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A Purely ModelTheoretic Semantics for Disjunctive Logic Programs with Negation ⋆
"... Abstract. We present a purely modeltheoretic semantics for disjunctive logic programs with negation, building on the infinitevalued approach recently introduced for normal logic programs [9]. In particular, we show that every disjunctive logic program with negation has a nonempty set of minimal in ..."
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Abstract. We present a purely modeltheoretic semantics for disjunctive logic programs with negation, building on the infinitevalued approach recently introduced for normal logic programs [9]. In particular, we show that every disjunctive logic program with negation has a nonempty set of minimal infinitevalued models. Moreover, we show that the infinitevalued semantics can be equivalently defined using Kripke models, allowing us to prove some properties of the new semantics more concisely. In particular, for programs without negation, the new approach collapses to the usual minimal model semantics, and when restricted to normal logic programs, it collapses to the wellfounded semantics. Lastly, we show that every (propositional) program has a finite set of minimal infinitevalued models which can be identified by restricting attention to a finite subset of the truth values of the underlying logic. 1
Strong Equivalence of Logic Programs under the InfiniteValued Semantics
"... We consider the notion of strong equivalence [4] of normal propositional logic programs under the infinitevalued semantics [7] (which is a purely modeltheoretic semantics that is compatible with the wellfounded one). We demonstrate that two such programs are strongly equivalent under the infinite ..."
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We consider the notion of strong equivalence [4] of normal propositional logic programs under the infinitevalued semantics [7] (which is a purely modeltheoretic semantics that is compatible with the wellfounded one). We demonstrate that two such programs are strongly equivalent under the infinitevalued semantics if and only if they are logically equivalent in the infinitevalued logic of [7]. In particular, we show that strong equivalence of normal propositional logic programs is decidable, and more specifically coNPcomplete. Our results have a direct implication for the wellfounded semantics since, as we demonstrate, if two programs are strongly equivalent under the infinitevalued semantics, then they are also strongly equivalent under the wellfounded semantics. Keywords: Formal Semantics, Negation in Logic Programming, Strong Equivalence. 1
A Extensional HigherOrder Logic Programming
"... We propose a purely extensional semantics for higherorder logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of ..."
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We propose a purely extensional semantics for higherorder logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixedpoint of an immediate consequence operator. We also propose an SLDresolution proof system which is proven sound and complete with respect to the minimum Herbrand model semantics. In other words, we provide a purely extensional theoretical framework for higherorder logic programming which generalizes the familiar theory of classical (firstorder) logic programming.
Decision problems for partial specifications:
"... Partial specifications allow approximate models of systems such as Kripke structures, or labeled transition systems to be created. Using the abstraction possible with these models, an avoidance of the statespace explosion problem is possible, whilst still retaining a structure that can have propert ..."
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Partial specifications allow approximate models of systems such as Kripke structures, or labeled transition systems to be created. Using the abstraction possible with these models, an avoidance of the statespace explosion problem is possible, whilst still retaining a structure that can have properties checked over it. A single partial specification abstracts a set of systems, whether Kripke, labeled transition systems, or systems with both atomic propositions and named transitions. This thesis deals in part with problems arising from a desire to efficiently evaluate sentences of the modal µcalculus over a partial specification. Partial specifications also allow a single system to be modeled by a number of partial specifications, which abstract away different parts of the system. Alternatively, a number of partial specifications may represent different requirements on a system. The thesis also addresses the question of whether a set of partial specifications is consistent, that is to say, whether a single system exists that is abstracted by each member of the set. The effect of nominals, special atomic propositions true on only one state in a system, is also considered on the problem of the consistency of many partial specifications. The thesis also addresses the question of whether the systems a partial specification abstracts are all abstracted by a second partial specification, the problem of inclusion. The thesis demonstrates how commonly used “specification patterns ” – useful properties specified in the modal µcalculus, can be efficiently evaluated over partial specifications, and gives upper and lower complexity bounds on the problems related to sets of partial specifications. 3 4
Infinite Games and WellFounded Negation ⋆
"... Abstract. We present a new characterization of the wellfounded semantics [vGRS91] using an infinite twoplayer game of perfect information) [GS53]. Our game generalizes the standard game for ordinary logic programming [vE86]. Our proof of correctness is based on a refined version of the game, with ..."
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Abstract. We present a new characterization of the wellfounded semantics [vGRS91] using an infinite twoplayer game of perfect information) [GS53]. Our game generalizes the standard game for ordinary logic programming [vE86]. Our proof of correctness is based on a refined version of the game, with degrees of winning and losing, which, as we demonstrate, corresponds exactly to the infinitevalued characterization of the wellfounded semantics that we have recently introduced [RW05]. 1