Coalgebraic Automata Theory: Basic Results

Abstract

Abstract. We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects.Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.

Coalgebraic Logic and Synthesis of Mealy Machines

“…The same process has been recently generalized to F -coalgebras in [10]. In this paper, we use a different approach.…”

Equational Coalgebraic Logic

“…This observation, which is closely linked to fundamental automata-theoretic constructions, lies at the heart of the theory of the modal μ-calculus, and has many applications, see for instance [5,27]. Generalizing the link between fix-point logics and automata theory to the coalgebraic level of generality, Kupke & Venema [15] generalized some of these observations to show that many fundamental results in automata theory are really theorems of universal coalgebra.…”

The Logic of Exact Covers: Completeness and Uniform Interpolation

“…A similar argument can be made for the distributive laws (see e.g. [21]) that underlie our treatment of bisimulation quantifiers. On the other hand, this simplicity comes at a price: the loss of monotonicity.…”

“…It is known that X ∈ Base(X ) and moreover Base(X ) is the smallest subset of P(Σ) with this property [21,Proposition 6.7]. Using the notion of base, we can characterise satisfiability of ∇-formulae as follows.…”