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 = {s, f} ∪ Qsf , K = ( Q, Σ, δ, S, F ). (1) Qsf = { q ∈ Q | q > max(s, f) } , 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 ∈ Σ; ∅ ∅ a • for each regular expressions r and s, we set p −→ r, p ∈ {s} ∪ Qsf , r ∈ {f} ∪ 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({ q | q ∈ Q }), transition graph of the considered automaton.
then automaton (2) (i.e., Kqmax ) defines the language, defined also by regular expression III.A SPECIALVERSIONOFTHEPROOF n a o∗ OF LEENE STHEOREM K ’ ρqmax = a ∈ Σ | qmax −→ qmax ; (3) Let us consider a special version of the proof of Kleene’s remark that anyway ρqmax 3 ε. 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 ∈ Σ | 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 ∈ Σ | s −→ s + ρs→q · ρq · ρq→s . (5) For some states q, p, r ∈ 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 ∈ S and f ∈ F , denote [ such that Ls→f = L(Ks→f ) + L(Ks→q) ·L(Kq) ·L(Kq→f ). q 2 International Journal of Open Information Technologies ISSN: 2307-8162 vol. 6, no. 7, 2018 where, for instance, the language defined by ρs→f is a subset we obtain, using (3)–(6) and some simplest equivalent trans- of the language defined by formations ([14] etc.), the following expressions: [ ∗ ρs→q · ρq · ρq→f . ρq2 = b , ρq2→q1 = {c}, ρq1→q2 = {a} q∈Q and However, the last record, as is easy to verify, usually gives a ∗ ∗ ρq = (ab c) , much larger number of components (“terms”) in the obtained 1 regular expression. and, therefore, R(K, τ ) = (ab∗c)∗ · (ab∗c)∗ · (ab∗c)∗. (7) IV. THE STAR-HEIGHT OF A FINITE AUTOMATON 1 In this section, we define the main object of this paper, Similarly, considering the function i.e., the star-height of a finite automaton It is important to τ (q ) = 2, τ (q ) = 1 , note that this definition is built on the basis of its transition 2 1 2 2 graph only, and is not associated with the marks of its edges, we obtain the following expressions: i.e., the letters. ρ = {ε}, ρ = {a}, ρ = {c} Firstly, let us consider some propositions; they are simply q1 q1→q2 q2→q1 the corollaries of the given before definitions of regular and expressions ρ (i.e., (3)–(5)). ∗ ρq2 = (ca + b) , Proposition 1: SH(ρqmax ) ≤ 1. Proposition 2: and, therefore, R(K, τ ) = {ε}· {ε} + a · (ca + b)∗ · c ·{ε}. (8) SH(ρqs→f ) = max SH(ρq) ≤ max SH(ρq). 2 q∈∆max(s,f)(s,f) q∈Qs→f Counting the star-height of regular expressions (7) and (8), Proposition 3: If ∆ 6= ∅, then s we obtain, that the star-height of considered automaton is SH(ρs) = max SH(ρq)+1 ≤ max SH(ρq)+1. equal to 1. q∈∆ q>s s Proposition 4: For automaton without useless and inac- In previous section, we fixed K and τ. Then below, the cessible states and given ordering function τ, obtained regular expression defining L(K) for the ordering function τ (i.e., (6)) will be denoted below by R(K, τ ). SH R(K, τ) = max SH(ρq). q∈Q Let for automaton (1), its set Q consists of n states. We are able to consider n! different injective functions τ (because V. THE FINITE AUTOMATON FOR THE GIVEN LANGUAGE for each pair of states q, r ∈ Q, only the value of predicate In this section, for a given regular language, the corre- τ(q) < τ(r) is important). These n! different functions have, sponding finite automaton is constructed. As we already said, generally speaking, different regular expressions defined by this problem was solved more than 60 years ago, but the (4) and (5). method we are considering has the property that the “reverse Definition 1: The star-height of automaton (1) is defined construction” (that is, the standard algorithm for constructing in the following way: a regular expression based on a given finite automaton) gives SH(K) = min SH(R(K, τ)), (after some simplest equivalent