DOCSLIB.ORG

Duality and Automata Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Duality and Automata Theory Duality and Automata Theory Duality and Automata Theory Mai Gehrke Universit´eParis VII and CNRS Joint work with Serge Grigorieff and Jean-Eric´ Pin Duality and Automata Theory Elements of automata theory a A finite automaton 1 2 b b a 3 a, b The states are 1; 2; 3 . f g The initial state is 1, the final states are 1 and 2. The alphabet is A = a; b The transitions are f g 1 a =22 a =33 a =3 · · · 1 b =32 b =13 b =3 · · · Duality and Automata Theory Elements of automata theory a Recognition by automata 1 2 b b a 3 a, b Transitions extend to words: 1 aba =2 , 1 abb =3 . · · The language recognized by the automaton is the set of words u such that 1 u is a final state. Here: · L( ) = (ab)∗ (ab)∗a A [ where means arbitrary iteration of the product. ∗ Duality and Automata Theory Elements of automata theory Rational and recognizable languages A language is recognizable provided it is recognized by some finite automaton. A language is rational provided it belongs to the smallest class of languages containing the finite languages which is closed under union, product and star. Theorem: [Kleene '54] A language is rational iff it is recognizable. Example: L( ) = (ab) (ab) a. A ∗ [ ∗ Duality and Automata Theory Connection to logic on words Logic on words To each non-empty word u is associated a structure u = ( 1; 2;:::; u ; <; (a)a A) M f j jg 2 where a is interpreted as the set of integers i such that the i-th letter of u is an a, and < as the usual order on integers. Example: Let u = abbaab then = ( 1; 2; 3; 4; 5; 6 ; <; (a; b)) Mu f g where a = 1; 4; 5 and b = 2; 3; 6 . f g f g Duality and Automata Theory Connection to logic on words Some examples The formula φ = x ax interprets as: 9 There exists a position x in u such that the letter in position x is an a. This defines the language L(φ) = A∗aA∗. The formula x y (x < y) ax by defines the language 9 9 ^ ^ A∗aA∗bA∗. The formula x y [(x < y) (x = y)] ax defines the language 9 8 _ ^ aA∗. Duality and Automata Theory Connection to logic on words Defining the set of words of even length Macros: (x < y) (x = y) means x y _ 6 y x y means x = 1 8 6 y y x means x = u 8 6 j j x < y z (x < z y z) means y = x + 1 ^ 8 ! 6 Let φ = X (1 = X u X x (x X x + 1 = X)) 9 2 ^ j j 2 ^ 8 2 $ 2 Then 1 = X, 2 X, 3 = X, 4 X,..., u X. Thus 2 2 2 2 j j 2 2 L(φ) = u u is even = (A )∗ f j j j g Duality and Automata Theory Connection to logic on words Monadic second order Only second order quantifiers over unary predicates are allowed. Theorem: (B¨uchi'60, Elgot '61) Monadic second order captures exactly the recognizable languages. Theorem: (McNaughton-Papert '71) First order captures star-free languages (star-free = the ones that can be obtained from the alphabet using the Boolean operations on languages and lifted concatenation product only). How does one decide the complexity of a given language??? Duality and Automata Theory The algebraic theory of automata Algebraic theory of automata Theorem: [Myhill '53, Rabin-Scott '59] There is an effective way of associating with each finite automaton, , a finite monoid, A (M ; ; 1). A · Theorem: [Sch¨utzenberger '65] A recognizable language is star-free if and only if the associated monoid is aperiodic, i.e., M is such that there exists n > 0 with xn = xn+1 for each x M. 2 Submonoid generated by x: xi+1 xi+2 1 x x2 x3 xi+p = xi ... xi+p 1 This makes starfreeness decidable! − Duality and Automata