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The Star-Height of a Finite Automaton and Some Related Questions

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The Star-Height of a Finite Automaton and Some Related Questions International Journal of Open Information Technologies ISSN: 2307-8162 vol. 6, no. 7, 2018 The star-height of a finite automaton and some related questions B. F. Melnikov Abstract—The paper is related to the star-height for non- was some improved using an approach of the game theory, deterministic finite automata, not for the star-height problem see [8]. And unlike the previous comment, the last proof was for regular languages. We describe an alternative proof of called “elegant”. Although it does improve the understanding Kleene’s theorem, and then our version of the reduction of some problems related to the star-height to nondeterministic of the proof of D. Kirsten, the number of automata being finite automata. For a given regular language, the corresponding examined due to this “elegance” does not decrease. finite automaton is constructed; the method we are considering The second of these mentioned problems is the generalized has the property that the reverse construction gives the original star-height problem. For now, it is unknown the answer regular expression. to the question whether or not this problem is decidable. The publication of this not-so-complicated problem has two goals, the both are related to the future development of the topic The author also does not know significant papers on this under consideration. First, we assume the future development of subject published after [9] (1992). The connection between the topic with the aim of describing a similar approach for gen- the generalized star-height problem and generalized pseudo- eralized regular expressions. Secondly, another generalization automata (see [10] for the last formalism) is to be considered is proposed, namely, consideration of the structures we describe in one of the following publications. for the so-called extended automata. Thus, the material of this article is necessary for the definition of extended generalized In this paper we describe, firstly, an alternative proof of pseudo-automata, which the author proposes to cite in the next Kleene’s theorem, and secondly, our version of the reduction publication. of some problems related to the star-height to nondeterminis- Keywords—nondeterministic finite automata, regular lan- tic finite automata. Both these topics represent an alternative guages, Kleene’s theorem, star-height problem. to the particular version of the reduction proposed back in 1963 in the above-mentioned paper [4]. As we already noted above, the subject of this paper is practically unrelated to I. INTRODUCTION AND MOTIVATION the complex star-height problems. However, the publication This paper summarizes some previous publications of the of this not-so-complicated problem has two goals, the both author. Among these publications, we note papers [1], [2], ones are related to the future development of the topic under [3] (in chronological order). consideration. First, we assume the future development of In [1], only some schemes of algorithms were published. the topic with the aim of describing a similar approach for In [3], an algorithm was given that relates Kleene’s the- generalized regular expressions (see [11] etc.) and general- orem and the star-height. This material has for now been ized automata (see [10]). Secondly, another generalization is published in Russian only; besides, due to the termination of proposed; namely, we shall describe the consideration of our Samara State University in 2015, the website of the journal structures for so-called extended automata (see [12]) and, is now unavailable. accordingly, extended pseudo-automata. Let us note, that all In the current paper, the author gives the English version concepts needed for such extensions are present in this paper. of [3], where also some noted misprints are corrected, and The structure of this paper is as follows. Section II briefly for some statements simpler proofs are given. describes the used notation. In Section III, we consider a And the paper [2] was published long ago, it contained special version of the proof of Kleene’s theorem. another algorithm that also describes the connection between In Section IV, we define the main object of this paper, i.e., Kleene’s theorem and the star-height. the star-height of a finite automaton. This definition is built The current paper is related to the star-height for nonde- on the basis of its transition graph only, and is not associated terministic finite automata, its subject is practically unrelated with the marks of its edges. to much more complex problems, i.e., to both “ordinary” In Section V, for a given regular language, the correspond- star-height