMS can be used to
compute parsimonious explanations for propositional Horn-clause theories.
Under diagnosis from first principles [Reiter, 1987b], or diagnosis from structure and behavior, the only information at hand is a description of some system,
say a physical device, together with an observation of that system’s behavior.
If this observation conflicts with intended system behavior, then the diagnostic
problem is to determine which components could by malfunctioning account
for the discrepancy. Since components can fail in various and often unpredictable ways, their normal or default behaviors should be described. These
descriptions fit the pattern of nonmonotonic reasoning. For example, an AND
gate in a digital circuit would have the description: Normally, an AND-gate’s
output is the Boolean AND function of its inputs. In diagnosis, such component descriptions are used in the following way: We first assume that all of
the system components are behaving normally. Suppose, however, the system
behavior predicted by this assumption conflicts with (i.e. is inconsistent with)
the observed system behavior. Thus some of the components we assume to be
behaving normally must really be malfunctioning. By retracting enough of the
original assumptions about correctly behaving components, we can remove the
inconsistency between the predicted and observed behavior. The retracted components yield a diagnosis. This approach to diagnosis from first principles was
called a consistency-based diagnosis, and it forms the basis for several diagnostic
reasoning systems (see also [de Kleer et al., 1992]).
From a logical point of view, the consistency-based diagnosis reduces to
classical consistency reasoning with respect to a certain abnormality theory, in
which abnormalities are minimized as in circumscription. This kind of diagnosis
is still not fully abductive, since it determines only what is a minimal set of
abnormalities that is consistent with the observed behavior. In other words, it
does not explain observations, but only excuses them.
In medical diagnosis, the type of reasoning involved is abductive in nature
and consists in explaining observations or the symptoms of a patient. Moreover,
as in general nonmonotonic reasoning, adding more information can cause a
previously held conclusion to become invalid. If it is known only that a patient
has a fever, the most reasonable explanation is that he has the flu. But if we
learn that he also has jaundice, then it becomes more likely that he has a disease
of the liver. The first, ‘procedural’ formalization of this reasoning was the setcovering model, proposed in [Reggia et al., 1985]. This model had a set of causes
and a set of symptoms, along with a relation that maps a cause to the set of
33
symptoms that it induces. Given an observation of symptoms for a particular
case, a diagnosis is a set of causes that covers all of the observed symptoms and
contains no irrelevant causes.
On the face of it, consistency-based and abductive diagnosis appear very
different. To begin with, rather than abducing causes that imply the observations, the consistency approach tries to minimize the extent of the causation
set by denying as many of its elements as possible. Moreover, in the abductive
framework, the causes have implications for the effects, while in the consistency
based systems, the most important information seems to be the implication of
the observations for possible causes.
Despite these differences, it is now known that, under certain conditions,
consistency-based explanations in the Clark completion of the domain theory
coincide with abductive explanations in the source theory.
Theorem. ([Poole, 1988b; Console et al., 1991]) Let Σ be a set of nonatomic
definite clauses whose directed graph of dependencies is acyclic, and let Π be
the Clark completion of Σ. Then the consistency-based explanations of an
observation O in Π are exactly the abductive explanations of O in Σ.
For more complicated domain theories, Clark completion does not give the
required closure to abductive explanations. For such more general cases the
correct generalization of Clark completion is explanatory closure (see [Konolige,
1992]).
Classical abduction has also been used as a practical proof method for circumscription (see, e.g., [Ginsberg, 1989]). [Levesque, 1989] suggested a knowledge level analysis of abduction in which the domain theory is represented as
the beliefs of an agent. Motivated primarily by the intractability of logic-based
abduction, this representation allowed for incomplete deduction in connecting
hypotheses to observations for which tractable abduction mechanisms can be
developed.
[Poole, 1988a] has developed the Theorist system in which abduction is used
as an inference method in default theories. In the spirit of David Israel, Poole
argued that there is nothing wrong with classical logic; instead, nonmonotonicity is a problem of how the logic is used. Theorist assumed the typical abductive
machinery: a set of hypotheses, a first-order background theory, and the concept
of explanation. A scenario is any subset of the hypotheses consistent with the
background theory; in the language of Theorist, an explanation for an observation O is a scenario that implies O. Theorist defined the notion of extension as a
set of propositions generated by a maximal consistent scenario. In other words,
by taking defaults as possible hypotheses, default reasoning has been reduced
to a process of theory formation.
Defaults of Poole’s abductive system corresponded to a simplest kind of Reiter’s default rules, namely normal defaults of the form : A/A. In addition, Poole
employed the mechanism of naming defaults (closely related to McCarthy’s abnormality predicates) that has allowed him to say, in particular, when a default
is inapplicable. The resulting system has been shown to capture many of the
representative capabilities of Reiter’s default logic in an almost classical logical
framework.
34
Systems such as the ATMS and Theorist are popular in many AI applications, because they are easy to understand and relatively easy to implement.
The use of abductive methods is growing within AI, and they are now a standard part of most AI representation and reasoning systems. In fact, the study
of abduction is one of the success stories of nonmonotonic reasoning, and it has
a major impact on the development of an application area.
Abductive Logic Programming. One of the important applications of abduction has been developed in the framework of logic programming. A comprehensive survey of the extension of logic programming to perform abductive
reasoning (referred to as abductive logic programming) can be found in [Kakas
et al., 1992] together with an extensive bibliography on abductive reasoning. As
their primary goal, the authors introduced an argumentation theoretic approach
to the use of abduction as an interpretation of negation-as-failure. Abduction
was shown to generalize negation-as-failure to include not only negative but also
positive hypotheses, and to include general integrity constraints. They showed
that abductive logic programming is related to the justification-based truth
maintenance system of Doyle and the assumption-based truth maintenance system of de Kleer.
Abductive logic programs are defined as pairs (Π, A), where Π is a logic program, and A a set abducible atoms. A formalization of abductive reasoning in
this setting is provided by the generalized stable semantics [Kakas and Mancarella, 1990], in which an abductive explanation of a query q is a subset S
of abducibles such that there exists a stable model of the program Π ∪ S that
satisfies q.
[Denecker and Schreye, 1992] developed a family of extensions of SLDNF
resolution for normal abductive programs. Further details and directions in
abductive logic programming can be found, e.g., in [Brewka and Konolige, 1993;
You et al., 2000; Lin and You, 2002].
It has been shown in [Inoue and Sakama, 1998] that abductive logic programs
under the generalized stable semantics are reducible to general disjunctive logic
programs under the stable semantics. The relevant transformation of abductive
programs can be obtained simply by adding to Π the program rules p, not p ←,
for any abducible atom p from A. This reduction has confirmed, in effect,
that general logic programs are inherently abductive and hence have the same
representation capabilities as abductive logic programs.
An abstract abductive framework. To end this section on abduction, we
will provide below a brief description of an abstract abductive system that is
sufficient for the main applications of abduction in AI. Further details on this
system and its expressive capabilities can be found in [Bochman, 2005].
An abductive system is a pair A = (Cn, A), where Cn is a supraclassical consequence relation, while A a distinguished set of propositions called abducibles.
A set of abducibles a ⊆ A is an explanation of a proposition A, if A ∈ Cn(a).
In applications, the consequence relation Cn is usually given indirectly by
35
a generating conditional theory ∆, in which case the corresponding abductive
system can be defined as (Cn∆ , A). Many abductive frameworks also impose
syntactic restrictions on the set of abducibles A (Poole’s Theorist being a notable
exception). Thus, A is often restricted to a set of special atoms (e.g., those
built from abnormality predicates ab), or to the corresponding set of literals.
The restriction of this kind is not essential, however. Indeed, for any abducible
proposition A we can introduce a new abducible propositional atom pA , and add
the equivalence A ↔ pA to the underlying theory. The new abductive system
will have much the same properties.
An abductive system (Cn, A) will be called classical if Cn is a classical
consequence relation. A classical abductive system can be safely equated with
a pair (Σ, A), where Σ is a set of classical propositions (the domain theory). An
example of such a system in diagnosis is [de Kleer et al., 1992], a descendant of
the consistency-based approach of [Reiter, 1987b].
In abductive systems, acceptance of propositions depends on existence of explanations, and consequently such systems sanction not only forward inferences
determined by the consequence relation, but also backward inferences from facts
to their explanations, and combinations of both. All these kinds of inference
can be captured formally by considering only theories of Cn that are generated
by the abducibles. This suggests the following notion:
Definition. The abductive semantics SA of an abductive system A is the set of
theories {Cn(a) | a ⊆ A}.
By restricting the set of theories to theories generated by abducibles, we
obtain a semantic framework containing more information. Generally speaking,
all the information that can be discerned from the abductive semantics of an
abductive system can be seen as abductively implied by the latter.
The information embodied in the abductive semantics can be made explicit
by using the associated Scott (multiple-conclusion) consequence relation, defined
as follows6 : for any sets b, c of propositions,
b `A c ≡ (∀a ⊆ A)(b ⊆ Cn(a) → c ∩ Cn(a) 6= ∅)
This consequence relation describes not only forward explanatory relations,
but also abductive inferences from propositions to their explanations. Speaking
generally, it describes the explanatory closure, or completion, of an abductive
system, and thereby captures abduction by deduction (cf. [Console et al., 1991;
Konolige, 1992]).
Example. The following abductive system describes a variant of the well-known
Pearl’s example. Assume that an abductive system A is determined by the set
∆ of rules
Rained ` Grasswet
Sprinkler ` Grasswet
Rained ` Streetwet,
6 A Tarski consequence relation of this kind has been used for the same purposes in [Lobo
and Uzcátegui, 1997].
36
and the set of abducibles Rained, ¬Rained, Sprinkler, ¬Sprinkler, ¬Grassswet.
Since Rained and ¬Rained are abducibles, Rained is an independent (exogenous) parameter, and similarly for Sprinkler. However, since only ¬ Grassswet
is an abducible, non-wet grass does not require explanation, but wet grass does.
Thus, any theory of SA that contains Grasswet should contain either Rained,
or Sprinkler, and consequently we have
Grasswet `A Rained, Sprinkler.
Similarly, Streetwet implies in this sense both its only explanation Rained
and a collateral effect Grasswet.
3.6.2
Causal reasoning
The last example of the preceding section also illustrates that one of the central
applications of abductive inference consists in generating causal explanations:
reasons connecting causes and their effects. In this case, the hypotheses, or
abducibles, represent primitive causes, observations are about their effects, and
the background theory encodes the relation between them. This is just one of the
many ways in which the notion of causation that once has been expelled from
exact sciences (see [Russell, 1957]) reappears as an important representation
tool in Artificial Intelligence. Moreover, the theories of causality emerging in AI
are beginning to illuminate in turn the actual role of causality in our reasoning.
As a mater of fact, causal considerations play an essential role in abduction
in general. They determine, in particular, the very choice of abducibles, as well
as the right form of descriptions and constraints (even in classical first-order
representations). As has been shown already in [Darwiche and Pearl, 1994],
system descriptions that do not respect the natural causal order of things can
produce inadequate predictions and explanations.
The intimate connection between causation and abduction has become especially vivid in the abductive approach to diagnosis (see especially [Cox and
Pietrzykowski, 1987; Poole, 1994b; Konolige, 1994]). As has been acknowledged
in these studies, reasoning about causes and effects should constitute a logical
basis for diagnostic reasoning. Unfortunately, the absence of an adequate logical formalization for causal reasoning has relegated the latter to the role of an
informal heuristic background, with classical logic serving as the representation
language.
Abduction and diagnosis are not the only areas of AI in which causality has
emerged. Thus, causality is an essential part of general qualitative reasoning;
see, e.g., [Iwasaki and Simon, 1986; Nayak, 1994], which goes back to Herbert
Simon’s important work [Simon, 1952]. Judea Pearl and his students and associates have developed a profound research program in the study of causation
derived from the use of causal diagrams in reasoning about probabilities known
as Bayesian Belief Networks. A detailed description of the emerged theory of
causality and its applications can be found in [Pearl, 2000]. See also [Halpern,
1998; Halpern and Pearl, 2001a; 2001b; Eiter and Lukasiewicz, 2004].
37
But perhaps the most significant development concerning the role of causation in our reasoning has occurred in of one of the central fields of logical AI,
reasoning about action and change. The starting point in this development was
the discovery of the Yale Shooting Problem [Hanks and McDermott, 1987].
We are told that a gun is loaded at time 1, and that the gun is fired at Fred
at time 5. Loading a gun causes the gun to be loaded, and firing a loaded gun
at an individual causes the person to be dead. In addition, the fluents alive and
loaded persist as long as possible; that is, these fluents stay true unless an action
occurs that is abnormal with respect to these fluents. Thus, a person who is
alive tends to remain alive, and a gun that is loaded tends to remain loaded.
What can we conclude about Fred’s status at time 6? Although common sense
argues that Fred is dead at time 6, existed nonmonotonic representations of this
story supported two models. In one model (the expected model), the fluent
loaded persists as long as possible. Therefore, the gun remains loaded until it is
fired at Fred, and Fred dies. In this model, at time 5, shooting is abnormal with
respect to Fred’s being alive. In the other, unexpected model, the fluent alive
persists as long as possible (i.e., Fred is alive after the shooting). Therefore,
the fluent loaded did not persist; somehow the gun has become unloaded. That
is, in some situation between 2 and 5, the empty action Wait was abnormal
with respect to the gun being loaded. This existence of multiple extensions has
created a genuine problem. Hanks and McDermott in fact argued that the Yale
Shooting Problem underscored the inadequacy of the whole logicist temporal
reasoning.
Early solutions to the Yale Shooting problem has been based on chronological
minimization, but they quickly lost favor mainly due to their inability to handle
backward temporal reasoning. As was rightly pointed out in [McDermott, 1987],
the main source of the problem was that the previous nonmonotonic logics drew
conclusions that minimized disturbances instead of avoiding disturbances with
unknown causes. And the emerged alternative approaches has begun to employ
various forms of causal reasoning.
[Lifschitz, 1987b] was the beginning of a sustained line of research in the
causal approach. The causal-based approach argued that we expect Fred to die
because there is an action that causes Fred’s death, but there is no action that
causes the gun to become unloaded. All the causal representations formalize this
principle in some way. Lifschitz has provided also a solution to the qualification
problem in this framework. Several modifications of Lifschitz’s solution have
been suggested (see, e.g., [Baker, 1989]).
A broadly explanatory approach to nonmonotonic reasoning was pursued
by Hector Geffner in [Geffner, 1992]. In the last chapters of his book, Geffner
argued for the necessity of incorporating causation as part of the meaning of
default conditionals. He used a unary modal operator Cp meaning ‘p is caused’
and expressed causal claims as conditionals of the form p → Cq. Using this
language, he formalized a simple causal solution to the Yale Shooting Problem.
In [Geffner, 1992], however, the causal theory was only sketched; in particular
the Ramification Problem was left untouched.
One of the first causal approaches, Motivated Action Theory [Morgenstern
38
and Stein, 1994], was based on the idea of preferring models in which unexpected
actions do not happen. A description of a problem scenario in MAT consisted of
a theory and a partial chronicle description. The theory contained causal rules
and persistence rules. Causal rules described how actions change the world;
persistence rules described how fluents remain the same over time. Central to
MAT was the concept of motivation. Intuitively, an action is motivated if there
is a “reason” for it to happen. The effects of causal chains are motivated in
this sense. A model is preferred if it has as few unmotivated actions as possible.
This kind of reasoning was clearly nonmonotonic, and it allowed for a solution
to the Yale Shooting Problem. In the expected model, where the gun remains
loaded and Fred dies, there are no unmotivated actions. In the unexpected
model, there is an unmotivated action – the Unload action. Thus we prefer the
expected models over the unexpected models. However, MAT falled short as
a solution to the frame problem due to its persistence rules – which are just
another form of frame axioms.
