Coalgebraic Minimization of Automata by Initiality and Finality

Abstract

International audienceDeterministic automata can be minimized by partition refinement (Moore's algorithm, Hopcroft's algorithm) or by reversal and determinization (Brzozowski's algorithm). In the coalgebraic perspective, the first approach can be phrased in terms of a minimization construction along the final sequence of a functor, whereas a crucial part of the second approach is based on a reachability construction along the initial sequence of another functor. We employ this coalgebraic perspective to establish a precise relationship between the two approaches to minimization, and show how they can be combined. Part of these results are extended to an approach for language equivalence of a general class of systems with branching, such as non-deterministic automata

Minimisation in Logical Form

“…In particular, our picture formalises the role of trace logic in the minimisation algorithms. Some of the technical details of this part are known from [20,39,46,58] -precise connections are detailed throughout the sections and in Section 6.…”

“…Our paper focuses on comparing and unifying our earlier approaches from [16] and [20] under a common umbrella, but we hasten to remark that the concept of minimisation via logic presented in section 3.3 is already in [58]. At its core, [58] uses a dual adjunction that is lifted to a dual adjunction between coalgebras and algebras. A logic is then used to provide a construction for obtaining observable coalgebras.…”

“…This is esssentially what we call Algo1. The setting of [58] is more general as no assumptions are made on the specific shape of the algebra and coalgebra functors involved. Instead the necessary functor requirements are axiomatised.…”

“…Instead the necessary functor requirements are axiomatised. One achievement of [58] is to generalise the setup in [16] from dual equivalences to dual adjunctions. The central contribution in [58] is to combine the duality-based framework with coalgebraic partition-refinement [2] such that a logicbased treatment of Brzozowski and partition refinement is obtained.…”

“…With algebras corresponding to the syntactic, deductive side of logical systems, and topological spaces to their semantics, Stone-type dualities provide a powerful mathematical framework for studying various properties of logical systems. More recently, it has also been fruitfully explored in more algorithmic applications, notably in understanding minimisation of various types of automata [2,16,20,21,31,48,58]. Among these, [16] and [20] had striking similarities in the approach yet it was not clear whether the differences could be harmonised in a uniform way.…”