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33
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 36 (22 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.
Inductive Situation Calculus
 Artificial Intelligence
, 2004
"... see [2]. Temporal reasoning has always been a major test case for knowledge representation formalisms. In this paper, we develop an inductive variant of the situation calculus using the Logic for NonMonotone Inductive Definitions (NMID). This is an extension of classical logic that allows for unifo ..."
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Cited by 33 (21 self)
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see [2]. Temporal reasoning has always been a major test case for knowledge representation formalisms. In this paper, we develop an inductive variant of the situation calculus using the Logic for NonMonotone Inductive Definitions (NMID). This is an extension of classical logic that allows for uniform representation of various forms of definitions, including monotone inductive definitions and nonmonotone forms of inductive definitions such as iterated induction and induction over wellfounded posets [1]. Here, we demonstrate an application of NMIDlogic. The aim is twofold. First, we illustrate the role of NMIDlogic and nonmonotone inductive definitions for knowledge representation by presenting a variant of the situation calculus which we call inductive situation calculus. We show that ramification rules can be naturally modeled through a nonmonotone iterated inductive definition. Second, we illustrate the use of our recently developed modularity techniques for NMIDlogic in order to translate a theory of the inductive situation calculus into a classical logic theory of Reiter’s situation calculus [3].
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 16 (9 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
SAT(ID): Satisfiability of propositional logic extended with inductive definitions
 In Proceedings of the 11th conference on Theory and Applications of Satisfiability Testing, SAT 2008, volume 4996 of Lecture Notes in Computer Science
, 2008
"... Abstract. We investigate the satisfiability problem, SAT(ID), of an extension of propositional logic with inductive definitions. We demonstrate how to extend existing SAT solvers to become SAT(ID) solvers, and provide an implementation on top of MiniSat. We also report on a performance study, in ..."
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Cited by 14 (6 self)
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Abstract. We investigate the satisfiability problem, SAT(ID), of an extension of propositional logic with inductive definitions. We demonstrate how to extend existing SAT solvers to become SAT(ID) solvers, and provide an implementation on top of MiniSat. We also report on a performance study, in which our implementation exhibits the expected benefits: full use of the underlying SAT solver’s potential. 1
Representing Causal Information about a Probabilistic Process
"... Abstract. We study causal information about probabilistic processes, i.e., information about why events occur. A language is developed in which such information can be formally represented and we investigate when this suffices to uniquely characterize the probability distribution that results from s ..."
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Cited by 13 (4 self)
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Abstract. We study causal information about probabilistic processes, i.e., information about why events occur. A language is developed in which such information can be formally represented and we investigate when this suffices to uniquely characterize the probability distribution that results from such a process. We examine both detailed representations of temporal aspects and representations in which time is implicit. In this last case, our logic turns into a more finegrained version of Pearl’s approach to causality. We relate our logic to certain probabilistic logic programming languages, which leads to a clearer view on the knowledge representation properties of these language. We show that our logic induces a semantics for disjunctive logic programs, in which these represent nondeterministic processes. We show that logic programs under the wellfounded semantics can be seen as a language of deterministic causality, which we relate to McCain & Turner’s causal theories. 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 8 (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
What’s in a model? Epistemological analysis of logic programming
 Proceedings of the 9th International Conference on PrinICLP 2012 A Tarskian Informal Semantics for ASP ciples of Knowledge Representation and Reasoning
, 2004
"... It is commonly believed that the meaning of a formal declarative knowledge representation language is determined by its formal semantics. This is not quite so. This paper shows an epistemological ambiguity that arises in the context of logic programming. Several different logic programming formali ..."
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Cited by 8 (3 self)
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It is commonly believed that the meaning of a formal declarative knowledge representation language is determined by its formal semantics. This is not quite so. This paper shows an epistemological ambiguity that arises in the context of logic programming. Several different logic programming formalisms and semantics have been proposed. Hence, logic programming can be seen as an overlapping family of formal logics, each induced by a pair of a formal syntax and a formal semantics. We would expect that (a) each such pair has a unique declarative reading and (b) for a program in the intersection of several formal LP logics with the same formal semantics in each of them, its declarative reading is the same in each of them. I show in this paper that neither (a) nor (b) holds. The paper investigates the causes and the consequences of this phenomenon and points out some directions to overcome the ambiguity.
Data integration using IDlogic
 In Proc. 16th Int. Conf. on Advanced Information Systems Engineering (CAiSE’04), LNCS
, 2004
"... Abstract. IDLogic is a knowledge representation language that extends firstorder logic with nonmonotone inductive definitions. This paper introduces an IDLogic based framework for database schema integration. It allows us to to uniformly represent and reason with independent source databases ..."
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Cited by 6 (4 self)
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Abstract. IDLogic is a knowledge representation language that extends firstorder logic with nonmonotone inductive definitions. This paper introduces an IDLogic based framework for database schema integration. It allows us to to uniformly represent and reason with independent source databases that contain information about a common domain, but may have different schemas. The IDLogic theories that are obtained are called mediatorbased systems. We show that these theories properly capture the common methods for data integration (i.e., globalas view and localasview with either exact or partial definitions), and apply on them a robust abductive inference technique for query answering. 1
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 6 (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
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 6 (4 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.