DOCSLIB.ORG

On the Decomposition of Generalized Semiautomata

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Decomposition of Generalized Semiautomata WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS Merve Nur Cakir, Mehwish Saleemi, DOI: 10.37394/23209.2021.18.6 Karl-Heinz Zimmermann On the Decomposition of Generalized Semiautomata MERVE NUR CAKIR, MEHWISH SALEEMI, KARL-HEINZ ZIMMERMANN Department of Computer Engineering Hamburg University of Technology 20171 Hamburg GERMANY Abstract: - Semiautomata are abstractions of electronic devices that are deterministic finite-state machines having inputs but no outputs. Generalized 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 making 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 deterministic semiautomaton. Key-Words: - Semiautomaton, stochastic automaton, monoid, Birkhoff-von Neumann Received: May 31, 2021. Revised: May 4, 2021. Accepted: May 22, 2021. Published: May 29, 2021 1 Introduction ity [5, 16]. It is well-known that each stochastic au- The theory of discrete stochastic systems has been tomaton can be decomposed into a sequential product initiated by the work of Shannon [14] and von Neu- of a dependent source and deterministic semiautoma- mann [10]. While Shannon has considered memory- ton [2]. This result makes use in part of the celebrated less communication channels and their generalization theorem of Birkhoff-von Neumann that each doubly by introducing states, von Neumann has studied the stochastic matrix can be represented as a convex com- synthesis of reliable systems from unreliable compo- bination of permutation matrices. In this paper, it will nents. The fundamental work of Rabin and Scott [12] be shown that each generalized semiautomaton can be about deterministic finite-state automata has led to partitioned into a sequential product of a generalized two generalizations. First, the generalization of tran- dependent source and a deterministic semiautomaton. sition functions to conditional distributions studied by Carlyle [3] and Starke [15]. This in turn yields a gen- Notation. Let X be a set. The set of all mappings eralization of discrete-time Markov chains in which on X, T (X) = ff j f : X ! Xg, forms a monoid the chains are governed by more than one transition under function composition (fg)(x) = g(f(x)), x 2 probability matrix. Second, the generalization of reg- X, and the identity function idX : X ! X : x 7! x ular sets by introducing stochastic automata as de- is the identity element. The monoid T (X) is called scribed by Rabin [11]. the full transformation monoid of X. By the work of Turakainen [16], stochastic accep- tors can be viewed equivalently as generalized au- 2 Semiautomata tomata in which the ”probability” is neglected. This Semiautomata are abstractions of electronic devices leads to a more accessible approach to stochastic au- which are deterministic finite-state machines having tomata [5]. input but no output [7, 9]. On the other hand, the class of nondeterministic A (deterministic) semiautomaton (SA) is a triple automata [13] can be generalized to monoidal au- tomata, where the input alphabet corresponds to an A = (S; Σ; fδx j x 2 Σg) arbitrary monoid instead of a free monoid [8, 9, 17]. This leads to the class of monoidal automata whose where languages are closed under a smaller set of operations • S is the non-empty finite set of states, when compared with regular languages. A first step into the study of automata theory are • Σ is the set of input symbols, semiautomata which are abstractions of electronic de- • δ : S ! S is a (partial) mapping for each x 2 Σ. vices that are deterministic finite-state machines hav- x ing inputs but no outputs [7, 9]. Generalized semi- Let Σ∗ denote the free monoid over the alphabet Σ. automata are obtained from stochastic semiautomata By the universal property of free monoids [4, 9], the by dropping the restrictions imposed by probabil- mapping δ :Σ ! T (S): x 7! δx extends uniquely to E-ISSN: 2224-3402 34 Volume 18, 2021 WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS Merve Nur Cakir, Mehwish Saleemi, DOI: 10.37394/23209.2021.18.6 Karl-Heinz Zimmermann ∗ a monoid homomorphism δ :Σ ! T (S): u 7! δu • S is the non-empty finite set of states, such that for each word u = x : : : x 2 Σ∗, 1 k • Σ is the input alphabet, and δ = δ ··· δ (1) u