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46
Recursive Aggregates in Disjunctive Logic Programs: Semantics and Complexity
 In Proceedings of European Conference on Logics in Artificial Intelligence (JELIA
, 2004
"... Abstract. The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modeling power of ASP, in terms of concise problem representations. While many important problems can be encoded using nonrecursive aggregates, som ..."
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Cited by 92 (11 self)
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Abstract. The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modeling power of ASP, in terms of concise problem representations. While many important problems can be encoded using nonrecursive aggregates, some relevant examples lend themselves for the use of recursive aggregates. Previous semantic definitions typically agree in the nonrecursive case, but the picture is less clear for recursion. Some proposals explicitly avoid recursive aggregates, most others differ, and many of them do not satisfy desirable criteria, such as minimality or coincidence with answer sets in the aggregatefree case. In this paper we define a semantics for disjunctive programs with arbitrary aggregates (including monotone, antimonotone, and nonmonotone aggregates). This semantics is a fully declarative, genuine generalization of the answer set semantics for disjunctive logic programming (DLP). It is defined by a natural variant of the GelfondLifschitz transformation, and treats aggregate and nonaggregate literals in a uniform way. We prove that our semantics guarantees the minimality (and therefore the incomparability) of answer sets, and demonstrate that it coincides with the standard answer set semantics on aggregatefree programs. Finally we analyze the computational complexity of this language, paying particular attention to the impact of syntactical restrictions on programs. 1
A new perspective on stable models
 In Proceedings of International Joint Conference on Artificial Intelligence (IJCAI
, 2007
"... The definition of a stable model has provided a declarative semantics for Prolog programs with negation as failure and has led to the development of answer set programming. In this paper we propose a new definition of that concept, which covers many constructs used in answer set programming (includi ..."
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Cited by 66 (30 self)
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The definition of a stable model has provided a declarative semantics for Prolog programs with negation as failure and has led to the development of answer set programming. In this paper we propose a new definition of that concept, which covers many constructs used in answer set programming (including disjunctive rules, choice rules and conditional literals) and, unlike the original definition, refers neither to grounding nor to fixpoints. Rather, it is based on a syntactic transformation, which turns a logic program into a formula of secondorder logic that is similar to the formula familiar from the definition of circumscription. 1
Ultimate Wellfounded and Stable Semantics for Logic Programs With Aggregates (Extended Abstract)
 In Proceedings of ICLP01, LNCS 2237
, 2001
"... is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate wellfounded model will be 2valued and will coincide with the least fixpoint of TP . This is not the case for the standard wellfounded semantics. For example in the sta ..."
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Cited by 45 (7 self)
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is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate wellfounded model will be 2valued and will coincide with the least fixpoint of TP . This is not the case for the standard wellfounded semantics. For example in the standard wellfounded model of the program: # p. p. p is undefined while the associated TP operator is monotone and p is true in the ultimate wellfounded model. One disadvantage of using the ultimate semantics is that it has a higher computational cost even for programs without aggregates. The complexity goes one level higher in the polynomial hierarchy to # 2 for the wellfounded model and to 2 for a stable model which is also complete for this class [2]. Fortunately, by adding aggregates the complexity does not increase further. To give an example of a logic program with aggregates we consider the problem of computing the length of the shortest path between two nodes in a direc
Aggregate Functions in Disjunctive Logic Programming: Semantics, . . .
"... Disjunctive Logic Programming (DLP) is a very expressive formalism: it allows to express every property of nite structures that is decidable in the complexity class ). Despite the high expressiveness of DLP, there are some simple properties, often arising in realworld applications, which ..."
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Cited by 41 (4 self)
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Disjunctive Logic Programming (DLP) is a very expressive formalism: it allows to express every property of nite structures that is decidable in the complexity class ). Despite the high expressiveness of DLP, there are some simple properties, often arising in realworld applications, which cannot be encoded in a simple and natural manner. Among these, properties requiring to apply some arithmetic operators (like sum, times, count) on a set of elements satisfying some conditions, cannot be naturally expressed in DLP. To overcome this de ciency, in this paper we extend DLP by aggregate functions. We formally de ne the semantics of the new language, named DLP . We show the usefulness of the new constructs on relevant knowledgebased problems. We analyze the computational complexity of DLP , showing that the addition of aggregates does not bring a higher cost in that respect. We provide an implementation of the DLP language in DLV{ the stateoftheart DLP system { and report on experiments which con rm the usefulness of the proposed extension also for the eciency of the computation.
Specifying normgoverned computational societies
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2007
"... Electronic markets, dispute resolution and negotiation protocols are three types of application domains that can be viewed as open agent societies. Key characteristics of such societies are agent heterogeneity, conflicting individual goals and unpredictable behaviour. Members of such societies may f ..."
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Cited by 38 (10 self)
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Electronic markets, dispute resolution and negotiation protocols are three types of application domains that can be viewed as open agent societies. Key characteristics of such societies are agent heterogeneity, conflicting individual goals and unpredictable behaviour. Members of such societies may fail to, or even choose not to, conform to the norms governing their interactions. It has been argued that systems of this type should have a formal, declarative, verifiable, and meaningful semantics. We present a theoretical and computational framework being developed for the executable specification of open agent societies. We adopt an external perspective and view societies as instances of normative systems. In this paper we demonstrate how the framework can be applied to specifying and executing a contractnet protocol. The specification is formalised in two action languages, the C+ language and the Event Calculus, and executed using respective software implementations, the Causal Calculator and the Society Visualiser. We evaluate our executable specification in the light of the presented case study, discussing the strengths and weaknesses of the employed action languages for the specification of open agent societies.
