J. Jaar, J.-L. Lassez, and J.W. Lloyd. Completeness of the Negation as Failure rule. Proc. IJCAI'83, pp. 500-506, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is more difficult to answer. Negation as failure is incomplete in general [13] i.e. completeness results for SLDNF resolution, where no equational theory is considered, are obtained only for restricted classes of programs. For instance, it is complete for definite programs and normal ground goals [30] and for hierarchical normal programs and normal goals provided they are allowed [30, 49] In the latter, case the conditions attached to programs and goals ensure that the SLDNF tree of a program and a goal exists and is finite and that a derivation of a goal with respect to a program never ....

....completeness results for SLDNF resolution, where no equational theory is considered, are obtained only for restricted classes of programs. For instance, it is complete for definite programs and normal ground goals [30] and for hierarchical normal programs and normal goals provided they are allowed [30, 49]. In the latter, case the conditions attached to programs and goals ensure that the SLDNF tree of a program and a goal exists and is finite and that a derivation of a goal with respect to a program never flounders. We need similar conditions for the E programs considered in this paper. But there ....

J. Jaffar, J.-L. Lassez, and J. Lloyd. Completeness of the Negation as Failure Rule. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), pages 500--506, 1983.

....which contains definite programs and allowed programs. Our third result is that SLDNF resolution is sound and complete for decomposable programs and admissible goals with respect to three valued models of the completion [Theorem 89, page 64] This theorem extends the completeness results of [4, 13, 18]. In the proof of the completeness theorem we use proof theoretic methods. If a goal is a three valued consequence of the completion then it is provable in the sequent calculus for the completion without axioms of the form #, A, A. The complexity of the proof can be bounded by a partial ....

....of the theorem there is no condition on the goal L 1 , L n . In [18] Kunen shows that for call consistent programs and strict goals three valued consequences of the completion and classical consequences are the same. Therefore Theorem 89 is a strengthening of the completeness results in [4, 13, 18]. Chapter 5 A sound and complete semantics for ESLDNFS resolution Chapter 4 was on ESLDNF resolution, this chapter is on ESLDNFS resolution. We will introduce a new semantics for the completion of a program. We will introduce threevalued Kripke structures. The main theorem will be that ....

J. Ja#ar, J.-L. Lassez, and J. W. Lloyd. Completeness of the negation as failure rule. In Proceedings of the 8th International Joint Conference on Artificial Intelligence IJCAI-83, pages 500--506, Karlsruhe, 1983.

....i# P is a regular program, and if this is the case then # is an # goal for P i# it is a regular goal for P . We come to the main theorem of this section. It corresponds to the completeness of the negation as failure rule for definite programs which was proved by Ja#ar, Lassez and Lloyd in [7]. Theorem 22 Let P be an # program. If the goal L 1 , L q is not finitely failed then there exists a countable three valued structure M with (1) M is a model of comp(P ) 2) #(L 1 # . # L q ) is not true in M, 3) if an atom A is true in M then some instance of A ....

J. Ja#ar, J.-L. Lassez, and J. W. Lloyd. Completeness of the negation as failure rule. In Proceedings of the 8th International Joint Conference on Artificial Intelligence IJCAI-83, pages 500--506, Karlsruhe, 1983.

....like the class of decomposable programs. This class contains the definite programs and the allowed programs. We show that SLDNF resolution is sound and complete for decomposable programs with respect to three valued models of the completion. This theorem extends the completeness results of [3, 13, 18]. It is also possible to prove the completeness of SLDNF resolution for programs which have the cut property directly, using model theoretic methods (see [32] The 2 proof theoretic method, however, which we will use in this paper, gives more insight. The motivation to use proof theory for ....

....(2) of the theorem there is no condition on L 1 , L r . In [18] Kunen shows that for call consistent programs and strict goals three valued consequences of the completion and classical consequences are the same. Therefore Theorem 7. 23 is a strengthening of the completeness results in [3, 13, 18]. 8. The calculus of Negation as Failure and SLDNFS resolution The calculus of Negation as Failure is an intuitionistic fragment of C3(P ) in the language with , #, # and # only. #, # denote finite sets of equations; finite sets of equations or formulas are denoted by #, #. The objects which ....

