Everything is Knowable - Computer Science Intranet
... update; the matter is also taken up in van Ditmarsch and Kooi (2006) and Qian (2002) and more recently in Holliday and Icard (2010). We will formally define public announcements in section 5. The word ‘unsuccessful update’ is not coincidental. Another philosophical root of the dynamic turn in logics ...
... update; the matter is also taken up in van Ditmarsch and Kooi (2006) and Qian (2002) and more recently in Holliday and Icard (2010). We will formally define public announcements in section 5. The word ‘unsuccessful update’ is not coincidental. Another philosophical root of the dynamic turn in logics ...
![Propositions as [Types] - Research Showcase @ CMU](http://s1.studyres.com/store/data/005730189_1-e85fa7d3c7cfa08d9a3b8e96a27d7888-300x300.png)
Propositions as [Types] - Research Showcase @ CMU
... the composition g ◦ f , whereas for a completely water-tight interpretation equality is required. There are several standard ways of resolving this problem, most notably by interpreting the type theory in a suitable fibered category [Jac99], and then applying technical results pertaining to these [H ...
... the composition g ◦ f , whereas for a completely water-tight interpretation equality is required. There are several standard ways of resolving this problem, most notably by interpreting the type theory in a suitable fibered category [Jac99], and then applying technical results pertaining to these [H ...
Classical first-order predicate logic This is a powerful extension of
... But predicate logic has machinery to vary the arguments to bought. This allows us to express properties of the relation ‘bought’. The machinery is called quantifiers. ...
... But predicate logic has machinery to vary the arguments to bought. This allows us to express properties of the relation ‘bought’. The machinery is called quantifiers. ...
Conjunctive normal form - Computer Science and Engineering
... contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem with k>2) while 2-SAT is known to have solutions in polynomial time. As a consequence,[3] the task of converting a formula into a DNF, preserving satisfiability, is NP-hard; dually, converting into CNF, preserving valid ...
... contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem with k>2) while 2-SAT is known to have solutions in polynomial time. As a consequence,[3] the task of converting a formula into a DNF, preserving satisfiability, is NP-hard; dually, converting into CNF, preserving valid ...
Introduction to first order logic for knowledge representation
... functions, . . . . real world entities. In everyday communication, we are not referring to such mathematical models, but, especially in science, in order to show that a certain argumentation is correct, people provide mathematical models that describes in an abstract and concise manner the specific ...
... functions, . . . . real world entities. In everyday communication, we are not referring to such mathematical models, but, especially in science, in order to show that a certain argumentation is correct, people provide mathematical models that describes in an abstract and concise manner the specific ...
An Overview of Intuitionistic and Linear Logic
... Constructivism is a point of view concerning the concepts and methods used in mathematical proofs, with preference towards constructive concepts and methods. It emerged in the late 19th century, as a response to the increasing use of abstracts concepts and methods in proofs in mathematics. Kronecker ...
... Constructivism is a point of view concerning the concepts and methods used in mathematical proofs, with preference towards constructive concepts and methods. It emerged in the late 19th century, as a response to the increasing use of abstracts concepts and methods in proofs in mathematics. Kronecker ...
Belief Revision in non
... revision operators for non-classical logics. This is based on the following components: i) a sound and complete classical logic axiomatisation of the semantics of the object logic L, ii) a domain-dependent notion of “acceptability” for theories of L and iii) a classical AGM belief revision operation ...
... revision operators for non-classical logics. This is based on the following components: i) a sound and complete classical logic axiomatisation of the semantics of the object logic L, ii) a domain-dependent notion of “acceptability” for theories of L and iii) a classical AGM belief revision operation ...
Taming method in modal logic and mosaic method in temporal logic
... *Some logics aren’t ”well behaved” in some desirable characteristics. As an example of failure in some characteristics we have undecidability of classic FOL (First Order Logic) or undecidability of several versions of AL (Arrow Logic) which is a branch of Modal Logic; *Andréka (’95) proposed relati ...
... *Some logics aren’t ”well behaved” in some desirable characteristics. As an example of failure in some characteristics we have undecidability of classic FOL (First Order Logic) or undecidability of several versions of AL (Arrow Logic) which is a branch of Modal Logic; *Andréka (’95) proposed relati ...
