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Elements of automata theory:

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Bibliographic Details
Main Author:	Sakarovitch, Jacques 1947- (Author)
Format:	Book
Language:	German English French
Published:	Cambridge Cambridge Univ. Press 2009
Edition:	1. publ.
Subjects:	Automata math Automata theory Formal language Machine theory Automatentheorie
Online Access:	Inhaltsverzeichnis
Item Description:	Aus d. Franz. übers. - Orig. ersch. 2003
Physical Description:	XXIV, 758 S. graph. Darst.
ISBN:	9780521844253

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MARC


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adam_text	TABLE OF CONTENTS Preface to the English edition xv Preface xvii M. Pascal s division machine 1 0 Fundamental structures 7 1 Relations ...................................... 9 2 Monoids ....................................... 14 3 Words and languages ................................ 18 4 Free monoids .................................... 24 5 Semirings ...................................... 27 6 Matrices ....................................... 30 7 Lexicon of graph theory .............................. 33 8 Complexity and decidability ............................ 34 Solutions to the exercises ............................... 39 Notes & references ................................... 46 The three stages of rationality 1 The simplest possible machine. .. 49 1 What is an automaton ? ............................. 51 1.1 First definitions - first examples ..................... 51 States, transitions, etc. - Computations, recognised languages etc. - Transposition and left-right duality 1.2 Basic constructions, basic properties ................... 60 Union - Cartesian product - Quotient (of a language) 1.3 The graph perspective ........................... 66 Trim automata ~ The empty and the inñnite - Criteria for recognisabil- ty 1.4 Some supplementary definitions ...................... 74 Unambiguous automata - Complete automata - Deterministic au¬ tomata - Automata with spontaneous transitions 2 Rational languages ................................. 82 2.1 Rational operations ............................ 82 Product of languages - Star of a language - national operations 2.2 Rational languages ............................. 86 vii TABLE OF CONTENTS 2.3 Rational is recognisable .......................... 87 Normalised automata - Closure under product and star - Standard automata 2.4 Recognisable is rational .......................... 94 The McNaughton-Yamada algorithm, or algorithm MNY- The state elimination method — Solving equations The functional perspective ............................. 101 3.1 Prom transitions to the transition function ............... 102 3.2 Deterministic automata .......................... 104 Reformulation of the definition - Determinisation - The case of one- ietter alphabets - Complement of recognisable languages 3.3 Minimisation ................................ Ill The automaton of quotients of a language. ..- .. -is minimal - Compu¬ tation of the minimal automaton - Another minimisation method 3.4 Return to the Star Lemma ........................ 118 Block iteration and block simpliñcation - Ramsey s Theorem Proof of Theorem 3.3 Rational expressions ................................ 123 4.1 Rational expressions and languages .................... 124 Rational expressions over an alphabet — Rational expressions over a set of variables 4.2 Rational identities ............................. 128 Classical identities - A formal computation 4.3 Expressions for the behaviour of a finite automaton .......... 133 The state elimination and equation solution methods - The В MC and MN Y algorithms, identica] orders - The В MC and MN Y aigorithms, distinct orders 4.4 Derivation of expressions ......................... 138 Derivatives of an expression - A theorem of J. Brzozowski — Derivative automaton Prom expressions to automata ........................... 145 5.1 The standard automaton of an expression ................ 145 Direct construction ~ Thompson s construction 5.2 The derived term automaton ....................... 149 Derived terms - A theorem of V. Antimirov 5.3 String matching .............................. 152 Automaton for fínding a word - Searching by sliding window - Imple¬ mentation with a default successor Star height ..................................... 157 6.1 Two heights and a degree ......................... 158 Star height of an expression - Star height of a language - Loop com¬ plexity of an automaton 6.2 Eggan s Theorem .............................. Ig2 From expressions to automata - From automata to expressions: calcu¬ lating the index- Not so fast 6.3 An infinite hierarchy ............................ 167 6.4 Generalised star height ........................... 170 A field of automata ................................. 