Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-29T20:25:41.934Z Has data issue: false hasContentIssue false
  • English
  • Français

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: sergiortm@gmail.com
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: pmr@math.ist.utl.pt
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Harel, D., Kozen, D. and Tiuryn, J.Dynamic Logic (MIT Press, 2000).CrossRefGoogle Scholar
[2]Emerson, E. A. Temporal and modal logic. In van Leeuwen, J. (editor), Handbook of Theoretical Computer Science, vol. B. (MIT Press, 1990), pp. 955–1072.Google Scholar
[3]Hughes, G. E. and Cresswell, M. J.An Introduction to Modal Logic (Methuen & Co. Ltd., 1968).Google Scholar
[4]Hughes, G. E. and Cresswell, M. J.A Companion to Modal Logic (Methuen & Co. Ltd., 1984).Google Scholar
[5]Johnstone, P. T.Stone Spaces (Cambridge University Press, 1982).Google Scholar
[6]Joyal, A. and Tierney, M. An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc., vol. 309. (American Mathematical Society, 1984).CrossRefGoogle Scholar
[7]Lawson, M. V.Inverse Semigroups—The Theory of Partial Symmetries (World Scientific, 1998).CrossRefGoogle Scholar
[8]Moerdijk, I. and Mrčun, J.Introduction to Foliations and Lie Groupoids. (Cambridge University Press, 2003).CrossRefGoogle Scholar
[9]Mulvey, C. J. Quantales. In Hazewinkel, M. (editor), The Encyclopaedia of Mathematics third supplement. (Kluwer Academic Publishers, 2002), pp. 312–314.Google Scholar
[10]Mulvey, C. J. and Resende, P.A noncommutative theory of Penrose tilings. Internat. J. Theoret. Phys. 44 (2005), 655–689.CrossRefGoogle Scholar
[11]Paseka, J. and Rosický, J. Quantales. In Coecke, B., Moore, D. and Wilce, A. (editors), Current Research in Operational Quantum Logic: Algebras, Categories and Languages. Fund. Theories Phys., vol. 111. (Kluwer Academic Publishers, 2000), pp. 245–262.CrossRefGoogle Scholar
[12]Paterson, A. L. T.Groupoids, Inverse Semigroups and Their Operator Algebras (Birkhäuser, 1999).CrossRefGoogle Scholar
[13]Resende, P.Étale groupoids and their quantales. Adv. Math. 208 (2007), 147–209.CrossRefGoogle Scholar
[14]Resende, P.Tropological systems are points of quantales. J. Pure Appl. Algebra 173 (2002), 87–120.CrossRefGoogle Scholar
[15]Rosenthal, K.Quantales and Their Applications (Longman Scientific & Technical, 1990).Google Scholar