Sound and Complete Axiomatizations of Coalgebraic Language Equivalence

Abstract

Coalgebras provide a uniform framework for studying dynamical systems, including several types of automata. In this article, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalized powerset construction that determinizes coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor FT , where T is a monad describing the branching of the systems (e.g., non-determinism, weights, probability, etc.), has as a quotient the rational fixpoint of the determinized type functor F , a lifting of F to the category of T -algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain nondeterministic automata in which we recover Rabinovich’s sound and complete calculus for language equivalence.

Towards a Coalgebraic Chomsky Hierarchy

“…We conclude this section with an easy corollary of Theorem 6.5 and Proposition 2.2 contrasting the results in [21,3]. Corollary 6.6.…”

“…We plan to derive a sound calculus of reactive expressions extending [3] and explore the boundaries for completeness (by Corollary 6.6 completeness is only possible for specific choices of T); establish a converse to Theorem 7.3;capture further language and complexity classes, such as the context-sensitive languages (equivalently NSPACE(n)). Capturing various classes of machines under the umbrella of coalgebra results in standard tools such as bisimulation proof methods becoming available for those classes of machines and their language semantics.…”

“…22.4] showed that for any monad T on a category with powers there is a bijective correspondence between Eilenberg-Moore algebras on B and monad morphisms from T to T B . Now, given a T-automaton ( ), we define a T B -automaton 3 …”

“…We build on previous work on coalgebraic modelling and monad-based semantics. Most of the applications of coalgebra to automata and formal languages however addressed rational models (regular languages, rational power series) from which we note [31] (regular languages and finite automata), [12] (bialgebraic view of Kleene algebra and regular expressions), [34,21,23,3] (coalgebraic regular expressions). More recently, some further generalizations were proposed.…”

A New Foundation for Finitary Corecursion

“…This correspondence has been generalized to locally finitely presentable categories [8,20], where finitely presentable objects play the role of finite sets, leading to the notion of rational fixpoint that provides final semantics to all models with finitely presentable carrier [31]. It is known that the rational fixpoint is fully abstract (identifies all behaviourally equivalent states) as long as finitely presentable objects agree with finitely generated objects in the base category [12,Proposition 3.12]. While this is the case in some categories (e.g.…”