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Bisimilarity is not Borel


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.

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

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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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