Towards an algebraic characterization of rational word functions

Introduction In formal language theory, several models characterize regular languages: finite au-tomata, congruences of finite index, monadic second-order logic (MSO), see e.g. [Büc60, Elg61, Tra61]. Further connections exist between monoid varieties and logical fragments of MSO, for instance aperiodic monoids have been shown to recognize exactly the word languages defined by first-order (FO) formulas [Sch65, MP71]. When considering word relations instead of languages, automata are replaced by transducers, that is automata with outputs associated with transitions. A transducer is an automaton that reads an input word and returns an output word obtained by concatenating the outputs of the transitions. However, several important properties do not generalize from automata to transducers. For instance, the well known equivalence between deterministic and non-deterministic one-way automata, as well as the equivalence between one-way and two-way automata, do not transfer to transducers. One property that is preserved is the equivalence between automata and MSO formulas: it has been shown [EH01] that MSO word transductions and two-way transducers define the same class of word functions called regular functions. A recently introduced model of computation, streaming string transducers (SST), has been shown to compute the same class of regular functions [A ˇ C10]. Two recent related results are the equivalence between FO transductions and aperiodic SSTs [FKT14] and the equivalence between FO transductions and aperiodic two-way transducers [CD15]. However it is not known if one can decide if a given regular function is FO definable. Our contribution: Our result deals with functions that are definable by one-way word transducers. These functions are known in the literature as rational functions. The notion of minimal automaton goes beyond minimizing the state space. Indeed to decide whether a regular language satisfies some algebraic property, like aperiodicity, it suffices to consider the minimal automaton. Therefore in order to have an algebraic characterization of rational functions, we need a notion similar to the one of minimal automata for transductions. For the class of functions defined by deterministic transducers, such a notion exists [Cho03] and this minimal transducer 1 enjoys, among deterministic transducers, the same kind of minimality properties. In an attempt to obtain a similar notion for rational functions, we study the model of bimachines [Sch61] which has been shown to be a canonical model for rational functions (see e.g. [BB79]). We describe a canonical bimachine, introduced by [RS91], and show that this representation preserves, similarly to the minimal automaton for languages, …