problem and generalized star-height problem for ing finite automaton is constructed. Of course, this problem regular languages. was solved more than 60 years ago, but the method we are For these problems for the languages, let us only make considering has the property that the “reverse construction” some remarks. The first of mentioned problems, i.e., the gives the original regular expression. “ordinary” star-height problem, was set in 1963 in [4] and In Conclusion (Section VI), we formulate the directions solved in 1988 in [5]; in [6], however, this solution was called for the further research. “extremely difficult”. In 2005, there was published the much more understand- able solution of D. Kirsten, see [7]. In 2015, this solution II. PRELIMINARIES Received May 12, 2018. Boris F. Melnikov, Russian State Social University (email: bf-melnikov@ We shall not describe in detail the notation used in this yandex.ru). paper; they are exactly the same as those used in [13]. Like 1 International Journal of Open Information Technologies ISSN: 2307-8162 vol. 6, no. 7, 2018 there, the “main” automaton under consideration will be where denoted by Qs!f = fs; fg [ Qsf ; K = ( Q; Σ; δ; S; F ): (1) Qsf = f q 2 Q j q > max(s; f) g ; We shall use the classical definition of the star-height of and δs!f is constructed in the following way: the regular expression, see [7], [11]. We define it (denoting a by SH(r) for expression r) by the induction in the following p −! r δ way: s!f ∗ if and only if • SH( ) = SH( ) = SH(a) = 0 for each a 2 Σ; ? ? a • for each regular expressions r and s, we set p −! r; p 2 fsg [ Qsf ; r 2 ffg [ Qsf : δ SH((r + s)) = SH((r · s)) = max( SH(r); SH(s)); We shall construct regular expressions corresponding lan- guages of automata of the type (2) by the induction on the • for each regular expression r (where r 6= ?), we set value τ(min(s; f)). We shall denote these expressions by SH((r∗)) = SH(r) + 1: ρs!f ; if s = f, we shall denote the automaton and the expression by Ks and ρs. Remark that sometimes, another definition is considered: (Let us remark, that we consider only expressions, ob- SH(?∗) is defined as 1. There is easy to prove, that such tained in the way described below. Certainly, each automaton difference is important (i.e., two different definitions for has also infinitely many other corresponding regular expres- expressions give two different values of the star-height of sions. But we shall not consider them in this section.) the regular languages) for some finite languages only. Thus, let us consider the induction formulated before. Its Besides, some new notation will be described as necessary; basis is the following. If most of these notations are associated with paths in the min(s; f) = qmax = max(f q j q 2 Q g); transition graph of the considered automaton. then automaton (2) (i.e., Kqmax ) defines the language, defined also by regular expression III. A SPECIAL VERSION OF THE PROOF n a o∗ OF LEENE S THEOREM K ’ ρqmax = a 2 Σ j qmax −! qmax ; (3) Let us consider a special version of the proof of Kleene’s remark that anyway ρqmax 3 ". theorem. For the given automaton (1), let us consider some Step of induction. If s 6= f, then we write expression injective “ordering” function defining language of automaton Ks!f in the following way: + τ : Q ! R : n a o [ ρs!f = a 2 Σ j s −! f + ρs!q ·ρq ·ρq!f (4) We shall also write, e.g., p < r meaning τ(p) < τ(r); also q>max(s;f) we shall use notation “max”, meaning (for expression, [ symbolizes +’s). And if s = f, then we ( p; if τ(p) ≥ τ(r) write expression defining language of automaton Ks in the max(p; r) = r; if τ(p) < τ(r) ; following way: ∗ n a o [ etc. Let us fix K and τ in this section. ρs = a 2 Σ j s −! s + ρs!q · ρq · ρq!s : (5) For some states q; p; r 2 Q, where p ≥ q and r ≥ q, let q>s us consider some simple path from p to r, whose sequence By the hypothesis of inductions, all the expressions of the of vertices is right parts of (4) and (5) are already constructed. (p; q1; q2; : : : ; qs; r); For some pair s 2 S and f 2 F , denote [ such that Ls!f = L(Ks!f ) + L(Ks!q) ·L(Kq) ·L(Kq!f ): q<min(s;f) s ≥ 0 and (8i 2 f1; : : : ; sg)(q > q): i Then we can write language of the given automaton (1) (i.e., (We allow p = r. In this case, i.e., if p = r, it is a simple L(K)) in the following way: loop.) We shall denote the set of all such paths by ∆q(p; r). [ L(K) = L(Ks) ·Ls!f ·L(Kf ): We also set s2S; f2F q 0 0 ∆ = f q is a state of a path of ∆q(q; q) and q 6= q g : Therefore, the corresponding regular expression is [ [ Some more notation: ρs · ρs!f + ρs!q · ρq · ρq!f · ρf : s2S; f2F • if ∆q(p; r) 6= ?, then we shall write Vq(p; r); q<min(s;f) (6) • otherwise, Vq(p; r); • if both Vq(p; r) and Vq(r; p), then we shall write We shall everywhere (i.e., both in this paper and in the Wq(p; r).
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