A first systematic causal representation of reasoning about action and change
has been developed by Fangzhen Lin in [Lin, 1995; 1996]. The representation
has allowed, in particular, to handle natural causes in addition to actions, and
provided a natural solution to the ramification problem. Lin’s representation
was formulated in a purely classical first-order language, but employed a (reified)
distinction between facts that hold in a situation versus facts that are caused in
it. The adequate models have been obtained by minimizing (i.e., circumscribing)
the caused facts.
A formalization of this causal reasoning in action theories has been suggested
in [McCain and Turner, 1997] in the framework of what they called causal theories. A causal theory is a set of causal rules that express causal relations among
propositions. The inertia (or persistence) principle is also expressed as a kind
of a causal rule. Then a true proposition can be caused either because it is
the direct or indirect effect of an action, or because it involves the persistence
of a caused proposition. Initial conditions are also considered to be caused,
by stipulation. Finally, the nonmonotonic semantics of a causal theory is determined by causally explained models, namely the models that both satisfy
the causal rules and such that every fact holding in them is caused by some
causal rule. In other words, in causally explained models the caused propositions coincide with the propositions that are true, and this must be the only
possibility consistent with the extensional part of the model. The resulting
nonmonotonic formalism has been shown to provide a plausible and efficient
solution for both the frame and ramification problem. See [Lifschitz, 1997;
Turner, 1999; Giunchiglia et al., 2004] for a detailed exposition of this theory
and applications in representing action domains. Related causal approaches
to representing actions and change have been suggested in [Thielscher, 1997;
Schwind, 1999; Zhang and Foo, 2001], to mention only a few.
The logical foundations of causal reasoning of this kind have been formulated
in [Bochman, 2003c; 2004a] in the framework of an inference system for causal
rules originated in input/output logics of [Makinson and van der Torre, 2000].
Formally, a causal inference relation is a binary relation ⇒ on the set of classical
39
propositions satisfying the following postulates:
(Strengthening)
(Weakening)
If A ⇒ B and B C, then A ⇒ C;
If A ⇒ B and A ⇒ C, then A ⇒ B ∧ C;
(And)
(Or)
If A B and B ⇒ C, then A ⇒ C;
If A ⇒ C and B ⇒ C, then A ∨ B ⇒ C.
(Cut)
(Truth)
(Falsity)
If A ⇒ B and A ∧ B ⇒ C, then A ⇒ C;
t ⇒ t;
f ⇒ f.
From a logical point of view, the most significant ‘omission’ of the above set
is the absence of the reflexivity postulate A ⇒ A. It is precisely this feature of
causal inference that creates a possibility of nonmonotonic reasoning. Causal
inference relations can be given a standard possible worlds semantics. Namely,
given a relational possible worlds model (W, R, V ), where W is a set of possible
worlds, R a binary accessibility relation on W , and V a valuation function, the
validity of causal rules can be defined as follows:
Definition. A rule A ⇒ B is valid in a possible worlds model (W, R, V ) if, for
any α, β ∈ W such that αRβ, if A holds in α, then B holds in β.
Causal inference relations are determined by possible worlds models in which
the relation R is quasi-reflexive, that is, αRβ holds only if αRα.
Causal rules are extended to rules with sets of propositions
V in premises by
stipulating that, for a set u of propositions, u ⇒ A holds if a ⇒ A for some
finite a ⊆ u. C(u) denotes the set of propositions explained by u:
C(u) = {A | u ⇒ A}
The production operator C plays the same role as the usual derivability
operator for consequence relations. In particular, it is a monotonic operator,
that is, u ⊆ v implies C(u) ⊆ C(v). Still, it does not satisfy inclusion, that is,
u ⊆ C(u) does not in general hold.
InVcausalWinference relations, any causal rule is reducible to a set of clausal
rules li ⇒ lj , where li , lj are classical literals. In addition, any rule A ⇒ B
is equivalent to a pair of rules A ∧ ¬B ⇒ f and A ∧ B ⇒ B.
The rules A ∧ B ⇒ B are explanatory rules. Though logically trivial, they
play an important explanatory role in causal reasoning by saying that, if A
holds, B is self-explanatory (and hence does not require explanation). Such
rules has been used in [McCain and Turner, 1997] for representing inertia or
persistence claims (see an example at the end of this section). On the other
hand, the rule A ∧ ¬B ⇒ f is a constraint that does not have an explanatory
content, but imposes a factual restriction A→B on the set of interpretations.
Causal inference relations determine also a natural nonmonotonic semantics, and provide thereby a logical basis for a particular form of nonmonotonic
reasoning.
40
Definition. A nonmonotonic semantics of a causal inference relation is the set
of all its exact worlds — maximal deductively closed sets u of propositions such
that u = C(u).
Exact worlds are worlds that are fixed points of the production operator C.
An exact world describes a model that is closed with respect to the causal rules
and such that every proposition in it is caused by other propositions accepted
in the model. Accordingly, they embody an explanatory closure assumption,
according to which any accepted proposition should also have explanation for
its acceptance. Such an assumption is nothing other than the venerable principle
of Universal Causation (cf. [Turner, 1999]).
The nonmonotonic semantics for causal theories is indeed nonmonotonic in
the sense that adding new rules to the causal relation may lead to a nonmonotonic change of the associated semantics, and thereby of derived information.
This happens even though causal rules themselves are monotonic, since they
satisfy the postulate of Strengthening (the Antecedent).
The nonmonotonic semantics of causal theories coincides with the semantics suggested in [McCain and Turner, 1997]. Moreover, it has been shown
in [Bochman, 2003c] that causal inference relations constitute a maximal logic
adequate for this kind of nonmonotonic semantics.
Causal inference relations and their variations have turned out to provide a
new general-purpose formalism for nonmonotonic reasoning with representation
capabilities stretching far beyond reasoning about action. In particular, it has
been shown in [Bochman, 2004a] that they provide a natural logical representation of abduction, based on treating abducibles as self-explanatory propositions satisfying reflexivity A ⇒ A. In addition, it has been shown in [Bochman,
2004b] that causal reasoning provides a precise interpretation for general logic
programs. Namely, any program rule c, not d ← a, not b can be translated as a
causal rule d, ¬b ⇒ ∧a →∨c. Then the reasoning underlying the stable model semantics for logic programs (see above) is captured by augmenting the resulting
causal theory with the causal version of the Closed World Assumption stating
that all negated atoms are self-explanatory:
Default Negation ¬p ⇒ ¬p, for any propositional atom p.
The causal nonmonotonic semantics of the resulting causal theory will correspond precisely to the stable semantics of the source logic program. Moreover,
unlike known embedding of logic programs into other nonmonotonic formalisms,
namely default and autoepistemic logics, the causal interpretation of logic programs turns out to be bi-directional in the sense that any causal theory is
reducible to a general logic program.
As can be shown, a world α is an exact world of a causal inference relation
if and only if, for any propositional atom p,
p ∈ α if and only if α ⇒ p
and
¬p ∈ α if and only if α ⇒ ¬p.
41
By the above description, the exact worlds of a causal relation are determined
ultimately by rules of the form A ⇒ l, where l is a literal. Such rules are called
determinate. McCain and Turner have established an important connection
between the nonmonotonic semantics of a determinate causal theory and Clark’s
completion of the latter. A finite causal theory ∆ is definite, if it consists of
determinate rules, or rules A ⇒ f , where f is a falsity constant. A completion
of such a theory is the set of all classical formulas
_
_
p ↔ {A | A ⇒ p ∈ ∆}
¬p ↔ {A | A ⇒ ¬p ∈ ∆}
for any propositional atom p, plus the set {¬A | A ⇒ f ∈ ∆}. Then the classical models of the completion precisely correspond to exact worlds of ∆ (see
[Giunchiglia et al., 2004]).
The completion formulas embody two kinds of information. As (forward)
implications from right to left, they contain the material implications corresponding to the causal rules from ∆. In addition, left-to-right implications
state that a literal belongs to the model only if one of its causes is also in
the model. These implications reflect the impact of causal descriptions using
classical logical means. In other words, the completion of a causal theory is a
classical logical theory that embodies the required causal content, so it obliterates in a sense the need in causal representation. In fact, a classical theory
of actions and change based directly on such a completion has been suggested
by Ray Reiter as a simple solution to the frame problem — see [Reiter, 1991;
2001]. This solution can now be reformulated as follows.
Reiter’s simple solution. The following toy representation contains the
main ingredients of causal reasoning in temporal domains. The temporal behavior of a propositional fluent F is described using two propositional atoms F0
and F1 saying, respectively, that F holds now and F holds in the next moment.
C − ⇒ ¬F1
C + ⇒ F1
F0 ∧ F1 ⇒ F1
F0 ⇒ F0
¬F0 ∧ ¬F1 ⇒ ¬F1
¬F0 ⇒ ¬F0 .
The first pair of causal rules describes the actions or natural factors that
can cause F and, respectively, ¬F (C + and C − normally describe the present
situation). Second, we have a pair of inertia axioms, purely explanatory rules
stating that if F holds (does not hold) now, then it is self-explanatory that it
will hold (resp., not hold) in the next moment. The last pair of initial axioms
states that F0 is an exogenous parameter.
The above causal theory is determinate, and its completion is as follows:
F1 ↔ C + ∨ (F0 ∧ F1 )
¬F1 ↔ C − ∨ (¬F0 ∧ ¬F1 ).
These formulas are equivalent to the conjunction of ¬(C + ∧ C − ) and
F1 ↔ C + ∨ (F0 ∧ ¬C − ).
42
The above formulas provide an abstract description of Reiter’s simple solution: the first formula corresponds to his consistency condition, while the last
one - to the successor state axiom for F .
4
Preferential Nonmonotonic Reasoning
As a theory of a rational use of assumptions, the main problem nonmonotonic
reasoning deals with is that assumptions are often incompatible with one another, or with known facts. In such cases of conflict we must have a reasoned
choice. The preferential approach follows here the slogan “Choice presupposes
preference”. According to this approach, the choice of assumptions should be
made by forming the space of options for choice and establishing preference relations among them. This makes preferential approach a special case of a general
methodology that is at least as old as the decision theory and theory of social
choice.
McCarthy’s circumscription can be seen as the ultimate origin of the preferential approach. A generalization of this approach was initiated by Gabbay
in [Gabbay, 1985] on the logical side, and by Shoham [Shoham, 1988] on the AI
side. A first overview of the preferential approach has been given in [Makinson,
1994]. A detailed description of the approach as we see it now can be found in
[Bochman, 2001; Schlechta, 2004; Makinson, 2005]; see also the paper of Karl
Schlechta in this volume.
Both the Closed World Assumption and circumscription can be seen as working on the principle of preferring interpretations in which positive facts are minimized; this idea was pursued already in [Lifschitz, 1985]. Generalizing this idea,
[Shoham, 1988] argued that any form of nonmonotonicity necessarily involves
minimality of one kind or another. He argued also for a shift in emphasis from
syntactic characterizations in favor of semantic ones. Namely, the relevant nonmonotonic entailment should be defined in terms of truth in all those models
of a given axiomatization minimal with respect to some application dependent
criterion. The ability to characterize such minimality criteria axiomatically is
not essential. In effect, on Shoham’s view, an axiomatization of an application
domain coupled with a characterization of its preferred minimal models is a
sufficient specification of the required entailments.
Shoham defined a model preference logic by using an arbitrary preference
ordering of the interpretations of a language.
Definition. An interpretation i is a preferred model of A if it satisfies A and
there is no better interpretation j > i satisfying A. A preferentially entails B
(written A |∼ B) iff all preferred models of A satisfy B.
In support of his conclusions, Shoham offered his own theory of temporal
minimization, as well as a minimal model semantics for a certain simplification
of Reiter’s default logic.
Shoham’s approach was very appealing, and suggested a unifying perspective
on nonmonotonic reasoning. This treatment of nonmonotonicity was also similar
43
to the earlier modal semantic theories of conditionals and counterfactuals, which
have been studied in the philosophical literature — see, e.g., [Stalnaker, 1968;
Lewis, 1973]. Just as the nonmonotonic entailment, counterfactual conditionals
do not satisfy monotonicity, that is, A > C does not imply A ∧ B > C. The
interrelations between these two theories have become an important theme.
Thus, [Delgrande, 1987; 1988] employed conditional logics for reasoning about
typicality.
The paper [Kraus et al., 1990] constituted a turning point in the development of the preferential approach. Based on Gabbay’s earlier description of
generalized inference relations, given in [Gabbay, 1985], and on semantic ideas
of [Shoham, 1988], the authors described both semantics and axiomatization for
a range of nonmonotonic inference relations. They also strongly argued that
preferential conditionals provide a more adequate and versatile formalization of
the notion of normality than, say, default logic. [Kraus et al., 1990] has established the logical foundations for a research program that has attracted many
researchers, both in AI and in logic. It has been found, in particular, that the
new approach to nonmonotonic reasoning is intimately connected also with the
influential theory of belief change suggested in [Alchourrón et al., 1985]. As a
consequence, an alternative but equivalent formalization of nonmonotonic inference relations was developed in [Gärdenfors and Makinson, 1994] based on
expectation ordering of classical formulas.
It is worth mentioning here, however, that despite the general enthusiasm
with this new approach, there have also been voices of doubt. Already [Reiter,
1987a] has warned against overly hasty generalizations when it comes to nonmonotonic reasoning. Moreover, Reiter has rightly argued that, for the purposes
of representing nonmonotonic reasoning, these preferential logics have two fatal
flaws; they are (globally) monotonic and extremely weak. On Reiter’s view,
nonmonotonicity is achieved in these logics by pragmatic considerations affecting how the logic is used, which destroys the principled semantics on which these
logics were originally based.
4.1
Epistemic States
Semantic interpretation constitutes one of the main components of a viable reasoning system, monotonic or not. A formal inference engine, though important,
can be effectively used for representing and solving reasoning tasks only if its
basic notions have clear meaning allowing to discern them from a description of
a situation at hand.
The standard semantics of preferential inference relations is based on abstract possible worlds models in which worlds are ordered by a preference relation. A more specific semantic interpretation for such inference relations, suitable for nonmonotonic reasoning, can be obtained, however, based on preference
ordering on sets of default assumptions. This strategy was pursued by Hector
Geffner in [Geffner, 1992] in the context of an ambitious general project in
nonmonotonic reasoning, which showed also how to apply the preferred model
approach to particular reasoning problems. This more specific interpretation
44
also subsumes the approach to nonmonotonic reasoning employed in Poole’s
Theorist system [Poole, 1988a] (see above).
It has been suggested in [Bochman, 2001] that a general representation
framework for preferential nonmonotonic reasoning can be given in terms of
epistemic states, defined below.
Definition. An epistemic state is a triple (S, l, ≺), where S is a set of admissible
belief states, ≺ a preference relation on S, while l is a labeling function assigning
a deductively closed belief set to every state from S.
On the intended interpretation, admissible belief states are generated as logical closures of allowable combinations of default assumptions. Such states are
taken to be the options for choice. The preference relation on admissible belief
states reflects the fact that not all admissible combinations of defaults constitute equally preferred options for choice. For example, defaults are presumed
to hold, so an admissible belief state generated by a larger set of defaults is
normally preferred to an admissible state generated by a smaller set of defaults.
In addition, not all defaults are born equal, so they may have some priority
structure that imposes, in turn, additional preferences among belief states (see
below).
Epistemic states guide our decisions what to believe in particular situations.
They are epistemic, however, precisely because they say nothing directly about
what is actually true, but only what is believed (or assumed) to hold. This makes
epistemic states relatively stable entities; change in facts and situations will not
necessary lead to change in epistemic states. The actual assumptions made in
particular situations are obtained by choosing preferred admissible belief states
that are consistent with the facts.
