Minimization via Duality

Abstract

Abstract. We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object.

Domains of Commutative C*-Subalgebras

“…But also noncommutative approximately finite-dimensional C*-algebras have been used as operational semantics of probabilistic languages [11], [12]. Furthermore, C*-algebras find applications in computer science in minimization of automata [13], and via graph theory: any directed graph gives rise to a C*-algebra which contains almost all information about the graph [14]. Additionally, C*-algebras give semantics for linear logic [15] and geometry of interaction [16].…”

“…Thus we can speak about the domain of commutative C*-subalgebras entirely in the language of C*-algebras. The interpretation of this construction in terms of transition systems is unclear, but it might give rise to a similar construction as the Brzozowski minimization of an automaton [13], and hence provides interesting material for further study.…”

Stone Duality for Markov Processes

“…The theory of Stone-type dualities for transition systems has been investigated at length by Bonsangue and Kurz [8] and there have been many recent investigations of Stone-type dualities from the viewpoint of coalgebra and automata theory [30][31][32]. Recent very interesting work by Jacobs [33] has explored convex dualities for probability and quantum mechanics.…”

Brzozowski’s Algorithm (Co)Algebraically

“…Yet another approach, but somewhat similar in spirit to ours, can be found in [5], where a Stone-like duality between automata and their logical characterization is taken as a basis for Brzozowski's algorithm. The precise connection between that approach and the present paper, remains to be better understood.…”

“…Our present approach (as well as that of [5]) takes a deterministic automaton as a starting point. Recently [15], we have presented a generalisation of the subset construction for the determinisation of many different types of automata.…”