171 TABLE OF CONTENTS ix 7.1 The Rabin-Scott model .......................... 171 7.2 Two-way automaton ............................ 172 7.3 Moore and Mealy machines ........................ 174 8 A crop of properties ................................ 175 Solutions to the exercises ............................... 179 Notes & references ................................... 214 II The power of algebra 217 1 Automata and rational sets ............................ 219 1.1 Automata over a monoid ......................... 219 1.2 Rational sets ................................ 220 The semiring 9ß(M) - Rational operations and subsets - Rational ex¬ pressions - Image under morphism - Intersection and inverse morphism 1.3 Behaviour of finite automata ....................... 225 1.4 Unambiguous rational sets ......................... 228 Definitions - The family URat 2 Actions and recognisable sets ........................... 231 2.1 Actions on a set .............................. 232 Definition - Matrix representation of actions — Subsets recognised by an action 2.2 Recognisable here, recognisable there ................... 238 Consistency - Kleene s Theorem - Automaton of an action 2.3 Elementary operations on recognisable subsets ............. 243 Boolean operations - Inverse morphism - Quotient - Morphism and product 2.4 Minimisation ................................ 246 Action morphisms - Minimal action - Syntactic congruence and monoid 2.5 Algebra at work .............................. 251 Two examples - Recognisable subsets included in a product 3 Morphisms and coverings ............................. 255 3.1 Automata morphisms ........................... 255 Definitions and examples - Conformai morphisms - iocal properties 3.2 Quotients of automata ........................... 261 Out-surjective morphisms - Totally surjective morphisms - Moore s al¬ gorithm 3.3 Automata coverings ............................ 264 From locai to globa] - Product of an automaton with an action - The Coloured Transition Lemma 3.4 The Schiitzenberger covering ....................... 270 4 Universal automaton ................................ 273 TABLE OF CONTENTS 4.1 Factorisations ................................ 275 2-factorisations - Sub-factorisations and factorisations - Morphisms and factorisations 4.2 Universal automata of a subset ...................... 279 Definitions and examples - Properties - Universal automaton relative to a generating set - Universality of universal automata 4.3 Construction of the universal automaton ................. 286 Expansion of a deterministic automaton - Extraction of the universal automaton 4.4 Language approximations ......................... 291 5 The importance of being well ordered ...................... 293 5.1 Well quasi-orderings ............................ 293 5.2 Derivations ................................. 297 Preparations - Proof of Theorem 5.4 6 Rationals in the free group ............................ 301 6.1 Recognisable and rational in groups ................... 301 Recognisable subsets Rational subgroiips ■ Fatou property 6.2 Description of the free group ....................... 305 Dyck congruence and Dyck words Shamir congruence and parenthetic words - Shnplifícations Reduction associated with a simplifìcation - Unambiguous factorisation induced by a reduction 6.3 Rationals of the free group ........................ 314 Rationals of simplifìcatìon monoids Return to the free group 6.4 Buchi systems ............................... 319 7 Rationals in conimutative monoids ........................ 323 7.1 The natural order on A® ......................... 323 The free commutative monoid - Dickson s Lemma 7.2 The lexicographic order on Nfc ...................... 326 Congruences of ΝΛ - Lexicographic decomposition 7.3 Subtractive submonoids and affine sets .................. 330 7.4 Semi-linear and semi-simple sets ..................... 333 7.5 Rationals of N* ............................... 335 The Freedom Lemma - Positive solutions of Diophantine linear systems - Semi-simple subsets of Z* - Proof of Theorems 7.3 and 7.4 7.6 Rationals of commutative monoids .................... 341 8 Star height of group languages .......................... 342 Solutions to the exercises ............................... 34g N otes & references ................................... 372 III The pertinente of enumeration 375 1 Formal power series on a graded monoid ..................... 379 1.1 Formal power series over M with coefficients in К ........... 379 Operations on K{{AI}) Support of a series - characteristic series - Hadamard product - Scalar product 1.2 Graded monoids .............................. 333 TABLE OF CONTENTS xi 1.3 Topology on Kpi» ............................ 385 Distance Distance on K((M)} ~ Summable families - Continuous шог- phisms 2 K-automata and K-rational power series ..................... 392 2.1 Star of a power series ........................... 