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24
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
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.
The Second Answer Set Programming Competition
"... Abstract. This paper reports on the Second Answer Set Programming Competition. The competitions in areas of Satisfiability checking, PseudoBoolean constraint solving and Quantified Boolean Formula evaluation have proven to be a strong driving force for a community to develop better performing syste ..."
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Cited by 27 (6 self)
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Abstract. This paper reports on the Second Answer Set Programming Competition. The competitions in areas of Satisfiability checking, PseudoBoolean constraint solving and Quantified Boolean Formula evaluation have proven to be a strong driving force for a community to develop better performing systems. Following this experience, the Answer Set Programming competition series was set up in 2007, and ran as part of the International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR). This second competition, held in conjunction with LPNMR 2009, differed from the first one in two important ways. First, while the original competition was restricted to systems designed for the answer set programming language, the sequel was open to systems designed for other modeling languages, as well. Consequently, among the contestants of the second competition were a CLP(FD) team and three model generation systems for (extensions of) classical logic. Second, this latest competition covered not only satisfiability problems but also optimization ones. We present and discuss the setup and the results of the competition. 1
Satisfiability checking for PC(ID
 In LPAR
, 2005
"... Abstract. The logic FO(ID) extends classical first order logic with inductive definitions. This paper studies the satisifiability problem for PC(ID), its propositional fragment. We develop a framework for model generation in this logic, present an algorithm and prove its correctness. As FO(ID) is an ..."
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Cited by 7 (2 self)
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Abstract. The logic FO(ID) extends classical first order logic with inductive definitions. This paper studies the satisifiability problem for PC(ID), its propositional fragment. We develop a framework for model generation in this logic, present an algorithm and prove its correctness. As FO(ID) is an integration of classical logic and logic programming, our algorithm integrates techniques from SAT and ASP. We report on a prototype system, called MidL, experimentally validating our approach. 1
Grounding for model expansion in kguarded formulas with inductive definitions
 In IJCAI
, 2007
"... Mitchell and Ternovska [2005] proposed a constraint programming framework based on classical logic extended with inductive definitions. They formulate a search problem as the problem of model expansion (MX), which is the problem of expanding a given structure with new relations so that it satisfies ..."
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Cited by 7 (4 self)
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Mitchell and Ternovska [2005] proposed a constraint programming framework based on classical logic extended with inductive definitions. They formulate a search problem as the problem of model expansion (MX), which is the problem of expanding a given structure with new relations so that it satisfies a given formula. Their longterm goal is to produce practical tools to solve combinatorial search problems, especially those in NP. In this framework, a problem is encoded in a logic, an instance of the problem is represented by a finite structure, and a solver generates solutions to the problem. This approach relies on propositionalisation of highlevel specifications, and on the efficiency of modern SAT solvers. Here, we propose an efficient algorithm which combines grounding with partial evaluation. Since the MX framework is based on classical logic, we are able to take advantage of known results for the socalled guarded fragments. In the case of kguarded formulas with inductive definitions under a natural restriction, the algorithm performs much better than naive grounding by relying on connections between kguarded formulas and tree decompositions. 1
Logic Programming for Knowledge Representation
, 2007
"... This note provides background information and references to the tutorial on recent research developments in logic programming inspired by need of knowledge representation. ..."
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Cited by 7 (0 self)
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This note provides background information and references to the tutorial on recent research developments in logic programming inspired by need of knowledge representation.
Reducing inductive definitions to propositional satisfiability
 In International Conference on Logic Programming (ICLP’05
, 2005
"... Abstract. The FO(ID) logic is an extension of classical firstorder logic with a uniform representation of various forms of inductive definitions. The definitions are represented as sets of rules and they are interpreted by twovalued wellfounded models. For a large class of combinatorial and searc ..."
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Cited by 5 (4 self)
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Abstract. The FO(ID) logic is an extension of classical firstorder logic with a uniform representation of various forms of inductive definitions. The definitions are represented as sets of rules and they are interpreted by twovalued wellfounded models. For a large class of combinatorial and search problems, knowledge representation in FO(ID) offers a viable alternative to the paradigm of Answer Set Programming. The main reasons are that (i) the logic is an extension of classical logic and (ii) the semantics of the language is based on wellunderstood principles of mathematical induction. In this paper, we define a reduction from the propositional fragment of FO(ID) to SAT. The reduction is based on a novel characterization of twovalued wellfounded models using a set of inequality constraints on level mappings associated with the atoms. We also show how the reduction to SAT can be adapted for logic programs under the stable model semantics. Our experiments show that when using a state of the art SAT solver both reductions are competitive with other answer set programming systems — both direct implementations and SAT based. 1
Complexity of expanding a finite structure and related tasks
 The 8th Int. Workshop on Logic and Comput. Complexity (LCC
, 2006
"... The authors of [MT05] proposed a declarative constraint programming framework based on classical logic extended with nonmonotone inductive definitions. In the framework, a problem instance is a finite structure, and a problem specification is a formula defining the relationship between an instance ..."
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Cited by 4 (4 self)
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The authors of [MT05] proposed a declarative constraint programming framework based on classical logic extended with nonmonotone inductive definitions. In the framework, a problem instance is a finite structure, and a problem specification is a formula defining the relationship between an instance and its solutions. Thus, problem solving amounts to expanding a finite structure with new relations, to satisfy the formula. We present here the complexities of model expansion for a number of logics, alongside those of satisfiability and model checking. As the task is equivalent to witnessing the existential quantifiers in ∃SO model checking, the paper is in large part of a survey of this area, together with some new results. In particular, we describe the combined and data complexity of FO(ID), firstorder logic extended with inductive definitions [DT04] and the guarded and kguarded logics of [AvBN98] and [GLS01]. 1
Model Expansion as a Framework for Modelling and Solving Search Problems
"... We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite mod ..."
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Cited by 3 (3 self)
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We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework. 1.
A Method for Solving NP Search Based on Model Expansion and Grounding
, 2007
"... The logical task of model expansion (MX) has been proposed as a declarative constraint programming framework for solving search and decision problems. We present a method for solving NP search problems based on MX for first order logic extended with inductive definitions and cardinality constraints ..."
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Cited by 2 (0 self)
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The logical task of model expansion (MX) has been proposed as a declarative constraint programming framework for solving search and decision problems. We present a method for solving NP search problems based on MX for first order logic extended with inductive definitions and cardinality constraints. The method involves grounding, and execution of a propositional solver, such as a SAT solver. Our grounding algorithm applies a generalization of the relational algebra to construct a ground formula representing the solutions to an instance. We demonstrate the practical feasibility of our method with an implementation, called MXG. We present axiomatizations of several NPcomplete benchmark problems, and experimental results comparing the performance of MXG with stateoftheart Answer Set programming (ASP) solvers. The performance of MXG is competitive with, and often better than, the ASP solvers on the problems studied.