Matrix Graph Grammars Item Type Thesis Authors Perez Velasco, Pedro Pablo Citation Matrix Graph Grammars 2008-01, Download date 07/10/2021 13:37:24 Link to Item http://hdl.handle.net/10150/105736 MATRIX GRAPH GRAMMARS by Pedro Pablo P´erez Velasco Version 1.00 c Copyright by Pedro Pablo P´erez Velasco 2007, 2008 To my family VII ACKNOWLEDGEMENTS These lines are particularly pleasant to write. After all those years, I have a quite long list of people that have contributed to this book in one way or another. Unfortunately, I will not be able to include them all. Apologizes for the absences. First of all my family. Gema, with neverending patience and love, always supports me in every single project that I undertake. My unbounded love and gratitude. Hard to return, though I’ll try. My two daughters, Sof´ıa and Diana, make every single moment worthy. I’m absolutely grateful for their existence. My brothers Alex´ and Nina, now living in Switzerland, with whom I shared so many moments and that I miss so much. My parents, always supporting also with patience and love, worried if this boy would become a man (am I?). Juan, my thesis supervisor, whose advice and interest is invaluable. He has been actively involved in this project despite his many responsibilities. Also, I would like to thank the people at the series of seminars on complexity theory at U.A.M., headed by Roberto Moriy´on, for their interest on Matrix Graph Grammars. Many friends have stoically stood some chats on this topic affecting interest. Thank you very much for your friendship. KikeSim, GinHz, Alvaro´ Iglesias, Jaime Guerrero, ... All those who have passed by are not forgotten: People at ELCO (David, Fabrizio, Juanjo, Juli´an, Lola, ...), at EADS/SIC (Javier, Sergio, Roberto, ...), at Isban, at Banco Santander. Almost uncountable. I am also grateful to those that have worked on the tools used in this dissertation: Emacs and microEmacs, MikTeX, TeTeX, TeXnicCenter, OpenOffice and Ubuntu. I would like to highlight the very good surveys available on different topics on math- VIII ematics at the web, in particular at websites http://mathworld.wolfram.com and http://en.wikipedia.org, and the anonymous people behind them. Last few years have been particularly intense. A mixture of hard work and very good luck. I feel that I have received much more than I’m giving. In humble return, I will try to administer http://www.mat2gra.info, with freely available information on Matrix Graph Grammars such as articles, seminars, presentations, posters, one e-book (this one you are about to read) and whatever you may want to contribute with. Contents 1 Introduction ................................................... ..... 1 1.1 HistoricalOverview.............................. ................. 2 1.2 Motivation ...................................... ................ 6 1.3 BookOutline ..................................... ............... 11 2 Background and Theory ............................................ 15 2.1 Logics.......................................... ................. 15 2.2 CategoryTheory .................................. ............... 19 2.3 GraphTheory ..................................... .............. 26 2.4 TensorAlgebra................................... ................ 31 2.5 Functional Analysis.............................. ................. 34 2.6 GroupTheory ..................................... .............. 37 2.7 Summary and Conclusions........................... .............. 39 3 Graph Grammars Approaches ...................................... 41 3.1 Double PushOut (DPO).............................. ............. 42 3.1.1 Basics ........................................ ............ 42 3.1.2 Sequentialization and Parallelism . .............. 44 3.1.3 Application Conditions......................... ............. 46 3.1.4 Adhesive HLR Categories ......................... .......... 48 3.2 Other Categorical Approaches...................... ................ 49 X Contents 3.3 Node Replacement................................. ............... 53 3.4 Hyperedge Replacement............................ ............... 56 3.5 MSOLApproach .................................... ............. 60 3.6 Relation-Algebraic Approach...................... ................. 62 3.7 Summary and Conclusions........................... .............. 65 4 Matrix Graph Grammars Fundamentals ............................ 67 4.1 Characterization and Basic Concepts................ ................ 68 4.2 Completion...................................... ................ 74 4.3 Sequences and Coherence........................... ............... 79 4.4 Minimal and Negative Initial Digraphs . ............... 88 4.4.1 Minimal Initial Digraph ......................... ............ 