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Bisimilarity is not Borel

  • PEDRO SÁNCHEZ TERRAF (a1) (a2)
Abstract

We prove that the relation of bisimilarity between countable labelled transition systems (LTS) is Σ1 1-complete (hence not Borel), by reducing the set of non-well orders over the natural numbers continuously to it.

This has an impact on the theory of probabilistic and non-deterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes.

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Partially supported by CONICET, ANPCyT project PICT 2012-1823, SeCyT-UNC project 05/B284, and EU 7FP grant agreement 295261 (MEALS). Part of this work was presented at Dagstuhl Seminar 12411 on Coalgebraic Logics.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

V. Danos , J. Desharnais , F. Laviolette and P. Panangaden (2006). Bisimulation and cocongruence for probabilistic systems. Information and Computation 204 (4) 503523.

J. Desharnais , A. Edalat and P. Panangaden (2002). Bisimulation for labelled Markov processes. Information and Computation 179 (2) 163193.

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M. Kracht (1999). Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics, Elsevier.

K.G. Larsen and A. Skou (1991). Bisimulation through probabilistic testing. Information and Computation 94 (1) 128.

Y.N. Moschovakis (2009). Descriptive Set Theory, 2nd edition, Mathematical Surveys and Monographs, American Mathematical Society.

J.J.M.M. Rutten (2000). Universal coalgebra: A theory of systems. Theoretical Computer Science 249 (1) 380.

P. Sánchez Terraf (2011). Unprovability of the logical characterization of bisimulation. Information and Computation 209 (7) 10481056.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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