4.1.1
Prioritization
An explicit construction of epistemic states generated by default bases provides
us with characteristic properties of epistemic states arising in particular reasoning contexts. An epistemic state is base-generated by a set ∆ of propositions
with respect to a classical Tarski consequence relation Th if
• the set of its admissible states is the set P(∆) of subsets of ∆;
• l is a function assigning each Γ ⊆ ∆ a theory Th(Γ);
• the preference order is monotonic on P(∆): if Γ ⊂ Φ, then Γ ≺ Φ.
The preference order on admissible belief states is usually derived in some
way from priorities among individual defaults. This task turns out to be a
special case of a general problem of combining a set of preference relations into
a single ‘consensus’ preference order. Let us suppose that the set of defaults
∆ is ordered by some priority relation C which will be assumed to be a strict
partial order: α C β will mean that α is prior to β.
Recall that defaults are beliefs we are willing to hold insofar as it is consistent
to do so. Hence any default δ determines a primary preference relation 4δ on
45
P(∆) by which admissible belief sets containing the default are preferred to
belief sets that do not contain it:
Γ 4δ Φ ≡ if δ ∈ Γ then δ ∈ Φ
Each 4δ is a weak order having just two equivalence classes, namely sets
of defaults that contain δ, and sets that don’t. In this setting, the problem
of finding a global preference order amounts to constructing an operator that
maps a set of preference relations {4δ | δ ∈ ∆} to a single preference relation 4
on P(∆). As has been shown in [Andreka et al., 2002], any finitary operator of
this kind satisfying the so-called Arrow’s conditions is definable using a priority
graph (N, C, v), where C is a priority order on a set of nodes N , and v is a
labeling function assigning each node a preference relation. The priority graph
determines a single resulting preference relation via the lexicographic rule, by
which t is weakly preferred to s overall if it is weakly preferred for each argument preference, except possibly those for which there is a prior preference that
strictly prefers t to s:
s 4 t ≡ ∀i ∈ N (s 4v(i) t ∨ ∃j ∈ N (j C i ∧ s ≺v(j) t))
In our case, the prioritized base (∆, C) can be viewed as a priority graph in
which every node δ is assigned a preference relation 4δ . Consequently, we can
apply the lexicographic rule and arrive at the following definition:
Γ 4 Φ ≡ (∀α ∈ Γ\Φ)(∃β ∈ Φ\Γ)(β C α)
Γ 4 Φ holds when, for each default in Γ \ Φ, there is a prior default in Φ \ Γ.
The corresponding strict preference Γ ≺ Φ is defined as Γ 4 Φ ∧ Γ 6= Φ.
[Lifschitz, 1985] was apparently the first to use this construction in prioritized circumscription, while [Geffner, 1992] employed it for defining preference
relations among sets of defaults (see also [Grosof, 1991]).
4.2
Nonmonotonic inference and its kinds
In particular situations, we restrict our attention to admissible belief sets that
are consistent with the facts, and choose preferred among them. The latter are
used to support the assumptions and conclusions we make about the situation
at hand. Accordingly, all kinds of nonmonotonic inference relations, described
below, presuppose a two-step selection procedure: for a current evidence A, we
consider admissible belief states that are consistent with A and choose preferred
elements in this set.
An admissible belief state s ∈ S will be said to be compatible with a proposition A, if ¬A does not belong to its belief set, that is, ¬A ∈
/ l(s). The set of
all admissible states that are compatible with A will be denoted by hAi.
A skeptical inference (or prediction) with respect to an epistemic state is
obtained when we infer only what is supported by each of the preferred states.
In other words, B will be a skeptical conclusion from the evidence A in an
epistemic state E if each preferred admissible belief set in E that is consistent
with A, taken together with A itself, implies B.
46
Definition. B is a skeptical consequence of A (notation A |∼ B) in an epistemic
state if A → B is supported by all preferred belief states in hAi.
A set of conditionals A |∼ B that are valid in an epistemic state E will be
called a skeptical inference relation determined by E.
The above definition generalizes the notion of prediction from [Poole, 1988a],
as well as an expectation-based inference of [Gärdenfors and Makinson, 1994].
But in fact, it is much older. While the semantics of nonmonotonic inference
from [Kraus et al., 1990] derives from the possible worlds semantics of Stalnaker
and Lewis, the above definition can be traced back to the era before the discovery
of possible worlds, namely to Frank Ramsey and John S. Mill:
“In general we can say with Mill that ‘If p then q’ means that q
is inferrable from p, that is, of course, from p together with certain
facts and laws not stated but in some way indicated by the context.”
[Ramsey, 1978, page 144]
This definition has also been used in the ‘premise-based’ semantics for counterfactuals [Veltman, 1976; Kratzer, 1981] (see also [Lewis, 1981]).
A credulous inference (or explanation) with respect to an epistemic state is
obtained by assuming that we can reasonably infer (or explain) conclusions that
are supported by at least one preferred belief state consistent with the facts. In
other words, B will be a credulous conclusion from A if at least one preferred
admissible belief set in E that is consistent with A, taken together with A itself,
implies B. In still other words,
Definition. B is a credulous consequence of A in an epistemic state if A→B is
supported by at least one preferred belief state in hAi.
The set of conditionals that are credulously valid in an epistemic state E
forms a credulous inference relation determined by E. The above definition
constitutes a generalization of the corresponding definition of explanation in
Poole’s abductive system [Poole, 1988a]. Credulous inference is only one, though
important, instance of a broad range of non-skeptical inference relations (see
[Bochman, 2003a]).
4.3
Syntactic characterizations
As we mentioned, [Gabbay, 1985] was a starting point of the approach to nonmonotonic reasoning based on describing associated inference relations. This
approach was primarily designed to capture the skeptical view of nonmonotonic
reasoning. We have seen, however, that many reasoning tasks in AI, such as
abduction and diagnosis, are based on a credulous understanding of nonmonotonic reasoning. A common ground for both skeptical and credulous inference
can be found in the logic of conditionals suggested in [van Benthem, 1984].
The main idea behind van Benthem’s approach was that a conditional can
be seen as a generalized quantifier representing a relation between the respective
sets of instances or situations supporting and refuting it. A situation confirms
47
a conditional A |∼ B if it supports the classical implication A→B, and refutes
A |∼ B if it supports A→¬B. Then the validity of a conditional in a set of
situations is determined by appropriate, ‘favorable’ combinations of confirming
and refuting instances. Whatever they are, we can assume that adding new
confirming instances to a valid conditional, or removing refuting ones, cannot
change its validity. Accordingly, we can accept the following principle:
If all situations confirming A |∼ B confirm also C |∼ D, and all situations refuting C |∼ D refute also A |∼ B, then validity of A |∼ B
implies validity of C |∼ D.
The above principle is already sufficient for justifying the rules of the basic inference relation, given below. It supports also a representation of conditionals in terms of expectation relations. More exactly, we can say that a
conditional is valid if the set of its confirming situations is sufficiently good
compared with the set of its refuting situations. Accordingly, A |∼ B can be
defined as A→¬B < A→B for an appropriate expectation relation <, while the
above principle secures that this is a general expectation relation as defined in
[Bochman, 2001].
A basic inference relation B satisfies the following postulates:
Reflexivity
A |∼ A
Left Equivalence
Right Weakening
Antecedence
Deduction
If A ↔ B and A |∼ C, then B |∼ C
If A |∼ B and B C, then A |∼ C
If A |∼ B, then A |∼ A ∧ B
If A ∧ B |∼ C, then A |∼ B → C
Very Cautious Monotony
If A |∼ B ∧ C, then A ∧ B |∼ C
A set Γ of conditionals implies a conditional α with respect to basic inference
if α belongs to the least basic inference relation containing Γ. Note, however,
that all the above postulates involve at most one conditional premise. As a
result, the basic entailment boils down to a derivability relation among single
conditionals. The following theorem describes this derivability relation in terms
of the classical entailment.
Theorem. A |∼ B B C |∼ D if and only if either C D, or A → B C → D
and C→¬D A→¬B.
Theorem 4.3 justifies the principle stated earlier: A |∼ B implies C |∼ D if
and only if either all situations confirm C |∼ D, or else all confirming instances
of the former are confirming instances of the latter and all refuting instances of
the latter are refuting instances of the former.
Basic inference relation is a weak inference system, first of all because it does
not allow to combine different conditionals. Nevertheless, it is in a sense complete so far as we are interested in derivability among individual conditionals.
48
More exactly, basic derivability captures exactly the one-premise derivability of
both skeptical and credulous inference relations.
As was noted already in [Gabbay, 1985], a characteristic feature of sceptical
inference is the validity of the following postulate:
(And) If A |∼ B and A |∼ C, then A |∼ B ∧ C.
Indeed, in the framework of basic inference, And is all we need for capturing
precisely the preferential inference relations from [Kraus et al., 1990]. Such
inference relations provide a complete axiomatization of skeptical inference with
respect to epistemic states. An important special case of preferential inference,
rational inference relations, are determined by linearly ordered epistemic states;
they are obtained by adding further
(Rational Monotony) If A |∼ B and A | ¬C, then A ∧ C |∼ B.
In contrast, credulous inference relations do not satisfy And. Still, they are
axiomatized as basic inference relations satisfying Rational Monotony.
Rational Monotony is not a ‘Horn’ rule, so it does not allow us to derive new
conditionals from given ones. In fact, credulous inference relations do not derive
much more conditionals than what can be derived already by basic inference
(see [Bochman, 2001]). This indicates that there should be no hope to capture
credulous nonmonotonic reasoning by derivability in some nonmonotonic logic.
Something else should be added to the above logical framework in order to
represent the relevant form of nonmonotonic reasoning. Though less evident,
the same holds for skeptical inference. Both these kinds of inference need to be
augmented with an appropriate globally nonmonotonic semantics that would
provide a basis for the associated systems of defeasible entailment, as described
in the next section.
4.4
Defeasible entailment
A theory of reasoning about default conditionals should occupy an important
place in the general theory of nonmonotonic reasoning. Thus, the question
whether a proposition B is derivable from an evidence A in a default base
is reducible to the question whether the conditional A |∼ B is derivable from
the base, so practically all reasoning problems about default conditionals are
reducible to the question what conditionals can be derived from a conditional
default base. The latter problem constitutes therefore the main task of a theory
of default conditionals (see [Lehmann and Magidor, 1992]).
For a skeptical reasoning, a most plausible understanding of default conditionals is obtained by treating them as skeptical inference rules in the framework
of epistemic states. Accordingly, preferential inference relations of [Kraus et al.,
1990] can be considered as a logic behind skeptical nonmonotonic reasoning; the
rules of the former should be taken for granted by the latter.
This does not mean, however, that nonmonotonic reasoning about default
conditionals is reducible to preferential derivability. Preferential inference is
49
severely sub-classical and does not allow us, for example, to infer “Red birds
fly” from “Birds fly”. In fact, this is precisely the reason why such inference
relations have been called nonmonotonic. Clearly, there are good reasons for not
accepting such a derivation as a logical rule for preferential inference; otherwise
“Birds fly” would imply also “Birds with broken wings fly” and even “Penguins
fly”. Still, this should not prevent us from accepting “Red birds fly” on the basis
of “Birds fly” as a reasonable nonmonotonic (or defeasible) conclusion, namely
a conclusion made in the absence of information against it. By doing this, we
would just follow the general strategy of nonmonotonic reasoning that involves
making reasonable assumptions on the basis of available information. Thus,
the logical core of skeptical inference, preferential inference relations, should be
augmented with a mechanism of making nonmonotonic conclusions. This kind of
reasoning will of course be defeasible, or globally nonmonotonic, since addition
of new conditionals can block some of the conclusions made earlier. Note in
this respect that, though preferential inference is (locally) nonmonotonic with
respect to premises of conditionals, it is nevertheless globally monotonic: adding
new conditionals does not change the validity of previous derivations.
On the semantic side, default conditionals are constraints on epistemic states
in the sense that the latter should make them skeptically valid. Still, usually
there is a huge number of epistemic states that satisfy a given set of conditionals,
so we have both an opportunity and necessity to choose among them. Our
guiding principle in this choice can be the same basic principle of nonmonotonic
reasoning, namely that the intended epistemic states should be as normal as is
permitted by the current constraints. By choosing particular such states, we
thereby will adopt conditionals that would not be derivable from a given base
by preferential inference alone.
The above considerations lead to a seemingly inevitable conclusion that default conditionals possess a clear logical meaning and associated logical semantics based on epistemic states (or possible worlds models), but they still lack a
globally nonmonotonic semantics that would provide an interpretation for the
associated defeasible entailment.
Actually, the literature on nonmonotonic reasoning is abundant with such
theories of defeasible entailment. A history of studies on this subject could
be summarized as follows. Initial formal systems, namely Lehmann’s rational closure [Lehmann, 1989; Lehmann and Magidor, 1992] and Pearl’s system
Z [Pearl, 1990], have turned out to be equivalent. This encouraging development has followed by a realization that both theories are insufficient for representing defeasible entailment, since they do not allow to make certain intended conclusions. Hence, they have been refined in a number of ways, giving such systems as lexicographic inference [Benferhat et al., 1993; Lehmann,
1995], and similar modifications of Pearl’s system [Goldszmidt et al., 1993;
Tan and Pearl, 1995]. Unfortunately, these refined systems have encountered an
opposite problem, namely, together with some desirable properties, they invariably produced some unwanted conclusions. All these systems have been based
on a supposition that defeasible entailment should form a rational inference relation. A more general approach in the framework of preferential inference has
50
been suggested in [Geffner, 1992].
Yet another, more syntactic, approach to defeasible entailment has been pursued in the framework of inheritance hierarchies (see [Horty, 1994]). Inheritance
reasoning deals with a quite restricted class of conditionals constructed from
literals. Nevertheless, in this restricted domain it has achieved a remarkably
close correspondence between what is derived and what is expected intuitively.
Accordingly, inheritance reasoning has emerged as an important test bed for
adjudicating proposed theories.
Despite the diversity, the systems of defeasible entailment have a lot in common, and take as a starting point a few basic principles. Thus, most of them
presuppose that intended models should be described, ultimately, in terms of
material implications corresponding to a given set of conditionals. More exactly, these classical implications should serve as defaults in the nonmonotonic
reasoning sanctioned by a default base. This idea can be made precise as follows:
Default base-generation. The intended epistemic states for a
default base B should be base-generated by the corresponding set
~ of material implications.
B
In other words, the admissible belief states of intended epistemic states for
~ and the preference order
a default base B should be formed by subsets of B,
should be monotonic on these subsets (see Section 4.1.1). In addition, it should
be required that all the conditionals from B should be skeptically valid in the
resulting epistemic state.
Already these constraints on intended epistemic states allow us to derive
“Red birds fly” from “Birds fly” for all default bases that do not contain conflicting information about redness. The constraints also sanction defeasible entailment across exception classes: if penguins are birds that normally do not
fly, while birds normally fly and have wings, then we are able to conclude that
penguins normally have wings, despite being abnormal birds. This excludes, in
effect, Pearl’s system Z and rational closure that cannot make such a derivation. Still, these requirements are quite weak and do not produce problematic
conclusions that plagued some stronger systems suggested in the literature.
Unfortunately, though the above constraints deal successfully with many examples of defeasible entailment, they are still insufficient for capturing some
important reasoning patterns. What is missing in our construction is a principled way of constructing a preference order on default sets. This problem
has turned out to be far from being trivial or univocal. As for now, two most
plausible solutions to this problem, suggested in the literature, are Geffner’s
conditional entailment and inheritance reasoning.
Conditional entailment [Geffner, 1992] determines a prioritization of default
bases by making use of the following relation among conditionals:
Definition. A conditional α dominates a set of conditionals Γ if the set of
implications {~Γ, α
~ } is incompatible with the antecedent of α.