393 Star in a topological semiring - Star of a proper series - Star of an arbitrary series 2.2 K-rational series .............................. 398 K-rational operations - Rational ^-expressions - Star of a matrix 2.3 Weighted automaton in a semiring .................... 402 K-automaton over M - Behaviour of a K-automaton Notes Some other defìnitions and examples 2.4 The Fundamental Theorem of finite automata .............. 409 Proper automata - proper families of series Statement and proof - Notes and corollaries 2.5 K-coverings - K-quotients ......................... 416 From coverings to ÌL-coverings - Matrix description - Co-WL-covering - Minimal IK-quotient 3 K-recognisable series ................................ 424 3.1 K-representations .............................. 424 3.2 Products .................................. 426 Tensor product of K-representations Hadamard product - Tensor product of series - Shuffle product 3.3 The Kleene-Schützenberger Theorem .................. 433 4 Series on a free monoid .............................. 438 4.1 A characterisation of recognisable series ................. 438 Quotients of series - Stable modules - The Kleene-Schtitzenberger The¬ orem revisited 4.2 Derivation of rational K-expressions ................... 443 Polynomials of K-expressions - K-derivatives of a DC-expression - De¬ rived terms - The derived term automaton 4.3 Series on a field ............................... 451 Raníc of a series - Reduced representation Linear recurrence - Effec¬ tive computations 4.4 Rational series and their supports .................... 463 Rationality of supports - The Rational Skimming Theorem, I - Unde- cidable questions 5 Series on an arbitrary monoid ........................... 470 5.1 Complete semirings, continuous semirings ................ 470 5.2 Star of a series ............................... 472 5.3 K-rational series .............................. 474 6 Rational subsets in free products ......................... 476 6.1 Free product of monoids .......................... 476 6.2 Bipartite automaton over a free product ................. 478 6.3 Bipartite deterministic automaton .................... 482 6.4 Minimal deterministic bipartite automaton ............... 484 7 A non-commutative linear algebra primer .................... 488 Solutions to the exercises ............................... 498 xii TABLE OF CONTENTS Notes & references................................... 519 Rationality in relations IV The richness of transducers 523 1 Rational relations: an introduction ........................ 525 1.1 Rational relations ............................. 525 Rational relations between free monoids - Raţionai relations between arbitrary monoids 1.2 Realisation by automata .......................... 529 1.3 Realisation by morphisms ......................... 531 Realisation - .Evaluation Theorem - Composition Theorem - Star Lemma 1.4 Recognisable relations ........................... 539 1.5 Realisation by representation ....................... 540 Real-time transducers - From real-time transducers to representations - Theorem of evaluation and composition of representations 1.6 The Rabin -Scott model .......................... 545 2 K-relations ..................................... 546 2.1 Definitions ................................. 548 The canonical isomorphism - ^.-relations - Support of relations - char¬ acteristic relations Continuity 2.2 Composition ................................ 553 2.3 Multiplicative K-relations ......................... 555 3 Rational K-relations ................................ 557 3.1 Reasonable semirings ............................ 558 Image of series under continuous morphisms - Image of seres under projections - ^-intersections 3.2 Realisation of rational K-relations .................... 561 Realisation by Wi-automaton - Realisation by K-representation - Real¬ isation by morphisms 3.3 Evaluation and Composition Theorems ................. 564 Using recognition by morphisms - Using recognition by representation 4 Equivalence of finite K-transducers ........................ 568 4.1 Equivalence of B-transducers, general case ................ 569 4.2 Equivalence of B-transducers. case of small alphabets ......... 571 4.3 Equivalence of N-transducers ....................... 574 5 Deterministic rational relations .......................... 577 5.1 Transducers with an endmarker ...................... 577 5.2 Deterministic transducers ......................... 578 . Defínition Uniqueness of computations - Almost an action 5.3 Deterministic relations ........................... 584 Deñnitions - Complement ~ Iteration Lemma 5.4 Geography of Rat Л хй І ........................ 588 TABLE OF CONTENTS xiii 5.5 Matrix representations........................... 590 Representation of a deterministic transducer - Representation of a de¬ terministic relation 5.6 An example: the map equivalence of a morphism ............ 