Overview of disjunctive logic programming

The field of disjunctive programming started approximately in 1982 and has reached its first decade. The first result in the field was the development of the Generalized Closed World Assumption (GCWA). Major results have been made in this field since 1986. An overview is presented of the developments that have taken place, which include model theoretic, proof theoretic and fixpoint semantics for disjunctive, and extended normal disjunctive theories including alternative forms of negation.Dedicated to Chitta Baral, José Alberto Fernández, Jorge Lobo and Arcot Rajasekar.     

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.     

Semantic Forcing in Disjunctive Logic Programs

We propose a semantics for disjunctive logic programs, based on the single notion of forcing. We show that the semantics properly extends, in a natural way, previous approaches. A fixpoint characterization is also provided. We also take a closer look at the relationship between disjunctive logic programs and disjunctive-free logic programs. We present certain criteria under which a disjunctive program is semantically equivalent with its disjunctive-free (shifted) version.     

A Fixpoint Semantics for Stratified Databases

Przmusinski extended the notion of stratified logic programs,developed by Apt,Blair and Walker,and by van Gelder,to stratified databases that allow both negative premises and disjunctive consequents.However,he did not provide a fixpoint theory for such class of databases.On the other hand,although a fixpoint semantics has been developed by Minker and Rajasekar for non-Horn logic programs,it is tantamount to traditional minimal model semantics which is not sufficient to capture the intended meaning of negation in the premises of clauses in stratified databases.In this paper,a fixpoint approach to stratified databases is developed,which corresponds with the perfect model semantics.Moreover,algorithms are proposed for computing the set of perfect models of a stratified database.     

Weak Generalized Closed World Assumption

Explicit representation of negative information in logic programs is not feasible in many applications such as deductive databases and artificial intelligence. Defining default rules which allow implicit inference of negated facts from positive information encoded in a logic program has been an attractive alternative to the explicit representation approach. There is, however, a difficulty associated with implicit default rules. Default rules such as the CWA and the GCWA, which closely model logical negation, are in general computationally intractable. This has led to the development of weaker definitions of negation such as the Negation-As-Failure (NF) and the Support-For-Negation (SN) rules which are computationally simpler. These are sound implementations of the CWA and the GCWA, respectively. In this paper, we define an alternative rule of negation based upon the fixpoint definition of the GCWA. This rule, called the Weak Generalized Closed World Assumption (WGCWA), is a weaker definition of the GCWA that allows us to implement a sound negation rule, called the Negation-As-Finite-Failure (NAFF), similar to the NF-rule and less cumbersome than the SN-rule. We present three definitions of the NAFF. Two declarative definitions similar to those for the NF-rule and one procedural definition based on SLI-resolution.     

Disjunctive LP+integrity constraints= stable model semantics

We show that stable models of logic programs may be viewed as minimal models of programs that satisfy certain additional constraints. To do so, we transform the normal programs into disjunctive logic programs and sets of integrity constraints. We show that the stable models of the normal program coincide with the minimal models of the disjunctive program thatsatisfy the integrity constraints. As a consequence, the stable model semantics can be characterized using theextended generalized closed world assumption for disjunctive logic programs. Using this result, we develop a bottomup algorithm for function-free logic programs to find all stable models of a normal program by computing the perfect models of a disjunctive stratified logic program and checking them for consistency with the integrity constraints. The integrity constraints provide a rationale as to why some normal logic programs have no stable models.     

Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation

Disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning. This paper focuses on stable model semantics, currently the most widely acknowledged semantics for disjunctive logic programs. After presenting a new notion of unfounded sets for disjunctive logic programs, we provide two declarative characterizations of stable models in terms of unfounded sets. One shows that the set of stable models coincides with the family of unfounded-free models (i.e., a model is stable iff it contains no unfounded atoms). The other proves that stable models can be defined equivalently by a property of their false literals, as a model is stable iff the set of its false literals coincides with its greatest unfounded set. We then generalize the well-founded operator to disjunctive logic programs, give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of function-free programs. The algorithm's soundness and completeness are proved and some complexity issues are discussed.     

Completeness of hyper-resolution via the semantics of disjunctive logic programs

We present a proof of completeness of hyper-resolution based on the fixpoint semantics of disjunctive logic programs. This shows that hyper-resolution can be studied from the point of view of logic programming.     

An alternative approach to the semantics of disjunctive logic programs and deductive databases

In this paper, we study a new semantics of logic programming and deductive databases. Thepossible model semantics is introduced as a declarative semantics of disjunctive logic programs. The possible model semantics is an alternative theoretical framework to the classical minimal model semantics and provides a flexible inference mechanism for inferring negation in disjunctive logic programs. We also present a proof procedure for the possible model semantics and show that the possible model semantics has an advantage from the computational complexity point of view.This is a revised and extended version of the paper [36] which was presented at the Tenth International Conference on Logic Programming, Budapest, 21–25 June 1993.     

A semantics for a class of non-deterministic and causal production system programs

We define a class of function-free rule-based production system (PS) programs that exhibit non-deterministic and/or causal behavior. We develop a fixpoint semantics and an equivalent declarative semantics for these programs. The criterion to recognize the class of non-deterministic causal (NDC) PS programs is based upon extending and relaxing the concept of stratification, to partition the rules of the program. Unlike strict stratification, this relaxed stratification criterion allows a more flexible partitioning of the rules and admits programs whose execution is non-deterministic or causal or both. The fixpoint semantics is based upon a monotonic fixpoint operator which guarantees that the execution of the program will terminate. Each fixpoint corresponds to a minimal database of answers for the NDC PS program. Since the execution of the program is non-deterministic, several fixpoints may be obtained. To obtain a declarative meaning for the PS program, we associate a normal logic program with each NDC PS program. We use the generalized disjunctive well-founded semantics to provide a meaning to the normal logic program Through these semantics, a well-founded state is associated with and a set of possible extensions, each of which are minimal models for the well-founded state, are obtained. We show that the fixpoint semantics for the NDC PS programs is sound and complete with respect to the declarative semantics for the corresponding normal logic program .This research is partially sponsored by the National Science Foundation under grant IRI-9008208 and by the Institute for Advanced Computer Studies.