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PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010

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PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Facultat de Lletres Departament de Filologies Romàniques Petri Net Controlled Grammars PhD Dissertation Presented by Sherzod Turaev Supervised by Jürgen Dassow Tarragona, 2010 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Supervisor: Prof. Dr. Jürgen Dassow Fakultät für Informatik Institut für Wissens- und Sprachverarbeitung Otto-von-Guericke-Universität Magdeburg D-39106 Magdeburg, Germany Tutor: Dr. Gemma Bel Enguix Grup de Recerca en Lingüística Matemàtica Facultat de Lletres Universitat Rovira i Virgili Av. Catalunya, 35 43002 Tarragona, Spain UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Acknowledgements It is a pleasure to thank many people who helped me to made this thesis possible. First of all I would like to express my sincere gratitude to my super- visor, Professor Jürgen Dassow. He helped me with his enthusiasm, his inspiration, and his efforts. During the course of this work, he provided with many helpful suggestions, constant encouragement, important advice, good teaching, good company, and lots of good ideas. It has been an honor to work with him. I would like to thank the head of our research group, Professor Car- los Martín-Vide, for his great effort in establishing the wonderful circum- stances of PhD School on Formal Languages and Applications, and for his help in receiving financial support throughout my PhD. This work was made thanks to the financial support of research grant 2006FI-01030 from AGAUR of Catalonia Government. I would like to thank all professors in our PhD school, especially Pro- fessors Tom Head, Erzsébet-Chuhaj Varjú, Victor Mitrana, Markus Holzer and Claudio Moraga, for providing solid knowledge and interest in formal languages and automata theory. I wish to express my cordial apprecia- tion to Professor Victor Mitrana who encouraged me to work on this topic, gave me constructive advice and recommended me to work with Profes- sor Dassow. Special thanks are due to my co-author, Dr. Ralf Stiebe, Otto-von-Guericke-University Magdeburg, for his ideas, efforts, useful dis- cussions and time spent on our papers. I would like to express my thanks to my colleagues at GRLMC, and my friends (Alexander Krassovitskiy, Areti Panou, Cˇatˇalin Tîrnˇaucˇa, Alexan- der Perekrestenko and Abhik Ghosh) for useful discussions and for making my staying in Tarragona enjoyable. I specially thank Dr. Gemma Bel En- guix, Lilica Voicu and Alexander Krassovitskiy for their assistance with all types of documentation and organization matters. 1 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 I am also grateful to all members of the Department of Knowledge and Language Engineering, Faculty of Computer Science, Otto-von-Guericke- University Magdeburg (in particular, to Dr. Bianca Truthe) for their kind and generous hospitality during my stays, and for their fruitful discussions, suggestions, ideas. I should also thank Professor Abdimajid Pulatov, National University of Uzbekistan, who has been a significant presence in my life. He taught me not only to conduct scientific research but also to think about life, politics, philosophy. I am always thankful for his wisdom and knowledge. Many thanks to my teacher Mr. Sayfiddin Qalandarov who taught me mathematics and always inspired me to progress in science. My heartiest thanks go to my best friends Jessica and Carlo who never let me feel that I was away from my family, and helped me in everything. I express my deep love for my wife Matluba, and our children Jaloliddin and Zahiriddin. Their patience, love and encouragement always upheld me. Lastly, and most importantly, I wish to thank my parents, Rustam Turaev and Qurbonoy Tursunova for their support and love. They always helped me and encouraged me to concentrate on my study. 2 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Contents Acknowledgements 1 1 Introduction 7 2 Preliminaries 17 2.1 FormalLanguages ....................... 17 2.1.1 GeneralNotations .. .. .. .. .. .. .. .. .. 17 2.1.2 Strings,LanguagesandOperations . 17 2.1.3 Grammars........................ 21 2.1.4 GrammarswithRegulatedWriting . 22 2.2 PetriNets............................ 26 2.2.1 BasicDefinitions .................... 26 2.2.2 SpecialPetriNets ................... 29 2.2.3 PetriNetLanguages . 33 3 Semi-Matrix Grammars 36 3.1 Introduction........................... 36 3.2 DefinitionandExamples. 37 3.3 Closure Properties . 