Stable semantics for disjunctive programs

We introduce the stable model semantics fordisjunctive logic programs and deductive databases, which generalizes the stable model semantics, defined earlier for normal (i.e., non-disjunctive) programs. Depending on whether only total (2-valued) or all partial (3-valued) models are used we obtain thedisjunctive stable semantics or thepartial disjunctive stable semantics, respectively. The proposed semantics are shown to have the following properties:

? For normal programs, the disjunctive (respectively, partial disjunctive) stable semantics coincides with thestable (respectively,partial stable) semantics.

? For normal programs, the partial disjunctive stable semantics also coincides with thewell-founded semantics.

? For locally stratified disjunctive programs both (total and partial) disjunctive stable semantics coincide with theperfect model semantics.

? The partial disjunctive stable semantics can be generalized to the class ofall disjunctive logic programs.

? Both (total and partial) disjunctive stable semantics can be naturally extended to a broader class of disjunctive programs that permit the use ofclassical negation.

? After translation of the programP into a suitable autoepistemic theory \( \hat P \) the disjunctive (respectively, partial disjunctive) stable semantics ofP coincides with the autoepistemic (respectively, 3-valued autoepistemic) semantics of \( \hat P \) .

     

Well-founded and stationary models of logic programs

We investigate the class ofstationary or partial stable models of normal logic programs. This important class of models includes all (total)stable models, and, moreover, thewell-founded model is always its smallest member. Stationary models have several natural fixed-point definitions and can be equivalently obtained as expansions or extensions of suitable autoepistemic or default theories. By taking a particular subclass of this class of models one can obtain different semantics of logic programs, including the stable semantics and the well-founded semantics. Stationary models can be also naturally extended to the class of all disjunctive logic programs. These features of stationary models designate them as an important class of models with applications reaching far beyond the realm of logic programming.Partially supported by the National Science Foundation grant #IRI-9313061.     

A new fixpoint semantics for general logic programs compared with the well-founded and the stable model semantics

We study a new fixpoint semantics for logic programs with negation. Our construction is intermediate between Van Gelder’s well-founded model and Gelfond and Lifschitz’s stable model semantics. We show first that the stable models of a logic programP are exactly the well-supported models ofP, i.e. the supported models with loop-free finite justifications. Then we associate to any logic programP a non-monotonic operator over the semilattice of justified interpretations, and we define in an abstract form its ordinal powers. We show that the fixpoints of this operator are the stable models ofP, and that its ordinal powers after some ordinala are extensions of the well-founded partial model ofP. In particular ifP has a well-founded model then that canonical model is also an ordinal power and the unique fixpoint of our operator. We show with examples of logic programs which have a unique stable model but no well-founded model that the converse is false. We relate also our work to Doyle’s truth maintenance system and some implementations of rule-based expert systems.     

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.     

The strong semantics for logic programs

Recently, the well-founded semantics of a logic programP has been strengthened to the well-founded semantics-by-case (WF_C) and this in turn has been strengthened to the extended well-founded semantics (WF_E). Both WF_C(P) and WF_E(P) have thelogical consequence property, namely, if an atomAj is true in the theory Th(P), thenAj is true in the semantics as well. However, neither WF_C nor WF_E has the GCWA property, i.e., if an atomAj is false in all minimal models ofP,Aj may not be false in WF_C(P) (resp. WF_E(P)). We extend the ideas in WF_C and WF_E to define a strong well-founded semantics WF_S which has the GCWA property. The strong semantics WF_S(P) is defined by combining GCWA with the notion ofderived rules. Here we use a new Type-III derived rules in addition to those used in WF_C and WF_E. The relationship between WF_S and WF_C is also clarified.     

