Automata and fixed point logic: A coalgebraic perspective

  • Authors:

  • Yde Venema

  • Affiliations:

  • Institute for Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, NL-1018 TV Amsterdam, The Netherlands

  • Venue:
  • Information and Computation - Special issue: Seventh workshop on coalgebraic methods in computer science 2004
  • Year:
  • 2006

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Abstract

This paper generalizes existing connections between automata and logic to a coalgebraic abstraction level. Let F: Set to Set be a standard functor that preserves weak pullbacks. We introduce various notions of F-automata, devices that operate on pointed F-coalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally, we show that the two approaches are equivalent in expressive power. We prove that any coalgebraic fixed point formula can be transformed into an F-automaton that accepts precisely those pointed F-coalgebras in which the formula holds. And conversely, we prove that any F-automaton can be converted into an equivalent fixed point formula that characterizes the pointed F-coalgebras accepted by the automaton.