Some Results on the Generalized Star-Height Problem
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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. -
Language and Automata Theory and Applications
LANGUAGE AND AUTOMATA THEORY AND APPLICATIONS Carlos Martín-Vide Characterization • It deals with the description of properties of sequences of symbols • Such an abstract characterization explains the interdisciplinary flavour of the field • The theory grew with the need of formalizing and describing the processes linked with the use of computers and communication devices, but its origins are within mathematical logic and linguistics A bit of history • Early roots in the work of logicians at the beginning of the XXth century: Emil Post, Alonzo Church, Alan Turing Developments motivated by the search for the foundations of the notion of proof in mathematics (Hilbert) • After the II World War: Claude Shannon, Stephen Kleene, John von Neumann Development of computers and telecommunications Interest in exploring the functions of the human brain • Late 50s XXth century: Noam Chomsky Formal methods to describe natural languages • Last decades Molecular biology considers the sequences of molecules formed by genomes as sequences of symbols on the alphabet of basic elements Interest in describing properties like repetitions of occurrences or similarity between sequences Chomsky hierarchy of languages • Finite-state or regular • Context-free • Context-sensitive • Recursively enumerable REG ⊂ CF ⊂ CS ⊂ RE Finite automata: origins • Warren McCulloch & Walter Pitts. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5:115-133, 1943 • Stephen C. Kleene. Representation of events in nerve nets and -
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. -
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. -
Some Results on the Generalized Star-Height Problem Jean-Eric Pin, Howard Straubing, Denis Thérien
Some results on the generalized star-height problem Jean-Eric Pin, Howard Straubing, Denis Thérien To cite this version: Jean-Eric Pin, Howard Straubing, Denis Thérien. Some results on the generalized star-height problem. Information and Computation, Elsevier, 1992, 101, pp.219-250. hal-00019978 HAL Id: hal-00019978 https://hal.archives-ouvertes.fr/hal-00019978 Submitted on 2 Mar 2006 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. May 12, 2002 Some results on the generalized star-height problem Jean-Eric Pin∗, H. Straubingyand D. Th´erienz [email protected], [email protected], [email protected] Abstract We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, com- plementation 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 commu- tative group by (Z=2Z)n. -
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]. -
Theme and Variations on the Concatenation Product
Theme and variations on the concatenation product Jean-Eric´ Pin1⋆ LIAFA, University Paris-Diderot and CNRS, France. Abstract. The concatenation product is one of the most important op- erations on regular languages. Its study requires sophisticated tools from algebra, finite model theory and profinite topology. This paper surveys research advances on this topic over the last fifty years. The concatenation product plays a key role in two of the most important results of automata theory: Kleene’s theorem on regular languages [23] and Sch¨utzen- berger’s theorem on star-free languages [60]. This survey article surveys the most important results and tools related to the concatenation product, including connections with algebra, profinite topology and finite model theory. The paper is organised as follows: Section 1 presents some useful algebraic tools for the study of the concatenation product. Section 2 introduces the main definitions on the product and its variants. The classical results are summarized in Section 3. Sections 4 and 5 are devoted to the study of two algebraic tools: Sch¨utzenberger products and relational morphisms. Closure properties form the topic of Section 6. Hierarchies and their connection with finite model theory are presented in Sections 7 and 8. Finally, new directions are suggested in Section 9. 1 The instruments This section is a brief reminder on the algebraic notions needed to study the con- catenation product: semigroups and semirings, syntactic ordered monoids, free profinite monoids, equations and identities, varieties and relational morphisms. More information can be found in [1–3,18,35,42,45]. 1.1 Semigroups and semirings If S is a semigroup, the set P(S) of subsets of S is also a semiring, with union as addition and multiplication defined, for every X, Y ∈P(S), by XY = {xy | x ∈ X,y ∈ Y } In this semiring, the empty set is the zero and for this reason, is denoted by 0. -
Languages of Generalised Star Height 1