The origins of this relation can be found already in [Adams, 1975], and it has
been used in practically all studies of defeasible entailment, including the notion
51
of preemption in inheritance reasoning. A suggestive reading of dominance says
that if α dominates Γ, it should have priority over at least one conditional in
Γ7 . Accordingly, a priority order on the default base is admissible if it satisfies
this condition. Then the intended models can be identified with epistemic states
that are generated by all admissible priority orders on the default base (using
the lexicographic rule - see above).
Conditional entailment has shown itself as a serious candidate on the role of
a general theory of defeasible entailment. Still, it does not capture inheritance
reasoning. The main difference between the two theories is that conditional
entailment is based on absolute priorities among defaults, while inheritance
hierarchies determine such priorities in a context-dependent way, namely in
presence of other defaults that provide a (preemption) link between two defaults
(see [Dung and Son, 2001]). Indeed, it has been shown in [Bochman, 2001]
that inheritance reasoning is representable by epistemic states that are basegenerated by default conditionals ordered by certain conditional priority orders.
Still, the corresponding construction could hardly be called simple or natural.
A more natural representation of inheritance reasoning has been given in
[Dung and Son, 2001] as an instantiation of an argumentation theory that belongs already to explanatory nonmonotonic formalisms, discussed in the next
section. Furthermore, Geffner himself has shown in [Geffner, 1992] that conditional entailment still does not capture some important derivations, and it
should be augmented with an explicit representation of causal reasoning. In
fact, the causal generalization suggested by Geffner in the last chapters of his
book has served as one of the inspirations for a causal theory of reasoning about
actions and change (see [Turner, 1999]).
5
Explanatory nonmonotonic reasoning
The ‘mainstream’ approach to nonmonotonic reasoning includes default and
modal nonmonotonic logics, logic programming, abductive and causal reasoning. We will call this approach explanatory nonmonotonic reasoning, since explanation can be seen as its basic ingredient. Propositions and facts may be not
only true or false in a model of a problem situation, but some of them are explainable (justified) by other facts and rules that are accepted. In the epistemic
setting, some of the propositions are derivable from other propositions using
rules that are admissible in the situation. In the objective setting, some of the
facts are caused by other facts and causal rules acting in the domain. Furthermore, explanatory nonmonotonic reasoning is based on very strong principles
of Explanation Closure or Causal Completeness (see [Reiter, 2001]), according
to which any fact holding in a model should be explained, or caused, by the
rules that describe the domain. Incidentally, it is these principles that make
explanatory reasoning nonmonotonic.
By the above description, abduction, that is, reasoning from facts to their
explanations, is an integral part of explanatory nonmonotonic reasoning. Ulti7 This
secures that α will be valid in the resulting epistemic state.
52
mate explanations, or abducibles, correspond not to normality defaults, but to
conjectures representing base causes or facts that do not require explanation;
we assume the latter only for explaining some evidence.
In some domains, explanatory formalisms adopt simplifying assumptions
that exempt, in effect, certain propositions from the burden of explanation.
Closed World Assumption [Reiter, 1978] is the most important assumption of
this kind. According to it, negative assertions do not require explanation. Nonmonotonic reasoning in databases and logic programming are domains for which
such an assumption turns out to be most appropriate. It is important to note
that the minimization principle that has been the source of the preferential approach can also be derived as a result of combining Explanation Closure with
the Closed World Assumption (see below). Consequently, it need not be viewed
as a principle of scaled preference of negative information; rather, it could be
seen as a by-product of the stipulation that negated propositions can be accepted without any further explanation, while positive assertions always require
explanation. This understanding allows us to explain why McCarthy’s circumscription, that is based on the principle of minimization, is subsumed also by
explanatory formalisms.
The above principles form an ultimate basis for all formal systems of explanatory nonmonotonic reasoning. They presuppose, however, a richer picture
of what is in the world than what is usually captured in logical models of the
latter. The traditional understanding of possible worlds in logic stems from
the Tractatus’ metaphysics where ‘[e]ach item can be the case or not the case
while everything else remains the same’ ([Wittgenstein, 1961]). Consequently,
there is no way of making an inference from one fact to another, and there is
no causal nexus to justify such an inference. The only restriction on the structure of the world is the principle of non-contradiction. Such worlds leave no
place for dependencies among facts and related notions, in particular for causation. Wittgenstein himself concluded that belief in such dependencies is a
superstition.
An alternative picture depicts the world not as a mere assemblage of unrelated facts, but as something that has a structure. This structure determines
dependencies among occurrent facts that serve as a basis for our explanatory
and causal claims. It is this structure that makes the world intelligible and,
what is especially important for AI, controllable. By this picture, explanatory
and causal relations form an integral part of understanding of and acting in the
world. Consequently, such relations should form an essential part of knowledge
representation, at least in Artificial Intelligence.
5.1
Biconsequence Relations
A uniform account of explanatory nonmonotonic formalisms can be given in the
framework of biconsequence relations described in this section. Biconsequence
relations are specialized consequence relations for reasoning with respect to a
pair of contexts. On the interpretation suitable for nonmonotonic reasoning, one
of these contexts is the main (objective) one, while the other context provides
53
assumptions, or explanations, that justify inferences in the main context. This
separation of inferences and their justifications creates a possibility of explanatory nonmonotonic reasoning.
The two contexts will be termed, respectively, the context of truth and the
context of falsity. In the truth context the propositions are evaluated as being
either true or non-true, while in the falsity context they can be false or nonfalse. As a benefit of this terminological decision, a bi-context reasoning can
also be interpreted as a reasoning with possibly inconsistent and incomplete
information. Furthermore, such a reasoning can also be viewed as a four-valued
reasoning (see [Belnap, 1977]).
A bisequent is an inference rule of the form a : b c : d, where a, b, c, d
are finite sets of propositions. We will employ two informal interpretations of
bisequents. According to the four-valued interpretation, it says
‘If all propositions from a are true and all propositions from b are
false, then either one of the propositions from c is true or one of the
propositions from d is false’.
According to the explanatory (or assumption) interpretation, it says
‘If no proposition from b is assumed, and all propositions from d
are assumed, then all propositions from a hold only if one of the
propositions from c holds’.
A biconsequence relation is a set of bisequents satisfying the rules:
Monotonicity
Reflexivity
Cut
a:bc:d
, if a ⊆ a0 , b ⊆ b0 , c ⊆ c0 , d ⊆ d0 ;
: b0 c0 : d0
a0
A:A:
and
: A : A;
a : b A, c : d A, a : b c : d
a:bc:d
a : b c : A, d a : A, b c : d
.
a:bc:d
A biconsequence relation can be seen as a fusion, or fibring, of two Scott
consequence relations much in the sense of [Gabbay, 1998]. This fusion gives a
syntactic expression to combining two independent contexts.
The definition of a biconsequence relation is extendable to arbitrary sets of
propositions by accepting the compactness requirement:
Compactness
u : v w : z iff a : b c : d,
for some finite sets a, b, c, d such that a ⊆ u, b ⊆ v, c ⊆ w and d ⊆ z.
For a set u of propositions, u will denote the set of propositions that do
not belong to u. A pair (u, v) of sets of propositions will be called a bitheory
of a biconsequence relation if u : v 1 u : v. A set u of propositions is a
(propositional) theory of , if (u, u) is a bitheory of .
54
Bitheories can be seen as pairs of sets that are closed with respect to the
bisequents of a biconsequence relation. A bitheory (u, v) of is positively minimal, if there is no bitheory (u0 , v) of such that u0 ⊂ u. Such bitheories play
an important role in describing nonmonotonic semantics.
By a bimodel we will mean a pair of sets of propositions. A set of bimodels
will be called a binary semantics.
Definition. A bisequent a : b c : d is valid in a binary semantics B, if, for
any (u, v) ∈ B, if a ⊆ u and b ⊆ v, then either c ∩ u 6= ∅, or d ∩ v 6= ∅.
The set of bisequents that are valid in a binary semantics forms a biconsequence relation. On the other hand, any biconsequence relation is determined in this sense by its canonical semantics defined as the set of bitheories of
. Consequently, the binary semantics provides an adequate interpretation of
biconsequence relations.
According to Belnap’s idea, the four truth-values {>, t, f , ⊥} of a four-valued
interpretation can be identified with the four subsets of the set {t, f } of classical
truth-values, namely {t, f }, {t}, {f } and ∅. Thus, > means that a proposition is
both true and false (i.e., contradictory), t means that it is ‘classically’ true (that
is, true without being false), f means that it is classically false, while ⊥ means
that it is neither true nor false (undetermined). This representation allows us
to see any four-valued interpretation ν as a pair of ordinary interpretations
corresponding, respectively, to independent assignments of truth and falsity to
propositions:
ν |= A
iff t ∈ ν(A)
ν =| A
iff
f ∈ ν(A).
Now, a bimodel (u, v) can be viewed as a four-valued interpretation, where
u is the set of true propositions, while v is the set of propositions that are not
false. Biconsequence relations provide in this sense a syntactic formalism for
four-valued reasoning.
A bisequent theory is an arbitrary set of bisequents. For any bisequent
theory ∆ there is a least biconsequence relation ∆ containing it that describes
the logical content of ∆. This allows us to extend the notions of a bitheory and
propositional theory to arbitrary bisequent theories.
Two kinds of bisequents are of special interest for nonmonotonic reasoning.
The first are default bisequents a:b c: without negative conclusions that are
related to rules of default logic. The second are autoepistemic bisequents :b c:d
that are related to autoepistemic logic. A default bisequent says that if no
proposition from b is assumed, then all propositions from a hold only if one
of the propositions from c holds. Such a bisequent can be viewed as a Scott
inference rule a ` c that is conditioned by a set of negative assumptions (i.e.,
absence of b’s). Thus, such bisequents involve full inference capabilities with
respect to the main context, but permit only negative assumptions. In contrast,
an autoepistemic bisequent :b c:d says that if no proposition from b is assumed,
and all propositions from d are assumed, then one of the propositions from c
holds. Such rules have rich assumption capabilities, but allow us to make only
unconditional assertions.
55
In addition to the above distinction, we are often interested in singular bisequents, namely bisequents a:b C:d having a single proposition C as a positive
conclusion. Such bisequents can be seen as counterparts of Tarski rules. The
formalism of default consequence relations, introduced in [Bochman, 1994], provided, in effect, a uniform description of singular default and autoepistemic
bisequents as species of default rules of the form a:b ` C. The difference of
default vs. autoepistemic rules has been reflected, however, as a difference in
corresponding logics for default rules. This common representation has given a
convenient basis for a comparative study of default and modal nonmonotonic
reasoning (see [Bochman, 1998c]).
An important feature of biconsequence relations is a possibility of imposing
structural constraints on the binary semantics by accepting additional structural
rules. Some of them play an important role in nonmonotonic reasoning. Thus,
a biconsequence relation is consistent, if it satisfies
Consistency
A:A
On the explanatory interpretation, Consistency says that no proposition can
be taken to hold without assuming it. This amounts to restricting the binary
semantics to consistent bimodels, that is, bimodels (u, v) such that u ⊆ v. On
the four-valued representation, Consistency requires that no proposition can be
both true and false, so it determines a semantic setting of partial logic (see,
e.g., [Blamey, 1986]) that usually deals only with possible incompleteness of
information.
A biconsequence relation is regular if it satisfies
Regularity
b:aa:b
: a : b
Regularity is a kind of an assumption coherence constraint. It says that a
coherent set of assumptions should be such that it is compatible with taking
these assumptions as actually holding. A semantic counterpart of Regularity is
a quasi-reflexive binary semantics in which, for any bimodel (u, v), (v, v) is also
a bimodel.
Any four-valued connective is definable in biconsequence relations via a pair
of introduction rules and a pair of elimination rules corresponding to the two
valuations of a four-valued interpretation. We are primarily interested, however,
in information a bi-context reasoning can give us about ordinary, classical truth
and falsity, so we restrict attention to classical connectives that are conservative
on the subset {t, f }. Such connectives give classical truth-values when their
arguments receive classical values t or f .
There are four natural connectives that are jointly sufficient for defining all
classical four-valued functions. The first is the well-known conjunction:
ν |= A ∧ B
ν =| A ∧ B
iff
ν |= A and ν |= B
iff ν =| A or ν =| B.
56
Next, there are two negation connectives that can be seen as two alternative
extensions of classical negation to the four-valued setting:
ν |= ¬A
iff ν 6|= A
ν =| ¬A iff
ν 6=| A
ν |= ∼A
iff ν =| A
ν =| ∼A
ν |= A.
iff
We will call ¬ and ∼ a local and global negation, respectively. Each of them
can be used together with the conjunction to define a disjunction:
A ∨ B ≡ ∼(∼A ∧ ∼B) ≡ ¬(¬A ∧ ¬B).
Finally, the unary connective A can be seen as a kind of a modal operator,
closely related to the modal operator L that will be used in the modal extension
of our formalism.
ν |= AA
iff ν 6=| A
ν =| AA
iff ν =| A.
From the classical point of view, the most natural subclass of the classical
four-valued connectives is formed by connectives that behave as ordinary classical connectives with respect to each of the two contexts. Connectives of this
kind will be called locally classical. The conjunction ∧ and local negation ¬
form a functionally complete basis for all such connectives.
Having the four-valued connectives at our disposal, we can transform bisequents into more familiar inference rules, and even to ordinary logical formulas.
Let ¬u denote the set {¬A | A ∈ u}, and similarly for ∼u, etc. Then any
bisequent a : b c : d is equivalent to each of the following:
a, ∼b : c, ∼d :
(1)
¬a, c : d, ¬b
(2)
Bisequents of the form (1) can be seen as ordinary sequents. In fact, this
is a common trick used for representing many-valued logics in the form of a
sequent calculus. If the language contains also conjunction (or, equivalently,
disjunction), the set of premises can be replaced by their conjunction, while
the set of conclusions - by their disjunction. Consequently, we can transform
bisequents into Tarski-type rules A ` B. Actually, the resulting system will
coincide with a (flat) theory of relevant entailment [Dunn, 1976].
An important alternative possibility arises from using the representation
(2). This time we can use the local connectives {∧, ¬} in order to reduce such
bisequents to that of the form A : B, where A and B are classical propositions.
The latter bisequents will correspond to rules B ⇒ A of production and causal
inference relations, discussed later.
5.2
Nonmonotonic Semantics
Nonmonotonic semantics of a biconsequence relation is a certain set of its theories. Namely, such theories are explanatory closed in the sense that presence
and absence of propositions in the main context is explained (i.e., derived) using the rules of the biconsequence relation when the theory itself is taken as the
assumption context.
57
5.2.1
Exact semantics
The notion of an exact theory of a biconsequence relation provides us with a
simplest and most general kind of nonmonotonic reasoning.
Definition. A theory u of a biconsequence relation is exact, if there is no
other set v 6= u such that (v, u) is a bitheory of . The set of exact theories will
be called an exact nonmonotonic semantics of .
Exact theories correspond to bitheories, for which the assumption context
determines itself as a unique objective state compatible with it. Such theories
can be given the following syntactic description.
Lemma. A set u of propositions is an exact theory of a biconsequence relation
if and only if, for any proposition A,
A ∈ u iff : u A : u
and
A∈
/ u iff A : u : u.
The exact semantics is extendable to arbitrary bisequent theories by considering their associated biconsequence relations. Regular biconsequence relations
constitute a maximal logic suitable for the exact nonmonotonic semantics.
The above definition of nonmonotonic semantics leaves us much freedom
in determining nonmonotonic consequences of a bisequent theory. The most
obvious skeptical choice consists in taking propositions that belong to all exact
theories. As a credulous alternative, however, we can consider propositions that
belong to at least one theory. An even more general understanding amounts to
a view that all the information that can be discerned from the nonmonotonic
semantics of a bisequent theory or a biconsequence relation can be seen as
nonmonotonically implied by the latter.
