Automata for Coalgebras: An Approach Using Predicate Liftings

Abstract

Universal Coalgebra provides the notion of a coalgebra as the natural mathematical generalization of state-based evolving systems such as (infinite) words, trees, and transition systems. We lift the theory of parity automata to this level of abstraction by introducing, for a set Λ of predicate liftings associated with a set functor $\mathcal{T}$, the notion of a Λ-automata operating on coalgebras of type $\mathcal{T}$. In a familiar way these automata correspond to extensions of coalgebraic modal logics with least and greatest fixpoint operators.

Our main technical contribution is a general bounded model property result: We provide a construction that transforms an arbitrary Λ-automaton $\mathbb{A}$ with nonempty language into a small pointed coalgebra $(\mathbb{S},s)$ of type $\mathcal{T}$ that is recognized by $\mathbb{A}$, and of size exponential in that of $\mathbb{A}$. $\mathbb{S}$ is obtained in a uniform manner, on the basis of the winning strategy in our satisfiability game associated with $\mathbb{A}$. On the basis of our proof we obtain a general upper bound for the complexity of the non-emptiness problem, under some mild conditions on Λ and $\mathcal{T}$. Finally, relating our automata-theoretic approach to the tableaux-based one of Cîrstea et alii, we indicate how to obtain their results, based on the existence of a complete tableau calculus, in our framework.

The research of the authors has been made possible by VICI grant 639.073.501 of the Netherlands Organization for Scientific Research (NWO).