Well-founded Semantics For Logic Programs Research Articles

On Signings and the Well-Founded Semantics

AbstractIn this note, we use Kunen’s notion of a signing to establish two theorems about the well-founded semantics of logic programs, in the case where we are interested in only (say) the positive literals of a predicate p that are consequences of the program. The first theorem identifies a class of programs for which the well-founded and Fitting semantics coincide for the positive part of p. The second theorem shows that if a program has a signing, then computing the positive part of p under the well-founded semantics requires the computation of only one part of each predicate. This theorem suggests an analysis for query answering under the well-founded semantics. In the process of proving these results, we use an alternative formulation of the well-founded semantics of logic programs, which might be of independent interest.

Routley Star and Hyperintensionality

We compare the logic HYPE recently suggested by H. Leitgeb as a basic propositional logic to deal with hyperintensional contexts and Heyting-Ockham logic introduced in the course of studying logical aspects of the well-founded semantics for logic programs with negation. The semantics of Heyting-Ockham logic makes use of the so-called Routley star negation. It is shown how the Routley star negation can be obtained from Dimiter Vakarelov’s theory of negation and that propositional HYPE coincides with the logic characterized by the class of all involutive Routley star information frames. This result provides a much simplified semantics for HYPE and also a simplified axiomatization, which shows that HYPE is identical with the modal symmetric propositional calculus introduced by G. Moisil in 1942. Moreover, it is shown that HYPE can be faithfully embedded into a normal bi-modal logic based on classical logic. Against this background, we discuss the notion of hyperintensionality.

Safe inductions and their applications in knowledge representation

In many knowledge representation formalisms, a constructive semantics is defined based on sequential applications of rules or of a semantic operator. These constructions often share the property that rule applications must be delayed until it is safe to do so: until it is known that the condition that triggers the rule will continue to hold. This intuition occurs for instance in the well-founded semantics of logic programs and in autoepistemic logic. In this paper, we formally define the safety criterion algebraically. We study properties of so-called safe inductions and apply our theory to logic programming and autoepistemic logic. For the latter, we show that safe inductions manage to capture the intended meaning of a class of theories on which all classical constructive semantics fail.

Local closed world reasoning with description logics under the well-founded semantics

An important question for the upcoming Semantic Web is how to best combine open world ontology languages, such as the OWL-based ones, with closed world rule-based languages. One of the most mature proposals for this combination is known as hybrid MKNF knowledge bases (Motik and Rosati, 2010 [52]), and it is based on an adaptation of the Stable Model Semantics to knowledge bases consisting of ontology axioms and rules. In this paper we propose a well-founded semantics for nondisjunctive hybrid MKNF knowledge bases that promises to provide better efficiency of reasoning, and that is compatible with both the OWL-based semantics and the traditional Well-Founded Semantics for logic programs. Moreover, our proposal allows for the detection of inconsistencies, possibly occurring in tightly integrated ontology axioms and rules, with only little additional effort. We also identify tractable fragments of the resulting language.

Well-founded semantics for defeasible logic

Fixpoint semantics are provided for ambiguity blocking and propagating variants of Nute’s defeasible logic. The semantics are based upon the well-founded semantics for logic programs. It is shown that the logics are sound with respect to their counterpart semantics and complete for locally finite theories. Unlike some other nonmonotonic reasoning formalisms such as Reiter’s default logic, the two defeasible logics are directly skeptical and so reject floating conclusions. For defeasible theories with transitive priorities on defeasible rules, the logics are shown to satisfy versions of Cut and Cautious Monotony. For theories with either conflict sets closed under strict rules or strict rules closed under transposition, a form of Consistency Preservation is shown to hold. The differences between the two logics and other variants of defeasible logic—specifically those presented by Billington, Antoniou, Governatori, and Maher—are discussed.

Nonmonotonic Trust Management for P2P Applications

Community decisions about access control in virtual communities are non-monotonic in nature. This means that they cannot be expressed in current, monotonic trust management languages such as the family of Role Based Trust Management languages (RT). To solve this problem we propose RT⊖, which adds a restricted form of negation to the standard RT language, thus admitting a controlled form of non-monotonicity. The semantics of RT⊖ is discussed and presented in terms of the well-founded semantics for Logic Programs. Finally we discuss how chain discovery can be accomplished for RT⊖.

Enhancing global SLS-resolution with loop cutting and tabling mechanisms

Global SLS-resolution is a well-known procedural semantics for top–down computation of queries under the well-founded model. It inherits from SLDNF-resolution the linearity property of derivations, which makes it easy and efficient to implement using a simple stack-based memory structure. However, like SLDNF-resolution it suffers from the problem of infinite loops and redundant computations. To resolve this problem, in this paper we develop a new procedural semantics, called SLTNF-resolution, by enhancing Global SLS-resolution with loop cutting and tabling mechanisms. SLTNF-resolution is sound and complete w.r.t. the well-founded semantics for logic programs with the bounded-term-size property, and is superior to existing linear tabling procedural semantics such as SLT-resolution.