transformation) exactly the τ∈T original regular expression. where T is the set of all the bijective functions of the type We shall not formulate these transformations strictly, since τ : Q → {1, . . . , n}. this is simply not necessary. See, e.g., the example before, It is important to remark, that for defining SH(K), we where we have obtained 3 factors (ab∗c)∗ in (7), but we can have used exactly the way of constructing regular expres- consider, e.g., τ = (ab∗c)∗. sions, which was given in previous subsection. (Certainly, Thus, we shall formulate the main proposition of the paper there exist other ways of constructing regular expressions by as follows. the given automaton.) Proposition 5: For each regular expression r, there exists Let us consider a simple well-known example; the tran- automaton sition graph for the language of regular expression (ab∗c)∗ Kr = ( Qr, Σ, δr,Sr,Fr ) is given on Fig. 1. (The agreement on the use of single and double circles as a designation of states of the transition and function τr for its states, such that: graph was given in [10].) (r1) L(Kr) = L(r); (r2) SH(Kr) ≤ SH(R(Kr, τr)) = SH(r). ? a - (Let us note once again, that we do not assert, that r = q1 q2 b R(K, τ).) c I Proof. We shall prove this proposition in the usual way, ? i.e., considering the usual process of constructing the given Fig. 1 regular expression; at the same time, we shall construct the equivalent automaton and corresponding function τ. To determine the star-height of this automaton, we have In addition to conditions (r1) and (r2) formulated before, to consider 2 functions τ; let us call them τ1 and τ2. For the we shall construct automata (let it be Kr) and corresponding function function (let τr), for which two additional requirements are τ1(q1) = 1, τ1(q2) = 2 , also met: 3 International Journal of Open Information Technologies ISSN: 2307-8162 vol. 6, no. 7, 2018 + (r3) there exists a value Tr ∈ R , such that: For expression (p · r), we can consider automaton – τ(s) < Tr for each s ∈ Sr; 0 K(p·r) = ( Qp ∪ Qr, Σ, δp ∪ δr ∪ δ ,Sp,Fr ), – τ(q) > T for each q∈ / S ; 0 r r where δ (q , a) 3 sr if and only if δ(q , a) 3 fp (r4) there is no edge of the type for all possible fp ∈ Fp , sr ∈ Sr , a ∈ Σ a 0 s1 −→ s2, (δ contains no other elements). The ordering function τ(p·r) δr can be defined in the following way: where s1, s2 ∈ Sr. ( τp(qp), if qp ∈ Qp ; Possible automata for the regular expressions ∅, ε and τ(p+r)(q) = a (for each a ∈ Σ) are given on Fig. 2–4 respectively. τr(qr) + Mp, if qr ∈ Qr . It is evident, that for each of these expressions, we have Like previous case, SH(Kr) = 0. Conditions (r3) and (r4) also hold. SH(R(K(p·r), τ(p·r))) = max SH(R(Kp, τp), SH(R(Kr, τr)) , - s - s ∅ a and conditions (r1), (r3) and (r4) also hold. And for expression (r∗), we can consider automaton a 0 0 0 0 - s - K(r∗) = ( Qr ∪ {q }, Σ, δr ∪ δ ,Sr ∪ {q },Fr ∪ {q } ), ε ? 0 where δ (q , a) 3 sr if and only if δ(q , a) 3 fr - - f∅ fa for all possible fr ∈ Fr , sr ∈ Sr , a ∈ Σ 0 (δ contains no other elements). The ordering function τ(r∗) can be defined in the following way: Fig. 2 Fig. 3 Fig. 4 τr(q), if q ∈ Qr ; Now, let us suppose that we already have automata cor- τ(r∗)(q) = 0 responding to given regular expressions p and r; we mean τr(q ) = min τr(q) / 2 . q∈Qr that we suppose, that already have automata, which define the same regular languages and have the same star-height. Conditions (r1), (r3) and (r4) are evident; let us prove (r2). Certainly, we can also suppose that we also have functions τ By the way of construction K(r∗), we obtain the following for these automata obtaining regular expressions which star- fact: each path of its transition graph, which belongs to K(r∗) height is equal to