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 106 (1992) 327-335 327 Elsevier Note More languages of generalised star height 1 J.M. Robson Department qf‘ Computer Science, GPO Box 4, Canberra, ACT 2601, Australia Communicated by D. Perrin Received September 1991 Revised January 1992 Abstract Robson, J.M., More languages of generalized star height 1, Theoretical Computer Science 106 (1992) 327-335. Suppose that membership of a word w in a language L is specified in terms of the number (modulo some integer k) of occurrences of a fixed word W’ as a factor of w. Then the language obtained for any M by adding arbitrary numbers of copies of a”’ anywhere in the words of L has generalized star height at most 1. 1. Introduction The “star height” of a regular language is defined as the minimum nesting depth of stars in a regular expression denoting the language. This definition can be interpreted in the context of regular expressions using only concatenation, union and star operators (the “restricted” star height), or of expressions using also intersection and complement operators (“generalised” star height). In this paper we are interested only in generalised star height and we will consistently use “star height” or even “height” to mean generalised star height. Schutzenberger [4] gave a simple characterisation of languages of star height 0 (or “star-free”): a language has star height 0 if and only if its syntactic monoid is aperiodic. Correspondence to: J.M. -
Star Height Via Games
Star height via games Mikołaj Bojanczyk´ (University of Warsaw) Abstract—This paper proposes a new algorithm deciding the I would like to thank Nathanael¨ Fijalkow, Denis Kuperberg, star height problem. As shown by Kirsten, the star height prob- Christof Loding¨ and the anonymous referees for their helpful lem reduces to a problem concerning automata with counters, comments. This work has been supported by the Polish NCN called limitedness. The new contribution is a different algorithm for the limitedness problem, which reduces it to solving a Gale- grant 2014-13/B/ST6/03595. Stewart game with an !-regular winning condition. I. REDUCTION TO LIMITEDNESS OF COST AUTOMATA The star height of a regular expression is its nesting depth This section introduces cost automata, and describes a of the Kleene star. The star height problem is to compute reduction of the star height problem to a problem for cost the star height of a regular language of finite words, i.e. the automata, called limitedness. The exact reduction used in this smallest star height of a regular expression that defines the paper was shown in [Kir05] and is included to make the paper language. For instance, the expression (a∗b∗)∗ has star height self-contained. The idea to use counters and limitedness in two, but its language has star height one, because it is defined the study of star height dates back to the original proof of by the expression (a + b)∗ of star height one, and it cannot Hashiguchi. be defined by a star-free expression. Here we consider regular Cost automata and the limitedness problem: The lim- expressions without complementation, in which case star-free itedness problem concerns an automaton model with coun- regular expressions define only finite languages. -
Varieties of Tree Languages Definable by Syntactic Monoids
Acta Cybernetica 17 (2005) 21–41. Varieties of Tree Languages Definable by Syntactic Monoids Saeed Salehi ∗ Abstract An algebraic characterization of the families of tree languages definable by syntactic monoids is presented. This settles a question raised by several authors. 1 Introduction A Variety Theorem establishing a bijective correspondence between general vari- eties of tree languages definable by syntactic monoids and varieties of finite monoids, is proved. This has been a relatively long-standing open problem, the most recent references to which are made by Esik´ [4] as “No variety theorem is known in the semigroup [monoid] approach” (page 759), and by Steinby [18] as “there are no general criteria for deciding whether or not a given GVTL [general variety of tree languages] can or cannot be defined by syntactic monoids” (page 41). The question was also mentioned in the last section of Wilke’s paper [21]. Most of the interesting classes of algebraic structures form varieties, and sim- ilarly, most of the interesting families of tree or string languages studied in the literature turn out to be varieties of some kind. The first Variety Theorem was proved by Eilenberg [3] who established a correspondence between varieties of finite monoids and varieties of regular (string) languages. It was motivated by charac- terizations of several families of languages by syntactic monoids or semigroups (see [3],[10]), above all by Sch¨utzenberger’s [15] theorem connecting star-free languages and aperiodic monoids. Eilenberg’s theorem has since been extended in various directions. One could mention Pin’s [11] Variety Theorem for positive varieties of string languages and varieties of ordered monoids, or Th´erien’s [19] extension that includes also varieties of congruences on free monoids.