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13
Beyond modalities: Sufficiency and mixed algebras
 In E. Orłowska & A. Szałas (Eds.), Relational Methods in Computer Science Applications, 277– 299
, 2000
"... this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed structure ..."
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this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed structures occur when modal and sufficiency operators arise from the same accessibility relation. In this paper we introduce the classes of sufficiency algebras and that of mixed algebras which include both a modal and a sufficiency operator, and study representation and duality theory for these classes of algebras. We also give examples for classes of firstorder definable frames, where such operators are required for a "modalstyle" axiomatisation. 2 Why sufficiency and mixed algebras?
Monotonic Modal Logics
, 2003
"... Monotonic modal logics form a generalization of normal modal logics... ..."
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Monotonic modal logics form a generalization of normal modal logics...
AXIOMS, ALGEBRAS, AND TOPOLOGY
"... This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations. ..."
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Cited by 9 (0 self)
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This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations.
Sahlqvist Formulas Unleashed in Polyadic Modal Languages
 Advances in Modal Logic
, 2000
"... We propose a generalization of Sahlqvist formulae to polyadic modal languages by representing modal polyadic languages in a combinatorial style and thus, in particular, developing what we believe to be the right approach to Sahlqvist formulae at all. The class of polyadic Sahlqvist formulae PSF defi ..."
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Cited by 8 (3 self)
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We propose a generalization of Sahlqvist formulae to polyadic modal languages by representing modal polyadic languages in a combinatorial style and thus, in particular, developing what we believe to be the right approach to Sahlqvist formulae at all. The class of polyadic Sahlqvist formulae PSF defined here expands essentially the so far known one. We prove firstorder definability and canonicity for the class PSF.
On canonical modal logics that are not elementarily determined. Logique et Analyse
, 2003
"... There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are als ..."
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Cited by 6 (5 self)
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There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are also constructed that extend K4 and are related to the proof of noncanonicity of the McKinsey axiom. We dedicate this paper to Max Cresswell, a pioneer in the study of canonicity, on the occasion of his 65th birthday. 1
SCAN is complete for all Sahlqvist formulae
 In Relational and KleeneAlgebraic Methods in Computer Science (RelMiCS 7
, 2004
"... Abstract. SCAN is an algorithm for reducing monadic existential secondorder logic formulae to equivalent simpler formulae, often firstorder logic formulae. It is provably impossible for such a reduction to firstorder logic to be always successful, even if there is an equivalent firstorder formul ..."
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Cited by 6 (3 self)
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Abstract. SCAN is an algorithm for reducing monadic existential secondorder logic formulae to equivalent simpler formulae, often firstorder logic formulae. It is provably impossible for such a reduction to firstorder logic to be always successful, even if there is an equivalent firstorder formula for a secondorder logic formula. In this paper we show that SCAN successfully computes the firstorder equivalents of all Sahlqvist formulae in the classical (multi)modal language. 1
Atom structures and Sahlqvist equations
 Algebra Universalis
, 1997
"... This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra A is determined by its atom structure At A. We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary ..."
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Cited by 5 (1 self)
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This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra A is determined by its atom structure At A. We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary to the positive result and a recent result by I. Hodkinson, we prove that the variety RRA of representable relation algebras, although canonical, cannot be axiomatised by Sahlqvist equations. 1 1
Boolean Algebras Arising From Information Systems
, 2002
"... this paper we continue the development of algebraic counterparts of logics arising from information systems which we have begun in [5] and extend some results to reasoning about relative relations ..."
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Cited by 5 (3 self)
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this paper we continue the development of algebraic counterparts of logics arising from information systems which we have begun in [5] and extend some results to reasoning about relative relations
Duality for Lattices with Operators: a modal logic approach.
, 2000
"... This thesis discusses a proposal for a duality theory for bounded lattices with operators, amalgamating work on distributive lattices and on lattices with operators. Introduction 1. This work belongs to the eld of algebraic logic. Algebraic logic has its roots in the 19th century, starting with th ..."
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This thesis discusses a proposal for a duality theory for bounded lattices with operators, amalgamating work on distributive lattices and on lattices with operators. Introduction 1. This work belongs to the eld of algebraic logic. Algebraic logic has its roots in the 19th century, starting with the work of Boole. Its main aim is to study logic from an algebraic perspective in order to translate problems in logic to (universal) algebraic questions. At the beginning of the twentieth century, Tarski introduced the connection between boolean algebras and classical propositional calculus. Duality theory serves as a tool for translating questions from one perspective (logic) to another one (universal algebra): the existence of a duality between two categories means that the two categories are essentially the same; this makes it possible to use one category in order to understand the other one. Duality theory between logic and universal algebra has its basis in Stone's duality for boolea...
Algebraic Polymodal Logic: A Survey
 LOGIC JOURNAL OF THE IGPL
, 2000
"... This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with ..."
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This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with