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16
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.
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 27 (16 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 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
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 9 (3 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
What's in a Model? Epistemological analysis of Logic Programming
 CeurWS
, 2003
"... The paper is an epistemological analysis of logic programming and shows an epistemological ambiguity. Many different logic programming formalisms and semantics have been proposed. Hence, logic programming can be seen as a family of formal logics, each induced by a pair of a syntax and a semantics ..."
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Cited by 8 (3 self)
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The paper is an epistemological analysis of logic programming and shows an epistemological ambiguity. Many different logic programming formalisms and semantics have been proposed. Hence, logic programming can be seen as a family of formal logics, each induced by a pair of a syntax and a semantics, and each having a different declarative reading. However, we may expect that (a) if a program belongs to different logics of this family and has the same formal semantics in these logics, then the declarative meaning attributed to this program in the different logics is equivalent, and (b) that one and the same logic in this family has not been associated with distinct declarative readings.
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
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
An algebraic account of modularity in IDlogic
 In Proc. LPNMR’05
, 2005
"... Abstract. IDlogic uses ideas from the field of logic programming to extend second order logic with nonmonotone inductive defintions. In this work, we reformulate the semantics of this logic in terms of approximation theory, an algebraic theory which generalizes the semantics of several nonmonoton ..."
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Cited by 4 (0 self)
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Abstract. IDlogic uses ideas from the field of logic programming to extend second order logic with nonmonotone inductive defintions. In this work, we reformulate the semantics of this logic in terms of approximation theory, an algebraic theory which generalizes the semantics of several nonmonotonic reasoning formalisms. This allows us to apply certain abstract modularity theorems, developed within the framework of approximation theory, to IDlogic. As such, we are able to offer elegant and simple proofs of generalizations of known theorems, as well as some new results. 1
Extending the role of causality in probabilistic modeling. http://www.cs.kuleuven.ac.be/∼joost/#research
, 2006
"... The remarkable success of Bayesian networks in probabilistic modeling seems to be at least in part due to the causal interpretation that can be given to such networks, i.e., the fact that the parents of a node can be seen as causally determining this node itself. For the most part, however, this cau ..."
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Cited by 4 (1 self)
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The remarkable success of Bayesian networks in probabilistic modeling seems to be at least in part due to the causal interpretation that can be given to such networks, i.e., the fact that the parents of a node can be seen as causally determining this node itself. For the most part, however, this causal interpretation remains an informal guideline. Indeed, it is not reflected in the formal semantics of Bayesian networks, which is expressed in terms of probabilistic independencies and conditional probabilities, rather than causal relations. In this paper, we propose a probabilistic modeling language that has causality at the heart of its fundamental constructs. We first show how this language can be used to express independencies similar to those expressed by a Bayesian network. We then examine how the fundamental causal principles of this language lead to different ways of modeling certain probabilistic relations and compare the two representations.
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.