The rational skimming theorem

We define the notion of K-covering of automata with multiplicity (in a semiring K) that extend the one of covering of automata. We make use of this notion, together with the Schützenberger construct that we have explained in a previous work and that we briefly recall here, in order to give a direct and constructive proof of a fundamental theorem on N-rational power series. In a previous work (cf. [4]), we have shown how a construction, proposed by Schützenberger (in [8] and [9]) in order to prove that rational functions are unambiguous, can be given a central position in the theory of relations and functions realized by finite automata. The other basic results such as the “Rational Cross-section Theorem”, the “Rational Uniformisation Theorem” (that is dual to the preceeding one), and the “Decomposition Theorem ” (of rational functions into sequential functions) appear then as direct and formal consequences of it. We have explained that this construction is indeed a construction on finite automata and we have described it in the framework of covering of automata — which is derived from the notion of covering of graphs that was proposed by Stallings ([10]) — and which makes (in our opinion) the whole subject much clearer. The purpose of the present communication is to extend the concept of covering to the one of K-covering that apply to automata with multiplicity in a semiring K. And to make use of this notion together with the Schützenberger construct quoted above, in order to establish another result, due to Schützenberger as well, and that we call the Rational Skimming Theorem. LTCI, UMR 5141 CNRS/ENST , Paris 1 Theorem 1 [7] If s is a N-rational power series on A, then the series s obtained from s by substracting 1 to every non-zero coefficient of s, i.e. the series s = s− supp s is a N-rational power series as well. This result is not new, by far. In [2, Theorem VI.11.1], it is obtained as the consequence of the Rational Cross-section Theorem quoted above (and of some other results such as the division theorem). In [6, Theorem II.8.6] and in [1, Theorem V.2.1], more direct proofs are given (the attribution to Schützenberger is made in the latter reference). The proof presented here is hopefully simpler than the preceeding ones and corresponds to an explicit construction on automata. A complete exposition of all that matter, K-coverings and their use in the theory of K-rational series will be found in [5]. 1 The Schützenberger covering We basically follow the definitions and notation of [2] which we use without further notice. Those that follow in this section and that are more original have been described in detail in [4]. A (finite) automaton over a finite alphabet A, A = 〈Q,A,E, I, T 〉, is a directed labelled graph where Q, I and T are respectively the (finite) sets of states, initial states and terminal states, and E is the set of labelled edges. The language accepted by A, that is the set of the labels of the successful computations in A, also called the behaviour of A, is denoted by |A|. A morphism φ from an automaton B = 〈R,A,F, J, U 〉 into an automaton A = 〈Q,A,E, I, T 〉 is indeed a pair of mappings (both denoted by φ): one between the set of states φ : R → Q, and one between the set of edges φ : F → E, which are consistent with the structure of the automata, that is, for every f in F : i) the origin of fφ is the image (by φ) of the origin of f ; ii) the label of fφ is equal to the label of f ; iii) and Jφ ⊆ I and Uφ ⊆ T . These conditions imply that the image of a successful computation in B is a successful computation in A, that their labels are equal, and thus that |B| ⊆ |A| holds. 2 For every state q of an automaton A = 〈Q,A,E, I, T 〉, let us denote by OutA(q) the set 1 of edges of A the origin of which is q, that is edges that are “going out” of q. One defines dually InA(q) as the set of edges of A the end of which is q, that is edges that are “going in” q. If φ is a morphism from B = 〈R,A,F, J, U 〉 into A = 〈Q,A,E, I, T 〉 then for every r in R, φmaps OutB(r) into OutA(rφ), and InB(r) into InA(rφ) . We say that φ is Out-surjective (resp. Out-bijective, Out-injective) if for every r in R the restriction of φ to OutB(r) is surjective onto OutA(rφ) (resp. bijective between OutB(r) and OutA(rφ), injective). Accordingly, we say that φ is In-surjective (resp. In-bijective, In-injective) if for every r in R the restriction of φ to InB(r) is surjective onto InA(rφ) (resp. bijective between InB(r) and InA(rφ), injective). Definition 1 Let B = 〈R,A,F, J, U 〉 and A = 〈Q,A,E, I, T 〉 ; a morphism φ : B → A is a covering (resp. a co-covering) if the following conditions hold: i) φ is Out-bijective (resp. In-bijective); ii) for every i in I, there exists a unique j in J such that jφ = i (resp. for every t in T , there exists a unique s in S such that sφ = t); iii) Tφ = U (resp. Iφ = J). Proposition 1 Any covering (resp. any co-covering) φ : B → A induces a bijection between the successful computations in B and those in A. Theorem & Definition 2 Let A be an automaton and Adet the determinized automaton of A. We call Schützenberger covering of A the accessible part S of Adet×A. Then: i) πA is a covering from S onto A. ii) πAdet is an In-surjective morphism from S onto Adet. We call immersion of A a sub-automaton of a covering of A. From all these definitions and result, one derives easily the result which is the basis of the present work. Corollary 2 Let A be an automaton on A. Then there exists an unambiguous automaton that is equivalent to A and that is a sub-automaton of a covering of A. Stallings denotes it “StarA(q)”. As the star is the common denomination for the generated submonoid, we cannot keep it, though it nicely conveys the idea of “a set of edges going out” of q. 3 Proof. Let S be the Schützenberger covering ofA. As πAdet is In-surjective from S onto Adet, one can delete edges in S (and possibly suppress the quality of being terminal to some of its states) in such a way that the subautomaton T that is obtained is a co-covering of Adet. The automaton T is then unambiguous — as there is a one-to-one correspondence between its successful computations and those of Adet — and equivalent to Adet, hence to A. The essence of this statement lies of course in the fact that the quoted unambiguous automaton is at the same time equivalent to and an immersion of A. For otherwise, the deterministic automaton Adet associated to A by the subset construction is obviously unambiguous and equivalent to A; but it can not be immersed in A: there is no relationships between the pathes in A and those in Adet. Example 1 : The Figure 1 represents an automaton A1 that accepts all words of {a, b} which contain at least one b (vertically, on the left), its determinized automaton A1det, the Schützenberger covering of A1, and the two possible immersions that can be derived from it. 2 p q b a a b b {p} {p, q} b a a b