x1 xk • Q is a collection of n × n nonnegative matrices Qx, x 2 Σ, where n is the number of states. and particularly δϵ = idS. The mapping δ is called the transition function of A. Its image T (A) = fδu j In view of the universal property of free u 2 Σ∗g is a submonoid of the full transformation monoids [4, 9], the mapping Q :Σ ! Rn×n : monoid T (S) generated by fδx j x 2 Σg. The semi- x 7! Qx extends uniquely to a monoid homomor- automaton A is also denoted by A = (S; M; δ) or phism Q :Σ∗ ! Rn×n such that for each word A A A ∗ A = (S ;M ; δ ). u = x1 : : : xk 2 Σ , A semiautomaton A = (S; Σ; δ) serves as a skele- Q = Q ··· Q (2) ton of a deterministic finite-state machine that is ex- u x1 xk actly in one state at a time. If the semiautomaton A is and particularly Qϵ = In is the n × n identity ma- in state s and reads the word u 2 Σ∗, it transits into trix. The mapping Q is called the transition func- 0 ∗ the state s = δu(s). tion of A. Its image T (A) = fQu j u 2 Σ g is a submonoid of the full transformation monoid T (S) Example 1. Consider the semiautomaton A = f j 2 g f g generated by Qx x Σ . The generalized semi- (S; Σ; δ) with state set S = 1; 2; 3 , input alpha- automaton A is also denoted by A = (S; Σ;Q) or bet Σ = fx; yg, and transition function δ given by the A = (SA; ΣA;QA). automaton graph in Fig. 1. The associated transfor- The state set S = fs1; : : : ; sng can be viewed as mation monoid is generated by the transformations Rn ( ) ( ) the standard basis for the Euclidean vector space , 1 2 3 1 2 3 where si is the basis vector whose ith coordinate is 1 δ = δ = : x 1 1 1 and y 2 2 3 and all others are 0. In this way, the (i; j)the entry of (u) (u) T the matrix Qu = (sij ) is given by sij = si Qusj. We have ( ) ( ) Proposition 1. Each deterministic semiautomaton is 1 2 3 1 2 3 a generalized automaton. δxx = ; δxy = ; ( 1 1 1 ) ( 2 2 2 ) Proof. Let A = (S; Σ; δ) be a deterministic semiau- 1 2 3 1 2 3 f g δ = ; δ = : tomaton and let S = s1; : : : ; sn . Define the gener- yx 1 1 1 yy 2 2 3 alized semiautomaton B = (S; Σ;Q), where for each x 2 Σ, the (i; j)th entry of Qx is 1 if δx(si) = sj Hence, the transformation monoid T (A) is given by and otherwise 0. Then the mapping T (A) ! T (B): f g } idS; δx; δy; δxy . δu 7! Qu is a monoid isomorphism. y A generalized semiautomaton A = (S; Σ;P ) is + x ?>=<89:;k ?>=<89:; y called stochastic if the matrices Px, x 2 Σ, are 6 1 >^ > 2 h > x stochastic, i.e., P is a matrix of nonnegative real >> x >> numbers such that each row sum is equal to 1. The x >> ?>=<89:; product of stochastic matrices is again a stochastic 3 h y matrix and so the transition monoid T (A) consists of ∗ the stochastic matrices Pu, u 2 Σ . In particular, the (i; j)th element p(sj j u; si) of the matrix Pu is the Figure 1: Semiautomaton. transition probability that the automaton enters state sj when started in state si and reading the word u. 3 Generalized Semiautomata Example 2. Let m ≥ 2 be an integer. Put Σ = f − g A Stochastic automata are a generalization of non- 0; : : : ; m 1 . The stochastic semiautomaton = (fs1; s2g; Σ;P ) given by deterministic finite state automata [5]. Generalized ( ) automata can be obtained from stochastic automata 1 m − x x P = ; x 2 Σ; by dropping the restrictions imposed by probabil- x m m − x − 1 x + 1 ity [5, 16, 17]. A generalized semiautomaton (GSA) is a triple is called m-adic semiautomaton. For each word u = ∗ x1 : : : xk 2 Σ , A = (S; Σ; fQx j x 2 Σg); ( ) k 1 m − wk wk Pu = k k ; where m m − wk − 1 wk + 1 E-ISSN: 2224-3402 35 Volume 18, 2021 WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS Merve Nur Cakir, Mehwish Saleemi, DOI: 10.37394/23209.2021.18.6 Karl-Heinz Zimmermann k−1 where wk = xkm + ::: + x2m + x1 and the en- A permutation matrix P is a square binary matrix 1 try mk wk corresponds in the m-adic representation which has exactly one entry of 1 in each row and each to 0:xk : : : x1. } column and 0’s elsewhere. By the Birkhoff-von Neu- mann theorem [6], for each n × n doubly stochas- A generalized semiautomaton A = (S; Σ;D) is ≥ tic matrixPP there exist real numbers α1; : : : ; αN called doubly stochastic if the matrices Dx, x 2 Σ, N 0 with αi = 1 and permutation matrices are doubly stochastic, i.e., Dx is a matrix of nonnega- i=1 tive real numbers such that each row and column sum P1;:::;PN such that is equal to 1.
