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12
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.
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].
Automata Theory for Reasoning about Actions
 In Proceedings of the International Joint Conference on Arti Intelligence
, 1999
"... In this paper, we show decidability of a rather expressive fragment of the situation calculus. We allow second order quantification over finite and infinite sets of situations. We do not impose a domain closure assumption on actions; therefore, infinite and even uncountable domains are allowed. The ..."
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Cited by 8 (0 self)
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In this paper, we show decidability of a rather expressive fragment of the situation calculus. We allow second order quantification over finite and infinite sets of situations. We do not impose a domain closure assumption on actions; therefore, infinite and even uncountable domains are allowed. The decision procedure is based on automata accepting infinite trees.
Representing von NeumannMorgenstern Games in the Situation Calculus
, 2002
"... Sequential von NeumannMorgernstern (VM) games are a very general formalism for representing multiagent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and comple ..."
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Cited by 4 (0 self)
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Sequential von NeumannMorgernstern (VM) games are a very general formalism for representing multiagent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and complete axiomatization in the situation calculus. This axiomatization allows us to represent gametheoretic reasoning and solution concepts such as Nash equilibrium. We discuss the application of various concepts from VM game theory to the theory of planning and multiagent interactions, such as representing concurrent actions and using the Baire topology to define continuous payoff functions.
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.
Marc Denecker
"... 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 logic has been proposed recently and is an extension of clas ..."
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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 logic has been proposed recently and is an extension of classical logic. It allows for a uniform represention of various forms of definitions, including monotone inductive definitions and nonmonotone forms of inductive definitions such as iterated induction and induction over wellfounded posets. In the NMIDaxiomatisation of the situation calculus, fluents and causality predicates are defined by simultaneous induction on the wellfounded poset of situations. The inductive approach allows us to solve the ramification problem for the situation calculus in a uniform and modular way. Our solution is among the most general solutions for the ramification problem in the situation calculus. Using previously developed modularity techniques, we show that the basic variant of the inductive situation calculus without ramification rules is equivalent to Reiterstyle situation calculus.
unknown title
"... 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 logic has been proposed recently and is an extension of classical ..."
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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 logic has been proposed recently and is an extension of classical logic. It allows for a uniform represention of various forms of definitions, including monotone inductive definitions and nonmonotone forms of inductive definitions such as iterated induction and induction over wellfounded posets. In the NMIDaxiomatisation of the situation calculus, fluents and causality predicates are defined by simultaneous induction on the wellfounded poset of situations. The inductive approach allows us to solve the ramification problem for the situation calculus in a uniform and modular way. Our solution
unknown title
"... 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 logic has been proposed recently and is an extension of classical ..."
Abstract
 Add to MetaCart
(Show Context)
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 logic has been proposed recently and is an extension of classical logic. It allows for a uniform represention of various forms of definitions, including monotone inductive definitions and nonmonotone forms of inductive definitions such as iterated induction and induction over wellfounded posets. In the NMIDaxiomatisation of the situation calculus, fluents and causality predicates are defined by simultaneous induction on the wellfounded poset of situations. The inductive approach allows us to solve