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Nonmonotonic Logic Default Logic Sergio Tessaris Default Rules I In default logic, nonmonotonic conclusions are represented in terms of special inference rules. I Let A(x), B1 (x), . . . , Bn (x), C (x) be first-order formulas, whose free variables are among x = x1 , . . . , xk . A default is an expression of form A(x) : B1 (x), . . . , Bn (x) C (x) I Intuitive meaning: I I for all objects x = x1 , . . . , xk , if A(x) is known and B1 (x), . . . , Bn (x) can be consistently assumed, then infer C (x). Notation: I I I A(x) is the prerequisite , B1 (x), . . . , Bn (x) are the justifications, and C (x) is the consequent Examples I Prototypical reasoning: Bird(x) : Flies(x) Flies(x) I I Typically, birds fly. No-risk reasoning: Accused(x) : Innocent(x) Innocent(x) I The accused is innocent unless proven otherwise. Examples (contd.) I Best-guess reasoning: CurrentSolution(x) : BestSolution(x) BestSolution(x) I I In the absence of evidence to the contrary, assume that the best solution found so far is the best one. Subjective autoepistemic reasoning: true : ¬HaveElderBrother ¬HaveElderBrother I I In the absence of evidence to the contrary, assume that you have no elder brother. Models CWA as well Special Cases ϕ : ψ1 , . . . ψn χ I Prerequisite-free if ϕ = true : ψ1 , . . . ψn χ I Justification-free if n = 0 I Normal default if n = 1 and ψ1 = χ ϕ:χ χ Default Theories I A default theory is a pair hW , Di where I I I I W is a set of closed first-order formulas D is a set of defaults Elements of W represent certain, but incomplete, information about the world (facts or axioms) Elements of D enable the derivation of plausible conclusions I they hold typically, but they are not necessarily true Semantics of Default Theories I Given the default ϕ : ψ1 , . . . ψn χ the intended meaning is If ϕ is known, and it is consistent to assume ψ1 , . . . ψn , then assume χ. I In which context is ϕ known? I With what ψ1 , . . . ψn should be consistent? I With only the set of facts in W ? Open and Closed Defaults I Given the default A(x) : B1 (x), . . . , Bn (x) C (x) we consider two cases closed no free variables open with free variables I I Original semantics: the set of all ground instances I I alternative semantics in this case free variables as “meta-variables” for ground terms Semantics for Default theories is given for closed theories Default Schema Example I Given the default schema Bird(x) : Flies(x) Flies(x) I and facts Bird(Tweety ), Bird(Jonathan) I Theory corresponds to Bird(Tweety ) : Flies(Tweety ) Flies(Tweety ) Bird(Jonathan) : Flies(Jonathan) Flies(Jonathan) Example: friend of a friend I Usually, the friend of my friend is my friend. Friend(x, y ) ∧ Friend(y , z) : Friend(x, z) Friend(x, z) I Let’s consider the following facts Friend(Tom, Bob), Friend(Bob, Sally ), Friend(Sally , Tina) I Using facts and default we can derive the following: Friend(Tom, Sally ), Friend(Bob, Tina) I But we cannot directly derive Friend(Tom, Tina) I This suggests an iterative approach Example: Nixon Diamond I Let’s consider the defaults Republican(x) : ¬Pacifist(x) ¬Pacifist(x) Quaker (x) : Pacifist(x) Pacifist(x) I With the facts Republican(Nixon), Quaker (Nixon) I Using both the defaults we would conclude Pacifist(Nixon) ∧ ¬Pacifist(Nixon) I “Applying” one of the rules should inhibit the other Extensions I Consider a deductively closed set of closed FOL