Theory The algebraic theory of automata Eilenberg-Reiterman theory Varieties of finite monoids Eilenberg Reiterman Varieties In good Profinite Decidability of languages identities cases A variety of monoids here means a class of finite monoids closed under homomorphic images, submonoids, and finite products Various generalisations: [Pin 1995], [Pin-Weil 1996], [Pippenger 1997], [Pol´ak 2001], [Esik 2002], [Straubing 2002], [Kunc 2003] Duality and Automata Theory Duality and automata I Eilenberg, Reiterman, and Stone Classes of monoids (1) (2) algebrasoflanguages equationaltheories (3) (1) Eilenberg theorems (2) Reiterman theorems (3) extended Stone/Priestley duality (3) allows generalisation to non-varieties and even to non-regular languages Duality and Automata Theory Duality and automata I Assigning a Boolean algebra to each language For x A and L A , define the quotient 2 ∗ ⊆ ∗ 1 x− L= u A∗ xu L (= x L) f 2 j 2 g f gn and 1 Ly− = u A∗ uy L (= L= y ) f 2 j 2 g f g Given a language L A , let (L) be the Boolean algebra of ⊆ ∗ B languages generated by 1 1 x− Ly− x; y A∗ j 2 NB! (L) is closed under quotients since the quotient operations B commute with the Boolean operations. Duality and Automata Theory Duality and automata I a Quotients of a recognizable language 1 2 b b a 3 L( ) = (ab) (ab) a A ∗ [ ∗ 1 a− L= u A∗ au L = (ba)∗b (ba)∗ f 2 j 2 g [ a, b 1 La− = u A∗ ua L = (ab)∗ f 2 j 2 g 1 b− L= u A∗ bu L = f 2 j 2 g ; NB! These are recognized by the same underlying machine. Since (L) is finite it is also closed under residuation with respect B to arbitrary denominators and is thus a bi-module for (A ). P ∗ For any K (L) and any S (A ) 2 B 2 P ∗ 1 S K = u− K (L) n 2 B u S \2 1 K=S = Ku− (L) 2 B u S \2 Duality and Automata Theory Duality and automata I (L) for a recognizable language B If L is recognizable then the generating set of (L) is finite since B all the languages are recognized by the same machine with varying sets of initial and final states. Duality and Automata Theory Duality and automata I (L) for a recognizable language B If L is recognizable then the generating set of (L) is finite since B all the languages are recognized by the same machine with varying sets of initial and final states. Since (L) is finite it is also closed under residuation with respect B to arbitrary denominators and is thus a bi-module for (A ). P ∗ For any K (L) and any S (A ) 2 B 2 P ∗ 1 S K = u− K (L) n 2 B u S \2 1 K=S = Ku− (L) 2 B u S \2 Theorem: For a recognizable language L, the dual space of the algebra ( (L); ; ; ()c; 0; 1; ; =) is the syntactic monoid of L. B \ [ n { including the product operation! Duality and Automata Theory Duality and automata I The syntactic monoid via duality For a recognizable language L, the algebra c ( (L); ; ; () ; 0; 1; ; =) (A∗) B \ [ n ⊆ P is the Boolean residuation bi-module generated by L. Duality and Automata Theory Duality and automata I The syntactic monoid via duality For a recognizable language L, the algebra c ( (L); ; ; () ; 0; 1; ; =) (A∗) B \ [ n ⊆ P is the Boolean residuation bi-module generated by L. Theorem: For a recognizable language L, the dual space of the algebra ( (L); ; ; ()c; 0; 1; ; =) is the syntactic monoid of L. B \ [ n { including the product operation! D(J(D)) Downset lattice Duality and Automata Theory Duality { the finite case Finite lattices and join-irreducibles x = 0 is join-irreducible iff x = y z (x = y or x = z) 6 _ ) J(D) D Join-irreducibles Duality and Automata Theory Duality { the finite case Finite