Load more

Recommended publications
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • Sage 9.4 Reference Manual: Monoids Release 9.4
    Sage 9.4 Reference Manual: Monoids Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Monoids 3 2 Free Monoids 5 3 Elements of Free Monoids 9 4 Free abelian monoids 11 5 Abelian Monoid Elements 15 6 Indexed Monoids 17 7 Free String Monoids 23 8 String Monoid Elements 29 9 Utility functions on strings 33 10 Hecke Monoids 35 11 Automatic Semigroups 37 12 Module of trace monoids (free partially commutative monoids). 47 13 Indices and Tables 55 Python Module Index 57 Index 59 i ii Sage 9.4 Reference Manual: Monoids, Release 9.4 Sage supports free monoids and free abelian monoids in any finite number of indeterminates, as well as free partially commutative monoids (trace monoids). CONTENTS 1 Sage 9.4 Reference Manual: Monoids, Release 9.4 2 CONTENTS CHAPTER ONE MONOIDS class sage.monoids.monoid.Monoid_class(names) Bases: sage.structure.parent.Parent EXAMPLES: sage: from sage.monoids.monoid import Monoid_class sage: Monoid_class(('a','b')) <sage.monoids.monoid.Monoid_class_with_category object at ...> gens() Returns the generators for self. EXAMPLES: sage: F.<a,b,c,d,e>= FreeMonoid(5) sage: F.gens() (a, b, c, d, e) monoid_generators() Returns the generators for self. EXAMPLES: sage: F.<a,b,c,d,e>= FreeMonoid(5) sage: F.monoid_generators() Family (a, b, c, d, e) sage.monoids.monoid.is_Monoid(x) Returns True if x is of type Monoid_class. EXAMPLES: sage: from sage.monoids.monoid import is_Monoid sage: is_Monoid(0) False sage: is_Monoid(ZZ) # The technical math meaning of monoid has ....: # no bearing whatsoever on the result: it's ....: # a typecheck which is not satisfied by ZZ ....: # since it does not inherit from Monoid_class.
