Syntactic Structures of Regular Languages
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Classifying Regular Languages Via Cascade Products of Automata
Classifying Regular Languages via Cascade Products of Automata Marcus Gelderie RWTH Aachen, Lehrstuhl f¨urInformatik 7, D-52056 Aachen [email protected] Abstract. Building on the celebrated Krohn-Rhodes Theorem we char- acterize classes of regular languages in terms of the cascade decomposi- tions of minimal DFA of languages in those classes. More precisely we provide characterizations for the classes of piecewise testable languages and commutative languages. To this end we use biased resets, which are resets in the classical sense, that can change their state at most once. Next, we introduce the concept of the scope of a cascade product of reset automata in order to capture a notion of locality inside a cascade prod- uct and show that there exist constant bounds on the scope for certain classes of languages. Finally we investigate the impact of biased resets in a product of resets on the dot-depth of languages recognized by this product. This investigation allows us to refine an upper bound on the dot-depth of a language, given by Cohen and Brzozowski. 1 Introduction A significant result in the structure theory of regular languages is the Krohn- Rhodes Theorem [7], which states that any finite automaton can be decomposed into simple \prime factors" (a detailed exposition is given in [4, 6, 9, 10]). We use the Krohn-Rhodes Theorem to characterize classes of regular lan- guages in terms of the decompositions of the corresponding minimal automata. In [8] this has been done for star-free languages by giving an alternative proof for the famous Sch¨utzenberger Theorem [11]. -
Automata, Semigroups and Duality
Automata, semigroups and duality Mai Gehrke1 Serge Grigorieff2 Jean-Eric´ Pin2 1Radboud Universiteit 2LIAFA, CNRS and University Paris Diderot TANCL’07, August 2007, Oxford LIAFA, CNRS and University Paris Diderot Outline (1) Four ways of defining languages (2) The profinite world (3) Duality (4) Back to the future LIAFA, CNRS and University Paris Diderot Part I Four ways of defining languages LIAFA, CNRS and University Paris Diderot Words and languages Words over the alphabet A = {a, b, c}: a, babb, cac, the empty word 1, etc. The set of all words A∗ is the free monoid on A. A language is a set of words. Recognizable (or regular) languages can be defined in various ways: ⊲ by (extended) regular expressions ⊲ by finite automata ⊲ in terms of logic ⊲ by finite monoids LIAFA, CNRS and University Paris Diderot Basic operations on languages • Boolean operations: union, intersection, complement. • Product: L1L2 = {u1u2 | u1 ∈ L1,u2 ∈ L2} Example: {ab, a}{a, ba} = {aa, aba, abba}. • Star: L∗ is the submonoid generated by L ∗ L = {u1u2 · · · un | n > 0 and u1,...,un ∈ L} {a, ba}∗ = {1, a, aa, ba, aaa, aba, . .}. LIAFA, CNRS and University Paris Diderot Various types of expressions • Regular expressions: union, product, star: (ab)∗ ∪ (ab)∗a • Extended regular expressions (union, intersection, complement, product and star): A∗ \ (bA∗ ∪ A∗aaA∗ ∪ A∗bbA∗) • Star-free expressions (union, intersection, complement, product but no star): ∅c \ (b∅c ∪∅caa∅c ∪∅cbb∅c) LIAFA, CNRS and University Paris Diderot Finite automata a 1 2 The set of states is {1, 2, 3}. b The initial state is 1. b a 3 The final states are 1 and 2. -
CAI 2017 Book of Abstracts.Pdf
Table of Contents Track 1: Automata Theory and Logic ........................... 1 Invited speaker: Heiko Vogler . 1 Languages and formations generated by D4 and D8: Jean-Éric Pin, Xaro Soler-Escrivà . 2 Syntactic structures of regular languages: O. Klíma, L. Polák . 26 Improving witnesses for state complexity of catenation combined with boolean operations: P. Caron, J.-G. Luque, B. Patrou . 44 Track 2: Cryptography and Coding Theory ..................... 63 Invited speaker: Claude Carlet . 63 A topological approach to network coding: Cristina Martínez and Alberto Besana . 64 Pairing-friendly elliptic curves resistant to TNFS attacks: G. Fotiadis, E. Konstantinou . 65 Collaborative multi-authority key-policy attribute-based encryption for shorter keys and parameters: R. Longo, C. Marcolla, M. Sala 67 Conditional blind signatures: A. Zacharakis, P. Grontas, A. Pagourtzis . 68 Hash function design for cloud storage data auditing: Nikolaos Doukas, Oleksandr P. Markovskyi, Nikolaos G. Bardis . 69 Method for accelerated zero-knowledge identification of remote users based on standard block ciphers: Nikolaos G. Bardis, Oleksandr P. Markovskyi, Nikolaos Doukas . 81 Determining whether a given block cipher is a permutation of another given block cipher— a problem in intellectual property (Extended Abstract): G. V. Bard . 91 Track 3: Computer Algebra ..................................... 95 Invited speaker: Michael Wibmer . 95 Interpolation of syzygies for implicit matrix representations: Ioannis Z. Emiris, Konstantinos Gavriil, and Christos Konaxis . 97 Reduction in free modules: C. Fürst, G. Landsmann . 115 Instructing small cellular free resolutions for monomial ideals: J. Àlvarez Montaner, O. Fernández-Ramos, P. Gimenez . 117 Low autocorrelation binary sequences (LABS): lias S. Kotsireas . 123 A signature based border basis algorithm: J. Horáček, M. -
