Definite programs General programs Definite programs General programs NAF, CWA comp ( P ) Programs do not have any negative consequences. But sometimes we need them. Negation in logic programming Ex .: Old example program P f on family relations. Summary We would like to derive ¬ child ( a, b ) when P f � | = child ( a, b ) . Two approaches TDDD08 tutorial 1. CWA (closed world assumption) P ⊢ CWA ¬ A whenever P � | = A ( A – ground atom) Włodzimierz Drabent i.e. M P | = ¬ A For TDDD08 Link¨ oping University (Sweden) 2. NAF (negation as finite failure) ( A – atom) P ⊢ NAF ¬ A whenever P � | = A Version 0.6 compiled on October 5, 2020 and an SLD-tree for A is finite (finitely failed tree) 1 / 18 3 / 18 Definite programs General programs Definite programs General programs NAF, CWA comp ( P ) Outline Difference CWA, NAF ◮ Negation and definite logic programs ◮ NAF, CWA ◮ Program completion Ex .: P = { p ← p } P ⊢ CWA ¬ p P �⊢ NAF ¬ p ◮ General logic programs ◮ SLDNF-resolution (generalizing NAF) ◮ Generalizing CWA ◮ Stratified programs & standard models ◮ Stable model semantics 2 / 18 4 / 18
Definite programs General programs NAF, CWA comp ( P ) Definite programs General programs NAF, CWA comp ( P ) CWA, NAF – declarative semantics Program completion, roughly t 1 ) ← � p ( � B 1 equivalent 1 to · · · p ( V 1 , . . . , V k ) ← β one axiom CWA characterized by the least Herbrand model M P t n ) ← � p ( � B n Then the completion contains p ( V 1 , . . . , V k ) ↔ β NAF characterized by the completion of the program, comp ( P ) (for each predicate p in the program) As = occurs in β , the completion includes axioms for equality, called CET CET assures that distinct ground terms are not equal 1 in a certain sense 5 / 18 7 / 18 Definite programs General programs NAF, CWA comp ( P ) Definite programs General programs NAF, CWA comp ( P ) Program completion comp ( P ) works as expected NAF characterized by the completion of the program, comp ( P ) comp ( P ) describes NAF and the standard positive answers: Main idea (roughly): Th .: (Soundness and completeness) comp ( P ) is P with ← converted into ↔ . For any definite program and query P, Q so that comp ( P ) also says that A is false when not implied by the program P ⊢ NAF ¬ Q iff comp ( P ) | = ¬ Q P | = Q iff comp ( P ) | = Q iff Q is a computed Ex .: p ( a ) ← q ( X ) p ( V ) ↔ V = a, ∃ X. q ( X ) ∨ answer of P � p ( f ( Y )) ← r ( Y ) ∃ Y. ( V = f ( Y ) , r ( Y )) 6 / 18 8 / 18
Definite programs General programs SLDNF Generalizing CWA Stratified SMS Definite programs General programs SLDNF Generalizing CWA Stratified SMS General logic programs Declarative/logic characterization of SLDNF-resolution SLDNF-resolution is sound w.r.t. comp ( P ) : Th .: P, Q - general program and query We would like to have negation in queries and programs. Q is an SLDNF-computed answer for P ⇒ comp ( P ) | = Q E.g. A query grandchild ( X , adam ) , ¬ descendant ( X , bill ) A finitely failed tree for Q ⇒ comp ( P ) | = ¬ Q General programs – But comp ( P ) does not describe SLDNF-resolution well also negated atoms (negative literals) allowed in clause bodies It implies too much ·· ⌢ ☞ A “right” semantics: comp ( P ) in a certain 3-valued logic NAF � SLDNF-resolution, natural generalization of SLD-resolution Equivalently, a modified completion (in standard logic, cf. old slides) CWA � . . . For such semantics, SLDNF-resolution sound, and complete for non-floundering queries. 9 / 18 11 / 18 Definite programs General programs SLDNF Generalizing CWA Stratified SMS Definite programs General programs SLDNF Generalizing CWA Stratified SMS SLDNF-resolution Generalizing CWA WFS (well-founded semantics) Like SLD-resolution: derivations, answers, SLDNF-trees general programs ↑ When ¬ A selected in a query Q , build an SLDNF-tree for A . 3 cases: CWA − → standard model definite programs stratified programs answer A obtained 2 A finitely fails otherwise ↓ ( A false, ¬ A true) ( A true, ¬ A false) ( A – ? ) Q Q Q stable model semantics | × general programs NAF � � no children, Q with ¬ A removed definite programs � � ↓ no children, “failed leaf” ↓ floundering ASP (answer set programming) 3v completion sem., (SLDNF-resolution) disjunctive programs A finitely fails = SLDNF-tree finite, no answers, no floundering (with rules H 1 ∨ · · · ∨ H m ← B 1 , . . . , B n ) general programs 2 most general answer 10 / 18 12 / 18
Definite programs General programs SLDNF Generalizing CWA Stratified SMS Definite programs General programs SLDNF Generalizing CWA Stratified SMS Stratified programs Generalizing CWA for stratified programs The standard model of P constructed strata-wise Main feature - no recursion through negation. Strata: P 1 , . . . , P n A sequence of minimal Herbrand models: Program divided into strata. M i – the standard model of P 1 ∪ · · · ∪ P i Each predicate defined in a single stratum. M i \ M i − 1 contains only predicate symbols defined in P i A predicate in a positive body literal – defined in the same or an earlier stratum Constructed similarly to least Herbrand model, M i = � ∞ j =1 ( T P i ) j ( M i − 1 ) A predicate in a negative body literal – defined in an earlier stratum Details, example – [Nilsson,Małuszyński] Ch.4.4 Another approach, example – sli.10c.neg. ∗ 13 / 18 15 / 18 Definite programs General programs SLDNF Generalizing CWA Stratified SMS Definite programs General programs SLDNF Generalizing CWA Stratified SMS Example, stratified program Stable model semantics (SMS) Main idea: Guess an Herbrand interpretation I . � go well together ( X, Y ) ← ¬ incompatible ( X, Y ) . P 3 = Use I to evaluate negative literals of P � P I . � incompatible ( X, Y ) ← ¬ likes ( X, Y ) . P I – definite program, called the reduct of P w.r.t. I . P 2 = incompatible ( X, Y ) ← ¬ likes ( Y, X ) . If the the least Herbrand model of P I is I likes ( X, Y ) ← harmless ( Y ) . A | P I | likes ( X, Y ) ← eats ( X, Y ) . M P I = { A ∈ B = A } = I P 1 = harmless ( rabbit ) . then I is a stable model of P . eats ( python, rabbit ) . P I is ground ( P ) without Details: ◮ each rule with some ¬ A , s.t. I | = A ◮ each ¬ A , s.t. I | = ¬ A 14 / 18 16 / 18
Definite programs General programs SLDNF Generalizing CWA Stratified SMS Stable model semantics (SMS) Stratified program: unique stable model = standard model 17 / 18 Definite programs General programs SLDNF Generalizing CWA Stratified SMS Stable model semantics (SMS) Different pragmatics than the rest of Logic Programming: LP: problem solutions – program answers SMS: problem solutions – stable models of programs SMS usable only if the universe finite no function symbols, no data structures 18 / 18
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