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16
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 44 (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
Uniform Semantic Treatment of Default and Autoepistemic Logics
 ARTIFICIAL INTELLIGENCE
, 2000
"... We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the latti ..."
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Cited by 41 (23 self)
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We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise onetoone correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.
Logic programming revisited: logic programs as inductive definitions
 ACM Transactions on Computational Logic
, 2001
"... Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iffdefinitions. A second approach was to ..."
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Cited by 35 (21 self)
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Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iffdefinitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a nonmonotonic reasoning formalism strongly related to Default Logic and Autoepistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent nonmonotone inductive definitions. It is argued that this thesis results in an alternative justification of the wellfounded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy to comprehend meaning
A Logic of NonMonotone Inductive Definitions and its Modularity Properties
, 2004
"... Wellknown principles of induction include monotone induction and dierent sorts of nonmonotone induction such as inationary induction, induction over wellordered sets and iterated induction. In this work, we de ne a logic formalizing induction over wellordered sets and monotone and iterated ..."
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Cited by 29 (20 self)
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Wellknown principles of induction include monotone induction and dierent sorts of nonmonotone induction such as inationary induction, induction over wellordered sets and iterated induction. In this work, we de ne a logic formalizing induction over wellordered sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for NonMonotone Inductive De nitions (NMIDlogic). The semantics of the logic is strongly inuenced by the wellfounded semantics of logic programming.
A logic of nonmonotone inductive definitions
 ACM transactions on computational logic
, 2007
"... Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated i ..."
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Cited by 28 (16 self)
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Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for NonMonotone Inductive Definitions (IDlogic). The semantics of the logic is strongly influenced by the wellfounded semantics of logic programming. This paper discusses the formalisation of different forms of (non)monotone induction by the wellfounded semantics and illustrates the use of the logic for formalizing mathematical and commonsense knowledge. To model different types of induction found in mathematics, we define several subclasses of definitions, and show that they are correctly formalized by the wellfounded semantics. We also present translations into classical first or second order logic. We develop modularity and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming, Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.
Partial Stable Models for Logic Programs with Aggregates
 In: LPNMR7. LNCS 2923
, 2004
"... We introduce a family of partial stable model semantics for logic programs with arbitrary aggregate relations. The semantics are parametrized by the interpretation of aggregate relations in threevalued logic. Any semantics in this family satisfies two important properties: (i) it extends the pa ..."
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Cited by 16 (0 self)
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We introduce a family of partial stable model semantics for logic programs with arbitrary aggregate relations. The semantics are parametrized by the interpretation of aggregate relations in threevalued logic. Any semantics in this family satisfies two important properties: (i) it extends the partial stable semantics for normal logic programs and (ii) total stable models are always minimal. We also give a specific instance of the semantics and show that it has several attractive features.
Translation of Aggregate Programs to Normal Logic Programs
 In ASP’03
, 2003
"... We define a translation of aggregate programs to normal logic programs which preserves the set of partial stable models. We then define the classes of definite and stratified aggregate programs and show that the translation of such programs are, respectively, definite and stratified logic progra ..."
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Cited by 11 (0 self)
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We define a translation of aggregate programs to normal logic programs which preserves the set of partial stable models. We then define the classes of definite and stratified aggregate programs and show that the translation of such programs are, respectively, definite and stratified logic programs. Consequently these two classes of programs have a single partial stable model which is twovalued and is also the wellfounded model. Our definition of stratification is more general than the existing one and covers a strictly larger class of programs.
On the relation between IDlogic and answer set programming
 In Logics in Artificial Intelligence, 9th European Conference (JELIA), volume 3229 of Lecture Notes in Computer Science
, 2004
"... Abstract. This paper is an analysis of two knowledge representation extensions of logic programming, namely Answer Set Programming and IDLogic. Our aim is to compare both logics on the level of declarative reading, practical methodology and formal semantics. At the level of methodology, we put forw ..."
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Cited by 10 (4 self)
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Abstract. This paper is an analysis of two knowledge representation extensions of logic programming, namely Answer Set Programming and IDLogic. Our aim is to compare both logics on the level of declarative reading, practical methodology and formal semantics. At the level of methodology, we put forward the thesis that in many (but not all) existing applications of ASP, an ASP program is used to encode definitions and assertions, similar as in IDLogic. We illustrate this thesis with an example and present a formal result that supports it, namely an equivalence preserving translation from a class of IDLogic theories into ASP. This translation can be exploited also to use the current efficient ASP solvers to reason on IDLogic theories and it has been used to implement a model generator for IDLogic. 1
Anyworld assumptions in logic programming
, 2005
"... Due to the usual incompleteness of information representation, any approach to assign a semantics to logic programs has to rely on a default assumption on the missing information. The stable model semantics, that has become the dominating approach to give semantics to logic programs, relies on the C ..."
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Cited by 8 (1 self)
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Due to the usual incompleteness of information representation, any approach to assign a semantics to logic programs has to rely on a default assumption on the missing information. The stable model semantics, that has become the dominating approach to give semantics to logic programs, relies on the Closed World Assumption (CWA), which asserts that by default the truth of an atom is false. There is a second wellknown assumption, called Open World Assumption (OWA), which asserts that the truth of the atoms is supposed to be unknown by default. However, the CWA, the OWA and the combination of them are extremal, though important, assumptions over a large variety of possible assumptions on the truth of the atoms, whenever the truth is taken from an arbitrary truth space. The topic of this paper is to allow any assignment (i.e. interpretation), over a truth space, to be a default assumption. Our main result is that our extension is conservative in the sense that under the “everywhere false ” default assumption (CWA) the usual stable model semantics is captured. Due to the generality and the purely algebraic nature of our approach, it abstracts from the particular formalism of choice and the results may be applied in other contexts as well.
Splitting an operator: An algebraic modularity result and its application to autoepistemic logic
 In Proceedings of International Workshop on NonMonotonic Reasoning
, 2004
"... It is well known that it is possible to split certain autoepistemic theories under the semantics of expansions, i.e. to divide such a theory into a number of different “levels”, such that the models of the entire theory can be constructed by incrementally constructing models for each level. Similar ..."
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Cited by 6 (3 self)
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It is well known that it is possible to split certain autoepistemic theories under the semantics of expansions, i.e. to divide such a theory into a number of different “levels”, such that the models of the entire theory can be constructed by incrementally constructing models for each level. Similar results exist for other nonmonotonic formalisms, such as logic programming and default logic. In this work, we present a general, algebraic theory of splitting under a fixpoint semantics. Together with the framework of approximation theory, a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of this approach, by applying our results to autoepistemic logic.