A Quantum Query Complexity Trichotomy for Regular Languages
A Quantum Query Complexity Trichotomy for Regular Languages Scott Aaronson∗ Daniel Grier† Luke Schaeffer UT Austin MIT MIT [email protected] [email protected] [email protected] Abstract We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Θ(1), Θ˜ (√n), or Θ(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complex- ity Θ(nc) for all computable c [1/2,1]. Our result implies an equivalent trichotomy for the approximate degree of regular∈ languages, and a dichotomy—either Θ(1) or Θ(n)—for sensi- tivity, block sensitivity, certificate complexity, deterministic query complexity, and randomized query complexity. The heart of the classification theorem is an explicit quantum algorithm which decides membership in any star-free language in O˜ (√n) time. This well-studied family of the regu- lar languages admits many interesting characterizations, for instance, as those languages ex- pressible as sentences in first-order logic over the natural numbers with the less-than relation. Therefore, not only do the star-free languages capture functions such as OR, they can also ex- press functions such as “there exist a pair of 2’s such that everything between them is a 0.” Thus, we view the algorithm for star-free languages as a nontrivial generalization of Grover’s algorithm which extends the quantum quadratic speedup to a much wider range of string- processing algorithms than was previously known. -
Quantum Conditional Strategies for Prisoners' Dilemmata Under The
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 May 2019 doi:10.20944/preprints201905.0366.v1 Peer-reviewed version available at Appl. Sci. 2019, 9, 2635; doi:10.3390/app9132635 Article Quantum conditional strategies for prisoners’ dilemmata under the EWL scheme Konstantinos Giannakis* , Georgia Theocharopoulou, Christos Papalitsas, Sofia Fanarioti, and Theodore Andronikos* Department of Informatics, Ionian University, Tsirigoti Square 7, Corfu, 49100, Greece; {kgiann, zeta.theo, c14papa, sofiafanar, andronikos}@ionio.gr * Correspondence: [email protected] (K.G.); [email protected] (Th.A.) 1 Abstract: Classic game theory is an important field with a long tradition of useful results. Recently, 2 the quantum versions of classical games, such as the Prisoner’s Dilemma (PD), have attracted a lot of 3 attention. Similarly, state machines and specifically finite automata have also been under constant 4 and thorough study for plenty of reasons. The quantum analogues of these abstract machines, like the 5 quantum finite automata, have been studied extensively. In this work, we examine some well-known 6 game conditional strategies that have been studied within the framework of the repeated PD game. 7 Then, we try to associate these strategies to proper quantum finite automata that receive them as 8 inputs and recognize them with probability 1, achieving some interesting results. We also study the 9 quantum version of PD under the Eisert-Wilkens-Lewenstein scheme, proposing a novel conditional 10 strategy for the repeated version of this game. 11 Keywords: quantum game theory, quantum automata, prisoner’s dilemma, conditional strategies, 12 quantum strategies 13 1. Introduction 14 Quantum game theory has gained a lot of research interest since the first pioneering works of the 15 late ’90s [1–5]. -
On the Decomposition of Generalized Semiautomata
On the Decomposition of Generalized Semiautomata Merve Nur Cakir and Karl-Heinz Zimmermann∗ Department of Computer Engineering Hamburg University of Technology 21071 Hamburg, Germany April 21, 2020 Abstract Semiautomata are abstractions of electronic devices that are de- terministic finite-state machines having inputs but no outputs. Gen- eralized semiautomata are obtained from stochastic semiautomata by dropping the restrictions imposed by probability. It is well-known that each stochastic semiautomaton can be decomposed into a sequential product of a dependent source and deterministic semiautomaton mak- ing partly use of the celebrated theorem of Birkhoff-von Neumann. It will be shown that each generalized semiautomaton can be partitioned into a sequential product of a generalized dependent source and a de- terministic semiautomaton. AMS Subject Classification: 68Q70, 20M35, 15A04 Keywords: Semiautomaton, stochastic automaton, monoid, Birkhoff- von Neumann. 1 Introduction arXiv:2004.08805v1 [cs.FL] 19 Apr 2020 The theory of discrete stochastic systems has been initiated by the work of Shannon [14] and von Neumann [10]. While Shannon has considered memory-less communication channels and their generalization by introduc- ing states, von Neumann has studied the synthesis of reliable systems from unreliable components. The fundamental work of Rabin and Scott [12] about deterministic finite-state automata has led to two generalizations. First, the generalization of transition functions to conditional distributions studied by Carlyle [3] and Starke [15]. This in turn yields a generalization of discrete- time Markov chains in which the chains are governed by more than one ∗Email: [email protected] 1 transition probability matrix. Second, the generalization of regular sets by introducing stochastic automata as described by Rabin [11]. -