Exact theories determine a truly nonmonotonic semantics, since adding new
bisequents to a bisequent theory may not only eliminate exact theories, but
also add new ones (or both). In other words, the set of exact theories does not
change monotonically with the growth of the set of bisequents.
Finally, if a bisequent theory contains only default bisequents a:b c:, then
any its exact theory will also be a minimal propositional theory.
5.2.2
Default semantics
A more familiar class of nonmonotonic models, extensions, correspond to extensions of default logic and stable models of logic programs.
Definition. A set u is an extension of a biconsequence relation , if (u, u) is
a positively minimal bitheory of . A default nonmonotonic semantics of a
biconsequence relation is the set of its extensions.
u is an extension if it is a theory of a biconsequence relation such that there
is no smaller set v ⊂ u such that (v, u) is a bitheory. Hence any exact theory of
a biconsequence relation is an extension, though not vice versa.
Let P r denote the set of all propositions of the language. The following
lemma provides a syntactic description of extensions.
58
Lemma. A set u is an extension of a biconsequence relation if and only if
u = {A | :u A, u:u} and either u 6= P r, or 1: P r.
By the above description, extensions are theories of a biconsequence relation
that explain only why they have the propositions they have. In other words,
for extensions we are relieved from the necessity of explaining why propositions
do not belong to the intended theory. This agreement constitutes the essence
of Reiter’s Closed World Assumption. Another consequence of the above description is that extensions are determined by the autoepistemic bisequents of
a biconsequence relation.
It turns out that the default nonmonotonic semantics is precisely an exact
semantics under a stronger logic of consistent biconsequence relations. The effect
of Consistency A : A amounts to an immediate refutation of any proposition
that is assumed not to hold. That is why absence of propositions from extensions
does not require explanation, only their presence. This makes the minimality
condition in the definition of extensions a consequence of the logical Consistency
postulate, instead of an independent ‘rationality principle’ of nonmonotonic
reasoning.
5.2.3
Bisequent interpretation of logic programs
Representations of general logic programs and their semantics in the logical
formalism of biconsequence relations establish the ways of providing (or, better,
restoring) a logical basis of logic programming.
Recall that a general logic program is a set of program rules not d, c ←
a, not b, where a, b, c, d are finite sets of propositional atoms. Such program rules
involve disjunctions and default negations in heads and subsume practically all
structural extensions of Prolog rules suggested in the literature.
A program rule not d, c ← a, not b can be directly interpreted as a bisequent
a : b c : d. Let bc(Π) denote the bisequent theory corresponding to a general
program Π under this interpretation. The following result shows that it provides an exact correspondence between stable models of a general program and
extensions of the associated bisequent theory.
Theorem. Stable models of a general logic program Π coincide with the extensions of bc(Π).
Moreover, the correspondence turns out to be bidirectional, since any bisequent in a four-valued language is logically reducible to a set of bisequents
without connectives; such bisequents can already be viewed as program rules.
5.3
Causal and Production Inference
As we mentioned earlier, in languages containing certain (four-valued) connectives, bisequents are reducible to much simpler rules. In particular, in a language with local classical connectives, bisequents are reducible to rules of the
form A ⇒ B, where A and B are classical logical formulas (cf. representation
59
(2) above). Such rules can be given an informal reading ‘A causes, or explains,
B’, and the resulting logical system will turn out to coincide with the system
causal inference described in Section 3.6.2. Speaking more formally, for a biconsequence relation in a local classical language, we can define the following
set of rules, called the production subrelation of :
⇒ = {A ⇒ B | B : A}
Then the production subrelation of a regular biconsequence relation will
form a causal inference relation. Moreover, for any causal inference relation ⇒
there is a unique regular biconsequence relation in the language with the local
connectives that has ⇒ as its production subrelation. By this correspondence, a
causal rule A ⇒ B can be seen as an assumption-based conditional saying that if
A is assumed, then B should hold. In this sense, the system of causal inference
constitutes a primary logical system of explanatory nonmonotonic reasoning,
whereas biconsequence relations form a structural counterpart of this logical
formalism. In particular, in the general correspondence between causal and biconsequence relations, the causal nonmonotonic semantics (see Definition 3.6.2)
corresponds to the exact nonmonotonic semantics of biconsequence relations.
As for biconsequence relations, the default nonmonotonic semantics of causal
theories can be obtained by imposing a causal postulate corresponding to the
Consistency postulate for biconsequence relations:
(Default Negation)
¬p ⇒ ¬p, for any propositional atom p.
Default Negation stipulates that negations of atomic propositions are selfexplanatory, and hence it provides a simple causal expression for Reiter’s Closed
World Assumption. As we already mentioned in Section 3.6.2, this kind of causal
inference can be used as a logical basis for logic programming.
Production inference. A useful generalization of causal inference is obtained
by dropping the postulate Or of causal inference that has allowed for reasoning
by cases. The resulting formalism has been called in [Bochman, 2004a] the
system of production inference. For this formalism, we can generalize also the
corresponding nonmonotonic semantics as follows:
Definition. A nonmonotonic production semantics of a production inference
relation is the set of all its exact theories, namely sets u of propositions such
that u = C(u).
As before, an exact theory describes an informational state in which every
proposition is explained by other propositions accepted in this state. Accordingly, restricting our universe of discourse to exact theories amounts to imposing
a kind of an explanatory closure assumption on intended models.
The nonmonotonic production semantics allows us to provide a formal representation of abductive reasoning that covers the main applications of abduction
in AI.
60
To begin with, abducibles can be identified with self-explanatory propositions
of a production relation, that is, propositions satisfying A ⇒ A. Then it turns
out that traditional abductive systems are representable via a special class of
abductive production inference relations that satisfy
(Abduction) If B ⇒ C, then B ⇒ A ⇒ C, for some abducible A.
It has been shown in [Bochman, 2005] that abductive inference relations provide a generalization of abductive reasoning in causal theories [Konolige, 1992;
Poole, 1994b], as well as of abduction in logic programming. On the other hand,
any production inference relation includes a greatest abductive subrelation, and
in many regular situations (e.g., when the production relation is well-founded)
the latter determines the same nonmonotonic semantics. Summing up, the general nonmonotonic semantics of a production relation is usually describable by
some abductive system, and vice versa.
5.4
Epistemic Explanatory Reasoning
Epistemic formalisms of default and modal nonmonotonic logics find their natural place in the framework of supraclassical biconsequence relations, defined
below. An epistemic understanding of biconsequence relations amounts to treating the main and assumption contexts, respectively, as the contexts of knowledge
and belief: propositions that hold in the main context can be viewed as known,
while propositions of the assumption context form the associated set of beliefs.
This understanding will later receive an explicit expression in the modal extension of the formalism. Even in a non-modal setting, however, the epistemic
reading implies that both contexts should correspond not to complete worlds,
but to incomplete deductive theories.
Definition. A biconsequence relation in a classical language is supraclassical,
if it satisfies
Supraclassicality If a A, then a : A : and : A : a.
Falsity
f:
and
: f.
In supraclassical biconsequence relations both contexts respect the classical
entailment. In addition, sets of positive premises and negative conclusions can
be replaced by their conjunctions, but positive conclusion sets and negative
premise sets are not replaceable in this way by classical disjunctions. Also, the
deduction theorem, contraposition, and disjunction in the antecedent are not
valid, in general, for each of the two contexts.
A semantics of supraclassical biconsequence relations is obtained from the
general binary semantics by requiring that bimodels are pairs of consistent deductively closed sets. Structural rules for biconsequence relations can also be
used in the supraclassical case. As before, Consistency will correspond to the
requirement that u ⊆ v, for any bimodel (u, v). Similarly, regular biconsequence
relations will be determined by quasi-reflexive binary semantics.
61
A supraclassical biconsequence relation will be called saturated, if it is consistent, regular, and satisfies the following postulate:
Saturation
A ∨ B, ¬A ∨ B : B.
For a deductively closed set u, let u⊥ denote the set of all maximal subtheories of u, plus u itself. Then a classical bimodel (u, v) will be called saturated,
if u ∈ v⊥. A classical binary semantics B will be called saturated if it is regular,
and all its bimodels are saturated. Such a semantics provides an adequate
interpretation for saturated biconsequence relations.
5.4.1
Classical nonmonotonic semantics
The notions of an exact theory and extension can be directly extended to supraclassical consequence relations, with the only, though important, qualification
that they form now deductively closed sets. Still, practically all the results
about such semantics remain valid for the supraclassical case.
The default nonmonotonic semantics of supraclassical biconsequence relations forms a generalization of default logic. Supraclassical biconsequence relations that are consistent and regular constitute a maximal logic adequate for
extensions. For such biconsequence relations, extensions are described as sets
satisfying the following fixpoint equality:
u = {A : u A : u}.
Thus, an extension is a set of formulas that are provable on the basis of
taking itself as the set of assumptions. As before, classical extensions of a
default bisequent theory will be minimal theories. Actually, default bisequent
theories under this nonmonotonic semantics give an exact representation for
the disjunctive default logic [Gelfond et al., 1991]. For singular default rules
a:b C:, it reduces to the original default logic of [Reiter, 1980].
The nonmonotonic semantics defined below constitutes an exact non-modal
counterpart of Moore’s autoepistemic logic.
Definition. A theory u of a supraclassical biconsequence relation is a classical
expansion of , if, for any v ∈ u⊥ such that v 6= u, the pair (v, u) is not a
bitheory of . The set of classical expansions determines the autoepistemic
semantics of .
Any extension of a supraclassical biconsequence relation will be a classical
expansion, though not vice versa. In fact, classical expansions can be precisely
characterized as extensions of saturated biconsequence relations.
The next result states important sufficient conditions for coincidence of classical expansions and extensions of a bisequent theory. A bisequent theory ∆
will be called positively simple, if positive premises and positive conclusions of
any bisequent from ∆ are sets of classical literals.
Theorem. If a bisequent theory is autoepistemic or positively simple, then its
classical expansions coincide with classical extensions.
62
Bisequents a:b c:d such that a, b, c, d are sets of classical literals, are logical
counterparts of program rules of extended logic programs with classical negation
(see [Gelfond and Lifschitz, 1991; Lifschitz and Woo, 1992]). The semantics of
such programs is determined by answer sets that coincide with extensions of
respective bisequent theories. Moreover, such bisequent theories are positively
simple, so by Theorem 5.4.1 extended logic programs obliterate the distinction
between extensions and classical expansions. This is the logical basis for a
possibility of representing extended logic programs also in autoepistemic logic
(see [Lifschitz and Schwarz, 1993]).
5.5
Modal Nonmonotonic Logics
A general representation of modal nonmonotonic reasoning can be given in the
framework of modal biconsequence relations. The role of the modal operator L
in this setting consists in reflecting assumptions and beliefs as propositions in
the main context.
Definition. A supraclassical biconsequence relation in a modal language will
be called modal if it satisfies the following postulates:
Positive Reflection
Negative Reflection
A : LA:,
: LA :A,
Negative Introspection
: A ¬LA:.
Any theory of a modal biconsequence relation is a modal stable set in the
sense of [Moore, 1985], and hence extensions and expansions of modal biconsequence relations will always be stable theories.
For a modal logic M, a modal biconsequence relation will be called an Mbiconsequence relation, if A: holds for every modal axiom A of M. A possible
worlds semantics for K-biconsequence relations is obtained in terms of Kripke
models having a last cluster, namely models of the form M = (W, R, F, V ),
where (W, R, V ) is an ordinary Kripke model, while F ⊆ W is a non-empty
last cluster of the model (see [Segerberg, 1971]). We will call such models
final Kripke models. The relevance of such semantics for modal nonmonotonic
reasoning has been shown in [Schwarz, 1992a].
For a final Kripke model M = (W, R, F, V ), let |M | denote the set of modal
propositions that are valid in M , while kM k the set of propositions valid in the
S5-submodel of M generated by F . For a set S of final Kripke models, we define
the binary semantics BS = {(|M |, kM k) | M ∈ S}, and say that a bisequent
a:b c:d is valid in S, if it is valid in BS .
Biconsequence relations described in the next definition play a crucial role
in a modal representation of extension-based nonmonotonic reasoning.
Definition. A modal biconsequence relation is an F-biconsequence relation, if
it is regular and satisfies
F
A, LA→B : B.
63
F-biconsequence relations provide a concise representation for the modal
logic S4F obtained from S4 by adding the axiom (A ∧ M LB)→L(M A ∨ B).
A semantic characterization of S4F [Segerberg, 1971] is given in terms of final
Kripke models (W, R, F, V ), such that αRβ iff either β ∈ F , or α ∈
/ F.
Any bisequentVa:b c:d of an F-biconsequence
relation
is
already
reducible to
W
a modal formula (La ∪ L¬Lb) → (Lc ∪ L¬Ld). Consequently, any bisequent
theory ∆ in such a logic is reducible to an ordinary modal theory that we will
˜
denote by ∆.
5.5.1
Modal nonmonotonic semantics
By varying the underlying modal logic, we obtain a whole range of modal nonmonotonic semantics.
Definition. A set of propositions is an M-extension (M-expansion) of a bisequent theory ∆, if it is an extension (resp. expansion) of the least M-biconsequence
relation containing ∆.
If ∆ is a plain modal theory, then M-extensions of ∆ coincide with Mexpansions in the sense of [Marek et al., 1993]8 . Recall, however, that modal
extensions and expansions are modal stable theories, and hence they are determined by their objective subsets. This opens a possibility of reducing modal
nonmonotonic reasoning to a nonmodal one, and vice versa.
For any set u of propositions, let uo denote the set of all non-modal propositions in u. Similarly, if is a modal biconsequence relation, o will denote its
restriction to the non-modal sub-language. Then we have
Theorem. If is a modal biconsequence relation, and u a stable theory, then
u is an extension of if and only if uo is an extension of o .
According to the above result, the objective biconsequence relation o embodies all the information about the modal nonmonotonic semantics of . In
other words, the net effect of modal reasoning in biconsequence relations can be
measured by the set of derived objective bisequents. Consequently, non-modal
supraclassical biconsequence relations turn out to be sufficiently expressive to
capture modal nonmonotonic reasoning.
In the other direction, in modal F-biconsequence relations any bisequent
˜ This allows us to use ordinary
theory ∆ is reducible to a usual modal theory ∆.
modal logical formalisms for representing non-modal nonmonotonic reasoning.
Thus, the following result generalizes the corresponding result of [Truszczyński,
1991] about a modal embedding of default theories.
Theorem. If ∆ is an objective bisequent theory, then classical extensions of ∆
˜
are precisely objective parts of S4F-extensions of ∆.
8 This creates, of course, an unfortunate terminological discrepancy, which is compensated,
however, by the conformity of our terminology with the rest of this paper.
64
We end with considering modal expansions. Two kinds of expansions are
important for a general description. The first is stable expansions of Moore’s
autoepistemic logic. They coincide with M-expansions for any modal logic
M in the range 5 ⊆ M ⊆ KD45. The second kind of expansions is reflexive
expansions of Schwarz’ reflexive autoepistemic logic [Schwarz, 1992b]. They
coincide with M-expansions for any modal logic in the range KT ⊆ M ⊆ SW5.
Both kinds of expansions can be computed either by transforming a bisequent
theory into a modal theory and finding its modal extensions, or by reducing it to
an objective bisequent theory and computing its classical expansions. Finally,
‘normal’ expansions in general can be viewed as a combination of these two
kinds of expansions:
Theorem. A set of propositions is a K-expansion of a modal bisequent theory
∆ if and only if it is both a stable and reflexive expansion of ∆.
6
Conclusions
The following passage from [Reiter, 1987a] remains surprisingly relevant today,
almost twenty years later:
Nonmonotonicity appears to be the rule, rather than the exception, in much of what passes for human commonsense reasoning.
