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An algebraic generalization of Kripke structures†

Published online by Cambridge University Press:  01 November 2008

SÉRGIO MARCELINO  and
PEDRO RESENDE
SÉRGIO MARCELINO
Affiliation:
SQIG-Instituto de Telecomunicações, UTL-Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. e-mail: [email protected]
PEDRO RESENDE
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, UTL-Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. e-mail: [email protected]
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Abstract

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 549 - 577
Copyright
Copyright © Cambridge Philosophical Society 2008

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