91 4.4.2 Negative Initial Digraph........................ ............. 97 4.5 Composition and Compatibility ..................... ............... 104 4.6 Summary and Conclusions........................... .............. 109 5 Matching ................................................... ........ 111 5.1 Match and Extended Match ........................... ............ 112 5.2 Marking......................................... ................ 120 5.3 Initial Digraph Set and Negative Digraph Set . .............. 123 5.4 Internal and External ε-productions ................................ 127 5.5 Summary and Conclusions........................... .............. 130 6 Sequentialization and Parallelism ................................... 133 6.1 G-Congruence.................................... ................ 134 6.2 Sequentialization – Grammar Rules .................. ............... 148 6.3 Sequential Independence – Derivations . ................ 153 6.4 Explicit Parallelism............................. .................. 155 6.5 Summary and Conclusions........................... .............. 159 7 Restrictions on Rules ............................................... 161 7.1 Graph Constraints and Application Conditions . ............... 162 7.2 Extending Derivations ............................ ................ 172 Contents XI 7.3 Functional Representation ........................ ................. 181 7.4 MovingConditions ................................ ............... 191 7.5 From Simple Digraphs to Multidigraphs . .............. 199 7.6 Summary and Conclusions........................... .............. 206 8 Reachability ................................................... ..... 209 8.1 CrashCoursein Petri Nets .......................... .............. 210 8.2 MGG Techniques for Petri Nets ....................... ............. 213 8.3 Fixed Matrix Graph Grammars ........................ ............ 215 8.4 Floating Matrix Graph Grammars..................... ............. 221 8.4.1 External ε-production....................................... 222 8.4.2 Internal ε-production ....................................... 224 8.5 Summary and Conclusions........................... .............. 226 9 Conclusions and Further Research .................................. 229 9.1 Summary and Short Term Research..................... ............ 229 9.2 Main Contributions............................... ................ 232 9.3 Long Term Research Program......................... ............. 233 A Case Study ................................................... ...... 237 A.1 Presentation of the Scenario ....................... ................ 238 A.2 Sequences ....................................... ................ 245 A.3 Initial Digraph Sets and G-Congruence............... ............... 250 A.4 Reachability .................................... ................. 255 A.5 Graph Constraints and Application Conditions . ............... 260 A.6 Derivations ..................................... ................. 266 References ................................................... ........... 269 Index ................................................... ................ 275 List of Figures 1.1 Main Steps in a Grammar Rule Application . ............ 3 1.2 Partial Diagram of Problem Dependencies ............. ............... 9 1.3 Confluence ....................................... ................ 10 2.1 UniversalProperty ............................... ................. 21 2.2 Product, Cone and Universal Cone .................... .............. 21 2.3 Pushoutand Pullback .............................. ............... 22 2.4 Pushoutas Gluing ofSets ........................... ............... 23 2.5 Initial Pushout.................................. .................. 24 2.6 VanKampenSquare................................. .............. 25 2.7 Three, Four and Five Nodes Simple Digraphs . ............. 27 3.1 Example of Simple DPO Production.................... ............. 42 3.2 Direct Derivation as DPO Construction ............... ............... 43 3.3 Parallel Independence............................ .................. 44 3.4 Sequential Independence .......................... ................. 45 3.5 Generic Application Condition Diagram . ................ 47 3.6 Gluing Condition................................. ................. 48 3.7 SPO Direct Derivation ............................. ................ 50 3.8 SPO Weak Parallel Independence..................... ............... 50 3.9 SPO Weak Sequential Independence ................... .............. 50 XIV List of Figures 3.10 Sequential and Parallel Independence.............. ................... 51 3.11 SPB Replication Example .......................... ................ 52 3.12