The formal study of such reasoning patterns and their applications
has made impressive, and rapidly accelerating progress. Nevertheless, much remains to be done. ... [M]any more non-toy examples
need to be thoroughly explored in order for us to gain a deeper
understanding of the essential nature of nonmonotonic reasoning.
The ultimate quest, of course, is to discover a single theory embracing all the seemingly disparate settings in AI where nonmonotonic reasoning arises. Undoubtedly, there will be surprises en route,
but AI will profit from the journey, in the process becoming much
more the science we all wish it to be.
Despite clear success, twenty five years of nonmonotonic reasoning research
have shown that we need deep breath and long term objectives in order to make
nonmonotonic reasoning a viable tool for the challenges posed by AI. There
is still much to be done in order to meet the actual complexity of reasoning
tasks required by the latter. In particular, the quest for a unified theory of
nonmonotonic reasoning still has not been accomplished.
The relation between the two principal paradigms of nonmonotonic reasoning, preferential and explanatory one, has emerged as the main theoretical
problem for a future development of the field. By an inspiring analogy, in
nonmonotonic reasoning we have both a global relativity theory of preferential
reasoning and a local quantum mechanics of explanatory reasoning. So, what
we need is a unified theory of nonmonotonic reality. As in physics, however,
this unified theory is not going to emerge as a straightforward juxtaposition of
these components.
65
References
[Adams, 1975] E. W. Adams. The Logic of Conditionals. Reidel, Dordrecht,
1975.
[Alchourrón et al., 1985] C. Alchourrón, P. Gärdenfors, and D. Makinson. On
the logic of theory change: Partial meet contraction and revision functions.
Journal of Symbolic Logic, 50:510–530, 1985.
[Alferes and Pereira, 1992] J. J. Alferes and L. M. Pereira. On logic program
semantics with two kinds of negation. In K. R. Apt, editor, Proc. Joint Int.
Conf. and Symp. on Logic Programming, pages 574–589, Cambridge, Mass.,
1992. MIT Press.
[Amati et al., 1994] G. Amati, L. Carlucci Aiello, and F. Pirri. Defaults as
restrictions on classical hilbert-style proofs. Journal of Logic, Language and
Information, 3:303–326, 1994.
[Amati et al., 1997] G. Amati, L. Carlucci Aiello, and F. Pirri. Definability and
commonsense reasoning. Artificial Intelligence, 93:169–199, 1997.
[Andreka et al., 2002] H. Andreka, M. Ryan, and P.-Y. Schobbens. Operators
and laws for combining preference relations. Journal of Logic and Computation, 12:13–53, 2002.
[Antonelli, 1999] G. A. Antonelli. A directly cautious theory of defeasible consequence for default logic via the notion of general extension. Artificial Intelligence, 109:71–109, 1999.
[Antoniou and Wang, 2006] G. Antoniou and K. Wang. Default logic, 2006.
This volume.
[Antoniou, 1997] G. Antoniou. Nonmonotonic Reasoning. MIT Press, Cambridge, Mass., 1997.
[Apt et al., 1988] K. Apt, H. Blair, and A. Walker. Towards a theory of declarative knowledge. In J. Minker, editor, Foundations of Deductive Databases
and Logic Programming, pages 89–148. Morgan Kaufmann, San Mateo, CA,
1988.
[Babovich et al., 2000] Y. Babovich, E. Erdem, and V. Lifschitz. Fages’ theorem
and answer set programming. In Proceedings of the Nonmonotonic Reasoning
Workshop, NMR-2000, 2000.
[Baker, 1989] A. B. Baker. A simple solution to the Yale shooting problem. In
R. J. Brachman, H. J. Levesque, and R. Reiter, editors, KR’89: Principles of
Knowledge Representation and Reasoning, pages 11–20. Morgan Kaufmann,
San Mateo, California, 1989.
66
[Baral and Subrahmanian, 1991] C. Baral and V. S. Subrahmanian. Dualities
between alternative semantics for logic programming and nonmonotonic reasoning (extended abstract). In Proc. 1st Int. Worksop on Logic Programming
and Nonmonotonic Reasoning, pages 69–86, 1991.
[Baral, 2003] C. Baral. Knowledge Representation, Reasoning and Declarative
Problem Solving. Cambridge UP, 2003.
[Belnap, 1977] N. D. Belnap, Jr. A useful four-valued logic. In M. Dunn and
G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8–41. D.
Reidel, 1977.
[Benferhat et al., 1993] S. Benferhat, C. Cayrol, D. Dubois, J. Lang, and
H. Prade. Inconsistency management and prioritized syntax-based entailment. In R. Bajcsy, editor, Proceedings Int. Joint Conf. on Artificial Intelligence, IJCAI’93, pages 640–645, Chambery, France, 1993. Morgan Kaufmann.
[Besnard, 1989] P. Besnard. An Introduction to Default Logic. Springer, 1989.
[Blamey, 1986] S. Blamey. Partial logic. In D. M. Gabbay and F. Guenthner,
editors, Handbook of Philosophical Logic, Vol. III, pages 1–70. D. Reidel, 1986.
[Bochman, 1994] A. Bochman. On the relation between default and modal
consequence relations. In Proc Int. Conf. on Principles of Knowledge Representation and Reasoning, pages 63–74, 1994.
[Bochman, 1995a] A. Bochman. Default consequence relations as a logical
framework for logic programs. In Proc. Int. Conf. on Logic Programming
and Nonmonotonic Reasoning, pages 245–258. Springer, 1995.
[Bochman, 1995b] A. Bochman. Modal nonmonotonic logics demodalized. Annals of Mathematics and Artificial Intelligence, 15:101–123, 1995.
[Bochman, 1995c] A. Bochman. On bimodal nonmonotonic logics and their
unimodal and non-modal equivalents. In Proc. IJCAI’95, pages 1518–1524,
1995.
[Bochman, 1996a] A. Bochman. On a logical basis of general logic programs. In
Proc. Workshop on Nonmonotonic Extensions of Logic Programming, Lecture
Notes in AI, 1996. Springer Verlag.
[Bochman, 1996b] A. Bochman. On a logical basis of normal logic programs.
Fundamenta Informaticae, 28:223–245, 1996.
[Bochman, 1998a] A. Bochman. A logical foundation for logic programming
I: Biconsequence relations and nonmonotonic completion. Journal of Logic
Programming, 35:151–170, 1998.
67
[Bochman, 1998b] A. Bochman. A logical foundation for logic programming II:
Semantics of general logic programs. Journal of Logic Programming, 35:171–
194, 1998.
[Bochman, 1998c] A. Bochman. On the relation between default and modal
nonmonotonic reasoning. Artificial Intelligence, 101:1–34, 1998.
[Bochman, 2001] A. Bochman. A Logical Theory of Nonomonotonic Inference
and Belief Change. Springer, 2001.
[Bochman, 2003a] A. Bochman. Brave nonmonotonic inference and its kinds.
Annals of Mathematics and Artificial Intelligence, 39:101–121, 2003.
[Bochman, 2003b] A. Bochman. Collective argumentation and disjunctive logic
programming. Journal of Logic and Computation, 9:55–56, 2003.
[Bochman, 2003c] A. Bochman. A logic for causal reasoning. In Proceedings
IJCAI’03, Acapulco, 2003. Morgan Kaufmann.
[Bochman, 2004a] A. Bochman. A causal approach to nonmonotonic reasoning.
Artificial Intelligence, 160:105–143, 2004.
[Bochman, 2004b] A. Bochman. A causal logic of logic programming. In
D. Dubois, C. Welty, and M.-A. Williams, editors, Proc. Ninth Conference on
Principles of Knowledge Representation and Reasoning, KR’04, pages 427–
437, Whistler, 2004.
[Bochman, 2005] A. Bochman. Explanatory Nonmonotonic Reasoning. World
Scientific, 2005.
[Bonatti and Olivetti, 2002] P. A. Bonatti and N. Olivetti. Sequent calculi for
propositional nonmonotonic logics. ACM Trans. Comput. Log., 3:226–278,
2002.
[Bondarenko et al., 1997] A. Bondarenko, P. M. Dung, R. A. Kowalski, and
F. Toni. An abstract, argumentation-theoretic framework for default reasoning. Artificial Intelligence, 93:63–101, 1997.
[Bossu and Siegel, 1985] G. Bossu and P. Siegel. Saturation, nonmonotonic reasoning and the closed-world assumption. Artificial Intelligence, 25:13–63,
1985.
[Brewka and Gottlob, 1997] G. Brewka and G. Gottlob. Well-founded semantics for default logic. Fundamenta Informaticae, 31:221–236, 1997.
[Brewka and Konolige, 1993] G. Brewka and K. Konolige. An abductive framework for general logic programs and other nonmonotonic systems. In Proc.
IJCAI, pages 9–17, 1993.
[Brewka et al., 1997] G. Brewka, J. Dix, and K. Konolige. Nonmonotonic Reasoning: An Overview. CSLI Publications, Stanford, 1997.
68
[Brewka, 1991] G. Brewka. Cumulative default logic: In defense of nonmonotonic inference rules. Artificial Intelligence, 50:183–205, 1991.
[Brewka, 1992] G. Brewka. A framework for cumulative default logics. Technical
Report TR-92-042, ICSI, Berkeley, CA, 1992.
[Cabalar, 2001] P. Cabalar. Well-founded semantics as two-dimensional hereand-there. In Proceedings of ASP 01, 2001 AAAI Spring Symposium Series,
pages 15–20, Stanford, 2001.
[Chen, 1994] J. Chen. Relating only knowing to minimal belief and negation
as failure. Journal of Experimental and Theoretical Artificial Intelligence,
6:409–429, 1994.
[Chesñevar et al., 2000] C. I. Chesñevar, A. G. Marguitman, and R. P. Loui.
Logical models of argument. ACM Computing Surveys, 32:337–383, 2000.
[Clark, 1978] K. Clark. Negation as failure. In H. Gallaire and J. Minker,
editors, Logic and Data Bases, pages 293–322. Plenum Press, 1978.
[Colmerauer et al., 1973] A. Colmerauer, H. Kanoui, R. Pasero, and P. Roussel.
Un systeme de communication hommemachine. Technical report, Groupe de
Intelligence Artificielle Universitae de AixMarseille II, Marseille, 1973.
[Console et al., 1991] L. Console, D. Theseider Dupre, and P. Torasso. On the
relationship between abduction and deduction. Journal of Logic and Computation, 1:661–690, 1991.
[Cox and Pietrzykowski, 1987] P. T. Cox and T. Pietrzykowski. General diagnosis by abductive inference. In Proc. IEEE Symposium on Logic Programming,
pages 183–189, 1987.
[Darwiche and Pearl, 1994] A. Darwiche and J. Pearl. Symbolic causal networks. In Proceedings AAAI’94, pages 238–244, 1994.
[de Kleer et al., 1992] J. de Kleer, A. K. Mackworth, and R. Reiter. Characterizing diagnoses and systems. Artificial Intelligence, 52:197–222, 1992.
[de Kleer, 1986] J. de Kleer. An assumption-based TMS. Artificial Intelligence,
28:127–162, 1986.
[Delgrande, 1987] J. P. Delgrande. A first-order conditional logic for prototypical properties. Artificial Intelligence, 33:105–130, 1987.
[Delgrande, 1988] J. Delgrande. An approach to default reasoning based on a
firstorder conditional logic: revised report. Artificial Intelligence, 36:63–90,
1988.
[Denecker and Schreye, 1992] M. Denecker and D. De Schreye. SLDNFA: an
abductive procedure for normal abductive programs. In Proc. Joint Int.l
Conf. and Symp. on Logic Programming, pages 686–702, 1992.
69
[Denecker et al., 2001] M. Denecker, M. Bruynooghe, and V. W. Marek. Logic
programming revisited: Logic programs as inductive definitions. ACM Trans.
Comput. Log., 2:623–654, 2001.
[Denecker et al., 2003] M. Denecker, V. W. Marek, and M. Truszczyński. Uniform semantic treatment of default and autoepistemic logics. Artificial Intelligence, 143:79–122, 2003.
[Dix, 1991] J. Dix. Classifying semantics of logic programs. In A. Nerode,
W. Marek, and V. S. Subrahmanian, editors, Proceedings 1st International
Workshop on Logic Programming and Nonmonotonic Reasoning, pages 166–
180, Cambridge, Mass., 1991. MIT Press.
[Dix, 1992] J. Dix. Classifying semantics of disjunctive logic programs. In
K. Apt, editor, Proc. Joint Int. Conf. and Symp. on Logic Programming,
ICSLP’92, pages 798–812. MIT Press, 1992.
[Doherty et al., 1995] P. Doherty, W. Lukaszewicz, and A. Szalas. Computing
circumscription revisited: Preliminary report. In Proceedings Int. Joint Conf.
on Artificial Intelligence, pages 1502–1508, 1995.
[Doyle, 1979] J. Doyle. A truth maintenance system. Artificial Intelligence,
12:231–272, 1979.
[Doyle, 1994] J. Doyle. Reasoned assumptions and rational psychology. Fundamenta Informaticae, 20:3573, 1994.
[Dung and Son, 2001] P. M. Dung and T. C. Son. An argument-based approach
to reasoning with specificity. Artificial Intelligence, 133:35–85, 2001.
[Dung et al., 2006] P. M. Dung, R. A. Kowalski, and F. Toni. Dialectic proof
procedures for assumption-based, admissible argumentation. Artificial Intelligence, 170:114159, 2006.
[Dung, 1992] P. M. Dung. Acyclic disjunctive programs with abductive procedure as proof procedure. In Proc. Int. Conference on Fifth Generation
Computer Systems, pages 555–561. ICOT, 1992.
[Dung, 1995a] P. M. Dung. An argumentation-theoretic foundation for logic
programming. J. of Logic Programming, 22:151–177, 1995.
[Dung, 1995b] P. M. Dung. On the acceptability of arguments and its fundamental role in non-monotonic reasoning, logic programming and n-persons
games. Artificial Intelligence, 76:321–358, 1995.
[Dunn, 1976] J. M. Dunn. Intuitive semantics for first-degree entailment and
coupled trees. Philosophical Studies, 29:149–168, 1976.
[Eiter and Lukasiewicz, 2004] T. Eiter and T. Lukasiewicz. Complexity results
for explanations in the structural-model approach. Artificial Intelligence,
154(1-2):145–198, April 2004.
70
[Elkan, 1990] C. Elkan. A rational reconstruction of nonmonotonic truth maintenance systems. Artificial Intelligence, 43:219–234, 1990.
[Erdem and Lifschitz, 2003] E. Erdem and V. Lifschitz. Tight logic programs.
Theory and Practice of Logic Programming, 3:499–518, 2003.
[Etherington and Reiter, 1983] D. Etherington and R. Reiter. On inheritance
hierarchies with exceptions. In Proceedings of AAAI-83, pages 104–108, 1983.
[Etherington et al., 1985] D. Etherington, R. Mercer, and R. Reiter. On the
adequacy of predicate circumscription for closed world reasoning. Computational Intelligence, 1:11–15, 1985.
[Etherington et al., 1991] D. Etherington, S. Kraus, and D. Perlis. Nonmonotonicity and the scope of reasoning. Artificial Intelligence, 52:221–261, 1991.
[Etherington, 1987] D. Etherington. Reasoning with Incomplete Information.
Research Notes in AI. Pitman, London, 1987.
[Fages, 1994] F. Fages. Consistency of Clark’s completion and existence of stable models. Journal of Methods of Logic in Computer Science, 1:51–60, 1994.
[Finger, 1987] J. J. Finger. Exploiting Constraints in Design Synthesis. Ph.D.
dissertation, Department of Computer Science, Stanford University, Stanford,
California, 1987.
[Fitting et al., 1992] M. C. Fitting, W. Marek, and M. Truszczyński. The pure
logic of necessitation. Journal of Logic and Computation, 2:349–373, 1992.