The meaning of negative premises in transition system specifications II

This paper reviews several methods to associate transition relations to transition system specifications with negative premises in Plotkin's structural operational style. Besides a formal comparison on generality and relative consistency, the methods are also evaluated on their taste in determining which specifications are meaningful and which are not. Additionally, this paper contributes a proof theoretic characterisation of the well-founded semantics for logic programs.

Iterative belief revision in extended logic programming

Extended logic programming augments conventional logic programming with both default and explicit negation. Several semantics for extended logic programs have been proposed that extend the well-founded semantics for logic programs with default negation (called normal programs). We show that two of these extended semantics are intractable; both Dung's grounded argumentation semantics and the well-founded semantics of Alferes et al. are NP-hard. Nevertheless, we also show that these two semantics have a common core, a more restricted form of the grounded semantics, which is tractable and can be computed iteratively in quadratic time. Moreover, this semantics is a representative of a rich class of tractable semantics based on a notion of iterative belief revision.

Iterative belief revision in extended logic programming

Extended logic programming augments conventional logic programming with both default and explicit negation. Several semantics for extended logic programs have been proposed that extend the well-founded semantics for logic programs with default negation (called normal programs). We show that two of these extended semantics are intractable; both Dung's grounded argumentation semantics and the well-founded semantics of Alferes et al. are NP-hard. Nevertheless, we also show that these two semantics have a common core, a more restricted form of the grounded semantics, which is tractable and can be computed iteratively in quadratic time. Moreover, this semantics is a representative of a rich class of tractable semantics based on a notion of iterative belief revision.

Well-Founded Semantics for Extended Logic Programs with Dynamic Preferences

The paper describes an extension of well-founded semantics for logic programs with two types of negation. In this extension information about preferences between rules can be expressed in the logical language and derived dynamically. This is achieved by using a reserved predicate symbol and a naming technique. Conflicts among rules are resolved whenever possible on the basis of derived preference information. The well-founded conclusions of prioritized logic programs can be computed in polynomial time. A legal reasoning example illustrates the usefulness of the approach.

An extended Petri net model for normal logic programs

This paper presents an application of the concepts of siphons (deadlocks) and inhibitor arcs in Petri net theory to logic programs with negations. More specifically, an extended Petri net is used to model function-free normal logic programs. In this model, because of the presence of inhibitor arcs, the arbitrary applications of firing rule may cause a contradictory situation. We suggest two directions to avoid contradictions: greedy and secure applications of firing rule. We choose the secure application and show that this is a direct translation of the well-founded semantics in the net model. Furthermore, we show that the greatest unfounded set corresponds to the greatest siphon in Petri net theory when we delete the transitions disabled by the secure application of firing rule, and that the property of siphon simplifies the computation of well-founded semantics for logic programs. We also propose the reduced-Petri-net method by which we can reduce an extended Petri net to a Petri net without inhibitor arcs and compute the well-founded model by iterative applications of this transformation using conventional application of firing rule. >

On the relations between stable and well-founded semantics of logic programs

We study the relations between stable and well-founded semantics of logic programs. 1. We show that stable semantics can be defined in the same way as well-founded semantics based on the basic notion of unfounded sets. Hence, stable semantics can be considered as “two-valued well-founded semantics”. 2. An axiomatic characterization of stable and well-founded semantics of logic programs is given by a new completion theory, called strong completion. Similar to the Clark's completion, the strong completion can be interpreted in either two-valued or three-valued logic. We show that ◦ Two-valued strong completion specifies the stable semantics. ◦ Three-valued strong completion specifies the well-founded semantics. 3. We study the equivalence between stable semantics and well-founded semantics. At first, we prove the equivalence between the two semantics for strict programs. Then we introduce the bottom-stratified and top-strict condition generalizing both the stratifiability and the strictness, and show that the new condition is sufficient for the equivalence between stable and well-founded semantics. Further, we show that the call-consistency condition is sufficient for the existence of at least one stable model.

Generalized disjunctive well-founded semantics for logic programs

Generalized disjunctive well-founded semantics (GDWFS) is a refined form of the generalized well-founded semantics (GWFS) of Baral, Lobo and Minker, to disjunctive logic programs. We describe fixpoint, model theoretic and procedural characterizations of GDWFS and show their equivalence. The fixpoint semantics is similar to the fixpoint semantics of the GWFS, except that it iterates over state-pairs (a pair of sets; one a set of disjunctions of atoms and the other a pair of conjunctions of atoms), rather than partial interpretations. The model theoretic semantics is based on a dynamic stratification of the program. The procedural semantics is based on SLIS refutations, + trees and SLISNF trees.

Three-valued nonmonotonic formalisms and semantics of logic programs

We introduce three-valued extensions of major nonmonotonic formalisms and we prove that the recently proposed well-founded semantics of logic programs is equivalent, for arbitrary logic programs, to three-valued forms of McCarthy's circumscription, Reiter's closed world assumption, Moore's autoepistemic logic and Reiter's default theory. This result not only provides a further justification of the well-founded semantics as a natural extension of the perfect model semantics from the class of stratified programs to the class of all logic programs, but it also establishes the class of all logic programs as a large class of theories, for which natural forms of all four nonmonotonic formalisms coincide. It also paves the way for using efficient computation methods, developed for logic programming, as inference mechanisms for nonmonotonic reasoning.