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Journal of Logic and Computation Advance Access published online on December 21, 2007
Journal of Logic and Computation, doi:10.1093/logcom/exm085
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© The Author 2007. Published by Oxford University Press on behalf of the Association of Physicians. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original papers

A Note on Bisimulation Quantifiers and Fixed Points over Transitive Frames

Giovanna D'Agostino

University of Udine, Department of Mathematics and Computer Science, Viale delle Scienze 206, 33100 Udine, Italy E-mail: dagostin{at}dimi.uniud.it

Giacomo Lenzi

University of Pisa, Department of Mathematics, Via Buonarroti 2, 56127 Pisa, Italy E-mail: lenzi{at}mail.dm.unipi.it

Received 28 November 2005.


  Abstract

We consider three basic questions regarding the extension of modal logic with a special kind of propositional quantifiers, known as bisimulation quantifiers, over arbitrary classes of frames: bisimulation invariance, uniform interpolation, and expressive power. In particular:

– we discuss the relation between bisimulation invariance of bisimulation quantifiers and the semantical notion of amalgamation of the class of frames;
– we consider a strong form of interpolation, uniform interpolation, and its relation with the closure under bisimulation quantifiers;
– we compare bisimulation quantifiers logic with the better known extension of modal logic with extremal fixed points.
In this article we show that the answers to these questions that are valid for the class of all frames do not generalize to arbitrary classes, but they do generalize if we restrict to classes of (finite) transitive or (finite) transitive and reflexive frames.

Keywords: Fixed points; bisimulation quantifiers; Mu-calculus; transitive frames; uniform interpolation


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