40 3.4 GenerativeCapacity ...................... 43 4 Extended cf Petri Net Controlled Grammars 46 4.1 Introduction........................... 46 4.2 Context-FreePetriNets . 50 4.3 k-Petri Net Controlled Grammars . 52 4.3.1 DefinitionsandExamples . 52 4.3.2 HierarchyResults. 60 4.3.3 Closure Properties . 71 4.4 PetriNetswithChainsandCycles . 80 4.4.1 Chain Control . 80 3 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 CONTENTS 4.4.2 CyclicControl ..................... 83 4.4.3 Supervised Cyclic Control . 86 4.4.4 Grammars,LanguagesandExamples. 87 5 Arbitrary Petri Net Controlled Grammars 92 5.1 Introduction........................... 92 5.2 GrammarsandTheirLanguages . 93 5.3 The Effect of Labeling on the Computational Power . 96 5.4 The Effect of Final Markings on the Generative Power . 102 6 Grammars Controlled by Special Petri Nets 113 6.1 Introduction........................... 113 6.2 GrammarsandTheirLanguages . 114 6.3 Results: LabelingStrategies . 116 6.4 Results:FinalMarkings . 131 7 Capacity-Bounded Grammars 143 7.1 Introduction........................... 143 7.2 Capacity-Bounded Grammars . 144 7.3 The Power of Capacity-Bounded GS Grammars . 149 7.4 Capacity-Bounded Context-Free Grammars . 154 7.5 ControlbyPetri NetswithPlace Capacities . 158 8 Conclusions and Further Research 165 Bibliography 170 4 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 List of Figures 1.1 Citricacidcycle ........................ 10 1.2 A simplified Petri net representation of citric acid cycle . 12 2.1 APetrinet ........................... 28 2.2 Astatemachine ........................ 29 2.3 Ageneralizedstatemachine . 30 2.4 Amarkedgraph ........................ 30 2.5 Ageneralizedmarkedgraph. 30 2.6 Acasualnet........................... 31 2.7 Afree-choicenet ........................ 31 2.8 Anextendedfree-choicenet . 32 2.9 Anasymmetricchoicenet . 32 2.10 The hierarchy of Petri net classes . 33 2.11AlabeledPetrinet....................... 35 2.12 The relationship of the families of Petri net languages . 35 4.1 A cf Petri net N ........................ 51 4.2 A 1-Petri net N ½ ........................ 54 4.3 A 2-Petri net N ¾ ........................ 55 Þ 4.4 A -Petri net N Þ ........................ 83 4.5 A -Petri net N ........................ 85 × 4.6 An -Petri net N × ....................... 88 5.1 A labeled Petri net N ..................... 95 6.1 A labeled state machine N ×Ñ ................. 115 6.2 A labeled marked graph N Ñ ................. 116 6.3 The hierarchy of language families generated by Petri net controlledgrammars .. ... .. .. .. .. .. .. .. .. 142 5 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 LIST OF FIGURES 7.1 The hierarchy of language families generated by grammars withboundedcapacities . 164 6 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 1 Introduction Formal language theory, introduced by Noam Chomsky in the 1950s as a tool for a description of natural languages [12, 13, 14], has also been widely involved in modeling and investigating phenomena such as generative (pro- duction) processes appearing in computer science, artificial intelligence and other related fields. In formal language theory a model for a phenomenon is usually constructed by representing it as a set of words, i.e., a language over a certain alphabet, and defining a generative mechanism, i.e., a grammar which identifies exactly the words of this set. With respect to the forms of their rules, grammars and their languages are divided into four classes of Chomsky hierarchy: recursively enumerable or type 0, context-sensitive or type 1, context-free or type 2 and regular or type 3. Context-free grammars are the most investigated type of Chomsky hi- erarchy which, in addition, have good mathematical properties and are extensively used in many applications of formal languages. However, they cannot cover all aspects which occur in modeling of phenomena. On the other hand, context-sensitive grammars, the next level in Chomsky hierar- chy, are too powerful to be used in applications of formal languages, and have bad features, for instance, for context-sensitive grammars, the empti- ness problem is undecidable and the existing algorithms for the membership problem, thus for the parsing, have exponential complexities. Moreover, such concepts as a derivation tree, which is an important tool for the anal- 7 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 ysis of context-free languages, cannot be transformed to context-sensitive grammars. Therefore, it is of interest to consider “intermediate” gram-
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