Acyclic programs

We study here a natural subclass of the locally stratified programs which we call acyclic. Acyclic programs enjoy several natural properties. First, they terminate for a large and natural class of general goals, so they could be used as terminating PROLOG programs. Next, their semantics can be defined in several equivalent ways. In particular we show that the immediate consequence operator of an acyclic programP has a unique fixpointM _p , which coincides with the perfect model ofP, is the unique Herbrand model of the completion ofP and can be identified with the unique fixpoint of the 3-valued immediate consequence operator associated withP. The completion of an acylic programP is shown to satisfy an even stronger property: addition of a domain closure axiom results in a theory which is complete and decidable with respect to a large class of formulas including the variable-free ones. This implies thatM _p is recursive. On the procedural side we show that SLS-resolution and SLDNF-resolution for acyclic programs coincide, are effective, sound and (non-floundering) complete with respect to the declarative semantics. Finally, we show that various forms of temporal reasoning, as exemplified by the so-called Yale Shooting Problem, can be naturally described by means of acyclic programs.     

The Least Fixpoint Transformation for Disjunctive Logic Programs

The paradigm of disjunctive logic programming(DLP)enhances greatly the expressive power of normal logic programming(NLP)and many(declarative)semantics have been defined for DLP to cope with various problems of knowledge representation in artificial intelligence.However,the expressive ability of the semantics and the soundness of program transformations for DLP have been rarely explored.This paper defines an immediate consequence operatro T^GP for each disjunctive program and shows that T^GP has the least and computable fixpoint Lft(P),Lft is,in fact,a program transformation for DLP,which transforms all disjunctive programs into negative programs.It is shown that Lft preserves many key semantics,including the disjunctive stable models,well-founded model,disjunctive argunent semantics DAS,three-valued models,ect.Thic means that every disjunctive program P has a unique canonical form Lft(P)with respect to these semantics.As a result,the work in this paper provides a unifying framework for studying the expressive ability of various semantics for DLP On the other hand,the computing of the above semantics for negative programs is ust a trivial task,therefore,Lft(P)is also an optimization method for DLP.Another application of Lft is to derive some interesting semantic results for DLP.     

Autoepistemic logics as a unifying framework for the semantics of logic programs

In this paper, it is shown that a three-valued autoepistemic logic provides an elegant unifying framework for some of the major semantics of normal and disjunctive logic programs and logic programs with classical negation, namely, the stable semantics, the well-founded semantics, supported models, Fitting's semantics, Kunen's semantics, the stationary semantics, and answer sets. For the first time, so many semantics are embedded into one logic. The framework extends previous results—by Gelfond, Lifschitz, Marek, Subrahmanian, and Truszczynski —on the relationships between logic programming and Moore's autoepistemic logic. The framework suggests several new semantics for negation-as-failure. In particular, we will introduce the epistemic semantics for disjunctive logic programs. In order to motivate the epistemic semantics, an interesting class of applications called ignorance tests will be formalized; it will be proved that ignorance tests can be defined by means of the epistemic semantics, but not by means of the old semantics for disjunctive programs. The autoepistemic framework provides a formal foundation for an environment that integrates different forms of negation. The role of classical negation and various forms of negation-by-failure in logic programming will be briefly discussed.     

The expressiveness of locally stratified programs

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence, to obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of logically stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleene's hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets. In particular, a set is definable in the well-founded semantics by a programP whose well-founded partial model is total iff it is hyperarithmetic.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by NSF Grant IRI-9012902 and partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by NSF Grant IRI-8905166 and partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

On stratified disjunctive programs

We address the problem of a consistent fixpoint semantics for general disjunctive programs restricted to stratifiable programs which do not recurse through negative literals. We apply the nonmonotonic fixpoint theory developed by Apt, Blair and Walker to a closure operatorT ^c and develop a fixpoint semantics for stratified disjunctive programs. We also provide an iterative definition for negation, called the Generalized Closed World Assumption for Stratified programs (GCWAS), and show that our semantics captures this definition. We develop a model-theoretic semantics for stratified disjunctive programs and show that the least state characterized by the fixpoint semantics corresponds to a stable-state defined in a manner similar to the stable-models of Gelfond and Lifschitz. We also discuss a weaker stratification semantics for general disjunctive programs based on the Weak Generalized Closed World Assumption.     

