The lexicographic covering

In this paper, we describe a general method for building coverings, that we call lexicographic covering. The name comes from the fact that the scheme although more general than that is particularly fitted to the case where the transitions of an automaton are given an ordering at each state and thus the computations in the automaton may be lexicographic ordered. The lexicographic covering allows then to choose, among computations with the same input label, the smallest one in the lexicographic ordering. This method generalizes both the Schützenberger covering and the construction by H. Johnson of a lexicographic uniformization of deterministic rational relation. It allows us to construct an automaton that realizes a rational multi-layer skimming of N-rational series with a low complexity that outperforms the currently conjectured one. One very characteristic feature of automata theory is, as Eilenberg wrote in the preface of his treatise, that all arguments and proofs are constructive and that a statement asserting that something exists is of no interest unless it is accompanied by an algorithm for producing this “something”. We would like to add on this, and we think that a statement asserting that a set, or a series, is rational is much more interesting if it is accompanied by the effective construction of the automaton that accepts the set, or realizes the series. Many such constructions fit into a common overall scheme: starting from an automaton A, an expansion of A is performed and yields a larger automaton B, which is a covering of A; this means that the computations of B and those of A are in a 1 − 1 correspondence and that they have, roughly speaking, the same “structure”. In the larger B, it is so to say easier to distinguish between the computations and the proof of the property aimed at by the construction amounts to an adequate choice within these computations. For instance, the intersection of a rational set R with a recognizable set K is rational and this classical result may be revisited by the construction of the product of the automaton that accepts R by the action that recognizes K. A particular kind of construction of coverings, implicit in a paper by Schützenberger [9], has been used systematically by the first author under the name of Schützenberger covering [5], in order to produce new and clearer proofs of older results, for instance the Rational Uniformization Theorem and the Rational Skimming Theorem, that are recalled below. Of course, other kinds of coverings are used, explicitely or implicitely, in many papers in the field of automata theory. An interesting case is the construction due to H. Johnson to prove that the lexicographic selection of a deterministic rational relation is a deterministic rational function [2, 3]. It bears obvious similarity with the Schützenberger covering but is at the same time very different. ∗CNRS / ENST †ENST 1 In this paper, we describe a general method for building coverings, that we call lexicographic covering. The name comes from the fact that the scheme — although more general than that — is particularly fitted to the case where the transitions of an automaton are given an ordering at each state and thus the computations in the automaton may be lexicographic ordered. The lexicographic covering allows then to choose, among computations with the same input label, the smallest one in the lexicographic ordering. This method is more general and proves to be more accurate and versatile than the Schützenberger covering it encompasses anyway. We give here an evidence of its potentiality with a rational multi-layer skimming theorem. Let us first recall what we coined Rational Skimming Theorem. Theorem 1 (Schützenberger [8]). Let s be a N-rational series. The series s − supp s is N-rational. Let A be a N-automaton realizing the series s. The proof in [7] consists in building a covering C of A, based on the Schützenberger covering construction, such that a subautomaton C′ is an unambiguous automaton for the support of s and another subautomaton C′′ realizes s − supp s. If the dimension of A is n, the dimension of C (and of C′′) is n2 . Of course, this procedure can be iterated, but it yields for the series s . k an automaton whose dimension is a tower of k exponentials. As it was written in [7], the work of A. Weber on the decomposition of k-valued transducers ([10]) leads to think that a double exponentiation should be sufficient but that was still a conjecture. Here we prove, with the method of lexicographic covering, that a single exponentiation is indeed sufficient and that there exists an automaton of dimension n (k + 1) which realizes s . k. More precisely, we prove the following statement. Theorem 2. Let A be a finite N-automaton with n states realizing the series s. There exists an infinite N-covering B of A such that for every integer k > 0 , there exists a finite N-quotient Bk of B with the following properties: 1. Bk is a N-covering of A with at most n (k + 1) states; 2. for every i, 0 ≤ i < k, there exists an unambiguous subautomaton B k of Bk that recognizes the support of s . i; 3. there exists a subautomaton Ck of Bk whose behaviour is s . k. The paper is organized as follows. The first section is a reminder of the notion of covering and give two examples of the use of Schützenberger covering. The second and third sections give examples of lexicographic selections of computations, one for the Johnson’s construction, one for another construction for the Rational Skimming Theorem. They serve as introduction for the general definition of the lexicographic covering given in Section 4. The rational multilayer skimming theorem is proved at Section 5. 1 What are the coverings about? As usual, an automaton will be just a directed graph A = (Q, M, E, I, T ), where Q is a set of states, I and T are subsets of initial and final states, respectively, and E is a set of transitions labelled by elements of some monoid M . A computation of length n is a sequence of consecutive transitions p0 m1 −−→ p1 . . . mn −−→ pn, also denoted by c : p0 m −→ A pn, and it is successful if p0 ∈ I and pn ∈ T . The behaviour of A is the subset |A| of the labels of all the