Explaining Behavioural Inequivalence Generically in Quasilinear Time

Authors Thorsten Wißmann , Stefan Milius , Lutz Schröder

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Author Details

Thorsten Wißmann
  • Radboud University, Nijmegen, The Netherlands
Stefan Milius
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Lutz Schröder
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

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Thorsten Wißmann, Stefan Milius, and Lutz Schröder. Explaining Behavioural Inequivalence Generically in Quasilinear Time. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time 𝒪((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was 𝒪(m n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Program analysis
  • bisimulation
  • partition refinement
  • modal logic
  • distinguishing formulae
  • coalgebra


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