    [Show full text]
  • Monoids (I = 1, 2) We Can Define 1  C 2000 M
    Geometria Superiore Reference Cards Push out (Sommes amalgam´ees). Given a diagram i : P ! Qi Monoids (i = 1; 2) we can define 1 c 2000 M. Cailotto, Permissions on last. v0.0 u −!2 Send comments and corrections to [email protected] Q1 ⊕P Q2 := coker P −! Q1 ⊕ Q2 u1 The category of Monoids. 1 and it is a standard fact that the natural arrows ji : Qi ! A monoid (M; ·; 1) (commutative with unit) is a set M with a Q1 ⊕P Q2 define a cocartesian square: u1 composition law · : M × M ! M and an element 1 2 M such P −−−! Q1 that: the composition is associative: (a · b) · c = a · (b · c) (for ? ? all a; b; c 2 M), commutative: a · b = b · a (for all a; b 2 M), u2 y y j1 and 1 is a neutral element for the composition: 1 · a = a (for Q2 −! Q1 ⊕P Q2 : all a 2 M). j2 ' ' More explicitly we have Q1 ⊕P Q2 = (Q1 ⊕ Q2)=R with R the A morphism (M; ·; 1M ) −!(N; ·; 1N ) of monoids is a map M ! N smallest equivalence relation stable under product (in Q1 ⊕P of sets commuting with · and 1: '(ab) = '(a)'(b) for all Q2) and making (u1(p); 1) ∼R (1; u2(p)) for all p 2 P . a; b 2 M and '(1 ) = 1 . Thus we have defined the cat- M N In general it is not easy to understand the relation involved in egory Mon of monoids. the description of Q1 ⊕P Q2, but in the case in which one of f1g is a monoid, initial and final object of the category Mon.
    [Show full text]
  • 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].
    [Show full text]
  • 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].
    [Show full text]
  • Event Structures and Trace Monoids
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 91 (1991) 285-313 285 Elsevier Event structures and trace monoids Brigitte Rozoy* L.I.U.C. Universitt. 14032 CAEN Cedex, France P.S. Thiagarajan** The Institute of Mathematical Sciences, Madras 600 113, India Communicated by M. Nivat Received May 1989 Abstract Rozoy, B. and P.S. Thiagarajan, Event structures and trace monoids, Theoretical Computer Science 91 (1991) 285-313. Event structures are a poset-based model for describing the behaviour of distributed systems. They give rise to a well-understood class of Scott domains. Event structures are also related to Petri nets in a fundamental way. Trace monoids are a string-based formalism for describing the behaviour of distributed systems. They have an independent theory rooted in the theory of formal languages. Here we obtain a representation of trace monoids in terms of a subclass of labelled event structures. Rozoy, B. and P.S. Thiagarajan, Event structures and trace monoids, Theoretical Computer Science 91 (1991) 2855313. Les structures d’evenements peuvent itre vus comme un modele de representation du comportement des systemes distribues, base sur les ensembles partiellement ordonnes. IIs sont associes de facon classique a une classe de domaines de Scott, et aux reseaux de Petri. Les mono’ides de traces constituent aussi un moddle de description du fonctionnement des systemes distribues, a partir dun formalisme de theorie des langages. Cet article etablit entre eux des theoremes de representation. 0. Introduction In recent years, languages accompanied by a commutability relation (over the alphabet of the language) have received a good deal of attention.
    [Show full text]
  • Bi-Atomic Classes of Positive Semirings
    BI-ATOMIC CLASSES OF POSITIVE SEMIRINGS NICHOLAS R. BAETH, SCOTT T. CHAPMAN, AND FELIX GOTTI Abstract. A subsemiring of R is called a positive semiring provided that it contains 1 and consists of nonnegative numbers. Here we study factorizations in both the additive monoid pS; q and the multiplicative monoid pSzt0u; ¤q. In particular, we investigate when, for a positive semiring S, both pS; q and pSzt0u; ¤q have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP ñ BFP and FFP ñ BFP ñ ACCP ñ atomicity holds. We construct classes of positive semirings S wherein the additive and multiplicative structures both satisfy each of these properties and give examples to show that, in general, none of the implications in the previous chain is reversible. 1. Introduction The atomic structure of positive monoids (i.e., additive submonoids of the nonnegative cone of the real line) has been the subject of much recent investigation. The simplest positive monoids are numerical monoids (i.e., additive monoids consisting of nonnegative integers, up to isomorphism), and their additive structure has been investigated by many authors during the last three decades (see [4, 23] and references therein). In addition, the atomicity of Puiseux monoids (i.e., additive monoids consisting of nonnegative rational numbers) has been the subject of various papers during the last four years (see [14, 15] and references therein). Puiseux monoids have also been studied in connection to commutative rings [20] and factorizations of matrices [6].
    [Show full text]
  • 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.