J. Ja#ar, J.-L. Lassez, and J. W. Lloyd. Completeness of the negation as failure rule. In Proceedings of the 8th International Joint Conference on Artificial Intelligence IJCAI-83, pages 500--506, Karlsruhe, 1983.

....the main theorem of this section. A variant of it is proved in [27] in the context of three valued logic. Its proof is a generalization of the proof of Kunen in [20] The proof which we give below is more direct and it does not use three valued logic. It is based on the completeness proofs in [16] and [26] In some sense Lemma 4.2 above is a generalization of Theorem 3 of [25] and Theorem 4.5 below is a strong generalization of Lemma 2 of [25] Theorem 4.5 Let P be a program and (L 1 , L r ) be a goal in C (P ) which is not in N(P ) Let V be a set of variables which does not ....

J. Ja#ar, J.-L. Lassez, and J. W. Lloyd. Completeness of the negation as failure rule. In Proceedings of the 8th International Joint Conference on Artificial Intelligence IJCAI-83, pages 500--506, Karlsruhe, 1983.

....constraints are added to the formalism: P ; TD j= q iff fair execution of q finitely fails Here P is the Clark completion of P [5] Since P contains embedded existential quantifiers, the extension of this result cannot be established with the techniques already discussed. The construction of [20] used to establish this result is specific to TH , but the construction of [59] is of a form that can be extended to other constraint domains. There are many other results that extend from logic programming to CLP, some only if the constraint domain satisfies a certain expressiveness property. See ....

J. Jaffar, J-L. Lassez & J.W. Lloyd, Completeness of the Negation as Failure Rule, Proc. IJCAI-83, 500--506, 1983.

....failed executions) to a canonical program. See also the work of Wallace [27] It has been shown that, for every program P and query Q, if P ; EFT 6j= Q then there is a model AP;Q of EFT [ DCA such that gm(P ; AP;Q ) j= 9 Q. Different constructions of such an algebra AP;Q are given in [10] and [28] Now let F be the free product of the algebras AP;Q where P and Q range over all possible programs and queries, and let v P;Q be a valuation on AP;Q that demonstrates gm(P ; AP;Q ) j= 9 Q. Then, for every P and Q, gm(P ; F) j= 9 Q, using the valuation hP;Q ffi vP;Q , where hP;Q ....

J. Jaffar, J-L. Lassez & J. Lloyd, Completeness of the Negation-as-Failure Rule, Proc. 8th. International Joint Conference on Artificial Intelligence, 1983.

.... a Sequent Calculus for Negation as Failure James Harland, Royal Melbourne Institute of Technology 1 Introduction We consider the problem of giving a sequent calculus interpretation of the Negation as Failure (NAF) rule[2, 10, 6]. In essence, this consists of giving proof rules which can be used to demonstrate that a given formula can be disproved, i.e. that there can be no proof of the formula. The notion of being able to determine the existence of such disproofs seems to be at the heart of the Negation as Failure rule, ....

J. Jaffar, J-L. Lassez and J. Lloyd, Completeness of the Negation as Failure Rule, Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, West Germany, 1983.

....2. 3 Answering Queries Several query answering methods have been suggested for stratified programs in the literature: in particular, SLDNF resolution [Cla78] and XOLDT resolution [TS86] War91] SLDNF resolution, though sound [Cla78] is only complete for a subclass of stratified programs [JLL83]. Various practical Prolog systems have been developed based on SLDNF resolution. To answer queries with respect to programs with a multiple number of stable models, several approaches have been suggested [PAA91a, BNNS94, FLMS93, SZ90, IKH92, WC93, EK89] in the literature. Warren s XOLDT ....

J. Jaffar, J-L. Lassez, and J. Lloyd. Completeness of the negation as failure rule. In Proc. of IJCAI-83, volume 1, pages 500--506, 1983.

....a top down procedural semantics known as Negation as Failure, which when combined with SLD resolution [29] is referred to as SLDNF resolution. This method is sound with respect to the completion of the program, and is complete for Horn programs (possibly with negative subgoals in the goal only) [9]. Another approach was taken by Przymusinski [22] Przymusinski defined the class of perfect models of a program, and argued that the semantics of the program be given by the logical consequences of the (unique) perfect model. For locally stratified programs (and hence also for stratified ....

