Handbook of Weighted Automata

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1 Manfred Droste Werner Kuich Heiko Vogler Editors Handbook of Weighted Automata 4.1 Springer

2 Contents Part I Foundations Chapter 1: Semirings and Formal Power Series Manfred Droste and Werner Kuich 3 1 Introduction 3 2 Monoids and Semirings 5 3 Formal Power Series 12 4 Matrices 17 5 Cycle-Fme Linear Equations 22 References 26 Chapter 2: Fixed Point Theory Zoltcin Esik 29 1 Introduction 29 2 Least Fixed Points 30 3 Conway Theories 38 4 Iteration Theories 41 5 Unique Fixed Points 44 6 Fixed Points of Linear Functions Inductive *-Semirings Complete Semirings Iterative Semirings 53 7 Fixed Points of Affine Functions Complete Semiring-Semimodule Pairs Bi-inductive Semiring-Semimodule Pairs 61 References 62 xi

3 xii Contents Part II Concepts of Weighted Recognizability Chapter 3: Finite Automata Zoltdn Esik and Werner Kuich 69 1 Introduction 69 2 Finite Automata over Semirings Finite Automata over Arbitrary Power Series Semirings Finite Automata over Conway Semirings Finite Linear Systems 81 3 Finite Automata over Quemirings Semiring Semimodule Pairs and Quemirings Finite Automata over Quemirings and a Kleene Theorem Finite Linear Systems over Quemirings 99 References 103 Chapter 4: Rational and Recognisable Power Series Jacques Sakarovitch Introduction Rational Series and Weighted Rational Expressions Series over a Graded Monoid Rational Series Weighted Automata The Behaviour of a Weighted Automaton The Fundamental Theorem of Automata Conjugacy and Covering of Automata Recognisable Series and Representations The Family of Recognisable Series Other Products on Recognisable Series Series on a Product of Monoids Series over a Free Monoid The Representability Theorem Reduced Representations Applications of the Reduction of Representations 161 Support of Rational Series 165 Notes General Sources Notes to Sect. 2: Rational Series Notes to Sect. 3: Weighted Automata Notes to Sect. 4: Recognisable Series Notes to Sect. 5: Series over a Free Monoid Not es to Sect. 6: Support of Rational Series 171 References 171

4 Contents xiii Chapter 5: Weighted Automata and Weighted Logics Manfred Droste and Paul Gastin Introduction MSO-Logic and Weighted Automata Weighted Logics Unambiguous Formulas Definability Equals Recognizability Locally Finite Semirings Weighted Automata an Infinite Words Weighted Logics an Infinite Words Conclusions and Open Problems 206 References 208 Chapter 6: Weighted Automata Algorithms Mehryar Mohri Introduction Preliminaries Semirings Weighted Transducers and Automata Shortest-Distance Algorithms All-Pairs Shortest-Distance Problems Single-Source Shortest-Distance Problems Rational Operations Elementary Unary Operations Fundamental Binary Operations Composition Intersection Difference Optimization Algorithms Epsilon-Removal Determinization Weight Pushing Minimization Synchronization Conclusion 250 References 250 Part III Weighted Discrete Structures Chapter 7: Algebraic Systems and Pushdown Automata Ion Petre and Arto Salomaa Introduction Auxiliary Notions and Results Algebraic Power Series 261

5 xiv Contents 3.1 Definition and Basic Reductions Interconnections with Context-Free Grammars Normal Forms Theorems of Shamir and Chomsky Schützenberger Transductions Pushdown Automata Pushdown Transition Matrices S((Z*))-Pushdown Automata Equivalence with Algebraic Systems Other Topics 287 References 288 Chapter 8: Lindenmayer Systems Juha Honkala Introduction Iterated Morphisms and Rational Series Lindenmayerian Algebraic Series DOL Power Series Other Power Series Generalizations of L Systems 307 References 309 Chapter 9: Weighted Tree Automata and Tree Transducers Zoltdn Fülöp and Heiko Vogler Introduction Preliminaries General Notation Trees Algebraic Concepts Tree Series Weighted Tree Automata BottomLup Tree Automata Recognizable Tree Series Closure Properties Support of Recognizable Tree Series Determinization of Weighted Tree Automata Pumping Lemmata and Decidability Finite Algebraic Characterizations of Recognizable Tree Series Equational Tree Series Rational Tree Series MSO-Definable Tree Series Other Models Related to Recognizable Tree Series Further Results IO-Substitution and Tree Series Transformations Weighted Tree Transducers Tree Transducers 364

6 Contents xv 5.2 The Basic Model Restricted Models Composition and Decomposition The Inclusion Diagram of Some Fundamental wtt Classes Hierarchies Further Models of Weighted Tree Transducers Further Results 394 References 394 Chapter 10: Traces, Series-Parallel Posets, and Pictures: A Weighted Study Ina Fichtner, Dietrich Kuske, and Ingmar Meinecke Introduction Traces Weighted Distributed Systems Other Formalisms: Presentations, Expressions, and Logics Relating the Formalisms History and Overview Series-Parallel Posets Series-Parallel Posets and Bisemirings Weighted Branching Automata Rationality The Hadamard Product History and Overview Pictures Pictures and Weighted Picture Automata Other Formalisms: Expressions and Logics Relating the Formalisms Decidability Issues History and Overview 445 References 446 Part IV Applications Chapter 11: Digital Image Compression Jürgen Albert and Jarkko Kari Introduction Image Types Weighted Finite Automata and Multi-resolution Images Drawing WFA Images An Encoding Algorithm Practical Image Compression Using WFA Weighted Finite Transducers (WFT) 469 Parametric Weighted Finite Automata (PWFA) 472

7 xvi Contents 9 Conclusions and Open Problems 476 References 477 Chapter 12: Fuzzy Languages George Rahonis Introduction Lattices and Fuzzy Languages Fuzzy Recognizability over Bounded Distributive Lattices Fuzzy Recognizability over Finite Words Fuzzy Recognizability over Infinite Words Multi-valued MSO Logic Fuzzy Languages: An Overview Fuzzy Languages over e-monoids Fuzzy Languages over Residuated Lattices Fuzzy Automata with Outputs Fuzzy Abstract Families of Languages Fuzzy Tree Languages Applications 510 References 513 Chapter 13: Model Checking Linear-Time Properties of Probabilistic Systems Christel Baier, Marcus Größer, and Frank Ciesinski Introduction Markov Decision Processes Maximal Reachability Probabilities Model Checking w-regular Properties Partial Order Reduction Partially Observable MDPs Conclusion Appendix 560 References 563 Chapter 14: Applications of Weighted Automata in Natural Language Processing Kevin Knight and Jonathan May Background WFST Techniques for. Natural Language Processing Example 1: Transliteration Example 2: Translation Language Modeling Applications of Weighted String Automata Language Translation Speech Recognition Lexical Processing Tagging 585

8 Contents xvii 3.5 Summarization Optical Character Recognition Applications of Weighted Tree Automata Open Problems Conclusion 591 References 591 Index 597

Finite-State Transducers in Language and Speech Processing

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