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Reformulating the Situation Calculus and the Event Calculus in the General Theory of Stable Models and in Answer Set Programming
"... Circumscription and logic programs under the stable model semantics are two wellknown nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis ..."
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Circumscription and logic programs under the stable model semantics are two wellknown nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis of a family of action languages, such as language A and several of its descendants. Based on the discovery that circumscription and the stable model semantics coincide on a class of canonical formulas, we reformulate the situation calculus and the event calculus in the general theory of stable models. We also present a translation that turns the reformulations further into answer set programs, so that efficient answer set solvers can be applied to compute the situation calculus and the event calculus. 1.
Connecting firstorder ASP and the logic FO(ID) through reducts
 In: Correct Reasoning: Essays on LogicBased AI in Honor of Vladimir Lifschitz
, 2012
"... on his 65th birthday! Abstract. Recently, an answerset programming (ASP) formalism of logic programing with the answerset semantics has been extended to the full firstorder setting. Earlier an extension of firstorder logic with inductive definitions, the logic FO(ID), was proposed as a knowledge ..."
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on his 65th birthday! Abstract. Recently, an answerset programming (ASP) formalism of logic programing with the answerset semantics has been extended to the full firstorder setting. Earlier an extension of firstorder logic with inductive definitions, the logic FO(ID), was proposed as a knowledge representation formalism and developed as an alternative ASP language. We present characterizations of these formalisms in terms of concepts of infinitary propositional logic. We use them to find a direct connection between the firstorder ASP and the logic FO(ID) under some restrictions on the form of theories (programs) considered. 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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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
Game Solution, Epistemic Dynamics and FixedPoint Logics
, 2010
"... Current methods for solving games embody a form of “procedural rationality” that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of anal ..."
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Cited by 4 (3 self)
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Current methods for solving games embody a form of “procedural rationality” that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of analysis from recent years that look different conceptually, and find that they are all mathematically equivalent. This shows how an abstract logical perspective can bring out basic invariant structure in games. We then generalize this to an exploration of fixedpoint logics on finite trees that best fit gametheoretic equilibria. We end with some open questions that suggest a broader program for merging current computational logics with notions and results from game theory. This paper is largely a program for opening up an area: an extended version of the technical results will be found in the forthcoming dissertation [26].
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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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.
Grounding FO and FO(ID) with bounds
 J. Artif. Intell. Res. (JAIR
"... Grounding is the task of reducing a firstorder theory and finite domain to an equivalent propositional theory. It is used as preprocessing phase in many logicbased reasoning systems. Such systems provide a rich firstorder input language to a user and can rely on efficient propositional solvers to ..."
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Grounding is the task of reducing a firstorder theory and finite domain to an equivalent propositional theory. It is used as preprocessing phase in many logicbased reasoning systems. Such systems provide a rich firstorder input language to a user and can rely on efficient propositional solvers to perform the actual reasoning. Besides a firstorder theory and finite domain, the input for grounders contains in many applications also additional data. By exploiting this data, the size of the grounder’s output can often be reduced significantly. A common practice to improve the efficiency of a grounder in this context is by manually adding semantically redundant information to the input theory, indicating where and when the grounder should exploit the data. In this paper we present a method to compute and add such redundant information automatically. Our method therefore simplifies the task of writing input theories that can be grounded efficiently by current systems. We first present our method for classical firstorder logic (FO) theories. Then we extend it to FO(ID), the extension of FO with inductive definitions, which allows for more concise and comprehensive input theories. We discuss implementation issues and experimentally validate the practical applicability of our method. 1.
A Tarskian Informal Semantics for Answer Set Programming ∗
"... In their seminal papers on stable model semantics, Gelfond and Lifschitz introduced ASP by casting programs as epistemic theories, in which rules represent statements about the knowledge of a rational agent. To the best of our knowledge, theirs is still the only published systematic account of the i ..."
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In their seminal papers on stable model semantics, Gelfond and Lifschitz introduced ASP by casting programs as epistemic theories, in which rules represent statements about the knowledge of a rational agent. To the best of our knowledge, theirs is still the only published systematic account of the intuitive meaning of rules and programs under the stable semantics. In current ASP practice, however, we find numerous applications in which rational agents no longer seem to play any role. Therefore, we propose here an alternative explanation of the intuitive meaning of ASP programs, in which they are not viewed as statements about an agent’s beliefs, but as objective statements about the world. We argue that this view is more natural for a large part of current ASP practice, in particular the socalled GenerateDefineTest programs.
Comments on Modeling Languages for AnswerSet Programming
"... Strong emphasis on intuitive and direct modeling of application domains is one of the distinguishing features and major strengths of the answerset programming paradigm. It leads naturally to several key questions. Is there a need for standardizing such languages? What functionality should these lan ..."
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Strong emphasis on intuitive and direct modeling of application domains is one of the distinguishing features and major strengths of the answerset programming paradigm. It leads naturally to several key questions. Is there a need for standardizing such languages? What functionality should these languages support? Are there any general design requirements for them? This note attempts to propose some answers.
Model Expansion and the Expressiveness of FO(ID) and Other Logics
"... Model expansion problem is a question of determining, given a formula and a structure for a part of the vocabulary of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problemsolving paradigm based on model expansion by (Mitchell & Tern ..."
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Model expansion problem is a question of determining, given a formula and a structure for a part of the vocabulary of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problemsolving paradigm based on model expansion by (Mitchell & Ternovska, 2005; Mitchell, Ternovska, Hach, & Mohebali, 2006) posed the question of complexity of this problem for logics used in the paradigm. We discuss the complexity of the model expansion problem for a number of logics, alongside that of satisfiability and model checking. As the task is equivalent to witnessing leading existential secondorder quantifiers in a model checking setting, the paper is in large part a survey of this area together with some new results. In particular, we describe the combined and data complexity of model expansion for FO(ID) (Denecker & Ternovska, 2008), as well as guarded and kguarded logics of (Andréka, van Benthem, & Németi, 1998) and (Gottlob, Leone, & Scarcello, 2001).
Celestijnenlaan 200A – B3001 Heverlee (Belgium) On the equivalence between CPlogic
, 2008
"... We give a detailed proof of the fact that the probabilistic logics of Logic Programs with Annotated Disjunctions (LPADs) and CPlogic are equivalent. This report contains a detailed proof of the fact that Logic Programs with Annotated Disjunctions (LPADs) (6) and CPlogic (5) are equivalent. Before ..."
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We give a detailed proof of the fact that the probabilistic logics of Logic Programs with Annotated Disjunctions (LPADs) and CPlogic are equivalent. This report contains a detailed proof of the fact that Logic Programs with Annotated Disjunctions (LPADs) (6) and CPlogic (5) are equivalent. Before moving on to this proof, we first present some preliminaries from lattice theory and logic programming, and summarize the definition of LPADs and CPlogic. 1