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Products of Modal Logics, Part 1
 LOGIC JOURNAL OF THE IGPL
, 1998
"... The paper studies manydimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: pmorphisms, the finite depth method, normal forms, ..."
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Cited by 36 (1 self)
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The paper studies manydimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: pmorphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.
Why Are Modal Logics So Robustly Decidable?
"... Modal logics are widely used in a number of areas in computer science, in particular ..."
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Cited by 22 (1 self)
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Modal logics are widely used in a number of areas in computer science, in particular
Ascribing Beliefs to Resource Bounded Agents
, 2002
"... Logical approaches to reasoning about agents often rely on idealisations about belief ascription and logical omniscience which make it difficult to apply the results obtained to real agents. In this paper, we show how to ascribe beliefs and an ability to reason in an arbitrary decidable logic to an ..."
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Cited by 16 (7 self)
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Logical approaches to reasoning about agents often rely on idealisations about belief ascription and logical omniscience which make it difficult to apply the results obtained to real agents. In this paper, we show how to ascribe beliefs and an ability to reason in an arbitrary decidable logic to an agent in a computationally grounded way. We characterise those cases in which the assumption that an agent is logically omniscient in a given logic is `harmless' in the sense that it does not lead to making incorrect predictions about the agent, and show that such an assumption is not harmless when our predictions have a temporal dimension: `now the agent believes p', and the agent requires time to derive the consequences of its beliefs. We present a family of logics for reasoning about the beliefs of an agent which is a perfect reasoner in an arbitrary decidable logic L but only derives the consequences of its beliefs after some delay #. We investigate two members of this family in detail, L# in which all the consequences are derived at the next tick of the clock, and L # # in which the agent adds at most one new belief to its set of beliefs at every tick of the clock, and show that these are sound, complete and decidable.
Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization?
, 1999
"... . We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule ..."
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Cited by 15 (4 self)
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. We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cutformulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...
Labelled Modal Logics: Quantifiers
, 1998
"... . In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logic ..."
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Cited by 15 (2 self)
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. In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework. 1 Introduction Motivation Modal logic is an active area of research in computer science and artificial intelligence: a large number of modal logics have been studied and new ones are frequently proposed. Each new log...
Elimination of Predicate Quantifiers
 UWE REYLE, HANS JÜRGEN OHLBACH (EDS.): LOGIC, LANGUAGE AND REASONING  ESSAYS IN HONOUR OF DOV GABBAY
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Modal Quantification over Structured Domains
, 1997
"... this paper can be viewed both as a new semantics for generalized quantifiers and as a new look at standard firstorder quantification, bringing the latter closer to modal logic. The standard semantics for generalized quantifiers interprets a monadic generalized quantifier Q as a set of subsets of a ..."
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Cited by 6 (1 self)
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this paper can be viewed both as a new semantics for generalized quantifiers and as a new look at standard firstorder quantification, bringing the latter closer to modal logic. The standard semantics for generalized quantifiers interprets a monadic generalized quantifier Q as a set of subsets of a domain. For example, the quantifier "there are precisely two" is interpreted by the set of all subsets of the domain which contain precisely two elements. A formula Qx' is true in a model if the set of elements satisfying ' belongs to the interpretation of the quantifier; in our example, if there are precisely two elements satisfying '. The existential quantifier can be treated as a generalized quantifier, too: it is interpreted as the set of all nonempty subsets of the domain. The universal quantifier is interpreted by the singleton set containing the whole domain. The quantifiers listed so far are firstorder definable in the following sense: they can be defined using ordinary quantifiers and equality. Many interesting generalized quantifiers are not firstorder definable. The present study is motivated by the work of Michiel van Lambalgen (1991) on Gentzenstyle proof theory for the quantifiers "for many" (its dual is interpreted as a nonprincipal filter), "for uncountably many" and "for almost all" (the latter contains all subsets of the domain which have Lebesgue measure 1). All those quantifiers are not firstorder definable. They have Hilbertstyle axiomatizations, but until lately no one believed that they can have a reasonable Gentzenstyle proof theory. In order to devise such a proof theory, van Lambalgen used a translation of generalized quantifier formulas into a firstorder language enriched with a predicate R of indefinite arity.
