Characterisation of (Sub)sequential Rational Functions over a General Class Monoids

In this technical report we describe a general class of monoids for which (sub)sequential rational can be characterised in terms of a congruence relation in the flavour of Myhill-Nerode relation. The class of monoids that we consider can be described in terms of natural algebraic axioms, contains the free monoids, groups, the tropical monoid, and is closed under Cartesian. 1 Inroduction The problem to efficiently represent functions f : Σ → M that map words to some monoid arises in different areas of Natural Language Processing: Speech Recognition, Machine Translation, Parsing, Similarity Search. Finite state transducers are a natural extension of (classical) finite state automata that provide an efficient representation a special class of such functions called rational functions, [5, 2, 10, 11, 12, 13, 14]. As it is common for most kinds of computational devices, the notion of determinism plays an important role since it usually implies more efficient computation. In terms of automata and transducers, the determinism means strongly linear on-line algorithm for parsing the input. This motivates the interest in deterministic finite state transducers that are called (sub)sequential transducers [5, 14]. For (classical) finite state automata it is well known that deterministic automata are equivalent to non-deterministic automata. However, this is not the case for transducers and (sub)sequential transducers [3, 1, 13]. Actually, the latter are capable to represent only a proper class of rational functions called (sub)sequential rational functions. In this paper we consider the characterisation problem of (sub)sequential rational functions. There are two main streams of characterisations known in the literature. The first one characterises the class of (sub)sequential rational functions as rational