Languages of R-Trivial Monoids*
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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. -
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. -