the star-height of the given expressions (i.e., and does not belong to Kr, has to contain a state sr ∈ Sr. 0 00 giving minimum possible star-height). I.e., we can suppose Then for any states q, q , q ∈ Qr, where q∈ / Sr, we obtain 0 00 that conditions (r1–r4) hold. Let these automata be the coincidence of the sets ∆q(q , q ) for automata K(r∗) and Kr. And only for states sr ∈ Sr, we can obtain some Kp = ( Qp, Σ, δp,Sp,Fp ) and Kr = ( Qr, Σ, δr,Sr,Fr ); new paths of sets ∆sr (sr, q) and ∆sr (q, sr). We mean the and corresponding functions be paths which belong to automaton K(r∗) and does not belong to Kr. τp : Qp → {1,..., |Qp| } and τr : Qr → {1,..., |Qr| }. For three the following objects: Then we can use the following automata and ordering func- • automaton Kr; tions; let us remark in advance, that the transition functions • corresponding (defined in this section) function τr; are considered as the sets, and the facts of defining required • and some its state q, regular expressions by these automata are evident. we denote the defined in previous section value SH(ρq) For expression (p + r), we can consider automaton by SHr(q). Then by Proposition 3 and condition (r3) for automaton K(r∗), we obtain, that: K(p+r) = ( Qp ∪ Qr, Σ, δp ∪ δr,Sp ∪ Sr,Fp ∪ Fr ). SH(r∗)(q) = SHr(q), if q ∈ Qr \ Sr ; And considering values Tp, Tr and SH(r∗)(s) ≤ SHr(q) + 1, if s ∈ Sr ; 0 Mp = max τp(qp), SH(r∗)(q ) = 0 . qp∈Qp we obtain, that the ordering function can be the following Then by condition (r4) for automaton K(r∗), we obtain one: condition (r2). τp(sp), if sp ∈ Sp ; VI.CONCLUSION τr(sr) + Tp, if sr ∈ Sr ; Using Proposition 5, we can reformulate the star-height τ(p+r)(q) = τp(qp) + Tp + Tr, if qp ∈ Qp \ Sp ; problem for regular languages in the following way: τr(qr) + Tp + Tr + Mp, if qr ∈ Qr \ Sr . for the given regular language, we have to con- struct the equivalent finite automaton K having the Evidently, minimum possible star-height. SH(R(K(p+r), τ(p+r))) = After that, considering n! bijective functions of the type τ : Q → {1, . . . , n} (where n = |Q|), we construct regular max SH(R(Kp, τp)), SH(R(Kr, τr)) . expressions R(K, τ) and choose the expression having the Conditions (r1), (r3) and (r4) also hold. minimum possible star-height. 4 International Journal of Open Information Technologies ISSN: 2307-8162 vol. 6, no. 7, 2018 In addition to this (“natural”) continuation of the topic of [11] Salomaa A. Jewels of Formal Language Theory. Rockville (Maryland): this paper, there are others, not so obvious. As we said before, Computer Science Press, Inc. 1981, 144 p. [12] Melnikov B. Extended nondeterministic finite automata. Fundamenta the publication of this not-so-complicated problem has some Informaticae. 2010, vol. 104, no. 3. pp. 255–265. different goals, they are related to the future development of [13] Melnikov B. The complete finite automaton. International Journal of the topic under consideration here. Open Information Technologies. 2017, vol. 5, no. 10. pp. 9–17. [14] Melnikov B. and Sayfullina M. O nekotoryh algoritmah ekviva- • The first topic is the consideration of the generalized lentnogo preobrazovaniya nedeterminirovannyh konechnyh avtoma- star-height problem. (As we said before, the connec- tov [On some algorithms for the equivalent transformation of nondeterministic finite automata]. Izvestiya of Higher Educational tion between the generalized star-height problem and Institutions. Mathematics. 2009, no. 4, pp. 67–72. (in Russian, generalized pseudo-automata is to be considered in one https://elibrary.ru/item.asp?id=11749888) of the following publications.) [15] Melnikov B. Extended nondeterministic finite automata. Fundamenta Informaticae. 