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The Modal Logic of Inequality
 J. Symbolic Logic
, 1992
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Cited by 67 (8 self)
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Hybrid Logics
"... This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur ..."
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Cited by 62 (18 self)
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This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur
Derivation rules as antiaxioms in modal logic
 Journal of Symbolic Logic
, 1993
"... Abstract. We discuss a ‘negative ’ way of defining frame classes in (multi)modal logic, and address the question whether these classes can be axiomatized by derivation rules, the ‘nonξ rules’, styled after Gabbay’s Irreflexivity Rule. The main result of this paper is a metatheorem on completeness ..."
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Cited by 46 (4 self)
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Abstract. We discuss a ‘negative ’ way of defining frame classes in (multi)modal logic, and address the question whether these classes can be axiomatized by derivation rules, the ‘nonξ rules’, styled after Gabbay’s Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas, and Λ+ is the extension of Λ with a set of nonξ rules, then Λ+ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.
A Calculus of Transition Systems (towards Universal Coalgebra)
 In Alban Ponse, Maarten de Rijke, and Yde Venema, editors, Modal Logic and Process Algebra, CSLI Lecture Notes No
, 1995
"... By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). ..."
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Cited by 28 (1 self)
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By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). In this paper, some standard results from universal algebra (such as the three isomorphism theorems and facts on the lattices of subalgebras and congruences) are reformulated (using the afore mentioned correspondence) and proved for transition systems. AMS Subject Classification (1991): 68Q10, 68Q55 CR Subject Classification (1991): D.3.1, F.1.2, F.3.2 Keywords & Phrases: Transition system, bisimulation, universal coalgebra, universal algebra, congruence, homomorphism. Note: This paper will appear in `Modal Logic and Process Algebra', edited by Ponse, De Rijke and Venema [PRV95]. 2 Table of Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ...
Modal Logic: A Semantic Perspective
 ETHICS
, 1988
"... This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimul ..."
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This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.
Cutfree Display Calculi for Nominal Tense Logics
 Conference on Tableaux Calculi and Related Methods (TABLEAUX
, 1998
"... . We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Krac ..."
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Cited by 17 (7 self)
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. We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)(C7). Finally, we show a weak Sahlqviststyle theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cutfree display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...
Display Calculi for Logics with Relative Accessibility Relations
, 1998
"... We define cutfree display calculi for knowledge logics where an indiscernibility relation is associated to each set of agents, and where agents decide the membership of objects using this indiscernibility relation. To do so, we first translate the knowledge logics into polymodal logics axiomatised ..."
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Cited by 8 (4 self)
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We define cutfree display calculi for knowledge logics where an indiscernibility relation is associated to each set of agents, and where agents decide the membership of objects using this indiscernibility relation. To do so, we first translate the knowledge logics into polymodal logics axiomatised by primitive axioms and then use Kracht's results on properly displayable logics to define the display calculi. Apart from these technical results, we argue that Display Logic is a natural framework to define cutfree calculi for many other logics with relative accessibility relations. This paper has not been submitted elsewhere in identical or similar form Visit to A.R.P. supported by an Australian Research Council International Fellowship. y Supported by an Australian Research Council Queen Elizabeth II Fellowship. 1 Introduction Background. Formal logic has been used by various authors to analyse and reason about knowledge. The possibleworlds semantics for knowledge logics initia...
What is the Coalgebraic Analogue of Birkhoff's Variety Theorem?
 THEORETICAL COMPUTER SCIENCE
, 2000
"... Logical definability is investigated for certain classes of coalgebras related to statetransition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state s ..."
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Cited by 7 (4 self)
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Logical definability is investigated for certain classes of coalgebras related to statetransition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state set of A. The ultrafilter enlargement is the subcoalgebra A of A + whose states are ultrafilters. Boolean combinations of equations between terms of observable (or output) type are identified as a natural class of formulas for specifying properties of coalgebras. These observable formulas are permitted to have a single state variable, and form a language in which modalities describing the effects of state transitions are implicitly present. A and A + validate the same observable formulas. It is shown that a class of coalgebras is de nable by observable formulas iff the class is closed under disjoint unions, images of bisimulations, and (ultra) lter enlargements. (Closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms.) Moreover, every set of observable formulas has the same models as some set of conditional equations. Examples are