[Fitting, 1991] M. C. Fitting. Bilattices and the semantics of logic programming.
Journal of Logic Programming, 11:91–116, 1991.
[Gabbay et al., 1994] D. Gabbay, C. J. Hogger, and J. A. Robinson, editors.
Handbook of Logic in Artificial Intelligence and Logic Programming. Volume
3: Nonmonotonic Reasoning and Uncertain Reasoning. Oxford UP, Oxford,
1994.
[Gabbay, 1985] D. M. Gabbay. Theoretical foundations for non-monotonic reasoning in expert systems. In K. R. Apt, editor, Logics and Models of Concurrent Systems. Springer, 1985.
[Gabbay, 1998] D. M. Gabbay. Fibring Logics. Oxford University Press, 1998.
[Gärdenfors and Makinson, 1994] P. Gärdenfors and D. Makinson. Nonmonotonic inference based on expectations. Artificial Intelligence, 65:197–245,
1994.
[Geffner, 1992] H. Geffner. Default Reasoning. Causal and Conditional Theories. MIT Press, 1992.
71
[Gelfond and Lifschitz, 1988] M. Gelfond and V. Lifschitz. The stable model
semantics for logic programming. In R. Kowalski and K. Bowen, editors,
Proc. 5th International Conf./Symp. on Logic Programming, pages 1070–
1080, Cambridge, MA, 1988. MIT Press.
[Gelfond and Lifschitz, 1991] M. Gelfond and V. Lifschitz. Classical negation
and disjunctive databases. New Generation Computing, 9:365–385, 1991.
[Gelfond et al., 1991] M. Gelfond, V. Lifschitz, H. Przymusińska, and
M. Truszczyński. Disjunctive defaults. In Proc. Second Int. Conf. on Principles of Knowledge Representation and Reasoning, KR’91, pages 230–237,
Cambridge, Mass., 1991.
[Gelfond, 1994] M. Gelfond. Logic programming and reasoning with incomplete
information. Annals of Mathematics and Artificial Intelligence, 12:89–116,
1994.
[Ginsberg, 1987] M. L. Ginsberg, editor. Readings in Nonmonotonic Reasoning.
Morgan Kaufmann, Los Altos, Ca., 1987.
[Ginsberg, 1989] M. L. Ginsberg. A circumscriptive theorem prover. Artif.
Intell., 39:209–230, 1989.
[Ginsberg, 1993] M. Ginsberg. Essentials of Artificial Intelligence. Morgan
Kaufmann, 1993.
[Giunchiglia et al., 2004] E. Giunchiglia, J. Lee, V. Lifschitz, N. McCain, and
H. Turner. Nonmonotonic causal theories. Artificial Intelligence, 153:49–104,
2004.
[Goldszmidt et al., 1993] M. Goldszmidt, P. Morris, and J. Pearl. A maximum
entropy approach to nonmonotonic reasoning. IEEE Transactions on Pattern
Analysis and Machine Intelligence, 15:220–232, 1993.
[Gottlob, 1994] G. Gottlob. The power of beliefs or translating default logic into
standard autoepistemic logic. In G. Lakemeyer and B. Nebel, editors, Foundations of Knowledge Representation and Reasoning, volume 810 of LNAI,
pages 133–144. Springer, 1994.
[Grosof, 1991] B. N. Grosof. Generalising prioritization. In J. Allen, R. Fikes,
and E. Sandewall, editors, Proc. Second International Conference on Principles of Knowledge Representation and Reasoning (KR’91), pages 289–300.
Morgan Kaufmann, 1991.
[Halpern and Moses, 1985] Joseph Y. Halpern and Yoram Moses. Towards a
theory of knowledge and ignorance. In Krzysztof R. Apt, editor, Logics and
Models of Concurrent Systems, pages 459–476. Springer-Verlag, Berlin, 1985.
72
[Halpern and Pearl, 2001a] J. Y. Halpern and J. Pearl. Causes and explanations: A structral-model approach—part I: Causes. In Proc. Seventh Conf.
On Uncertainty in Artificial Intelligence (UAI’01), pages 194–202, San Francisco, CA, 2001. Morgan Kaufmann.
[Halpern and Pearl, 2001b] J. Y. Halpern and J. Pearl. Causes and explanations: A structral-model approach—part II: Explanations. In Proceedings Int.
Joint Conf. on Artificial Intelligence, IJCAI-01. Morgan Kaufmann, 2001.
[Halpern, 1997] J. Y. Halpern. A critical reexamination of default logic, autoepistemic logic, and only knowing. Computational Intelligence, 13:144–163,
1997.
[Halpern, 1998] J. Y. Halpern. Axiomatizing causal reasoning. In G. F. Cooper
and S. Moral, editors, Uncertainty in Artificial Intelligence, pages 202–210.
Morgan Kaufmann, 1998.
[Hanks and McDermott, 1987] S. Hanks and D. McDermott. Non-monotonic
logics and temporal projection. Artificial Intelligence, 33:379–412, 1987.
[Hewitt, 1969] C. E. Hewitt. PLANNER: A language for proving theorems in
robots. In First International Joint Conference on Artificial Intelligence,
pages 295–301, 1969.
[Horty, 1994] J. F. Horty. Some direct theories of nonmonotonic inheritance. In
D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Handbook of Logic
in Artificial Intelligence and Logic Programming 3: Nonmonotonic Reasoning
and Uncertain Reasoning. Oxford University Press, Oxford, 1994.
[Imielinski, 1987] T. Imielinski. Results on translating defaults to circumscription. Artificial Intelligence, 32:131–146, 1987.
[Inoue and Sakama, 1994] K. Inoue and C. Sakama. On positive occurrences
of negation as failure. In Proc. 4th Int. Conf. on Principles of Knowledge
Representation and Reasoning, KR’94, pages 293–304. Morgan Kauffman,
1994.
[Inoue and Sakama, 1998] K. Inoue and C. Sakama. Negation as failure in the
head. Journal of Logic Programming, 35:39–78, 1998.
[Israel, 1980] D. J. Israel. What’s wrong with non-monotonic logic. In Proceedings of the First Annual Conference of the American Association for Artificial
Intelligence, Stanford University, 1980.
[Iwasaki and Simon, 1986] Yumi Iwasaki and Herbert Simon. Causality in device behavior. Artificial Intelligence, 29(1):3–32, 1986.
[Jakobovits and Vermeir, 1999] H. Jakobovits and D. Vermeir. Robust semantics for argumentation frameworks. Journal of Logic and Computation, 9:215–
261, 1999.
73
[Janhunen, 1996] T. Janhunen. Representing autoepistemic introspection in
terms of default rules. In Proc. European Conf. on AI, ECAI-96, pages 70–
74, Budapest, 1996. John Wiley & Sons.
[Kakas and Mancarella, 1990] A. C. Kakas and P. Mancarella. Generalized stable models: A semantics for abduction. In Proc. European Conf. on Artificial
Intelligence, ECAI-90, pages 385–391, Stockholm, 1990.
[Kakas and Toni, 1999] A. C. Kakas and F. Toni. Computing argumentation in
logic programming. Journal of Logic and Computation, 9:515–562, 1999.
[Kakas et al., 1992] A. C. Kakas, R. A. Kowalski, and F. Toni. Abductive logic
programming. Journal of Logic and Computation, 2:719–770, 1992.
[Konolige and Myers, 1989] K. Konolige and K. L. Myers. Representing defaults
with epistemic concepts. Computational Intelligence, 5:32–44, 1989.
[Konolige, 1988] K. Konolige. On the relation between default and autoepistemic logic. Artificial Intelligence, 35:343–382, 1988.
[Konolige, 1989] K. Konolige. On the relation between autoepistemic logic and
circumscription. In Proc. IJCAI-89, pages 1213–1218, 1989.
[Konolige, 1992] K. Konolige. Abduction versus closure in causal theories. Artificial Intelligence, 53:255–272, 1992.
[Konolige, 1994] K. Konolige. Using default and causal reasoning in diagnosis.
Annals of Mathematics and Artificial Intelligence, 11:97–135, 1994.
[Kowalski and Toni, 1996] R. A. Kowalski and F. Toni. Abstract argumentation. Artificial Intelligence and Law, 4:275–296, 1996.
[Kratzer, 1981] A. Kratzer. Partition and revision: The semantics of counterfactuals. Journal of Philosophical Logic, 10:201–216, 1981.
[Kraus et al., 1990] S. Kraus, D. Lehmann, and M. Magidor. Nonmonotonic
reasoning, preferential models and cumulative logics. Artificial Intelligence,
44:167–207, 1990.
[Lee and Lifschitz, 2003] J. Lee and V. Lifschitz. Loop formulas for disjunctive
logic programs. In Proc. Nineteenth Int. Conference on Logic Programming,
pages 451–465, 2003.
[Lee and Lin, 2004] J. Lee and F. Lin. Loop formulas for circumscription. In
Proc AAAI-04, pages 281–286, San Jose, CA, 2004.
[Lehmann and Magidor, 1992] D. Lehmann and M. Magidor. What does a conditional knowledge base entail? Artificial Intelligence, 55:1–60, 1992.
74
[Lehmann, 1989] D. Lehmann. What does a conditional knowledge base entail?
In R. Brachman and H. J. Levesque, editors, Proc. 2nd Int. Conf. on Principles of Knowledge Representation and Reasoning, KR’89, pages 212–222.
Morgan Kaufmann, 1989.
[Lehmann, 1995] D. Lehmann. Another perspective on default reasoning. Annals of Mathematics and Artificial Intelligence, 15:61–82, 1995.
[Levesque and Lakemeyer, 2000] H. Levesque and G. Lakemeyer. The Logic of
Knowledge Bases. The MIT Press, 2000.
[Levesque, 1989] H. J. Levesque. A knowledge-level account of abduction. In
Proc. IJCAI, pages 1061–1067, 1989.
[Levesque, 1990] H. J. Levesque. All I know: A study in autoepistemic logic.
Artificial Intelligence, 42:263309, 1990.
[Lewis, 1973] D. Lewis. Counterfactuals. Harvard University Press, Cambridge,
Mass., 1973.
[Lewis, 1981] D. Lewis. Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic, 10:217–234, 1981.
[Lifschitz and Schwarz, 1993] V. Lifschitz and G. Schwarz. Extended logic programs as autoepistemic theories. In L. M. Pereira and A. Nerode, editors,
Proc. Second Int. Workshop on Logic Programming and Nonmonotonic Reasoning, pages 101–114. MIT Press, 1993.
[Lifschitz and Woo, 1992] V. Lifschitz and T. Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Proc. Third Int. Conf. on Principles of Knowledge Representation and Reasoning, KR‘92, pages 603–614.
Morgan Kauffman, 1992.
[Lifschitz et al., 1999] V. Lifschitz, L. R. Tang, and H. Turner. Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence,
25:369–389, 1999.
[Lifschitz et al., 2001] V. Lifschitz, D. Pearce, and A. Valverde. Strongly equivalent logic programs. ACM Transactions on Computational Logic, 2:526–541,
2001.
[Lifschitz, 1985] V. Lifschitz. Computing circumscription. In Proc. 9th Int.
Joint Conf. on Artificial Intelligence, IJCAI-85, pages 121–127. Morgan
Kaufmann, 1985.
[Lifschitz, 1987a] V. Lifschitz. Pointwise circumscription. In M. Ginsberg, editor, Readings in Non-Monotonic Reasoning, pages 179–193. Morgan Kaufmann, San Mateo, CA, 1987.
75
[Lifschitz, 1987b] Vladimir Lifschitz. Formal theories of action: Preliminary
report. In John McDermott, editor, Proceedings of the Tenth International
Joint Conference on Artificial Intelligence, Los Altos, California, 1987. Morgan Kaufmann.
[Lifschitz, 1991a] V. Lifschitz, editor. Artificial Intelligence and Mathematical
Theory of Computation, Papers in Honor of John McCarthy. Academic Press,
1991.
[Lifschitz, 1991b] V. Lifschitz. Nonmonotonic databases and epistemic queries.
In Proceedings Int. Joint Conf. on Artificial Intelligence, IJCAI-91, pages
381–386. Morgan Kaufmann, 1991.
[Lifschitz, 1994a] V. Lifschitz. Minimal belief and negation as failure. Artificial
Intelligence, 70:53–72, 1994.
[Lifschitz, 1994b] Vladimir Lifschitz. Circumscription. In Dov Gabbay, C. J.
Hogger, and J. A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 3: Nonmonotonic Reasoning and
Uncertain Reasoning, pages 298–352. Oxford University Press, 1994.
[Lifschitz, 1997] V. Lifschitz. On the logic of causal explanation. Artificial
Intelligence, 96:451–465, 1997.
[Lin and Shoham, 1989] F. Lin and Y. Shoham. Argument systems: A uniform
basis for nonmonotonic reasoning. In Proceedings of 1st Intl. Conference
on Principles of Knowledge Representation and Reasoning, pages 245–255,
Stanford, CA, 1989.
[Lin and Shoham, 1992] F. Lin and Y. Shoham. A logic of knowledge and justified assumptions. Artificial Intelligence, 57:271–289, 1992.
[Lin and You, 2002] F. Lin and J.-H. You. Abduction in logic programming:
A new definition and an abductive procedure based on rewriting. Artificial
Intelligence, 140:175–205, 2002.
[Lin and Zhao, 2002] F. Lin and Y. Zhao. ASSAT: Computing answer sets of a
logic program by SAT solvers. In Proceedings AAAI-02, 2002.
[Lin, 1995] F. Lin. Embracing causality in specifying the inderect effect of actions. In Proc. Int. Joint Conf. on Artificial Intelligence, IJCAI-95, pages
1985–1991, Montreal, 1995. Morgan Kaufmann.
[Lin, 1996] F. Lin. Embracing causality in specifying the indeterminate effects
of actions. In Proceedings AAAI-96, pages 670–676, 1996.
[Lobo and Uzcátegui, 1997] J. Lobo and C. Uzcátegui. Abductive consequence
relations. Artificial Intelligence, 89:149–171, 1997.
[Lobo et al., 1992] J. Lobo, J. Minker, and A. Rajasekar. Foundations of Disjunctive Logic Programming. MIT Press, Cambridge, Mass., 1992.
76
[Lukaszewicz, 1990] W. Lukaszewicz. Non-Monotonic Reasoning: Formalization of Commonsense Reasoning. Ellis Horwood, New York, 1990.
[Makinson and van der Torre, 2000] D. Makinson and L. van der Torre. Input/Output logics. Journal of Philosophical Logic, 29:383–408, 2000.
[Makinson, 1989] D. Makinson. General theory of cumulative inference. In
M. Reinfrank, editor, Nonmonotonic Reasoning, volume 346 of LNAI, pages
1–18. Springer, 1989.
[Makinson, 1994] D. Makinson. General patterns in nonmonotonic reasoning. In
D. M. Gabbay and Others, editors, Handbook of Logic in Artificial Intelligence
and Logic Programming, Vol. 3, Nonmonotonic and Uncertain Reasoning,
volume 2, pages 35–110. Oxford University Press, Oxford, 1994.
[Makinson, 2003] D. Makinson. Bridges between classical and nonmonotonic
logic. Logic Journal of the IGPL, 11:69–96, 2003.
[Makinson, 2005] D. Makinson. Bridges from Classical to Nonmonotonic Logic.
King’s College Publications, 2005.
[Marek and Subrahmanian, 1992] W. Marek and V. S. Subrahmanian. The relationship between stable, supported, default and autoepistemic semantics for
general logic programs. Theoretical Computer Science, 103:365–386, 1992.
[Marek and Truszczyński, 1989] W. Marek and M. Truszczyński. Relating autoepistemic and default logics. In Int. Conf. on Principles of Knowledge Representation and Reasoning, KR’89, pages 276–288, San Mateo, Calif., 1989.