Quantitative Temporal Logics: PSPACE and below - FB3
... More precisely, we prove three results. Our first result is that extending since/until logic of the real line with metric operators ‘sometime in at most n time units’, n coded in binary, is PS PACE-complete even without FVA. (Note that the logic without FVA is more general than with FVA in the sense ...
... More precisely, we prove three results. Our first result is that extending since/until logic of the real line with metric operators ‘sometime in at most n time units’, n coded in binary, is PS PACE-complete even without FVA. (Note that the logic without FVA is more general than with FVA in the sense ...
Classical first-order predicate logic This is a powerful extension
... But predicate logic has machinery to vary the arguments to bought. This allows us to express properties of the relation ‘bought’. The machinery is called quantifiers. ...
... But predicate logic has machinery to vary the arguments to bought. This allows us to express properties of the relation ‘bought’. The machinery is called quantifiers. ...
Insights into Modal Slash Logic and Modal Decidability
... atomic formulas (Free 1 [φ]) and those appearing independence indications (Free 2 [φ]). The former sets are recursively defined as in the case of first-order logic, the independence indications playing no role. To define the latter sets, put Free 2 [(∃x/W )φ] = W ∪ (Free 2 [φ] \ {x}), Free 2 [∀xφ] = ...
... atomic formulas (Free 1 [φ]) and those appearing independence indications (Free 2 [φ]). The former sets are recursively defined as in the case of first-order logic, the independence indications playing no role. To define the latter sets, put Free 2 [(∃x/W )φ] = W ∪ (Free 2 [φ] \ {x}), Free 2 [∀xφ] = ...
Document
... : Show that for all A M(P), every interpretation I: I |= P implies I |= A. Let us consider Herbrand interpretation IH = {A | A ground atom and I |= A}. Then, I |= P I |= A ← B1, ... , Bn for all A ← B1, ... , Bn ground(P) if I |= B1, ... , Bn then I |= A for all A ← B1, ... , Bn ground(P) ...
... : Show that for all A M(P), every interpretation I: I |= P implies I |= A. Let us consider Herbrand interpretation IH = {A | A ground atom and I |= A}. Then, I |= P I |= A ← B1, ... , Bn for all A ← B1, ... , Bn ground(P) if I |= B1, ... , Bn then I |= A for all A ← B1, ... , Bn ground(P) ...
Modal Languages and Bounded Fragments of Predicate Logic
... with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free variable is invari ...
... with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free variable is invari ...
A brief introduction to Logic and its applications
... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...
... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...
Introduction to Mathematical Logic
... A0 is the τ 0 -type reduct of A, denoted by A τ 0 . Of course, it may happen that A τ 0 = B τ 0 while A 6= B. Fixing the type τ , if not stated otherwise, all structures are of this type. The cardinality of the structure A, denoted as |A|, is the cardinality of its ground set. Definition 2.3 A ...
... A0 is the τ 0 -type reduct of A, denoted by A τ 0 . Of course, it may happen that A τ 0 = B τ 0 while A 6= B. Fixing the type τ , if not stated otherwise, all structures are of this type. The cardinality of the structure A, denoted as |A|, is the cardinality of its ground set. Definition 2.3 A ...
A Proof Theory for Generic Judgments
... that the sequent Γ0 , ∀xB −→ C is proved using the introduction of ∀ on the left from the premise Γ0 , B[t/x] −→ C, where t is some term. To reduce the rank of the cut formula ∀x.B between the sequents Γ −→ ∀x.B and Γ0 , ∀xB −→ C, the eigenvariable c in the sequent calculus proof Π(c) must be substi ...
... that the sequent Γ0 , ∀xB −→ C is proved using the introduction of ∀ on the left from the premise Γ0 , B[t/x] −→ C, where t is some term. To reduce the rank of the cut formula ∀x.B between the sequents Γ −→ ∀x.B and Γ0 , ∀xB −→ C, the eigenvariable c in the sequent calculus proof Π(c) must be substi ...