formulae E Definition A (closed) default ϕ : ψ1 , . . . ψn χ is applicable to E iff ϕ ∈ E and ¬ψ1 6∈ E , . . . ¬ψn 6∈ E I An extension represents a possible world view (Beliefs) I When a default is applicable, then we can use it to draw conclusions How do we select E ? I I I Fixed point definition: pick E and check whether is fine (original) Operational definition Extensions Desiderata An extension E for a closed default theory hW , Di should I I include the facts W be deductively closed I I I be closed under the application of defaults; i.e. if applicable, then χ ∈ E I I I do not stop unless forced to extension are maximal sets w.r.t. defaults minimal w.r.t. the above properties I I to draw more conclusions, classical reasoning should not be prevented i.e. Th(E ) ⊆ E i.e. how to stop exclude “unfounded” formulae Unfounded Extensions I Consider the theory W = {Mollusc} Mollusc : ShellBearer } D={ ShellBearer I The set {Mollusc, ¬ShellBearer } I satisfies, the first 4 properties What about ¬ShellBearer ? I we don’t want arbitrary unsupported facts ϕ:ψ1 ,...ψn χ is Example: Tweety I Consider the theory T = hW , Di: W = {Bird(Tweety )} Bird(Tweety ) : Flies(Tweety ) D= Flies(Tweety ) I Extension of T contains Bird(Tweety ) and Flies(Tweety ) Example: Tweety (contd.) I Consider the theory T 0 = hW 0 , Di where: W 0 = W ∪ {Penguin(Tweety ) ∀x(Penguin(x) → ¬Flies(x))} I Extension E 0 of T 0 must include W 0 and its deductive closure, so ¬Flies(Tweety ) as well I Flies(Tweety ) cannot be consistently assumed any more I Then E 0 = Th(W 0 ) Example: Nixon Diamond I Consider the theory T = hW , Di: W = {Republican(Nixon), Quaker (Nixon)} Republican(x) : ¬Pacifist(x) D = {δ1 = ¬Pacifist(x) Quaker (x) : Pacifist(x) δ2 = } Pacifist(x) I Defaults δ1 and δ2 are mutually conflicting I Two alternative extensions: E1 = {Republican(Nixon), Quaker (Nixon), ¬Pacifist(Nixon)} E2 = {Republican(Nixon), Quaker (Nixon), Pacifist(Nixon)} Definition of Extensions Given a closed default theory T = hW , Di, its semantics is defined by means of a set of closed formulae (extension) E . I given a set of formulae E , the closure operator ΓT (E ) is the smallest set S s.t. I I I W ⊆S Th(S) = S for all rules ϕ : ψ1 , . . . ψn χ in D, if ϕ ∈ S, and ¬ψ1 6∈ E , . . . ¬ψn 6∈ E ; then χ ∈ S I E is the “context” for testing the applicability of defaults Definition of Extensions (contd.) I E is an Extension of T iff E = ΓT (E ) (fixpoint of ΓT (·)) I ΓT (E ) exists for any E What about the fixpoint? I I I ΓT is monotonic in hW , Di but anti-monotonic in its argument;i.e. S1 ⊆ S2 implies ΓT (S2 ) ⊆ ΓT (S1 ) I I there can be multiple fixpoints fixpoint existence is not guaranteed! Operator ΓT I remember justification free rules: ϕ: χ they can be seen as classical rules χ ← ϕ I reduct of D relative to S DS ϕ : ϕ : ψ1 , . . . ψn DS = | ∈ D, ¬ψ1 6∈ S, . . . ¬ψn 6∈ S χ χ I Let `DS be the classical inference ` with the addition of rules in DS ThDS (W ) = {ϕ | ϕ is closed , W `DS ϕ} I ΓT can be defined as ΓT (S) = ThDS (W ) Bounding Principle I The set ThDS (W ) can be defined in terms of Th(·) I ThDS (W ) = Th(W S i≥0 Ei ) ϕ: E0 = χ | ∈ DS , W ` ϕ χ ϕ: Ei = χ | ∈ DS , W ∪ Ei−1 ` ϕ χ I This restricts the kind of derivations we can get, i.e. E = Th(W ∪ G ) n o ϕ:ψ1 ,...