2010, vol. 104, no. 3. pp. 255–265. • The second topic is the consideration of extended au- [16] Dolgov V. and Melnikov B. Postroenie universal’nogo konechnogo tomata, see [15]. avtomata . . . [The construction of a universal finite automa- • And the third topic is precisely the transition to the star- ton. Part I: From theory to practical algorithms]. Vestnik of Voronezh State University. 2013, no. 2, pp. 173–181. (in Russian, height problem for the languages. https://elibrary.ru/item.asp?id=20267924) For all such topics, problems close to those considered in this [17] Dolgov V., Melnikov B. and Melnikova A. Cikly grafa perehodov bazisnogo avtomata . . . [Cycles of the transition article are possible. Besides, it is very important to note that graph of a basic automaton and related questions]. Vestnik of the possible developments of the topic described here can Voronezh State University. 2016, no. 4, pp. 95–111. (in Russian, be treated completely independently of each other (“move- https://elibrary.ru/item.asp?id=27257800) ments” towards the complication: “vertically”, “horizontally” and “upwards”): for example, it is possible to consider the generalized star-height problem (i.e., the star-height problem for generalized regular expressions) with approach using extended pseudo-automata. The problems considered in this paper are also related to a topic that is set aside from the ones described here: we should to describe an algorithm, answering the question whether or not a proper subset of the set of edges forms the equivalent automaton. (See some related questions in [16], [17]. Certainly, we do not mean an exhaustive algorithm. The brute force method consists in this case in the complete application of the algorithm for constructing two canonical automata and their subsequent comparison.) The author hopes, that the joint application of algorithms mentioned above can give a faster than available determining the star-height of the given regular language. REFERENCES [1] Melnikov B. Ob odnoy klassifikacii kontekstno-svobodnyh . . . [On a classification of sequentional context-free languages and gram- mars]. Vestnik of Moscow University. Series 15: Computational Mathematics and Cybernetics. 1993, no. 3, pp. 64–69. (in Russian, https://elibrary.ru/title_about.asp?id=8373) [2] Melnikov B. and Vakhitova A. Some more on the finite automata. The Korean Journal of Computational and Applied Mathematics (Journal of Applied Mathematics and Computing). 1998, vol. 5, no. 3. pp. 495–506. [3] Melnikov B. O zvyozdnoy vysote regulyarnogo yazyka . . . [On the star- height of a regular language. Part III: The star-height of an automaton and the scheme of the transformation algorithm]. Heuristic algorithms and distributed computations. 2014, no. 3, pp. 60–76. (in Russian, https://elibrary.ru/item.asp?id=22376180) [4] Eggan L. Transitions graphs and the star height of regular events. Michigan Mathematical Journal. 1963, vol. 10, pp. 385–397. [5] Hashiguchi K. Algorithms for determining relative star height and star height. Information and Computation. 1988, vol. 78, pp. 124–169. [6] Perrin D. Finite Automata. Handbook of theoretical computer science, Vol.A. MIT Press Cambridge, MA, USA, 1990, 57 p. [7] Kirsten D. Distance desert automata and the star height problem. Informatique Theorique´ et Applications. 2005, vol. 39, no. 3. pp. 455– 509. [8] Bojanczyk M. Star height via games. ACM/IEEE Symposium on Logic in Computer Science (LICS). 2015, pp. 214–219. [9] Pin J.-E., Straubing H., Therien´ D. Some results on the generalized star- height problem. Information and Computation. 1992, vol. 101, no. 2, pp. 219–250. [10] Melnikov B. and Melnikova A. Pseudo-automata for generalized reg- ular expressions. 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