Modularity Aspects of Disjunctive Stable Models
, 2007
"... Practically all programming languages used in software engineering allow to split a program into several modules. For fully declarative and nonmonotonic logic programming languages, however, the modular structure of programs is hard to realise, since the output of an entire program cannot in general ..."
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Cited by 27 (8 self)
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Practically all programming languages used in software engineering allow to split a program into several modules. For fully declarative and nonmonotonic logic programming languages, however, the modular structure of programs is hard to realise, since the output of an entire program cannot in general be composed from the output of its component programs in a direct manner. In this paper, we consider these aspects for the stablemodel semantics of disjunctive logic programs (DLPs). We define the notion of a DLPfunction, where a welldefined input/output interface is provided, and establish a novel module theorem enabling a suitable compositional semantics for modules. The module theorem extends the wellknown splittingset theorem and allows also a generalisation of a shifting technique for splitting shared disjunctive rules among components.
Answer set based design of knowledge systems
 ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE
, 2006
"... The aim of this paper is to demonstrate that AProlog is a powerful language for the construction of reasoning systems. In fact, AProlog allows to specify the initial situation, the domain model, the control knowledge, and the reasoning modules. Moreover, it is efficient enough to be used for pra ..."
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Cited by 22 (12 self)
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The aim of this paper is to demonstrate that AProlog is a powerful language for the construction of reasoning systems. In fact, AProlog allows to specify the initial situation, the domain model, the control knowledge, and the reasoning modules. Moreover, it is efficient enough to be used for practical tasks and can be nicely integrated with programming languages such as Java. An extension of AProlog (CRProlog) allows to further improve the quality of reasoning by specifying requirements that the solutions should satisfy if at all possible. The features of AProlog and CRProlog are demonstrated by describing in detail the design of USAAdvisor, an AProlog based decision support system for the Space Shuttle flight controllers.
Declarative and computational properties of logic programs with aggregates
 In: Nineteenth International Joint Conference on Artificial Intelligence (IJCAI05
, 2005
"... We investigate the properties of logic programs with aggregates. We mainly focus on programs with monotone and antimonotone aggregates (LP A m,a programs). We define a new notion of unfounded set for LP A m,a programs, and prove that it is a sound generalization of the standard notion of unfounded s ..."
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Cited by 21 (8 self)
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We investigate the properties of logic programs with aggregates. We mainly focus on programs with monotone and antimonotone aggregates (LP A m,a programs). We define a new notion of unfounded set for LP A m,a programs, and prove that it is a sound generalization of the standard notion of unfounded set for aggregatefree programs. We show that the answer sets of an LP A m,a program are precisely its unfoundedfree models. We define a wellfounded operator WP for LP A m,a programs; we prove that its total fixpoints are precisely the answer sets of P, and its least fixpoint Wω P (∅) is contained in the intersection of all answer sets (if P admits an answer set). W ω P (∅) is
Answer sets for logic programs with arbitrary abstract constraint atoms
 J. ARTIFICIAL INTELLIGENCE RESEARCH
, 2007
"... In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (catoms). These approaches generalize the fixpointbased and the level mapping based answer set semantics of normal logic programs to the case of logic p ..."
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Cited by 21 (2 self)
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In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (catoms). These approaches generalize the fixpointbased and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of catoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpointbased semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negationasfailure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of catoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly wellsupported models, are generalizations of the notion of wellsupported models of normal logic programs to the case of programs with catoms. As for the case of fixpointbased semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) wellsupported models of a program, thus generalizing the theorem on the correspondence between stable models and wellsupported models of a normal logic program to the class of programs with catoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone catoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with catoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature.
Unfounded Sets for Disjunctive Logic Programs with Arbitrary Aggregates
 In Logic Programming and Nonmonotonic Reasoning, 8th International Conference (LPNMR’05), 2005
, 2005
"... Abstract. Aggregates in answer set programming (ASP) have recently been studied quite intensively. The main focus of previous work has been on defining suitable semantics for programs with arbitrary, potentially recursive aggregates. By now, these efforts appear to have converged. On another line of ..."
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Cited by 17 (3 self)
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Abstract. Aggregates in answer set programming (ASP) have recently been studied quite intensively. The main focus of previous work has been on defining suitable semantics for programs with arbitrary, potentially recursive aggregates. By now, these efforts appear to have converged. On another line of research, the relation between unfounded sets and (aggregatefree) answer sets has lately been rediscovered. It turned out that most of the currently available answer set solvers rely on this or closely related results (e.g., loop formulas). In this paper, we unite these lines and give a new definition of unfounded sets for disjunctive logic programs with arbitrary, possibly recursive aggregates. While being syntactically somewhat different, we can show that this definition properly generalizes all main notions of unfounded sets that have previously been defined for fragments of the language. We demonstrate that, as for restricted languages, answer sets can be crisply characterized by unfounded sets: They are precisely the unfoundedfree models. This result can be seen as a confirmation of the robustness of the definition of answer sets for arbitrary aggregates. We also provide a comprehensive complexity analysis for unfounded sets, and study its impact on answer set computation. 1