Chapter 9, Slide 1 1 There Are Many More Things to Learn About Finite Automata Than Are Covered in This Book
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INSTITUT FÜR INFORMATIK Finite Automata, Digraph Connectivity
T U M INSTITUT FÜR INFORMATIK Finite Automata, Digraph Connectivity, and Regular Expression Size Hermann Gruber Markus Holzer TUM-I0725 Dezember 07 TECHNISCHE UNIVERSITÄT MÜNCHEN TUM-INFO-12-I0725-0/1.-FI Alle Rechte vorbehalten Nachdruck auch auszugsweise verboten c 2007 Druck: Institut für Informatik der Technischen Universität München Finite Automata, Digraph Connectivity, and Regular Expression Size Hermann Gruber Institut für Informatik, Ludwig-Maximilians-Universität München, Oettingstraße 67, 80538 München, Germany [email protected] Markus Holzer Institut für Informatik, Technische Universität München, Boltzmannstraße 3, D-85748 Garching, Germany [email protected] Abstract We study some descriptional complexity aspects of regular expressions. In particular, we give lower bounds on the minimum required size for the conversion of deterministic finite automata into regular expressions and on the required size of regular expressions resulting from applying some basic language operations on them, namely intersection, shuffle, and complement. Some of the lower bounds obtained are asymptotically tight, and, notably, the examples we present are over an alphabet of size two. To this end, we develop a new lower bound technique that is based on the star height of regular languages. It is known that for a restricted class of regular languages, the star height can be determined from the digraph underlying the transition structure of the minimal finite automaton accepting that language. In this way, star height is tied to cycle rank, a structural complexity measure for digraphs proposed by Eggan and Büchi, which measures the degree of connectivity of directed graphs. This measure, although a robust graph theoretic notion, appears to have remained widely unnoticed outside formal language theory. -
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. -
Open Problems About Regular Languages, 35 Years Later Jean-Eric Pin
Open problems about regular languages, 35 years later Jean-Eric Pin To cite this version: Jean-Eric Pin. Open problems about regular languages, 35 years later. Stavros Konstantinidis; Nelma Moreira; Rogério Reis; Jeffrey Shallit. The Role of Theory in Computer Science - Essays Dedicated to Janusz Brzozowski, World Scientific, 2017, 9789813148208. 10.1142/9789813148208_0007. hal- 01614375v2 HAL Id: hal-01614375 https://hal.archives-ouvertes.fr/hal-01614375v2 Submitted on 27 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Open problems about regular languages, 35 years later Dedicated to Janusz A. Brzozowski for his 80th birthday Jean-Eric´ Pin ∗ IRIF, CNRS and Universit´eParis-Diderot, Case 7014, 75205 Paris Cedex 13, France. [email protected] October 2016 Abstract In December 1979, Janusz A. Brzozowski presented a selection of six open problems about regular languages and mentioned two other problems in the conclusion of his article. These problems have been the source of some of the greatest breakthroughs in automata theory over the past 35 years. This survey article summarizes the state of the art on these questions and the hopes for the next 35 years. -
Digraph Complexity Measures and Applications in Formal Language Theory
Digraph Complexity Measures and Applications in Formal Language Theory Hermann Gruber∗y knowledgepark AG, Leonrodstr. 68, D-80636 München, Germany Abstract We investigate structural complexity measures on digraphs, in par- ticular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular lan- guages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)3=2)- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O∗(1:9129n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the reg- ular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height prob- lem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concern- ing cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact expo- nential algorithm for bideterministic star height. Keywords: digraph, cycle rank, regular expression, star height, ordered coloring, vertex ranking arXiv:1111.5357v1 [cs.DM] 22 Nov 2011 MSC: 05C20 (primary), 68Q45, 68Q25 (secondary) ACM CCS: G2.2 Graph Theory, F2.2 Nonnumerical Algorithms and Problems, F4.3 Formal Languages ∗This paper is a completely revised and expanded version of research presented at the 4th Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS 2008), see [16].