Combinatorial Slice Theory MATEUS DE OLIVEIRA OLIVEIRA Doctoral Thesis Stockholm, Sweden 2013 TRITA-CSC-A 2013:13 ISSN-1653-5723 KTH School of Computer Science and Communication ISRN-KTH/CSC/A–13/13-SE SE-100 44 Stockholm ISBN 978-91-7501-933-8 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i datalogi tors- dagen den 12 dec 2013 klockan 10.00 i F3 , Kungl Tekniska högskolan, Lindstedtsvä- gen 26, Stockholm. © Mateus de Oliveira Oliveira, December 2013 Tryck: Universitetsservice US AB iii Abstract Slices are digraphs that can be composed together to form larger digraphs. In this thesis we introduce the foundations of a theory whose aim is to provide ways of defining and manipulating infinite families of combinatorial objects such as graphs, partial orders, logical equations etc. We give special attention to objects that can be represented as sequences of slices. We have successfully applied our theory to obtain novel results in three fields: concurrency theory, combinatorics and logic. Some notable results are: Concurrency Theory: • We prove that inclusion and emptiness of intersection of the causal behavior of bounded Petri nets are decidable. These problems had been open for almost two decades. • We introduce an algorithm to transitively reduce infinite families of DAGs. This algorithm allows us to operate with partial order languages defined via distinct formalisms, such as, Mazurkiewicz trace languages and message sequence chart languages. Combinatorics: • For each constant z ∈ N, we define the notion of z-topological or- der for digraphs, and use it as a point of connection between the monadic second order logic of graphs and directed width measures, such as directed path-width and cycle-rank. Through this connec- tion we establish the polynomial time solvability of a large number of natural counting problems on digraphs admitting z-topological orderings. Logic: • We introduce an ordered version of equational logic. We show that the validity problem for this logic is fixed parameter tractable with respect to the depth of the proof DAG, and solvable in polynomial time with respect to several notions of width of the equations being proved. In this way we establish the polynomial time provability of equations that can be out of reach of techniques based on comple- tion and heuristic search. iv Sammanfattning Slices är riktade grafer som kan sammansättas för att forma större rik- tade grafer. Avhandlingen beskriver en ny metod, ’slice-teori’, vars syfte är att hantera komplexa beräkningar på kombinatoriska objekt som exemplifieras med grafer, partialordningar och logiska ekvationer. Dessa objekt represen- teras som sekvenser av slices. Vi har framgångsrikt använt vår teori för att etablera nya resultat inom tre områden: concurrency-teori, kombinatorik och logik. Några viktiga resultat är som följer: Concurrency-teori: • Vi bevisar att inklusion och snittomhet för det kausala beteendet för begränsade Petrinät är avgörbara. Dessa problem uppstår vid verifiering av parallella system och har varit öppna i nästan två decennier. • Vi introducerar en algoritm för att transitivt reducera oändliga familjer av riktade acykliska grafer. Denna algoritm gör det möjligt att hantera partialordningsspråk definierade genom distinkta for- malismer som Mazurkiewicz-spår och meddelandesekvensdiagram. Kombinatorik: • Vi introducerar idén om en z-topologisk ordning för digrafer, med tillhörande komplexitetsmått zig-zag-tal, och använder den för att knyta ihop andra ordningens monadiska logik för grafer med riktade breddmått, såsom riktad stigbredd och cykelrangordning. Genom denna konstruktion bevisar vi att ett stort antal naturliga beräkn- ingsproblem med digrafer som tillåter en z-topologisk ordning är lösbara i polynomisk tid, närhelst z är fixerad. Logik: • Vi introducerar en ordnad variant av ekvationslogik. Vi visar att giltighetsproblemet för denna logik är fixparameterlösligt med avseende på djupet i den riktade acykliska bevisgrafen, och lösbar i polynomisk tid med avseende på flera mått på bredd hos ekvation- erna som finns med i beviset. På detta sätt etablerar vi bevisbarhet i polynomisk tid för ekvationer i ekvationslogik som kan ligga utom räckhåll för tekniker som baseras på komplettering och heuristisk sökning. v Acknowledgements In first place I thank my advisor, Karl Meinke for hiring me as a PhD student and for sharing with me his knowledge and enthusiasm for equational logic. I am also grateful to my co-advisor Stefan Arnborg, for his insightful comments on several aspects of slice theory. I am deeply grateful both to Karl and Stefan for several rounds of corrections, comments and suggestions to improve the readability of this thesis. I thank the Swedish Research Council