Iteration of Simple Formulas in Duration Calculus
, 1998
"... A special kind of smallest fixed point known as iteration is in most cases sufficient for the description of temporal computation processes in Duration Calculus[ZHR91]. In 1994 Dang and Wang introduced an extension of Duration Calculus with iteration [DW94]. They showed how to describe the behaviour ..."
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Cited by 6 (5 self)
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A special kind of smallest fixed point known as iteration is in most cases sufficient for the description of temporal computation processes in Duration Calculus[ZHR91]. In 1994 Dang and Wang introduced an extension of Duration Calculus with iteration [DW94]. They showed how to describe the behaviours of a practically significant class of timed automata in this extension, using socalled simple formulas. In this paper we present a complete system of axioms for iteration of simple formulas. We obtained our axioms by translating appropriately the schemata for iteration from the proof system of propositional dynamic logic ([Seg77], cf. e.g. [AGM92]), which is a wellknown formal system with iteration. We present this translation and the correspondence between the semantics of propositional dynamic logic and that of interval temporal logic that underlies it. The argument of completeness for the axioms for iteration in propositional dynamic logic relies on appropriate assignments to proposit...
An Implementation and Optimization of an Algorithm for Reducing Formulae in SecondOrder Logic.
, 1996
"... We have shown that the elimination algorithm (DLS) for reducing secondorder logic to firstorder logic by Doherty, Lukaszewicz and Szalas, can be implemented. In order to make the algorithm efficient when more than one predicate is eliminated some changes are suggested. The DLS algorithm requires t ..."
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Cited by 6 (0 self)
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We have shown that the elimination algorithm (DLS) for reducing secondorder logic to firstorder logic by Doherty, Lukaszewicz and Szalas, can be implemented. In order to make the algorithm efficient when more than one predicate is eliminated some changes are suggested. The DLS algorithm requires the input to be transformed into disjunctive form, however it can be shown that this requirement can be relaxed somewhat in order to avoid exponential growth of the formula. Other optimizations in order to keep the size of the resulting formula are also suggested. A module for handling nested abnormality theories is included to show that the algoritm is general and can be useful in many areas. Contents 1 Introduction 1 1.1 Intelligence and commonsense : : : : : : : : : : : : : : : : : : : : 1 1.2 Nonmonotonic reasoning : : : : : : : : : : : : : : : : : : : : : : 1 1.3 What this report covers : : : : : : : : : : : : : : : : : : : : : : : 2 1.4 Outline of the paper : : : : : : : : : : : : :...
Finite Model Theory and Finite Variable Logics
 University of Pennsylvania
, 1995
"... Finite Model Theory and Finite Variable Logics Eric Rosen Supervisor: Scott Weinstein In this dissertation, I investigate some questions about the model theory of finite structures. One goal is to better understand the expressive power of various logical languages, including firstorder logic (FO), ..."
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Cited by 5 (2 self)
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Finite Model Theory and Finite Variable Logics Eric Rosen Supervisor: Scott Weinstein In this dissertation, I investigate some questions about the model theory of finite structures. One goal is to better understand the expressive power of various logical languages, including firstorder logic (FO), over this class. A second, related, goal is to determine which results from classical model theory remain true when relativized to the class, F , of finite structures. As it is wellknown that many such results become false, I also consider certain weakened generalizations of classical results. I prove some basic results about the languages L k (9) and L k 1! (9), the existential fragments of the finite variable logics L k and L k 1! . I show that there are finite models whose L k (9)theories are not finitely axiomatizable. I also establish the optimality of a normal form for L k 1! (9), and separate certain fragments of this logic. I introduce a notion of a `generalized preser...