Morgan Kaufmann.
[Marek and Truszczyński, 1990] W. Marek and M. Truszczyński. Modal logic
for default reasoning. Annals of Mathematics and Artificial Intelligence,
1:275–302, 1990.
[Marek and Truszczyński, 1993] W. Marek and M. Truszczyński. Nonmonotonic Logic, Context-Dependent Reasoning. Springer, 1993.
[Marek et al., 1990] W. Marek, A. Nerode, and J. Remmel. A theory of nonmonotonic rule systems. Annals of Mathematics and Artificial Intelligence,
1:241–273, 1990.
[Marek et al., 1993] V. W. Marek, G. F. Schwarz, and M. Truszchinski. Modal
nonmonotonic logics: ranges, characterization, computation. Journal of
ACM, 40:963–990, 1993.
[McCain and Turner, 1997] N. McCain and H. Turner. Causal theories of action
and change. In Proceedings AAAI-97, pages 460–465, 1997.
77
[McCarthy and Hayes, 1969] J. McCarthy and P. Hayes. Some philosophical
problems from the standpoint of artificial intelligence. In B. Meltzer and
D. Michie, editors, Machine Intelligence, pages 463–502. Edinburg University
Press, Edinburg, 1969.
[McCarthy, 1959] John McCarthy. Programs with common sense. In Proceedings of the Teddington Conference on the Mechanization of Thought Processes,
pages 75–91, London, 1959. Her Majesty’s Stationary Office.
[McCarthy, 1980] J. McCarthy. Circumscription – a form of non-monotonic
reasoning. Artificial Intelligence, 13:27–39, 1980.
[McCarthy, 1986] J. McCarthy. Applications of circumscription to formalizing
common sense knowledge. Artificial Intelligence, 13:27–39, 1986.
[McDermott and Doyle, 1980] D. McDermott and J. Doyle.
logic. Artificial Intelligence, 13:41–72, 1980.
Nonmonotonic
[McDermott, 1982] D. McDermott. Nonmonotonic logic II: Nonmonotonic
modal theories. Journal of the ACM, 29:33–57, 1982.
[McDermott, 1987] Drew McDermott. Critique of pure reason. Computational
Intelligence, 3(3):149–160, 1987.
[Minker, 1982] J. Minker. On indefinite databases and the closed world assumption. In Proceedings of 6th Conference on Automated Deduction, pages
292–308, New York, 1982.
[Minker, 1993] Jack Minker. An overview of nonmonotonic reasoning and logic
programming. Journal of Logic Programming, 17:95–126, 1993.
[Minker, 2000] Jack Minker, editor. Logic-Based Artificial Intelligence. Kluwer
Academic Publishers, Dordrecht, 2000.
[Minsky, 1974] M. Minsky. A framework for representing knowledge. Tech.
Report 306, Artificial Intelligence Laboratory, MIT, 1974.
[Moore, 1985] R. C. Moore. Semantical considerations on non-monotonic logic.
Artificial Intelligence, 25:75–94, 1985.
[Morgenstern and Stein, 1994] L. Morgenstern and L. Stein. Motivated action
theory: a formal theory of causal reasoning. Artificial Intelligence, 71(1):1–42,
1994.
[Morgenstern, 1996] L. Morgenstern. The problem with solutions to the frame
problem. In K. M. Ford and Z. Pylyshyn, editors, The Robot’s Dilemma
Revisited: The Frame Problem in Artificial Intelligence, pages 99–133. Ablex
Publishing Co., Norwood, New Jersey, 1996.
[Morris, 1988] P. H. Morris. The anomalous extension problem in default reasoning. Artificial Intelligence, 35:383–399, 1988.
78
[Nayak, 1994] P. P. Nayak.
70:277–334, 1994.
Causal approximations.
Artificial Intelligence,
[Niemelä, 1992] I. Niemelä. A unifying framework for nonmonotonic reasoning.
In Proc. 10th European Conference on Artificial Intelligence, ECAI-92, pages
334–338, Vienna, 1992. John Wiley.
[Oikarinen and Janhunen, 2005] E. Oikarinen and T. Janhunen. circ2dlp translating circumscription into disjunctive logic programming. In C. Baral,
G. Greco, N. Leone, and G. Terracina, editors, Logic Programming and
Nonmonotonic Reasoning, 8th International Conference, LPNMR 2005, Diamante, Italy, September 5-8, 2005, Proceedings, volume 3662 of Lecture Notes
in Computer Science, pages 405–409. Springer, 2005.
[Pearce, 1997] D. Pearce. A new logical characterization of stable models and
answer sets. In J. Dix, L. M. Pereira, and T. Przymusinski, editors, NonMonotonic Extensions of Logic Programming, volume 1216 of LNAI, pages
57–70. Springer, 1997.
[Pearl, 1990] J. Pearl. System Z: A natural ordering of defaults with tractable
applications to default reasoning. In Proceedings of the Third Conference on
Theoretical Aspects of Reasoning About Knowledge (TARK’90), pages 121–
135, San Mateo, CA, 1990. Morgan Kaufmann.
[Pearl, 2000] J. Pearl. Causality. Cambridge UP, 2000.
[Perlis, 1986] D. Perlis. On the consistency of commonsense reasoning. Computational Intelligence, 2:180–190, 1986.
[Poole, 1985] D. Poole. On the comparison of theories: Preferring the most
specific explanation. In Proceedings Ninth International Joint Conference on
Artificial Intelligence, pages 144–147, Los Angeles, 1985.
[Poole, 1988a] D. Poole. A logical framework for default reasoning. Artificial
Intelligence, 36:27–47, 1988.
[Poole, 1988b] D. Poole. Representing knowledge for logic-based diagnosis. In
Proc. Int. Conf. on Fifth Generation Computer Systems, pages 1282–1290,
Tokyo, 1988.
[Poole, 1989a] D. Poole. Explanation and prediction: An architecture for default and abductive reasoning. Computational Intelligence, 5:97–110, 1989.
[Poole, 1989b] D. Poole. What the lottery paradox tells us about default reasoning. In R. J. Brachman, H. J. Levesque, and R. Reiter, editors, Proceedings
of the First Int. Conf. on Principles of Knowledge Representation and Reasoning, pages 333 – 340. Morgan Kaufmann, 1989.
[Poole, 1994a] D. Poole. Default logic. In D. M. Gabbay, C. J. Hogger, and
J. A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic
Programming, volume 3, pages 189–215. Oxford University Press, 1994.
79
[Poole, 1994b] D. Poole. Representing diagnosis knowledge. Annals of Mathematics and Artificial Intelligence, 11:33–50, 1994.
[Przymusinska and Przymusinski, 1994] H. Przymusinska and T. Przymusinski.
Stationary default extensions. Fundamenta Informaticae, 21:67–87, 1994. (In
print).
[Przymusinski, 1988] T. C. Przymusinski. On the declarative semantics of stratified deductive databases and logic programs. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 193–216. Morgan
Kauffman, 1988.
[Przymusinski, 1990] T. C. Przymusinski. The well-founded semantics coincides
with the three-valued stable semantics. Fundamenta Informaticae, 13:445–
464, 1990.
[Przymusinski, 1991a] T. C. Przymusinski. Semantics of disjunctive logic programs and deductive databases. In Proc. Second Int. Conf. On Deductive and
Object-Oriented Databases, pages 85–107. Springer, 1991.
[Przymusinski, 1991b] T. C. Przymusinski. Three-valued nonmonotonic formalisms and semantics of logic programs. Artificial Intelligence, 49:309–343,
1991.
[Przymusinski, 1994] T. C. Przymusinski. A knowledge representation framework based on autoepistemic logic of minimal beliefs. In Proceedings AAAI94, 1994.
[Ramsey, 1978] F. P. Ramsey. Foundations. Routledge & Kegan Paul, London,
1978.
[Reggia et al., 1985] J. A. Reggia, D. S. Nau, and P. Y. Wang. A formal model
of diagnostic inference. i. problem formulation and decomposition. Inf. Sci.,
37:227–256, 1985.
[Reinfrank et al., 1989] M. Reinfrank, O. Dressler, and G. Brewka. On the
relation between truth maintenance and autoepistemic logic. In Proc. Int.
Joint Conf. on Artificial Intelligence, pages 1206–1212, 1989.
[Reiter and Criscuolo, 1981] R. Reiter and G. Criscuolo. On interacting defaults. In Proceedings of IJCAI81, pages 270–276, 1981.
[Reiter and de Kleer, 1987] R. Reiter and J. de Kleer. Formal foundations for
assumptionbased truth maintenance systems: preliminary report. In Proceedings of AAAI-87, pages 183–188, Seattle, Washington, 1987.
[Reiter, 1978] R. Reiter. On closed world data bases. In H. Gallaire and
J. Minker, editors, Logic and Data Bases, pages 119–140. Plenum Press, 1978.
[Reiter, 1980] R. Reiter. A logic for default reasoning. Artificial Intelligence,
13:81–132, 1980.
80
[Reiter, 1982] R. Reiter. Circumscription implies predicate completion (sometimes). In Proc. AAAI, pages 418–420, 1982.
[Reiter, 1987a] R. Reiter. Nonmonotonic reasoning. Annual Review of Computer Science, 2:147–186, 1987.
[Reiter, 1987b] R. Reiter. A theory of diagnosis from first principles. Artificial
Intelligence, 32:57–95, 1987.
[Reiter, 1991] R. Reiter. The frame problem in the situation calculus: A simple
solution (sometimes) and a completeness result for goal regression. In V. Lifschitz, editor, Artificial Intelligence and Mathematical Theory of Computation:
Papers in Honor of Lohn McCarthy, pages 318–420. Academic Press, 1991.
[Reiter, 2001] R. Reiter. Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamic Systems. MIT Press, 2001.
[Russell, 1957] B. Russell. Mysticism and Logic, chapter On the notion of cause,
pages 180–208. Allen & Unvin, London, 1957.
[Sandewall, 1972] E. Sandewall. An approach to the frame problem and its implementation. In Machine Intelligence, volume 7, pages 195–204. Edinburgh
University Press, 1972.
[Sandewall, 1994] Erik Sandewall. Features and Fluents: A Systematic Approach to the Representation of Knowledge About Dynamical Systems. Oxford
University Press, Oxford, 1994.
[Schlechta, 1997] K. Schlechta. Nonmonotonic Logics: Basic Concepts, Results,
and Techniques, volume 1187 of Lecture Notes in AI. Springer Verlag, 1997.
[Schlechta, 2004] Karl Schlechta. Coherent Systems, volume 2 of Studies in
Logic and Practical Reasoning. Elsevier, Amsterdam, 2004.
[Schubert, 1990] L. Schubert. Monotonic solution of the frame problem in the
situation calculus; an efficient method for worlds with fully specified actions.
In Henry Kyburg, Ronald Loui, and Greg Carlson, editors, Knowledge Representation and Defeasible Reasoning, pages 23–67. Kluwer Academic Publishers, Dordrecht, 1990.
[Schwarz and Truszczyński, 1994] G. Schwarz and M. Truszczyński. Minimal
knowledge problem: a new approach. Artificial Intelligence, 67:113–141, 1994.
[Schwarz, 1990] G. Schwarz. Autoepistemic modal logics. In R. Parikh, editor,
Theoretical Aspects of Reasoning about Knowledge, TARK-90, pages 97–109,
San Mateo, CA, 1990. Morgan Kaufmann.
[Schwarz, 1992a] G. Schwarz. Minimal model semantics for nonmonotonic
modal logics. In Proceedings LICS-92, pages 34–43, Santa Cruz, CA., 1992.
81
[Schwarz, 1992b] G. Schwarz. Reflexive autoepistemic logic. Fundamenta Informaticae, 17:157–173, 1992.
[Schwind, 1999] C. Schwind. Causality in action theories. Linköping Electronic
Articles in Computer and Information Science, 4(4), 1999.
[Segerberg, 1971] K. Segerberg. An Essay in Classical Modal Logic, volume 13
of Filosofiska Studier. Uppsala University, 1971.
[Shanahan, 1997] M. P. Shanahan. Solving the Frame Problem. The MIT Press,
1997.
[Shepherdson, 1988] J. C. Shepherdson. Negation in logic programming. In
J. Minker, editor, Deductive Databases and Logic Programming, pages 19–88.
M. Kaufmann, 1988.
[Shoham, 1988] Y. Shoham. Reasoning about Change. Cambridge University
Press, 1988.
[Simon, 1952] Herbert Simon. On the definition of the causal relation. The
Journal of Philosophy, 49:517–528, 1952.
[Stalnaker, 1968] R. Stalnaker. A theory of conditionals. In N. Rescher, editor,
Studies in Logical Theory. Basil Blackwell, Oxford, 1968.
[Stalnaker, 1993] Robert C. Stalnaker. A note on non-monotonic modal logic.
Artificial Intelligence, 64(2):183–196, 1993. Widely circulated in manuscipt
form, 1980 to 1992.
[Tan and Pearl, 1995] S.-W. Tan and J. Pearl. Specificity and inheritance in
default reasoning. In Proceedings Int. Joint Conf. on Artificial Intelligence,
IJCAI-95, pages 1480–1486, 1995.
[Thiele, 1990] H. Thiele. On generation of cumulative inference operators by
default deduction rules. In Nonmonotonic and Inductive Logic, Lecture Notes
in Computer Science, pages 100–137. Springer, 1990.
[Thielscher, 1997] M. Thielscher. Ramification and causality. Artificial Intelligence, 89:317–364, 1997.
[Thomason, 2003] Richmond Thomason. Logic and artificial intelligence. In
Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. 2003.
[Touretzky, 1986] D. S. Touretzky. The Mathematics of of Inheritance Systems.
Morgan Kaufmann, Los Altos, 1986.
[Truszczyński, 1991] M. Truszczyński. Modal interpretations of default logic.
In J. Myopoulos and R. Reiter, editors, Proceedings Int. Joint Conf. on Artificial Intelligence, IJCAI’91, pages 393–398, San Mateo, Calif., 1991. Morgan
Kaufmann.
82
[Turner, 1999] H. Turner. A logic of universal causation. Artificial Intelligence,
113:87–123, 1999.
[van Benthem, 1984] J. van Benthem. Foundations of conditional logic. Journal
of Philosophical Logic, 13:303–349, 1984.
[van Emden and Kowalski, 1976] M. H. van Emden and R. A. Kowalski. The
semantics of predicate logic as a programming language. J. of ACM, 23:733–
742, 1976.
[van Gelder et al., 1991] A. van Gelder, K. A. Ross, and J. S. Schlipf. The wellfounded semantics for general logic programs. J. ACM, 38:620–650, 1991.
[Veltman, 1976] F. Veltman. Prejudices, presuppositions and the theory of conditionals. In J. Groenendijk and M. Stokhof, editors, Amsterdam Papers on
Formal Grammar, volume 1, pages 248–281. Centrale Interfaculteit, Universiteit van Amsterdam, 1976.
[Wittgenstein, 1961] L. Wittgenstein. Tractatus Logico-Philosophicus. Routledge & Kegan Paul, London, 1961. English translation by D. F. Pears and
B. F. McGuinness.
[You et al., 2000] J.-H. You, L. Y. Yuan, and R. Goebel. An abductive apprach
to disjunctive logic programming. Journal of Logic Programming, 44:101–127,
2000.
[Zhang and Foo, 2001] D. Zhang and N. Foo. EPDL: A logic for causal reasoning. In Proc Int. Joint Conf. on Artificial Intelligence, IJCAI-01, pages
131–136, Seattle, 2001. Morgan Kaufmann.
[Zhang and Rounds, 1997] G.-Q. Zhang and W. C. Rounds. Defaults in domain
theory. Theor. Comput. Sci., 177:155–182, 1997.
83