VR, the European Union under ARTEMIS PROJECT 269335 MBAT (P.I. Karl Meinke) and the Department of Theoretical Computer Science (TCS/KTH) for financially supporting this research. I am grateful to my colleagues and friends at TCS/KTH for the nice atmosphere they create. I thank in particular Cenny Wenner for several nice discussions dur- ing the time we shared our office. I thank Karl Palmskog and Marcus Dicander for helping me with the translation of the abstract of this thesis into Swedish. I am also grateful to the Department of Theoretical Computer Science (TCS/CSC) for pro- viding financial support for the development of my course in Quantum Computing, which I enjoyed teaching. I dedicate this thesis to my beloved wife, Anna, who has been with me since the beginning of my journey as a doctoral student and to my daughter, Iris, who came this year to fulfill my life with happiness. Eu também gostaria de dedicar esta tese aos meus pais, Jaime e Rita, por todo o amor proporcionado aos seus filhos, e aos meus irmãos Vinicio e João por serem os irmãos maravilhosos que são. Finalmente eu gostaria de dedicar esta tese a Pedro, meu filho. Contents Contents vi I Introduction 1 1 Combinatorial Slice Theory 3 1.1 Slices.................................... 6 1.2 Slices in Concurrency Theory . 9 1.3 Slices in Combinatorial Graph Theory . 14 1.4 Slices in Equational Logic . 19 1.5 Main Contributions of this Thesis . 24 1.6 Overview of the Thesis . 26 II Slices 29 2 Slices 31 2.1 Slices and Slice Languages . 31 2.2 Operations On Slice Languages . 34 2.3 SliceGraphs................................ 38 2.4 Decidability Issues for Slice Languages . 43 3 Saturated Slice Languages 45 3.1 1-Saturation................................ 45 3.2 1-Saturated Slice Languages and Partial Orders . 49 3.3 z-Saturated Slice Languages . 50 4 Initial Applications 53 4.1 Sliced Graph Coloring . 53 4.2 Graphs that are unions of k directed paths . 55 4.3 Slice Languages and Hasse Diagrams . 57 4.4 Transitive Reduction of Slice Languages . 60 vi CONTENTS vii III Slices in Combinatorial Graph Theory 67 5 Directed Width Measures 69 5.1 Zig-Zag Number versus Other Digraph Width Measures . 71 5.2 Directed Path-Width, Directed Vertex Separation Number and Zig- ZagNumber................................ 72 6 The Monadic Second Order Logic of Graphs 75 6.1 Useful MSO Expressible Properties . 76 7 Counting Subgraphs Intersecting a z-Saturated Regular Slice Language 79 8 Slice Languages and MSO2 Properties 85 8.1 MSO2 and Saturated Slice Languages . 88 8.2 Proof of our Algorithmic Metatheorem . 89 9 Subgraph Homeomorphism 91 IV Partial Orders and Petri Nets 95 10 Petri Nets and their Partial Order Semantics 97 10.1 p/t-Nets.................................. 98 10.2Process .................................. 99 10.3 Causal vs Execution Semantics . 101 11 Interlaced Flows, Executions and Causal Orders 105 11.1 Characterization of Causal Orders and Executions in Terms of p- interlacedFlows. ............................. 108 12 Applications 111 12.1 Sliced Interlaced Flows . 111 12.2 Expressibility Theorem . 113 12.3 Verification of p/t-nets from Slice Languages . 114 12.4Synthesis .................................114 12.5 Model Checking the Partial Order Behavior of Petri Nets . 115 13 Translating Partial Order Formalisms into Slice Languages 117 13.1 Mazurkiewicz Traces . 118 13.2 Message Sequence Charts . 120 13.3 Application to Petri Nets . 124 viii CONTENTS V Slices in Equational Logic 127 14 Ordered Equational Logic 129 15 Slice Calculus 139 15.1 Slice Calculus Operators . 141 16 Provability in Ordered Equational Logic 147 17 Proof of the Congruence Operator 153 18 Proof Of the Transitivity Operator Lemma 157 19 Proof of the Substitution Operator Lemma 163 20ProofoftheRenamingOperatorLemma 167 20.1Notation..................................167 20.2 Proof of the Renaming Operator Lemma . 168 21UnitDecompositionsofaFixedEquation 175 VI Conclusions and Future Work 181 22 Conclusion 183 Bibliography 185 Part I Introduction 1 Chapter 1 Combinatorial Slice Theory In this thesis we introduce combinatorial slice theory, a theory aimed at describing how to manipulate and represent infinite families of combinatorial objects via for- mal languages over finite alphabets of slices. We call these formal languages slice languages. We will use slice theory to obtain novel algorithmic results in three fields: concurrency theory, combinatorial graph theory and logic. In the following subsections we will describe the main contributions of our theory to each of these fields. Partial Order Behavior of Petri nets Within concurrency theory slice languages will be used to represent infinite families of DAGs and infinite families of partial orders.