THE ABUNDANCE OF THE FUTURE A Paraconsistent Approach to
... this view has never been deeply investigated, and this is probably due to its contradictory flavor. Indeed, this position is naturally embodied by a paraconsistent semantics (Section 4). Can this possibly make sense or it is just a “weird” conceptual alternative among all the possible temporal logic ...
... this view has never been deeply investigated, and this is probably due to its contradictory flavor. Indeed, this position is naturally embodied by a paraconsistent semantics (Section 4). Can this possibly make sense or it is just a “weird” conceptual alternative among all the possible temporal logic ...
A Survey on Small Fragments of First-Order Logic over Finite
... then the complement is a polynomial of degree 1 since it is given as a∗ ∪ b∗ ba∗ . But Γ∗ \ Γ∗ abΓ∗ is not a polynomial as soon as Γ contains at least three letters. Indeed, consider (acb)∗ . Assume this subset is contained in a polynomial of degree k, then at least one factor acb in (acb)k+1 sits i ...
... then the complement is a polynomial of degree 1 since it is given as a∗ ∪ b∗ ba∗ . But Γ∗ \ Γ∗ abΓ∗ is not a polynomial as soon as Γ contains at least three letters. Indeed, consider (acb)∗ . Assume this subset is contained in a polynomial of degree k, then at least one factor acb in (acb)k+1 sits i ...
Modular Sequent Systems for Modal Logic
... System K + Ẋ. Figure 1 shows the set of rules from which we form our deductive systems. System K is the set of rules {∧, ∨, 2, k, ctr}. We will look at extensions of System K with sets of rules Ẋ ⊆ {ḋ, ṫ, ḃ, 4̇, 5̇}. The rules in Ẋ are called structural modal rules. The 5̇-rule is a bit specia ...
... System K + Ẋ. Figure 1 shows the set of rules from which we form our deductive systems. System K is the set of rules {∧, ∨, 2, k, ctr}. We will look at extensions of System K with sets of rules Ẋ ⊆ {ḋ, ṫ, ḃ, 4̇, 5̇}. The rules in Ẋ are called structural modal rules. The 5̇-rule is a bit specia ...
Expressiveness of Logic Programs under the General Stable Model
... a countable set of function variables. Every constant or variable is equipped with a natural number, called its arity. Nullary function constants and variables are called individual constants and variables, respectively. Nullary predicate constants are called propositional constants. Sometimes we do ...
... a countable set of function variables. Every constant or variable is equipped with a natural number, called its arity. Nullary function constants and variables are called individual constants and variables, respectively. Nullary predicate constants are called propositional constants. Sometimes we do ...
The Emergence of First
... quantifiers of functions when he treated the Principle of Mathematical Induction (1879, sections 11 and 26). To develop the general properties of infinite sequences, Frege both wanted and believed that he needed a logic at least as strong as what was later called second-order logic. Frege developed ...
... quantifiers of functions when he treated the Principle of Mathematical Induction (1879, sections 11 and 26). To develop the general properties of infinite sequences, Frege both wanted and believed that he needed a logic at least as strong as what was later called second-order logic. Frege developed ...
Supervaluationism and Classical Logic
... somewhat surprising claim that there’s actually such an n (they claim we know the existential generalization ‘there is an n that such and such’ even if there is no particular n of which we know that such and such). Many philosophers, however, find this claim something too hard to swallow and take it ...
... somewhat surprising claim that there’s actually such an n (they claim we know the existential generalization ‘there is an n that such and such’ even if there is no particular n of which we know that such and such). Many philosophers, however, find this claim something too hard to swallow and take it ...
Pebble weighted automata and transitive - LSV
... In this section we set up the notation and we recall some basic results on weighted automata and weighted logics. We refer the reader to [6,7] for details. Throughout the paper, Σ denotes a finite alphabet and Σ + is the free semigroup over Σ, i.e., the set of nonempty words. The length of u ∈ Σ + i ...
... In this section we set up the notation and we recall some basic results on weighted automata and weighted logics. We refer the reader to [6,7] for details. Throughout the paper, Σ denotes a finite alphabet and Σ + is the free semigroup over Σ, i.e., the set of nonempty words. The length of u ∈ Σ + i ...