lattices and join-irreducibles x = 0 is join-irreducible iff x = y z (x = y or x = z) 6 _ ) J(D) D(J(D)) D Join-irreducibles Downset lattice X {a, b} {a,c} {b, c} {a} {b} {c} ∅ Duality and Automata Theory Duality { the finite case The finite Boolean case In the Boolean case, the join-irreducibles( J) are exactly the atoms (At) and the downset lattice( ) of the atoms is just the power D set( ) P X {a, b} {a,c} {b, c} {a} {b} {c} ∅ Duality and Automata Theory Duality { the finite case The finite Boolean case In the Boolean case, the join-irreducibles( J) are exactly the atoms (At) and the downset lattice( ) of the atoms is just the power D set( ) P a b c Duality and Automata Theory Duality { the finite case The finite Boolean case In the Boolean case, the join-irreducibles( J) are exactly the atoms (At) and the downset lattice( ) of the atoms is just the power D set( ) P X {a, b} {a,c} {b, c} {a} {b} {c} a b c ∅ Duality and Automata Theory Duality { the finite case .mHBiv BM *i2;Q`B+H .mHBiv GQ;B+ Categorical Duality G2+im`2 R + DL | complete.G+ ě U+QKTH2i2 and completely M/ +QKTH2i2HvV distributive /Bbi`B#miBp2 lattices HiiB+2b rBi? with enoughGQ;B+ M/ HiiB+2b completely2MQm;? join DQBM irreducibles, B``2/m+B#H2b- with rBi? U+QKTH2i2V complete ?QKQKQ`T?BbKb homomorphisms .mHBiv P OS |SPa partiallyě T`iBHHv ordered Q`/2`2/ sets b2ib with rBi? order Q`/2` preserving T`2b2`pBM; KTb maps BMi`Q/m+iBQM aiQM2 `2T`2@ D b2MiiBQM .GX+ SPa 1H2K2Mib Q7 C iQTQHQ;v aiQM2 M/ S`B2biH2v (J(D))(C=(.))DJ= .
Load more

Recommended publications
  • 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
    [Show full text]
  • 2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (Chair), Bobby Kleinberg September 14Th, 2020
    2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (chair), Bobby Kleinberg September 14th, 2020 SIGACT Mission Statement: The primary mission of ACM SIGACT (Association for Computing Machinery Special Interest Group on Algorithms and Computation Theory) is to foster and promote the discovery and dissemination of high quality research in the domain of theoretical computer science. The field of theoretical computer science is the rigorous study of all computational phenomena - natural, artificial or man-made. This includes the diverse areas of algorithms, data structures, complexity theory, distributed computation, parallel computation, VLSI, machine learning, computational biology, computational geometry, information theory, cryptography, quantum computation, computational number theory and algebra, program semantics and verification, automata theory, and the study of randomness. Work in this field is often distinguished by its emphasis on mathematical technique and rigor. 1. Awards ▪ 2020 Gödel Prize: This was awarded to Robin A. Moser and Gábor Tardos for their paper “A constructive proof of the general Lovász Local Lemma”, Journal of the ACM, Vol 57 (2), 2010. The Lovász Local Lemma (LLL) is a fundamental tool of the probabilistic method. It enables one to show the existence of certain objects even though they occur with exponentially small probability. The original proof was not algorithmic, and subsequent algorithmic versions had significant losses in parameters. This paper provides a simple, powerful algorithmic paradigm that converts almost all known applications of the LLL into randomized algorithms matching the bounds of the existence proof. The paper further gives a derandomized algorithm, a parallel algorithm, and an extension to the “lopsided” LLL.