592 6 Synchronisation of transducers .......................... 595 6.1 Rational relations of bounded length discrepancy ............ 596 Definitions, notation and conventions - Characterisation of rational bid- relations - Translation into automata theoretic terms, and corollaries 6.2 Transducers of bounded lag ........................ 602 Lag in a computation or transducer - Resynchronisation algorithm for transducers - Composition of ¡etter-to-letter transducers 6.3 Synchronous relations ........................... 609 Another family of rational relations - Determinisation and minimisation - Geography of Rat A xB* II 7 Malcev-Neumann series .............................. 616 7.1 Order on the free group .......................... 617 On ordered groups - Representation of the free group - A detour via ordered rings - Order on the free group 7.2 Series on an ordered group ........................ 622 The semiring Kmo«G)> - Ordered semigroups - The fieid K»vo({G)) - A iast inclusion Solutions to the exercises ............................... 627 Notes & references ................................... 641 V The simplicity op functional transducers 643 1 Functionary ..................................... 645 1.1 Deciding functionality ........................... 645 An effective characterisation of functionality - Equivalence of rational functions 1.2 Sequential functions ............................ 651 Some unconventional terminology - Dual defìnitions - Composition 1.3 Pure sequential functions ......................... 658 1.4 Local functions ............................... 661 2 Uniformisation of rational relations ........................ 664 2.1 Proof of Theorem 2.1 (transducer version) ................ 666 2.2 Proof of Theorem 2.1 (representation version) .............. 667 Represent of S-immersions of an automaton - Semi-monomial matrices - Representation of S-uniformisations 2.3 Decomposition of rational functions ................... 673 The Weak Decomposition Theorem - The Strong Decomposition The¬ orem 2.4 The Rational Skimming Theorem II ................... 677 3 Cross-section of rational functions ........................ 679 TABLE OF CONTENTS 3.1 The rational cross-süction property .................... 680 Tiie Rational Cross-Section Theorem - The rational cross-section prop¬ erty for a monoid - Return to simplification monoids 3.2 Choosing the uniformisation (or the cross-section) ........... 685 Uniformisation of synchronous relations - Uniformisation of determin¬ istic relations - Тії. 3.3 back on the loom 4 Sequential functions ................................ 692 4.1 Two characterisations ........................... 692 Translations of a function - A functional characterisation - A quasi- topological point of view 4.2 Deciding sequentiality ........................... 699 4.3 Minimisation ................................ 704 Conjugation - Blockage of a sequential transducer - Reduction - Effec¬ tive computation 4.4 The (Great) Sequentiality Theorem ................... 711 Differential of a function - Proof of Theorem 4.5 iii) => і) - Proof of Theorem 4.5 ii) => iii) - Return to the Sequentiality Theorem 4.5 Pure sequential functions and local functions .............. 717 Solutions to the exercises ............................... 719 Notes & references ................................... 737 Bibliography 739 Index 749
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spelling	Sakarovitch, Jacques 1947- Verfasser (DE-588)129256285 aut Éléments de théorie des automates Elements of automata theory Jacques Sakarovitch 1. publ. Cambridge Cambridge Univ. Press 2009 XXIV, 758 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aus d. Franz. übers. - Orig. ersch. 2003 Automata math Automata theory Formal language Machine theory Automatentheorie (DE-588)4003953-5 gnd rswk-swf Automatentheorie (DE-588)4003953-5 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018669307&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Sakarovitch, Jacques 1947- Elements of automata theory Automata math Automata theory Formal language Machine theory Automatentheorie (DE-588)4003953-5 gnd
subject_GND	(DE-588)4003953-5
title	Elements of automata theory
title_alt	Éléments de théorie des automates
title_auth	Elements of automata theory
title_exact_search	Elements of automata theory
title_full	Elements of automata theory Jacques Sakarovitch
title_fullStr	Elements of automata theory Jacques Sakarovitch
title_full_unstemmed	Elements of automata theory Jacques Sakarovitch
title_short	Elements of automata theory
title_sort	elements of automata theory
topic	Automata math Automata theory Formal language Machine theory Automatentheorie (DE-588)4003953-5 gnd
topic_facet	Automata math Automata theory Formal language Machine theory Automatentheorie
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