Abstract

We consider bisimulation-invariant monadic second-order logic over various classes of finite transition systems. We present several combinatorial characterisations of when the expressive power of this fragment coincides with that of the modal μ-calculus. Using these characterisations we prove for some simple classes of transition systems that this is indeed the case. In particular, we show that, over the class of all finite transition systems with Cantor–Bendixson rank at most k, bisimulation-invariant ▪ coincides with Lμ.

Highlights

  • A characterisation of the bisimulation-invariant fragment of a given classical logic relates this logic to a suitable modal logic

  • The study of bisimulation-invariant fragments of classical logics was initiated by a result of van Benthem [2] who proved that the bisimulation-invariant fragment of first-order logic coincides with standard modal logic

  • We consider two logics in this paper: (i) monadic second-order logic (MSO), which is the extension of first-order logic by set variables and set quantifiers, and (ii) the modal μ-calculus (Lμ), which is the fixed-point extension of modal logic

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Summary

Introduction

A characterisation of the bisimulation-invariant fragment of a given classical logic relates this logic to a suitable modal logic. The main reason is that the original proof is based on automata-theoretic techniques and an essential ingredient is a reduction to trees, via the unravelling operation As this operation produces infinite trees, we cannot use it for formulae that are only bisimulation-invariant over finite transition systems. The former is already known and serves as an example of our techniques and to fix our notation for the second result, which is new. We characterise bisimulationinvariant monadic second-order logic over the class of all transition systems of a given Cantor–Bendixson rank

Bisimulation-invariance
Composition lemmas
Lassos
Hierarchical Lassos
Reductions
Finite Cantor–Bendixson rank
Conclusion
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