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Semantical Characterizations and Complexity of Equivalences in Answer Set Programming
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2007
"... In recent research on nonmonotonic logic programming, repeatedly strong equivalence of logic programs P and Q has been considered, which holds if the programs P ∪ R and Q ∪ R have the same answer sets for any other program R. This property strengthens the equivalence of P and Q with respect to answe ..."
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Cited by 28 (12 self)
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In recent research on nonmonotonic logic programming, repeatedly strong equivalence of logic programs P and Q has been considered, which holds if the programs P ∪ R and Q ∪ R have the same answer sets for any other program R. This property strengthens the equivalence of P and Q with respect to answer sets (which is the particular case for R =∅), and has its applications in program optimization, verification, and modular logic programming. In this article, we consider more liberal notions of strong equivalence, in which the actual form of R may be syntactically restricted. On the one hand, we consider uniform equivalence where R is a set of facts, rather than a set of rules. This notion, which is wellknown in the area of deductive databases, is particularly useful for assessing whether programs P and Q are equivalent as components of a logic program which is modularly structured. On the other hand, we consider relativized notions of equivalence where R ranges over rules over a fixed alphabet, and thus generalize our results to relativized notions of strong and uniform equivalence. For all these notions, we consider disjunctive logic programs in the propositional (ground) case as well as some restricted classes, providing semantical characterizations and analyzing the computational complexity. Our results, which naturally extend to answer set semantics for programs with strong negation, complement the results on strong
On acyclic and headcycle free nested logic programs
 Proceedings of 19th International Conference on Logic Programming (ICLP04), volume 3132 of Lecture Notes in Computer Science
, 2004
"... Abstract. We define the class of headcycle free nested logic programs, and its proper subclass of acyclic nested programs, generalising similar classes originally defined for disjunctive logic programs. We then extend several results known for acyclic and headcycle free disjunctive programs under ..."
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Cited by 5 (3 self)
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Abstract. We define the class of headcycle free nested logic programs, and its proper subclass of acyclic nested programs, generalising similar classes originally defined for disjunctive logic programs. We then extend several results known for acyclic and headcycle free disjunctive programs under the stablemodel semantics to the nested case. Most notably, we provide a propositional semantics for the program classes under consideration. This generalises different extensions of Fages ’ theorem, including a recent result by Erdem and Lifschitz for tight logic programs. We further show that, based on a shifting method, headcycle free nested programs can be rewritten into normal programs in polynomial time and space, extending a similar technique for headcycle free disjunctive programs. All this shows that headcycle free nested programs constitute a subclass of nested programs possessing a lower computational complexity than arbitrary nested programs, providing the polynomial hierarchy does not collapse. 1
Elimination of Disjunction and Negation in AnswerSet Programs under Hyperequivalence ⋆
"... Abstract. The study of different notions of equivalence is one of the cornerstones of current research in answerset programming. This is mainly motivated by the needs of program simplification and modular programming, for which ordinary equivalence is insufficient. A recently introduced equivalence ..."
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Cited by 3 (3 self)
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Abstract. The study of different notions of equivalence is one of the cornerstones of current research in answerset programming. This is mainly motivated by the needs of program simplification and modular programming, for which ordinary equivalence is insufficient. A recently introduced equivalence notion in this context is hyperequivalence, which includes as special cases strong, uniform, and ordinary equivalence. We study in this paper the question of replacing programs by syntactically simpler ones preserving hyperequivalence (we refer to such a replacement as a casting). In particular, we provide necessary and sufficient semantic conditions under which the elimination of disjunction, negation, or both, in programs is possible, preserving hyperequivalence. In other words, we characterise in modeltheoretic terms when a disjunctive logic program can be replaced by a hyperequivalent normal, positive, or Horn program, respectively. Furthermore, we study the computational complexity of the considered tasks and, based on similar results for strong equivalence developed in previous work, we provide methods for constructing the respective hyperequivalent programs. Our results contribute to the understanding of problem settings in logic programming in the sense that they show in which scenarios the usage of certain constructs are superfluous or not. 1
Characterising equilibrium logic and nested logic programs: Reductions and complexity
, 2009
"... Equilibrium logic is an approach to nonmonotonic reasoning that extends the stablemodel and answerset semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kind ..."
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Cited by 3 (2 self)
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Equilibrium logic is an approach to nonmonotonic reasoning that extends the stablemodel and answerset semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs into quantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of secondorder logic, and its formulas are usually referred to as quantified Boolean formulas (QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are the consistency problem, brave reasoning, and skeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions
Casting Away Disjunction and Negation under a Generalisation of Strong Equivalence with Projection ⋆
"... Abstract. In answerset programming (ASP), many notions of program equivalence have been introduced and formally analysed. A particular line of research in this direction aims at studying conditions under which certain syntactic constructs can be eliminated from programs preserving some given equiva ..."