ψn where G ⊆ χ | ∈D χ I This provides a way to select candidates for extensions Example: Nixon Diamond I Propositional version of the example, T is W ={q, r } q : p r : ¬p D= , p ¬p I meaning: I I I q Nixon is a Quaker r Nixon is a Republican p Nixon is a pacifist Example (contd.) I Four candidate extensions E1 =Th({q, r }) E2 =Th({q, r , p}) E3 =Th({q, r , ¬p}) E4 =Th({q, r , p, ¬p}) I I Construction of ΓT (Ei ): q: r: DE1 = , p ¬p q: DE2 = p r: DE3 = ¬p DE4 =∅ ΓT (E1 ) = Th({q, r , p, ¬p}) ΓT (E2 ) = Th({q, r , p}) ΓT (E3 ) = Th({q, r , ¬p}) ΓT (E4 ) = Th({q, r }) Extensions of T : E2 , E3 Example (contd.) I Extend W with additional knowledge: W 0 = W ∪ {s, ¬p ← s} = {q, r , s, ¬p ← s} I I Still 4 candidate extensions I I I Note that ¬p ∈ Th(W 0 ) E1 =Th(W 0 ) E3 = Th(W 0 ∪ {¬p}) E2 =Th(W 0 ∪ {p}) E4 = Th(W 0 ∪ {p, ¬p}) Note that E1 = E3 and E2 = E4 Construction of ΓT (Ei ): r: DE1 = ¬p DE2 =∅ E1 is the only extension ΓT (E1 ) = Th(W 0 ) ΓT (E2 ) = Th(W 0 ) Example: Non-Existence of Extensions I Consider the theory W ={p} p : ¬q D= q I Two candidate extensions E1 =Th({p}) E2 =Th({p, q}) I I Construction of ΓT (Ei ): p: DE1 = q DE2 =∅ ΓT (E1 ) = {p, q} ΓT (E2 ) = {p} There are no extensions for the given theory Example: Join Consistency of Justifications I Consider a theory with the defaults : p : ¬p , D= q r I This theory has the single extension Th({q, r }) I justifications doesn’t need to be consistent Example: Join Consistency of Justifications (contd.) I Consider two cars c1 and c2 , and you know that one of them is broken (but not which one); i.e. W = {broken(c1 ) ∨ broken(c2 )} I By default you assume that you can use your car, unless you know is broken : usable(c1 ) ∧ ¬broken(c1 ) : usable(c2 ) ∧ ¬broken(c2 ) , D= usable(c1 ) usable(c2 ) I Both defaults are applicable, and you get the extension Th(W ∪ {usable(c1 ), usable(c2 )}) Note on Joint Consistent Justification I Note that ϕ : ψ1 , . . . , ψn χ is not equivalent to ϕ : ψ1 ∧ . . . ∧ ψn χ I E.g. the sentence I If I don’t know p I assume q corresponds to I : p, ¬p q Which is different from : p ∧ ¬p q Semi-Recursive Characterisation of Extensions I Given a default theory T = hW , Di and set of closed formulae E , the sequence Ei , i ≥ 0 is E0 = W Ei+1 I n ϕ : ψ ,...ψ 1 n ∈ D, Ei ` ϕ, = Th(Ei ) ∪ χ | χ o ¬ψ1 6∈ E , . . . ¬ψn 6∈ E Definition E has a quasiinductive definition in T iff [ E= Ei i≥0 I Theorem: E is an extension of T iff it has a quasiinductive definition in T Semi-Recursive Characterisation of Extensions (contd.) I This definition replaces the ΓT (·) operator I Given a context E this definition is more intuitive However, it doesn’t provide an effective procedure I I The set E needs to be guessed anyway I Nevertheless it’s an useful tool to analyse properties of the framework I There is an alternative Operational definition due to G. Antoniou Maximality of Extensions I I Theorem: Let T = hW , Di be a closed theory and E , E 0 extensions of T . If E ⊆ E 0 , the E = E 0 . Proof I I S S E = i≥0 Ei and E 0 = i≥0 Ei0 by induction: Ei0 ⊆ Ei . Trivial for i = 0; assume Ei0 ⊆ Ei 0 Ei+1 Ei+1 I therefore S n ϕ : ψ ,...ψ 1 n = ∪ χ| ∈ D, Ei0 ` ϕ, χ o 0 0 ¬ψ1 6∈ E , . . . ¬ψn 6∈ E n ϕ : ψ ,...