J. Jaffar, J.-L. Lassez, and J. Lloyd. Completeness of the negation-as-failure rule. In Int'l Joint Conf. on Artificial Intelligence, pages 500--506, 1983.

....2. 3 Answering Queries Several query answering methods have been suggested for stratified programs in the literature: in particular, SLDNF resolution [Cla78] and XOLDT resolution [TS86] War91] SLDNF resolution, though sound [Cla78] is only complete for a subclass of stratified programs [JLL83]. Various practical Prolog systems have been developed based on SLDNF resolution. To answer queries with respect to programs with a multiple number of stable models, several approaches have been suggested [PAA91a, BNNS94, FLMS93, SZ90, IKH92, WC93, EK89] in the literature. Warren s XOLDT ....

J. Jaffar, J-L. Lassez, and J. Lloyd. Completeness of the negation as failure rule. In Proc. of IJCAI-83, volume 1, pages 500--506, 1983.

....sense that if a goal has a finite SLDNF derivation, then it is a logical consequence of the completed program. Jaffar, Lassez and Lloyd showed that SLDNF was complete (in the same sense) for Horn programs with non floundering queries consisting of a conjunction of positive and or negative literals [15]. SLDNF was further investigated for general logic programs by Lloyd [20] who coined the term SLDNF) Shepherdson [37, 38] q.v. for further bibliography) and others. This approach is logically impeccable, but does not address the issue of how the compiler or the interpreter of the general ....

....a one to one interpretation of the terms, so that q(c) cannot be made true by setting c = a or c = b. The original logical consequence approach essentially declares that only conclusions that are logical consequences (in the classical, 2 valued sense) of the completed program should be inferred [6, 15, 20, 37]. When the completed program is consistent, this approach implicitly defines a 3 valued interpretation: assign value true to instantiated atoms that are true in all (2 valued, not necessarily Herbrand) models of the completed program, false to instantiated atoms that are false in all models, and ....

J. Jaffar, J.-L. Lassez, and J. Lloyd. Completeness of the negation-as-failure rule. In Int'l Joint Conf. on Artificial Intelligence, pages 500--506, 1983.

....with the Negation as Failure rule has become known as SLDNFresolution, and is the basis of current Prolog. Apt and van Emden [AvE82] and Lassez and Maher [LM84] extended this work by providing a declarative semantics for the Negation as Failure rule using fixpoint theory. Jaffar, Lassez, and Lloyd [JLL83] showed that SLDNF resolution was complete for ground negated atoms and Horn programs. Later work [ABW88, CH85, GPP89] extended the Closed World Assumption and Negation as Failure to stratified programs as well, i.e. those programs which contain negative goals in clauses but do not recurse through ....

J. Jaffar, J.-L. Lassez, and J.W. Lloyd. Completeness of the negation as failure rule. In Proc. of the Eighth Int'l Joint Conf. on Artificial Intelligence, pages 500--506, Karlsruhe, West Germany, 1983. 157

.... P, i.e. for positive programs comp(P) does not introduce any new positive information (see [Ld] ffl SLDNF resolution (used in Prolog) is always sound with respect to Clark s semantics and for positive programs and queries as well as for recursion free programs it is also complete (see [JLL], Ld] Let us observe, that the second property seems very natural and important for positive programs. Indeed, non monotonic semantics of logic programs can be considered as built upon the following two important premises: a) It has to imply all formulae provable from P itself; b) It should ....

....The proof is given in the Appendix. 2 Theorem 6.5 Suppose that P is a stratified logic program. For any sentence F, if comp(P ) j= F , then PERF (P ) j= F: This implies that the perfect model semantics is strictly stronger than the semantics defined by Clark s completion. Proof: It is known (see [JLL] and [AVE] that M is a model of Clark s completion comp(P) of a program P iff M is a supported model of CET(P) By Proposition 6.1 every perfect model is supported and therefore every sentence F which is satisfied in all models of comp(P) is also satisfied in all models from PERF(P) To see that ....

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Jaffar, J., Lassez, J-L. and Lloyd, J., `Completeness of the Negation as Failure Rule', IJCAI-83, Karlsruhe, 1983, 500-506.