    [Show full text]
  • COMPUTER MODELS and AUTOMATA THEORY in BIOLOGY and MEDICINE Ion C
    Mathematical Modelling, Vol. 7, pp. 1513-1577,1986 Printed in the U,S.A. All rights reserved. Copyright C 1986 Pergamon Journals Ltd. REVIEW ARTICLE COMPUTER MODELS AND AUTOMATA THEORY IN BIOLOGY AND MEDICINE Ion C. Baianu University of Ulinois at Urbana Physical Chemistry and NMR Laboratories 567 Bevier Hall, 905 S. Goodwin Ave .Urbana, UIinois 61801 (Received 27 February 1985; revised 12 September 1985) 1. INTRODUCTION The applications of computers to biological and biomedical problem solving goes back to the very beginnings of computer science, automata theory [1], and mathematical biology [2]. With the advent of more versatile and powerful computers, biological and biomedical applications of computers have proliferated so rapidly that it would be virtually impossible to compile a comprehensive review of all developments in this field. Limitations of computer simulations in biology have also come under close scrutiny, and claims have been made that biological systems have limited information processing power [3]. Such general conjectures do not, however, deter biologists and biomedical researchers from developing new computer applications in biology and medicine. Microprocessors are being widely employed in biological laboratories both for automatic data acquisition/processing and modeling; one particular area, which is of great biomedical interest, involves fast digital image processing and is already established for routine clinical examinations in radiological and nuclear medicine centers, Powerful techniques for biological research are routinely employing dedicated, on-line microprocessors or array processors; among such techniques are: Fourier-transform nuclear magnetic resonance (NMR), NMR imaging (or tomography), x-ray tomography, x-ray diffraction, high performance liquid chromatography, differential scanning calorimetry and mass spectrometry.
    [Show full text]
  • A Mathematical Theory of Computation?
    A Mathematical Theory of Computation? Simone Martini Dipartimento di Informatica { Scienza e Ingegneria Alma mater studiorum • Universit`adi Bologna and INRIA FoCUS { Sophia / Bologna Lille, February 1, 2017 1 / 57 Reflect and trace the interaction of mathematical logic and programming (languages), identifying some of the driving forces of this process. Previous episodes: Types HaPOC 2015, Pisa: from 1955 to 1970 (circa) Cie 2016, Paris: from 1965 to 1975 (circa) 2 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 3 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 4 / 57 We today conflate: Types as an implementation (representation) issue Types as an abstraction mechanism Types as a classification mechanism (from mathematical logic) 5 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 6 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 7 / 57 Prehistory 1947 8 / 57 Goldstine and von Neumann [. ] coding [. ] has to be viewed as a logical problem and one that represents a new branch of formal logics. Hermann Goldstine and John von Neumann Planning and Coding of problems for an Electronic Computing Instrument Report on the mathematical and logical aspects of an electronic computing instrument, Part II, Volume 1-3, April 1947.
    [Show full text]
  • Introduction to Theory of Computation
    Introduction to Theory of Computation Anil Maheshwari Michiel Smid School of Computer Science Carleton University Ottawa Canada anil,michiel @scs.carleton.ca { } April 17, 2019 ii Contents Contents Preface vi 1 Introduction 1 1.1 Purposeandmotivation ..................... 1 1.1.1 Complexitytheory .................... 2 1.1.2 Computability theory . 2 1.1.3 Automatatheory ..................... 3 1.1.4 Thiscourse ........................ 3 1.2 Mathematical preliminaries . 4 1.3 Prooftechniques ......................... 7 1.3.1 Directproofs ....................... 8 1.3.2 Constructiveproofs. 9 1.3.3 Nonconstructiveproofs . 10 1.3.4 Proofsbycontradiction. 11 1.3.5 The pigeon hole principle . 12 1.3.6 Proofsbyinduction. 13 1.3.7 Moreexamplesofproofs . 15 Exercises................................. 18 2 Finite Automata and Regular Languages 21 2.1 An example: Controling a toll gate . 21 2.2 Deterministic finite automata . 23 2.2.1 Afirstexampleofafiniteautomaton . 26 2.2.2 Asecondexampleofafiniteautomaton . 28 2.2.3 A third example of a finite automaton . 29 2.3 Regularoperations ........................ 31 2.4 Nondeterministic finite automata . 35 2.4.1 Afirstexample ...................... 35 iv Contents 2.4.2 Asecondexample..................... 37 2.4.3 Athirdexample...................... 38 2.4.4 Definition of nondeterministic finite automaton . 39 2.5 EquivalenceofDFAsandNFAs . 41 2.5.1 Anexample ........................ 44 2.6 Closureundertheregularoperations . 48 2.7 Regularexpressions. .. .. 52 2.8 Equivalence of regular expressions and regular languages . 56 2.8.1 Every regular expression describes a regular language . 57 2.8.2 Converting a DFA to a regular expression . 60 2.9 The pumping lemma and nonregular languages . 67 2.9.1 Applications of the pumping lemma . 69 2.10Higman’sTheorem ........................ 76 2.10.1 Dickson’sTheorem .
