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Propositional Semantics for Disjunctive Logic Programs
 Annals of Mathematics and Artificial Intelligence
, 1994
"... In this paper we study the properties of the class of headcyclefree 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 th ..."
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Cited by 149 (2 self)
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In this paper we study the properties of the class of headcyclefree 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 NPcompleteness 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. 1 Introduction ...
Fixpoint semantics for logic programming  a survey
, 2000
"... The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. One source of this richness is the inherent nonmonotonicity of its negation, something that does not have close para ..."
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Cited by 106 (0 self)
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The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. One source of this richness is the inherent nonmonotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies.
The Family of Stable Models
, 1993
"... The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the well ..."
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Cited by 54 (4 self)
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The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the wellfounded model. There is also a dual largest stable model, S k P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models s k P and S k P just mentioned, yields the alternating fixed points of [29], denoted s t P and S t P here. From s t P and S t P in turn, s k P and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain co...
Default Reasoning Using Classical Logic
 Artificial Intelligence
, 1996
"... In this paper we show how propositional default theories can be characterized by classical propositional theories: for each finite default theory, we show a classical propositional theory such that there is a onetoone correspondence between models for the latter and extensions of the former. T ..."
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Cited by 22 (2 self)
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In this paper we show how propositional default theories can be characterized by classical propositional theories: for each finite default theory, we show a classical propositional theory such that there is a onetoone correspondence between models for the latter and extensions of the former. This means that computing extensions and answering queries about coherence, setmembership and setentailment are reducible to propositional satisfiability. The general transformation is exponential but tractable for a subset which we call 2DT  a superset of network default theories and disjunctionfree default theories. Consequently, coherence and setmembership for the class 2DT is NPcomplete and setentailment is coNPcomplete. This work paves the way for the application of decades of research on efficient algorithms for the satisfiability problem to default reasoning. For example, since propositional satisfiability can be regarded as a constraint satisfaction problem (CSP...
Twelve Definitions of a Stable Model
"... This is a review of some of the definitions of the concept of a stable model that have been proposed in the literature. These definitions are equivalent to each other, at least when applied to traditional Prologstyle programs, but there are reasons why each of them is valuable and interesting. A n ..."
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Cited by 18 (1 self)
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This is a review of some of the definitions of the concept of a stable model that have been proposed in the literature. These definitions are equivalent to each other, at least when applied to traditional Prologstyle programs, but there are reasons why each of them is valuable and interesting. A new characterization of stable models can suggest an alternative picture of the intuitive meaning of logic programs; or it can lead to new algorithms for generating stable models; or it can work better than others when we turn to generalizations of the traditional syntax that are important from the perspective of answer set programming; or it can be more convenient for use in proofs; or it can be interesting simply because it demonstrates a relationship between seemingly unrelated ideas.
WellFounded Semantics, Generalized
 In Proceedings of International Symposium on Logic Programming
, 1991
"... Classical fixpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, threevalued, semantics uses #P , which extends TP to Kleene's strong threevalued logic. Both these approaches generalize to cover logic programming systems based on a wide class of ..."
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Cited by 13 (2 self)
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Classical fixpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, threevalued, semantics uses #P , which extends TP to Kleene's strong threevalued logic. Both these approaches generalize to cover logic programming systems based on a wide class of logics, provided only that the underlying structure be that of a bilattice. This was presented in earlier papers. Recently wellfounded semantics has become influential for classical logic programs. We show how the wellfounded approach also extends naturally to the same family of bilatticebased programming languages that the earlier fixpoint approaches extended to. Doing so provides a natural semantics for logic programming systems that have already been proposed, as well as for a large number that are of only theoretical interest. And finally, doing so simplifies the proofs of basic results about the wellfounded semantics, by stripping away inessential details. 1 Introduction There hav...
A Hierarchy of Tractable Subsets for Computing Stable Models
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 1996
"... Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NPhard. In this paper we present a hierarchy of clas ..."
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Cited by 5 (0 self)
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Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NPhard. In this paper we present a hierarchy of classes of knowledge bases,\Omega 2 ; :::, with the following properties: first,\Omega 1 is the class of all stratified knowledge bases; second, if a knowledge base \Pi is k , then \Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in \Pi; third, for an arbitrary knowledge base \Pi, we can find the minimum k such that \Pi belongs in time polynomial in the size of \Pi; and, last, where K is the class of all knowledge bases, it is the case that i=1\Omega i = K, that is, every knowledge base belongs to some class in the hierarchy.
A Theory of Truth that prefers falsehood
 Journal of Philosophical Logic
, 1994
"... We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to h ..."
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Cited by 4 (0 self)
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We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to have many nice features. It is noted that a similar class of fixed points, preferring truth, can also be studied. The notion of intrinsic is shown to relativize to these two subclasses. The mathematical ideas presented here originated in investigations of socalled stable models in the semantics of logic programming. 1 Introduction Briefly stated, the job of a theory of truth is to assign truth values to sentences in a language allowing selfreference, in a way that respects intuition while avoiding paradox. Of course this can not be done in the framework of classical, twovalued logic because of liar sentences. Some generalization allowing partial truth assignments, or perhaps contradic...
Coalgebraic semantics for logic programming
 18th Worshop on (Constraint) Logic Programming, WLP 2004, March 0406
, 2004
"... www.dis.uniroma1.it / ¢ majkic/ Abstract. General logic programs with negation have the 3valued minimal Herbrand models based on the Kripke’s fixpoint knowledge revision operator and on Clark’s completion. Based on these results we deifine a new algebra £¥ ¤, (with the relational algebra embedded ..."
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Cited by 2 (1 self)
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www.dis.uniroma1.it / ¢ majkic/ Abstract. General logic programs with negation have the 3valued minimal Herbrand models based on the Kripke’s fixpoint knowledge revision operator and on Clark’s completion. Based on these results we deifine a new algebra £¥ ¤, (with the relational algebra embedded in it), and present an algorithmic transformation of logic programs into the system of tuplevariable equations which is a £ ¤coalgebra. The solution of any such system of equations (a £ ¤coalgebra) corresponds to the unique homomorphism from this £¦ ¤coalgebra into the final £ ¤coalgebra, which is just the coalgebraic semantics for logic programs. It is shown that such unique solution corresponds to the minimal Herbrand model of annotated version of logic programs and is closely related to the encapsulation of multivalued logic programs into the 2valued annotated logic programs. 1
Extending Answer Set Programming Proefschrift ingediend met het oog op het behalen van de graad
"... I would like to express my sincere gratitude to a number of people who have made the completion of this thesis possible. I would like to thank my promotor Dirk Vermeir for introducing me to the field of answer set programming in the first place and for helping me in almost every aspect of writing a ..."
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I would like to express my sincere gratitude to a number of people who have made the completion of this thesis possible. I would like to thank my promotor Dirk Vermeir for introducing me to the field of answer set programming in the first place and for helping me in almost every aspect of writing a thesis. He has helped me in pinning down a thesis subject, has pointed me to some valuable resources regarding answer set programming and has been a trustworthy sounding board for my ideas. I would also like to thank Jeroen Janssen and Steven Schockaert. We met at just the right time and they helped to focus the content of my thesis and brought me back on track by asking just the right questions at a time when I was lost. Special thanks go out to Jeroen Janssen for proofreading my thesis in such a short timeframe. The quality of this thesis would certainly not be on par with the current quality if it was not for his priceless input. I would like to thank my parents and brother for always being there and loving me unconditionally. Special thanks go out to my parents for their