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Cited by 1 (1 self)
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Abstract. In answerset programming (ASP), many notions of program equivalence have been introduced and formally analysed. A particular line of research in this direction aims at studying conditions under which certain syntactic constructs can be eliminated from programs preserving some given equivalence relation. In this paper, we continue this endeavour introducing novel conditions under which disjunction and negation can be eliminated from answerset programs under relativised strong equivalence with projection. This notion is a generalisation of the usual strongequivalence relation, as introduced by Lifschitz, Pearce, and Valverde, by allowing parametrisable context and output alphabets, which is an important feature in view of practical programming techniques like the use of local variables and modules. We provide modeltheoretic conditions that hold for a disjunctive logic program P precisely when there is a program Q, equivalent to P under our considered notion, such that Q is either positive, normal, or Horn, respectively. Moreover, we outline how such a Q, called a casting of P, can be obtained, and consider complexity issues. 1
Concrete Results on Abstract Rules
"... Abstract. There are many different notions of “rule ” in the literature. A key feature and main intuition of any such notion is that rules can be “applied ” to derive conclusions from certain premises. More formally, a rule is viewed as a function that, when invoked on a set of known facts, can prod ..."
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Abstract. There are many different notions of “rule ” in the literature. A key feature and main intuition of any such notion is that rules can be “applied ” to derive conclusions from certain premises. More formally, a rule is viewed as a function that, when invoked on a set of known facts, can produce new facts. In this paper, we show that this extreme simplification is still sufficient to obtain a number of useful results in concrete cases. We define abstract rules as a certain kind of functions, provide them with a semantics in terms of (abstract) stable models, and explain how concrete normal logic programming rules can be viewed as abstract rules in a variety of ways. We further analyse dependencies between abstract rules to recognise classes of logic programs for which stable models are guaranteed to be unique. 1
Complexity of SuperCoherence Problems in ASP ⋆
"... Abstract. Adapting techniques from database theory in order to optimize Answer Set Programming (ASP) systems, and in particular the grounding components of ASP systems, is an important topic in ASP. In recent years, the Magic Set method has received some interest in this setting, and a variant of it ..."
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Abstract. Adapting techniques from database theory in order to optimize Answer Set Programming (ASP) systems, and in particular the grounding components of ASP systems, is an important topic in ASP. In recent years, the Magic Set method has received some interest in this setting, and a variant of it, called DMS, has been proposed for ASP. However, this technique has a caveat, because it is not correct (in the sense of being queryequivalent) for all ASP programs. In recent work, a large fragment of ASP programs, referred to as supercoherent programs, has been identified, for which DMS is correct. An open question remained: How complex is it to determine whether a given program is supercoherent? This question turned out to be quite difficult to answer precisely. In this paper, we formally prove that deciding whether a propositional program is supercoherent is Π P 3complete in the disjunctive case, while it is Π P 2complete for normal programs. The hardness proofs are the difficult part in this endeavor: We proceed by characterizing the reductions by the models and reduct models which the ASP programs should have, and then provide instantiations that meet the given specifications. 1
Background
"... Description logics (DLs) and rules are formalisms that emphasize different aspects of knowledge representation: whereas DLs are focused on specifying and reasoning about conceptual knowledge, rules are focused on nonmonotonic inference. Many applications, however, require features of both DLs and ru ..."
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Description logics (DLs) and rules are formalisms that emphasize different aspects of knowledge representation: whereas DLs are focused on specifying and reasoning about conceptual knowledge, rules are focused on nonmonotonic inference. Many applications, however, require features of both DLs and rules. Developing a formalism that integrates DLs and rules would be a natural outcome of a large body of research in knowledge representation and reasoning of the last two decades; however, achieving this goal is very challenging and the approaches proposed thus far have not fully reached it. In this paper, we present a hybrid formalism of MKNF + knowledge bases, which integrates DLs and rules in a coherent semantic framework. Achieving seamless integration is nontrivial, since DLs use an openworld assumption, while the rules are based on a closedworld assumption. We overcome this discrepancy by basing the semantics of our formalism on the logic of minimal knowledge and negation as failure (MKNF) by Lifschitz. We present several algorithms for reasoning with MKNF + knowledge bases, each suitable to different kinds of rules, and establish tight complexity bounds.
ModelBased Recasting in AnswerSet Programming
"... Abstract. As wellknown, answerset programs do not satisfy the replacement property in general, i.e., programs P and Q that are equivalent may cease to be so when they are put in the context of some other program R, i.e., R ∪ P and R ∪ Q may have different (sets of) answer sets. Pearce et al. thus ..."
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Abstract. As wellknown, answerset programs do not satisfy the replacement property in general, i.e., programs P and Q that are equivalent may cease to be so when they are put in the context of some other program R, i.e., R ∪ P and R ∪ Q may have different (sets of) answer sets. Pearce et al. thus introduced strong equivalence for contextindependent equivalence, and proved that such equivalence holds between given programs P and Q iff P and Q are equivalent theories in the monotonic logic of hereandthere. In this article, we consider a related question: given a program P, does there exist some program Q from a certain class C of programs such that P and Q are equivalent under a given notion of equivalence? Furthermore, if the answer to this question is positive, how can we recast P to an equivalent program from C (i.e., construct such a Q)? In particular, we consider classes of programs that emerge by (dis)allowing disjunction and/or negation, and as equivalence notions we consider strong, uniform, and ordinary equivalence. Based on general modeltheoretic properties and a novel form of canonical programs, we develop semantic characterisations for the existence of such a program Q, determine the computational complexity of checking existence, and provide (local) rewriting rules for recasting.