Foundations of entity-relationship modeling

Database design methodologies should facilitate database modeling, effectively support database processing, and transform a conceptual schema of the database to a high-performance database schema in the model of the corresponding DBMS. The Entity-Relationship Model is extended to theHigher-orderEntity-RelationshipModel (HERM) which can be used as a high-level, simple and comprehensive database design model for the complete database information on the structure, operations, static and dynamic semantics. The model has the expressive power of semantic models and possesses the simplicity of the entity-relationship model. The paper shows that the model has a well-founded semantics. Several semantical constraints are considered for this 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.     

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.     

扩充析取逻辑程序的争论语义

该文探讨争论推理在扩充逻辑程序中的实现及其关系问题.基于“相干原理”,建立了扩充逻辑程序的争论推理框架,多种争论推理形式都可以嵌入其中.特别是提出了一种谨慎语义Acc.同时又定义了良基语义的一种合理扩充Mod,以处理较为大胆的推理形式.另外也研究了相关的理论性质.     

On Local Stratifiability of Logic Prograns and Databases

In this paper,we deal with the problem of verifying local stratifiability of logic programs and databases presented by Przymusinski.Necessary and sufficient condition for the local stratifiability of logic programs are presented and algorithms of performing the verification are developed.Finally,we prove that a database D B containing clauses with disjunctive consequents can be easily converted into a logic program P such that D B is locally statified iff P is locally stratified.     

Aggregation and negation-as-failure

Set-grouping and aggregation are powerful operations of practical interest in database query languages. An aggregate operation is a function that maps a set to some value, e.g., the maximum or minimum in the set, the cardinality of this set, the summation of all its members, etc. Since aggregate operations are typically non-monotonic in nature, recursive programs making use of aggregate operations must be suitably restricted in order that they have a well-defined meaning. In a recent paper we showed that partial-order clauses provide a well-structured means of formulating aggregate operations with recursion. In this paper, we consider the problem of expressing partial-order programs via negation-as-failure (NF), a well-known non-monotonic operation in logic programming. We show a natural translation of partial-order programs to normal logic programs: Anycost-monotonic partial-order programsP is translated to astratified normal program such that the declarative semantics ofP is defined as the stratified semantics of the translated program. The ability to effect such a translation is significant because the resulting normal programs do not make any explicit use of theaggregation capability, yet they are concise and intuitive. The success of this translation is due to the fact that the translated program is a stratified normal program. That would not be the case for other more general classes of programs thancost-monotonic partial-order programs. We therefore develop in stages a refined translation scheme that does not require the translated programs to be stratified, but requires the use of a suitable semantics. The class of normal programs originating from this refined translation scheme is itself interesting: Every program in this class has a clear intended total model, although these programs are in general neither stratified nor call-consistent, and do not have a stable model. The partial model given by the well-founded semantics is consistent with the intended total model and the extended well founded semantics,WFS ⁺, defines the intended model. Since there is a well-defined and efficient operational semantics for partial-order programs^{14, 15, 21)} we conclude that the gap between expression of a problem and computing its solution can be reduced with the right level of notation. Mauricio J. Osorio G., Ph.D.: He is an Associate Professor in the Department of Computer Systems Engineering, University of the Americas, Puebla, Mexico. He is the Head of the Laboratory of Theoretical Computer Science of the Center of Research (CENTIA), Puebla, Mexico. His research is currently funded by CENTIA and CONACYT (Ref. #C065-E9605). He is interested in Applications of Logic to Computer Science, with special emphasis on Logic Programming. He received his B.Sc. in Computer Science from the Universidad Autonoma de Puebla, his M.Sc. in Electrical Engineering from CINVESTAV in Mexico, and his Ph.D. from the State University of New York at Buffalo in 1995. Bharat Jayaraman, Ph.D.: He is an Associate Professor in the Department of Computer Science at the State University of New York at Buffalo. He obtained his bachelors degree in Electronics from the Indian Institute of Technology, Madras in 1975, and his Ph.D. from the University of Utah in 1981. His research interests are in Programming Languages and Declarative Modeling of Complex Systems. He has published over 50 research papers. He has served on the program committees of several conferences in the area of Programming Languages, and he is presently on the Editorial Board of the Journal of Functional and Logic Programming.     

Propositional semantics for disjunctive logic programs

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.