A Coalgebraic Perspective on Minimization and Determinization

Abstract

Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this paper, we use the coalgebraic view on systems to derive, in a uniform way, abstract procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization. First, we show that for coalgebras in categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. Then, we consider coalgebras in categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. We will apply our theory to the following examples: conditional transition systems and (non-deterministic) automata.

Minimization via Duality

“…e.g. [1]). This factorization structure can be used in order to define the minimization of a coalgebra.…”

“…In this case, given our assumptions on the category C and the functor T : C − → C, the minimization of a coalgebra (X, γ) ∈ Coalg(T) exists (cf. [1,Thm. 3.8]).…”

A description based on languages of the final non-deterministic automaton

“…Some recent descriptions of minimisations of non-deterministic automata have been presented by Brzozowski and Tamm [9] and Adámek, Bonchi et al [3]. We mention them here because they are based on the languages associated to every state of the automaton.…”

“…In [3], a coalgebraic point of view of this kind of description is presented. However, non-deterministic automata are considered there as coalgebras for the functor A × Id + 1 : Rel −→ Rel, where Rel denotes the category of sets and relations.…”

Algebra-coalgebra duality in brzozowski's minimization algorithm

“…We apply this algorithm to the following automaton (taken from[Adámek et al 2012a]): the language L = {wb | |w| b ≥ 1} consisting of all words ending in a b and containing at least two b's. Applying the lower inverse construction and taking reachability, we obtain the automaton ?>=< 89:; the reverse language rev(L) = {bw | |w| b ≥ 1}.…”