    [Show full text]
  • On Regularity and the Word Problem for Free Idempotent Generated Semigroups
    ON REGULARITY AND THE WORD PROBLEM FOR FREE IDEMPOTENT GENERATED SEMIGROUPS IGOR DOLINKA, ROBERT D. GRAY, AND NIK RUSKUCˇ Abstract. The category of all idempotent generated semigroups with a pre- scribed structure E of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over E, defined by a presentation over alphabet E, and denoted by IG(E). Recently, much effort has been put into investigating the structure of semigroups of the form IG(E), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(E). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗ represents a regular element; if in addition one assumes that all maxi- mal subgroups of IG(E) have decidable word problems, then the word problem in IG(E) restricted to regular words is decidable. On the other hand, we exhibit a biorder E arising from a finite idempotent semigroup S, such that the word problem for IG(E) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(E) to the subgroup membership problem in finitely presented groups. 1. Introduction In his foundational paper [28] Nambooripad made the fundamental observation that the set of idempotents E(S) of an arbitrary semigroup S carries the abstract structure of a so-called biordered set (or regular biordered set in the case of regular semigroups).
    [Show full text]
  • On Some Properties of the Free Monoids with Applications to Automata Theory*
    JOURNAL OF COMPUTER AND SYSTEM SCIENCES: 1, 13%154 (1967) On Some Properties of the Free Monoids with Applications to Automata Theory* YEHOSHAFAT GIVE'ON The Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138 INTRODUCTION Let T be any set, and denote by T* the free monoid generated by T. By definition, T* is characterized by the property that, for any function f from the set T into the carrier of any monoid M, there exists a unique monoid homornorphism f* : T* -+ M whose underlying function is an extension of the function f. This property of the free monoids is very useful in many applications where free monoids are encountered. Another property of the free monoids follows immediately from this definition. Namely, for any homomorphism h : T*-+ M 2 and any surjective homomorphism f : M 1 -~ M 2 there exists at least one homomorphism g : T* ~ M 1 such that Ylhm ~ M l ~ M a f commutes. This paper will be concerned with the following issues: (i) The above-mentioned derived property of the free monoids characterizes the free monoids in the category M of all monoids. This property should not be confused with either the defining property of the free monoids or with the property of the projective objects in 19I. On the other hand, our method of proof yields the analogous characterization of the free Abelian monoids in the category of Abelian monoids. (ii) Since every monoid is a homomorphic image of a free monoid, the above- derived property leads us to consider idempotent endomorphisms of free monoids.
    [Show full text]
  • Algebraic Characterization of Regular Languages
    Algebraic Characterization of Regular Languages Chloe Sheen Advised by Professor Steven Lindell A Thesis Presented in Partial Fulfillment of Bachelor's Degree Department of Computer Science Bryn Mawr College Abstract This paper presents the proof provided by Marcel-Paul Sch¨utzenberger, which connects regular language theory and algebraic automata theory. The two fundamental areas of study in mathematics and theoretical computer sci- ence hold a notable relationship, which motivated the discussion surrounding this topic. On a high-level perspective, Sch¨utzenberger's theorem bridges the two areas of study using languages recognized by aperiodic monoids to re- late them to star-free regular expressions. By outlining the essential ideas in both automata theory and group theory, we attempt to produce a more comprehensible document for both computer scientists and mathematicians to discuss this topic. Contents 1 Introduction 2 1.1 Background . .2 1.2 Motivations and Goals . .3 2 Notations, Definitions, and Examples 5 2.1 Finite Automata and Regular Languages . .5 2.2 Detailed Examples . .8 2.3 Algebra: Group Theory . 15 3 Sch¨utzenberger's theorem 20 4 Further Readings 27 5 Conclusion 28 1 Chapter 1 Introduction 1.1 Background Mathematics is the foundation of computing, and computing is often a key component in mathematical problem-solving. For instance, linear algebra is heavily used in computer graphics, graph theory forms the basis of network analysis, and number theory is used for cryptography. Such relationship between the two disciplines has inevitably resulted in the shared interest and involvement of both mathematicians and computer scientists in the study of formal languages.
    [Show full text]