ψ 1 n = Th(Ei ) ∪ χ | ∈ D, Ei ` ϕ, χ o ¬ψ1 6∈ E , . . . ¬ψn 6∈ E Th(Ei0 ) i≥0 Ei0 ⊆ S i≥0 Ei Extensions and Consistency In the following we consider closed theories without justification-free defaults, and Ei defined as before given an E . S I Theorem Let E = i≥0 Ei , then E is consistent iff W is consistent. I T has an inconsistent extension iff W is inconsistent. I I I If W is inconsistent, then Th(W ) is an extension of T If E is an inconsistent extension, then by the previous theorem W is inconsistent as well If T has an inconsistent extension, then it is the only extension of T . I I I let E an inconsistent extension of T , then W is inconsistent too an arbitrary extension E 0 must be inconsistent, since W ⊆ E 0 Th(E ) = Th(E 0 ) = the set of all formulae, so E = E 0 (extensions include their deductive closure) Proof of the Theorem ⇒ W = E0 ⊆ E , since E = S i≥0 Ei . ⇐ Assume E inconsistent, we know that E = Th(E ); therefore for every ϕ closed formula, ϕ ∈ E . I take Ei+1 = Th(Ei ) ∪ ∆i , where n o ϕ : ψ1 , . . . ψn ∆i = χ | ∈ D, Ei ` ϕ, ¬ψ1 6∈ E , . . . ¬ψn 6∈ E χ I I I ∆i = ∅, because D doesn’t contain any justification-free default S E = i≥0 Ei = Th(E0 ) = Th(W ) since E is inconsistent, then W must be inconsistent as well. Normal Default Theories I Normal defaults have the form ϕ:χ χ I A default theory T = hW , Di is normal iff all defaults in D are normal. I Normal theories enjoy stronger properties I They’re strictly less expressive than arbitrary theories Properties of Normal Default Theories I I Each closed normal default theory has an extension If a closed normal default theory T = hW , Di has two distinct extensions E , E 0 , then E ∪ E 0 is inconsistent. I I both extensions cannot be simultaneously accepted Semi-Monotonicity Let D, D 0 be sets of closed normal defaults s.t. D ⊆ D 0 , and let E be an extension of T = hW , Di. Then, T 0 = hW , D 0 i has an extension E 0 s.t. E ⊆ E 0 . Reasoning Tasks: Propositional Theories Given a default theory T = hW , Di, we consider the following reasoning tasks. I Existence does T have an extension? I I Brave reasoning a formula ϕ is a brave consequence of a default theory T iff there is an extension of T containing ϕ. I I problem is ΣP2 -complete Skeptical reasoning a formula ϕ is a skeptical consequence of a default theory T iff ϕ is in all extensions of T . I I problem is ΣP2 -complete (i.e. solvable by a Turing machine in class NP using an NP oracle) problem is ΠP2 -complete (i.e. solvable by a Turing machine in class coNP using an NP oracle) These results hold for propositional theories Complexity in FOL Default Theories I I Decidability for brave reasoning is guaranteed for closed normal theories on top of a decidable FOL fragment Decidability of the FOL fragment used don’t guarantee decidability in the case of open theories I I open default represent an (possibly) infinite set of closed defaults where variables are substituted with term of the Herbrand universe. Other results are based on related formalisms (e.g. Logic Programs with negation) References 1. Reiter, R. 1980. A logic for default reasoning. Artificial Intelligence 13, 81-132. 2. Antoniou, G. 1999. A tutorial on default logics. ACM Computing Surveys 31, 4 (Dec. 1999), 337-359. http://doi.acm.org/10.1145/344588.344602 3. W. Marek, A. Nerode, Nonmonotonic Reasoning, Encyclopedia of Computer Science and Technology. vol. 34, pages 281–289, Marcel Dekker, 1994. http://cs.engr.uky. edu/~marek/papers.dir/94.dir/encyclopaedia.pdf 4. M. Cadoli and M. Schaerf. A survey of complexity results for nonmonotonic logics. Journal of Logic Programming, 17, 127–160, 1993. http://citeseer.ist.psu.edu/cadoli93survey.html 5. V. Lifschitz, On open defaults. Computational Logic: Symposium Proceedings, pp. 80-95, 1990. http: //www.cs.utexas.edu/users/vl/mypapers/open.ps