A Categorical Approach to Syntactic Monoids 11
Logical Methods in Computer Science Vol. 14(2:9)2018, pp. 1–34 Submitted Jan. 15, 2016 https://lmcs.episciences.org/ Published May 15, 2018 A CATEGORICAL APPROACH TO SYNTACTIC MONOIDS JIRˇ´I ADAMEK´ a, STEFAN MILIUS b, AND HENNING URBAT c a Institut f¨urTheoretische Informatik, Technische Universit¨atBraunschweig, Germany e-mail address: [email protected] b;c Lehrstuhl f¨urTheoretische Informatik, Friedrich-Alexander-Universit¨atErlangen-N¨urnberg, Ger- many e-mail address: [email protected], [email protected] Abstract. The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D “ sets), the syntactic ordered monoids of Pin (D “ posets), the syntactic semirings of Pol´ak(D “ semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras or ordered algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids. 1. Introduction One of the successes of the theory of coalgebras is that ideas from automata theory can be developed at a level of abstraction where they apply uniformly to many different types of systems. -
Syntactic Semigroups
Syntactic semigroups Jean-Eric Pin LITP/IBP, CNRS, Universit´e Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE 1 Introduction This chapter gives an overview on what is often called the algebraic theory of finite automata. It deals with languages, automata and semigroups, and has connections with model theory in logic, boolean circuits, symbolic dynamics and topology. Kleene's theorem [70] is usually considered as the foundation of this theory. It shows that the class of recognizable languages (i.e. recognized by finite automata), coincides with the class of rational languages, which are given by rational expressions. The definition of the syntactic mono- id, a monoid canonically attached to each language, was first given in an early paper of Rabin and Scott [128], where the notion is credited to Myhill. It was shown in particular that a language is recognizable if and only if its syntactic monoid is finite. However, the first classification results were rather related to automata [89]. A break-through was realized by Schutzen-¨ berger in the mid sixties [144]. Schutzenb¨ erger made a non trivial use of the syntactic monoid to characterize an important subclass of the rational languages, the star-free languages. Schutzenb¨ erger's theorem states that a language is star-free if and only if its syntactic monoid is finite and aperiodic. This theorem had a considerable influence on the theory. Two other im- portant \syntactic" characterizations were obtained in the early seventies: Simon [152] proved that a language is piecewise testable if and only if its syntactic monoid is finite and J -trivial and Brzozowski-Simon [41] and in- dependently, McNaughton [86] characterized the locally testable languages. -
Typed Monoids – an Eilenberg-Like Theorem for Non Regular Languages
Electronic Colloquium on Computational Complexity, Report No. 35 (2011) Typed Monoids – An Eilenberg-like Theorem for non regular Languages Christoph Behle Andreas Krebs Stephanie Reifferscheid∗ Wilhelm-Schickard-Institut, Universität Tübingen, fbehlec,krebs,reiff[email protected] Abstract Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid. This leads to an algebraic description of regular and non regular languages. We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid. We prove an Eilenberg-like theorem for varieties of typed monoids as well as a similar correspondence for classes of languages with weaker closure properties. 1 Introduction We present an algebraic viewpoint on the study of formal languages and introduce a new algebraic structure to describe formal languages. We extend the known approach to use language recognition via monoids and morphisms by equipping the monoids with additional information to obtain a finer notion of recognition. In the concept of language recognition by monoids, a language L ⊆ Σ∗ is recognized by a monoid M if there exists a morphism h : Σ∗ ! M and a subset A of M such that L = h−1(A). The syntactic monoid, which is the smallest monoid recognizing L, has emerged as an important tool to study classes of regular languages. Besides problems applying this tool to non regular languages, even for regular languages this instrument lacks precision. One problem is, that not only the syntactic monoid but also the syntactic mor- phism, i.e. the morphism recognizing L via the syntactic monoid, plays an important role: Consider the two languages Lparity, Leven over the alphabet Σ = fa; bg where the first one consists of all words with an even number of a’s while the latter consists of all words of even length. -