....and equality axioms. He shows that every goal G with a finitely failed SLD tree is a logical consequence of comp(P) and equality axioms (actually his results extend to general programs using safe computation rules, i.e. rules that select only ground negative literals) Jaffar, Lassez and Lloyd [JLL83] prove the completeness result and show that if a goal G is a logical consequence of comp(P) then there is a finitely failed SLD tree for G. For a comprehensive description of negation in logic programs, see Shepherdson [She88] The excellent research monograph by John Lloyd [Llo87] pulls ....

....all of the positive disjuncts that can be derived from the program. If one were to form a set consisting of an atom from each disjunct in the fixpoint, one would obtain a minimal model of the program. We were also able to show, using the fixpoint operator and ideas borrowed from Lassez et al. [JLL83] how one can find those atoms that may be assumed false and showed that this gives the same result as the GCWA. We also addressed the problem of devising a procedural method [MR88] for finding answers to negated atoms. For this we used SLI resolution. To handle negation by failure, we modified SLI ....

J. Jaffar, J.-L. Lassez, and J.W. Lloyd. Completeness of the Negation as Failure Rule. In Proceedings Eighth International Joint Conference on Artificial Intelligence, pages 500--506, Karlsruhe, West Germany, 8-12 August 1983.

....iff P is a regular program, and if this is the case then Gamma is an goal for P iff it is a regular goal for P . We come to the main theorem of this section. It corresponds to the completeness of the negation as failure rule for definite programs which was proved by Jaffar, Lassez and Lloyd in [7]. Theorem 22 Let P be an program. If the goal L 1 ; L q is not finitely failed then there exists a countable three valued structure M with (1) M is a model of comp(P ) 2) 9(L 1 : L q ) is not true in M, 3) if an atom A is true in M then some instance of A succeeds with ....

J. Jaffar, J.-L. Lassez, and J. W. Lloyd. Completeness of the negation as failure rule. In Proceedings of the 8th International Joint Conference on Artificial Intelligence IJCAI-83, pages 500--506, Karlsruhe, 1983.

No context found.

J. Jaffar, J-L. Lassez & J.W. Lloyd, Completeness of the Negation as Failure Rule, Proc. IJCAI, Karlsruhe, 500--506, 1983.

....of the operational semantics (SLD resolution) with respect to success and to characterize finite failure. Clark [2] introduced the program completion as a logical semantics for finite failure and proved soundness of the operational semantics with respect to the completion. Jaffar et al. [9] proved completeness of the operational semantics with respect to the completion. Together these results provide an elegant algebraic, fixpoint and logical semantics for pure logic programs. The book of Lloyd [17] provides a detailed introduction to the semantics of logic programs. One natural ....

J. Jaffar, J.-L. Lassez and J.W. Lloyd. Completeness of the Negation as Failure Rule. Proc. IJCAI-83, 500--506, 1983.

....of the operational semantics (SLD resolution) with respect to success and to characterise finite failure. Clark [2] introduced the program completion as a logical semantics for finite failure and proved soundness of the operational semantics with respect to the completion. Jaffar et al. [9] proved completeness of the operational semantics with respect to the completion. Together these results provide an elegant algebraic, fixpoint and logical semantics for pure logic programs. The book of Lloyd [16] provides a good introduction to the semantics of logic programs. One natural ....

J. Jaffar, J.-L. Lassez and J.W. Lloyd. Completeness of the Negation as Failure Rule. Proc. IJCAI-83, 500--506, 1983.

No context found.

J. Jaar, J.-L. Lassez, and J.W. Lloyd. Completeness of the Negation as Failure rule. Proc. IJCAI'83, pp. 500-506, 1983.

No context found.

J. Jaffar, J.-L. Lassez and J. W. Lloyd, "Completeness of the negation as failure rule," in IJCAI-83 (Karlsruhe, 1983) pp. 500-506.

No context found.

J. Jaffar, J-L. Lassez, and J. Lloyd. Completeness of the negation as failure rule. In Proceedings AAAI-83, pages 500--506, Los Altos, CA, 1983. American Association for Artificial Intelligence, Morgan Kaufmann.

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