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The Structure of Graphs and New Logics for the Characterization of Polynomial Time

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The Structure of Graphs and New Logics for the Characterization of Polynomial Time The Structure of Graphs and New Logics for the Characterization of Polynomial Time DISSERTATION zur Erlangung des akademischen Grades Doctor Rerum Naturalium im Fach Informatik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät II Humboldt-Universität zu Berlin von M.Sc. Bastian Laubner geboren am 2. März 1982 in Berlin Gefördert durch die Deutsche Forschungsgemeinschaft im Rahmen des Graduiertenkollegs ‘Methods for Discrete Structures’ (GRK 1408) Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II: Prof. Dr. Peter Frensch Gutachter: 1. Prof. Dr. Martin Grohe 2. Prof. Dr. Erich Grädel 3. Prof. Dr. Anuj Dawar eingereicht am: 22. November 2010 Tag der mündlichen Prüfung: 23. Februar 2011 Zusammenfassung Die deskriptive Komplexitätstheorie ist der Bereich der theoretischen Informatik, der sich mit abstrakten Charakterisierungen von maschinellen Berechnungen durch mathematische Logiken beschäftigt. Ähnlich der gewöhnlichen Komplexitätstheorie ist das Ziel, Berechnungsprobleme anhand der Ressourcen zu klassifizieren, die für ihre Lösung benötigt werden. Anstelle von ma- schinellen Ressourcen betrachtet die deskriptive Komplexitätstheorie jedoch die logische Aus- drucksstärke, die benötigt wird, um eine Klasse von Problemen zu definieren. Der Zweck be- steht darin, alternative Charakterisierungen von Komplexitätsklassen zu finden, mit denen sich die Stärken und Schwächen von maschinellen Berechnungen ohne Bezugnahme auf das konkrete Berechnungsmodell untersuchen lassen. Diese Arbeit leistet einen Beitrag zu drei Teilgebieten der deskriptiven Komplexitätstheorie. Zunächst betrachten wir ein Missverhältnis zwischen maschinellen Berechnungen, deren Einga- be stets aus geordneten Wörtern besteht, und logischen Definitionen, deren abstrakte Betrachtung von Eingabestrukturen ohne Ordnung auskommt. Wir zeigen, dass ein kombinatorischer Graph- kanonisierungsalgorithmus mit einfach exponentieller Laufzeit von Corneil und Goldberg (1984) auf kantengefärbte Graphen ausgeweitet werden kann. Die Repräsentationsinvarianz dieses Al- gorithmus gestattet uns die Implementierung in der Logik „Choiceless Polynomial Time with Counting“ (CPT+C˜ ) bei Beschränkung auf logarithmisch große Fragmente solcher Graphen. Auf Strukturen, deren Relationen höchstens Stelligkeit 2 haben, charakterisiert CPT+C˜ damit gerade die Polynomialzeit-Eigenschaften (PTIME) von Fragmenten logarithmischer Größe. Der zweite Beitrag untersucht die deskriptive Komplexität von PTIME-Berechnungen auf ein- geschränkten Graphklassen. Wir stellen eine neuartige Normalform von Intervallgraphen vor, die sich in Fixpunktlogik mit Zählen (FP+C) definieren lässt, was bedeutet, dass FP+C auf dieser Graphklasse PTIME charakterisiert. Die Methoden, die wir dafür entwickeln, machen erstmals die modulare Zerlegung von Graphen nutzbar für eine systematische Ergründung des Verhält- nisses zwischen Logiken und Komplexitätsklassen. Wir führen anschließend das Konzept von „Non-Capturing“-Reduktionen ein, das uns zu einer Reihe von Graphklassen führt, auf denen das Einfangen von Komplexitätsklassen genau so schwer ist wie auf der Klasse aller Graphen. Schließlich adaptieren wir unsere Methoden, um einen kanonischen Beschriftungsalgorithmus für Intervallgraphen zu erhalten, der sich mit logarithmischer Platzbeschränkung (LOGSPACE) berechnen lässt. Wir ziehen den Schluss, dass das Intervallgraphen-Isomorphieproblem vollstän- dig ist für die Klasse LOGSPACE. Auf diese Art leisten unsere logischen Methoden einen Beitrag zur klassischen Komplexitätstheorie. Im dritten Teil dieser Arbeit beschäftigt uns die ungelöste Frage, ob es eine Logik gibt, die alle Polynomialzeit-Berechnungen charakterisiert. Wir führen eine Reihe von Ranglogiken ein, die die Fähigkeit besitzen, den Rang von Matrizen über Primkörpern zu berechnen. Wir zeigen, dass diese Ergänzung um lineare Algebra robuste Logiken hervor bringt, und dass diese die Aus- drucksstärke von FP+C und vielen anderen Logiken übertreffen, die im Zusammenhang mit dieser Frage betrachtet werden. Durch Anpassung einer Konstruktion von Hella (1996) gelingt es uns außerdem zu beweisen, dass Ranglogiken strikt an Ausdrucksstärke gewinnen, wenn wir die Zahl an Variablen erhöhen, die die betrachteten Matrizen indizieren. Wir rechtfertigen unsere spezi- elle Wahl von Rangberechnungen dadurch, dass die meisten klassischen Probleme der linearen Algebra entweder in Ranglogiken oder bereits in FP+C definierbar sind. Dann bauen wir eine Brücke zur klassischen Komplexitätstheorie, indem wir über geordneten Strukturen eine Reihe von Komplexitätsklassen zwischen LOGSPACE und PTIME durch Ranglogiken