    [Show full text]
  • A Combinatorial Approach to the Theory of O~-Automata
    INFORMATION AND CONTROL 48, 261-283 (1981) A Combinatorial Approach to the Theory of o~-Automata WOLFGANG THOMAS Mathematisches Institut der Universitdt, 7800 Freiburg, West Germany A combinatorial lemma is proved and used here to derive new results on ~o- automata and to give simpler proofs of known ones. In particular, we reprove McNaughton's fundamental theorem (characterizing the o~-regular sequence-sets), without having to construct a sophisticated w-automaton. The theorem is obtained by coding the behaviour of automata in a second-order language and a simple application of the lemma. In close analogy (now referring to a first-order language) a theory of counter-free co-automata is developed; it is shown that these automata are appropriate for characterizing the co-star-free sequence-sets. Finally, the lemma is applied in mathematical logic: Here new normal form theorems and also decidability results are proved for the first-order and the monadic second-order theory of certain structures over the ordering of natural numbers. 0. INTRODUCTION A central result in the theory of finite automata accepting co-sequences (short: co-automata) is the theorem of McNaughton (1966). It characterizes the sequence-sets accepted by co-automata as the co-regular sequence-sets. McNaughton's proof involves an intricate construction of an co-automaton; it shows "how a bounded memory (finite automaton) can be made to keep track of essential information, accruing in an ever growing past" (B/ichi (1977)). Modified and improved presentations of this construction have since been given in Rabin (1972), B/ichi (1973), Choueka (1974) and Trakhtenbrot and Barzdin (1973).
    [Show full text]
  • Combinatorics, Automata and Number Theory
    Combinatorics, Automata and Number Theory CANT Edited by Val´erie Berth´e LIRMM - Univ. Montpelier II - CNRS UMR 5506 161 rue Ada, 34392 Montpellier Cedex 5, France Michel Rigo Universit´edeLi`ege, Institut de Math´ematiques Grande Traverse 12 (B 37), B-4000 Li`ege, Belgium 2 Number representation and finite automata Christiane Frougny Univ. Paris 8 and LIAFA, Univ. Paris 7 - CNRS UMR 7089 Case 7014, F-75205 Paris Cedex 13, France Jacques Sakarovitch LTCI, CNRS/ENST - UMR 5141 46, rue Barrault, F-75634 Paris Cedex 13, France. Contents 2.1 Introduction 5 2.2 Representation in integer base 8 2.2.1 Representation of integers 8 2.2.2 The evaluator and the converters 11 2.2.3 Representation of reals 18 2.2.4 Base changing 23 2.3 Representation in real base 23 2.3.1 Symbolic dynamical systems 24 2.3.2 Real base 27 2.3.3 U-systems 35 2.3.4 Base changing 43 2.4 Canonical numeration systems 48 2.4.1 Canonical numeration systems in algebraic number fields 48 2.4.2 Normalisation in canonical numeration systems 50 2.4.3 Bases for canonical numeration systems 52 2.4.4 Shift radix systems 53 2.5 Representation in rational base 55 2.5.1 Representation of integers 55 2.5.2 Representation of the reals 61 2.6 A primer on finite automata and transducers 66 2.6.1 Automata 66 2.6.2 Transducers 68 2.6.3 Synchronous transducers and relations 69 2.6.4 The left-right duality 71 3 4 Ch.
