A coalgebraic perspective on linear weighted automata

Abstract

Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages. © 2012 Elsevier Inc. All rights reserved

A general account of coinduction up-to

“…We start by recalling from [9] the coalgebraic treatment of weighted automata. To simplify the presentation we assume the semiring (S, +, ·, 0, 1) to be commutative, but the presented results easily extend to the non-commutative case.…”

Bisimulation of Labeled State-to-Function Transition Systems of Stochastic Process Languages

“…An interesting direction of research combining coalgebra and quantities studies various types of weighted automata, including linear weighted automata, and associated notions of bisimulation and languages, as well as algorithms for these notions [6,25,5]. In particular, building on a result on bounded functors [11], it is shown in [5] for a functor involving functions of finite support over a field that the final coalgebra exists. Below, we have followed the scheme of [5] to obtain such a result for a functor induced by a FuTS.…”

“…In particular, building on a result on bounded functors [11], it is shown in [5] for a functor involving functions of finite support over a field that the final coalgebra exists. Below, we have followed the scheme of [5] to obtain such a result for a functor induced by a FuTS. The notions of equivalence addressed in this paper, as often in coalgebraic treatments of process relations, are all strong bisimilarities.…”

Coalgebraic Semantics of Heavy-Weighted Automata

“…In [4], such automata are used to give a compact representation of some combinatorial problems. One of our motivation here is to extend the coalgebraic setting existing for ordinary weighted automata [9,1] to study equivalence between heavyweighted automata. In particular, we are interested in producing a more compact representation of some well-shaped infinite weighted automata, generalizing the examples given in [4].…”

“…[8,9,1]. Our construction in Section 3 can be seen as an instance of the generalized determinization construction described in [12].…”