charakterisie- ren. Unsere Arbeit etabliert die stärkste unserer Ranglogiken als Kandidat für die Charakterisie- rung von PTIME und legt nahe, dass Ranglogiken genauer erforscht werden müssen, um weitere Fortschritte zu erzielen im Hinblick auf eine Logik für Polynomialzeit. iii Abstract Descriptive complexity theory is the area of theoretical computer science that is concerned with the abstract characterization of computations by means of mathematical logics. It shares the ap- proach of complexity theory to classify problems on the basis of the resources that are needed to solve them. Instead of computational resources, however, descriptive complexity theory considers the expressiveness necessary for a logic to define classes of problems. Its primary goal is to pro- vide alternative characterizations of computational complexity that give us a way to reason about the strengths and limitations of computational procedures without reference to the underlying machine model. This thesis is making contributions to three strands of descriptive complexity theory. First, we consider an incongruence between machine computations, which receive ordered strings as their input, and logics, whose abstract view on input structures omits the ordering. We show that a combinatorial, singly exponential-time graph canonization algorithm of Corneil and Goldberg (1984) can be extended to edge-colored directed graphs. The algorithm’s input invariance allows us to implement it in the logic Choiceless Polynomial Time with Counting (CPT+C˜ ) if we restrict our attention to logarithmic-sized fragments of such graphs. This means that on structures whose relations are of arity at most 2, the polynomial-time (PTIME) properties of logarithmic-sized fragments are precisely characterized by CPT+C˜ . The second contribution investigates the descriptive complexity of PTIME computations on re- stricted classes of graphs. We present a novel canonical form for the class of interval graphs which is definable in fixed-point logic with counting (FP+C), resulting in FP+C capturing PTIME on this graph class. The methods developed for this result may serve as the foundation for a systematic study of the interrelation of logics and complexity classes on the basis of modular decompositions of graphs. Then we introduce the notion of non-capturing reductions that lead us to a wide vari- ety of graph classes on which computational problems are equally hard to characterize as on the class of all graphs. We finally mold our methods into a canonical labeling algorithm for interval graphs which is computable in logarithmic space (LOGSPACE) and we draw the conclusion that interval graph isomorphism is complete for LOGSPACE. In this way, our methods developed for the logical domain make a contribution to classical complexity theory. In the final part of this thesis, we take aim at the open question whether there exists a logic which generally captures polynomial-time computations. We introduce a variety of rank logics with the ability to compute the ranks of matrices over (finite) prime fields. We argue that this introduction of linear algebra makes for a robust notion of a logic and that it increases the expres- siveness of FP+C and many other logics considered in the context of capturing PTIME. By adapt- ing a construction of Hella (1996), we establish that rank logics strictly gain in expressiveness when we increase the number of variables that index the matrices we consider. Rank computa- tions are the first natural operation for which this property is shown in the presence of recursion. Our particular choice of rank computations is justified by showing that most classic problems in linear algebra can either be defined by rank logics or are already definable in FP+C. Then we establish a direct connection to standard complexity theory by showing that in the presence of orders, a variety of complexity classes between LOGSPACE and PTIME can be characterized by suitably defined rank logics. Our exposition provides evidence that rank logics are a natural object to study and establishes the most expressive of our rank logics as a viable candidate for capturing PTIME, suggesting that rank logics need to be better understood if progress is to be made towards a logic for polynomial time. iv Acknowledgements I am indebted to my supervisor Martin Grohe for his guidance and support throughout the time I spent researching for this thesis. He brought to my attention most of the questions I investigated, worked closely together with me on some of them, and has helped me hone my understanding of all these problems in countless discussions. I am grateful to my coauthors Anuj Dawar, Bjarki Holm, Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky for the experience I gained during these collaborations. I would like to give particular thanks to Anuj Dawar and
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