    [Show full text]
  • A Theory of Regular Queries
    A Theory of Regular Queries Moshe Y. Vardi Rice University [email protected] ABSTRACT [22], he proposed using first-order logic as a declarative database A major theme in relational database theory is navigat- query language [23]. This led to the development of both ing the tradeoff between expressiveness and tractability for SEQUEL [16] and QUEL [54], as practical relational query query languages, where the query-containment problem is languages realizing Codd's idea, ultimately giving in 1986 considered a benchmark of tractability. The query class rise to the SQL standard [27], which has continued to evolve UCQ, consisting off unions of conjunctive queries, is a frag- over the past 30 years. ment of first-order logic that has a decidable query contain- Starting in the late 1970s, Codd's definition of first-order ment problem, but its expressiveness is limited. Extend- logic as \expressively complete" came under serious criti- ing UCQ with recursion yields Datalog, an expressive query cism [4], and various proposals emerged on how to extend language that has been studied extensively and has recently its expressive power [4, 6, 17], whose essence was to extend become popular in application areas such as declarative net- first-order logic with recursion, which was added to SQL in working. Unfortunately, Datalog has an undecidable query 1999 by way of common table expressions [29]. On the re- containment problem. Identifying a fragment of Datalog search side, the most popular relational language embodying that is expressive enough for applications but has a decid- recursion is Datalog [15], describe in more details below.
    [Show full text]
  • Arxiv:2001.09864V2 [Cs.FL]
    Provenance for Regular Path Queries G¨osta Grahne1 and Alex Thomo2 1 Concordia University, Montreal, Canada, [email protected] 2 University of Victoria, Victoria, Canada, [email protected] 1 Introduction It has been recognized that the result of a database query should be annotated with provenance, i.e. information about how, why, where, with what level of cer- tainty or security clearance, etc a particular fact of the query was derived. The seminal paper by Green, Tannen and Karvounarakis [4] convincingly showed that all major forms of provenance can be uniformly captured within the algebraic framework of semirings. Green et al. show that a suitably semiring-annotated positive (negation-free) relational algebra and datalog can capture the prove- nance of query results. Furthermore, the various data base semirings form a par- tial order where coarser (”smaller”) semirings can be obtained as homomorphic images of semirings with a finer grain of information. Green et al. also show that the annotated positive relational algebra and datalog form congruences within their semiring hierarchy. Regular path queries (RPQs) is the ubiquitous mechanism for querying graph databases [1]. RPQs are in essence regular expressions over the edge symbols. The answer to an RPQ on a given graph database is the set of pairs of objects (a,b), which are connected by paths spelling words in the language of the regular path query An annotated pair in the answer would naturally contain the set of words that spell paths between a and b. However, a finer grain of provenance can be obtained by annotating the words with the intermediate vertices of each path spelling the word.
    [Show full text]
  • Some Contributions to the Algebraic Theory of Automata
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura Some Contributions to the Algebraic Theory of Automata Enric Cosme i Ll´opez Supervised by Adolfo Ballester-Bolinches and Jean-Eric´ Pin Facultat de Ci`enciesMatem`atiques Universitat de Val`encia This dissertation is submitted for the degree of International Doctor in Mathematics September 2015 als meus pares i a la meua germana a Toni de l'Hostal i Agra¨ıments Els agra¨ıments s´oncomplicats quan tens tant a agra¨ır. M´es encara si ´esla part de la tesi que m´esexpectatives sol endur-se i que amb m´esentusiasme es llig la gent. Representem la resta de la tesi com una meta elevada i no ho ´estant; no ´escap obra suprema, ni principi ni final de res. Es´ treball fet, voluntats escrites i riuades de sentiments contraposats. Nom´escal pensar-la com un viatge honest cap endavant, cap al que anomenem ci`enciai que no deixa de ser una activitat humana, que beu de les nostres virtuts i dels nostres vicis. Per aix`os´on importants els agra¨ıments, perqu`e´esl'´uniclloc on no es parla del que has fet, sin´ode la gent que has conegut pel cam´ıi que t'ha ajudat a arribar fins al final. La tesi, al cap i a la fi, sempre ´esall`oque acompanya els agra¨ıments. Qui ho negue menteix. Agra¨ısc,aix´ı,tot el que he apr´esdel meu tutor, Adolfo Ballester.
