Bisimilarity is not Borel
Abstract
We prove that the relation of bisimilarity between countable labelled transition systems is -complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and nondeterministic 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.
Cite
@article{arxiv.1211.0967,
title = {Bisimilarity is not Borel},
author = {Pedro Sánchez Terraf},
journal= {arXiv preprint arXiv:1211.0967},
year = {2015}
}
Comments
20 pages, 1 figure; proof of Sigma_1^1 completeness added with extended comments. I acknowledge careful reading by the referees. Major changes in Introduction, Conclusion, and motivation for NLMP. Proof for Lemma 22 added, simpler proofs for Lemma 17 and Theorem 30. Added references. Part of this work was presented at Dagstuhl Seminar 12411 on Coalgebraic Logics