Results 1  10
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15
Extending Classical Logic with Inductive Definitions
, 2000
"... The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships to other logics in the area of nonmonotonic reasoning, logic programming and deductiv ..."
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Cited by 70 (46 self)
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The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships to other logics in the area of nonmonotonic reasoning, logic programming and deductive databases, and to show its application for knowledge representation by giving a typology of definitional knowledge.
The Event Calculus in Classical Logic  Alternative Axiomatisations
, 1999
"... We present several alternative classical logic axiomatisations of the Event Calculus, a narrative based formalism for reasoning about actions and change. We indicate the range of applicability and key characteristics of each alternative formulation. ..."
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Cited by 56 (1 self)
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We present several alternative classical logic axiomatisations of the Event Calculus, a narrative based formalism for reasoning about actions and change. We indicate the range of applicability and key characteristics of each alternative formulation.
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 56 (36 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.
Some alternative formulations of the event calculus
 Computer Science; Computational Logic; Logic programming and Beyond
, 2002
"... Abstract. The Event Calculus is a narrative based formalism for reasoning about actions and change originally proposed in logic programming form by Kowalski and Sergot. In this paper we summarise how variants of the Event Calculus may be expressed as classical logic axiomatisations, and how under ce ..."
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Cited by 53 (3 self)
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Abstract. The Event Calculus is a narrative based formalism for reasoning about actions and change originally proposed in logic programming form by Kowalski and Sergot. In this paper we summarise how variants of the Event Calculus may be expressed as classical logic axiomatisations, and how under certain circumstances these theories may be reformulated as “action description language ” domain descriptions using the Language E. This enables the classical logic Event Calculus to inherit various provably correct automated reasoning procedures recently developed for E. 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 37 (24 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].
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 16 (5 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
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 9 (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
A Preliminary Study on Reasoning About Causes
, 2003
"... This paper presents some preliminary work on causal reasoning about actions studying the causes of derived formulas in a given transition, in terms of subsets of the performed actions. We present a ..."
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Cited by 2 (2 self)
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This paper presents some preliminary work on causal reasoning about actions studying the causes of derived formulas in a given transition, in terms of subsets of the performed actions. We present a
Ramifications: An Extension and Correspondence Result for the Event Calculus
 JOURNAL OF LOGIC AND COMPUTATION
, 2007
"... Classical logic Event Calculus, and the special purpose logical action language E, are both well established formalisms for describing actions and change. However, there is yet to be an account of ramifications in Event Calculus sufficiently general to represent the classes of domains expressible in ..."
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Cited by 1 (0 self)
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Classical logic Event Calculus, and the special purpose logical action language E, are both well established formalisms for describing actions and change. However, there is yet to be an account of ramifications in Event Calculus sufficiently general to represent the classes of domains expressible in E. Indeed, an adequately general ramification theory constructed in any general purpose logical language still awaits. Therefore, under the motivation of creating a flexible ramification theory in a universal language, suitable for integration into a rich action theory, a new enhanced version of classical logic Event Calculus named ECR is proposed. ECR supports representation and reasoning about domains containing ramifications for classes of domains more general than those possible under previous general purpose language formulations. This article makes two main contributions. The first, ECR, is a narrativebased action formalism able to represent concurrent events, nondeterministic actions and indirect causal effects by virtue of an integrated solution to the frame and ramification problems. The formalism can reason about significant subclasses of domains containing both mutually interacting effects and cyclic causal dependencies. The formalism is elaboration tolerant and may be integrated with the standard variants of the Event Calculus. The second contribution is the definition of a semantic mapping between ECR and E, and a proof of soundness and completeness of the ECR theory with respect toE’s model theoretic specification.
Natural Events
"... This paper develops an inductive theory of predictive common sense reasoning. The theory provides the basis for an integrated solution to the three traditional problems of reasoning about change; the frame, qualification, and ramification problems. The theory is also capable of representing nondete ..."
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Cited by 1 (1 self)
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This paper develops an inductive theory of predictive common sense reasoning. The theory provides the basis for an integrated solution to the three traditional problems of reasoning about change; the frame, qualification, and ramification problems. The theory is also capable of representing nondeterministic events, and it provides a means for stating defeasible preferences over the outcomes of conflicting simultaneous events. 1.