    [Show full text]
  • Applying Abstract Algebraic Logic to Classical Automata Theory: an Exercise
    Applying Abstract Algebraic Logic to Classical Automata Theory: an exercise Lu´ısDescal¸co1, Alexandre Madeira2;1;3 and Manuel A. Martins1 1 Department of Mathematics, University of Aveiro 2 CCTC, University of Minho 3 Critical Software S.A Abstract. In [4], Blok and Pigozzi have shown that a deterministic finite au- tomaton can be naturally viewed as a logical matrix. Following this idea, we use a generalisation of the matrix concept to deal with other kind of automata in the same algebraic perspective. We survey some classical concepts of automata theory using tools from algebraic logic. The novelty of this approach is the understand- ing of the classical automata theory within the standard abstract algebraic logic theory. 1 Introduction There are several works in the literature about automata theory (AT) supported by results and arguments of different areas, such as, topology, algebra, quantum mechanics, etc. Following the recent works applying some tools of abstract algebraic logic (AAL) to computer sciences, namely in the formal specification and verification theory of software systems (cf. [10{12]), we survey some classical concepts of AT within an AAL perspective. AAL is an area of algebraic logic that focuses on the study of the relationship between logical equivalence and logical truth. More precisely, AAL is centered on the process of associating a class of algebras to a logical system (see [6]). A logical system, a deductive system as it has been called in the AAL field, is a pair formed by a signature Σ and a substitution-invariant consequence relation on the set of terms over Σ.
    [Show full text]
  • Introduction to the Theory of Complexity
    Introduction to the theory of complexity Daniel Pierre Bovet Pierluigi Crescenzi The information in this book is distributed on an “As is” basis, without warranty. Although every precaution has been taken in the preparation of this work, the authors shall not have any liability to any person or entity with respect to any loss or damage caused or alleged to be caused directly or indirectly by the information contained in this work. First electronic edition: June 2006 Contents 1 Mathematical preliminaries 1 1.1 Sets, relations and functions 1 1.2 Set cardinality 5 1.3 Three proof techniques 5 1.4 Graphs 8 1.5 Alphabets, words and languages 10 2 Elements of computability theory 12 2.1 Turing machines 13 2.2 Machines and languages 26 2.3 Reducibility between languages 28 3 Complexity classes 33 3.1 Dynamic complexity measures 34 3.2 Classes of languages 36 3.3 Decision problems and languages 38 3.4 Time-complexity classes 41 3.5 The pseudo-Pascal language 47 4 The class P 51 4.1 The class P 52 4.2 The robustness of the class P 57 4.3 Polynomial-time reducibility 60 4.4 Uniform diagonalization 62 5 The class NP 69 5.1 The class NP 70 5.2 NP-complete languages 72 v vi 5.3 NP-intermediate languages 88 5.4 Computing and verifying a function 92 5.5 Relativization of the P 6= NP conjecture 95 6 The complexity of optimization problems 110 6.1 Optimization problems 111 6.2 Underlying languages 115 6.3 Optimum measure versus optimum solution 117 6.4 Approximability 119 6.5 Reducibility and optimization problems 125 7 Beyond NP 133 7.1 The class coNP
    [Show full text]