Applications of Semigroups by P
Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 3 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Applications of Semigroups By P. Sreenivasulu Reddy & Mulugeta Dawud Samara University, Ethiopia Abstract- This section deals with the applications of semigroups in general and regular semigroups in particular. The theory of semigroups attracts many algebraists due to their applications to automata theory, formal languages, network analogy etc. In section 2 we have seen different areas of applications of semigroups. We identified some examples in biology, sociology etc. whose semigroup structures are nothing but regular, E-inversive and inverse semigroup etc. GJSFR -F Classification : FOR Code : MSC 2010: 16W22 ApplicationsofS emigroups Strictly as per the compliance and regulations of : © 2015. P. Sreenivasulu Reddy & Mulugeta Dawud. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Applications of Semigroups Notes P. Sreenivasulu Reddy α & Mulugeta Dawud σ 2015 Absract- This section deals with the applications of semigroups in general and regular semigroups in particular. The r ea theory of semigroups attracts many algebraists due to their applications to automata theory, formal languages, network Y analogy etc. In section 2 we have seen different areas of applications of semigroups. We identified some examples in biology, sociology etc. -
Syntactic Semigroup Problem for the Semigroup Reducts of Affine Near-Semirings Over Brandt Semigroups
SYNTACTIC SEMIGROUP PROBLEM FOR THE SEMIGROUP REDUCTS OF AFFINE NEAR-SEMIRINGS OVER BRANDT SEMIGROUPS JITENDER KUMAR AND K. V. KRISHNA Abstract. The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semi- + group problem for both the semigroup reducts of A (Bn), the affine near- semiring over a Brandt semigroup Bn. It is ascertained that both the semi- + group reducts of A (Bn) are syntactic semigroups. 1. Introduction The concept of syntactic semigroup (in particular, syntactic monoid) plays a vi- tal role in algebraic automata theory. In fact, a language is recognizable if and only if its syntactic monoid is finite (cf. [9, 10]). Sch¨utzenberger made a nontrivial use of syntactic semigroups to characterize an important subclass of recognizable lan- guages, viz., star-free languages. Sch¨utzenberger theorem states that a recognizable language is star-free if and only if its syntactic semigroup is finite and aperiodic [11]. Simon established that a recognizable language language is piecewise testable if and only if its syntactic monoid is J -trivial [12]. Syntactic semigroups have been studied in various contexts (e.g., [3, 4]). The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. Many authors have investigated the syntactic semigroup problem for various semigroups. Lallement and Milito proved that any finite semilattice of groups with identity is the syntactic monoid of a recognizable language [8]. In [2], Goralˇc´ıket al. observed that every finite inverse semigroup is syntactic. For a finite monoid M, an algorithm solving syntactic monoid problem for a large class of finite monoids in O(jMj3) time is obtained by Goralˇc´ıkand Koubek [1]. -
Some Results on the Generalized Star-Height Problem
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector INFORMATION AND COMPUTATION 101, 219-250 (1992) Some Results on the Generalized Star-Height Problem J. E. PIN,* H. STRAUBING,~ AND D. TH~RIEN~ LITP, Universiti de Paris VI. 4 place Jussieu, 75252 Paris Cedex, France We prove some results related to the generalized star-height problem. In this problem. as opposed to the restricted star-height problem, complementation is considered as a basic operator. We first show that the class of languages of star- height <n is closed under certain operations (left and right quotients, inverse alphabetic morphisms. injective star-free substitutions). It is known that languages recognized by a commutative group are of star-height 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by (L/2Z)“. In the same direction. we show that one of the languages that were conjectured to be of star-height 2 during the past ten years is in fact of star-height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form Mb (G r N), where M and N are aperiodic monoids, and G is a commutative group, then L is of star-height < 1. Finally we show that every rational language is the inverse image, under some morphism between free monoids. of a language of (resticted) star-height 1. c 1992 Academic Press. Inc. The determination of the star-height of a rational language is an old problem of formal language theory (see Brzozowski, 1980, for an historical survey).