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INAUGURAL DISSERTATION Zur Erlangung Der Doktorwürde Der INAUGURAL DISSERTATION zur Erlangung der Doktorwurde¨ der Naturwissenschaftlich-Mathematischen Gesamtfakultat¨ der Ruprecht-Karls-Universitat¨ Heidelberg vorgelegt von Diplom-Mathematiker Edgar Busse ( geb. Damir Serikovich Muldagaliev ) aus Almaty Tag der mundlichen¨ Prufung:¨ 27. April 2006 Thema Finite-State Genericity On the Diagonalization Strength of Finite Automata Gutachter: Prof. Dr. Klaus Ambos-Spies Prof. Dr. Frank Stephan Zusammenfassung Algorithmische Generizitatskonzepte¨ spielen eine wichtige Rolle in der Berechenbarkeits- und Komplexitatstheorie.¨ Diese Begriffe stehen in engem Zusammenhang mit grundle- genden Diagonalisierungstechniken, und sie wurden zur Erzielung starker Trennungen von Komplexitatsklassen¨ verwendet. Da fur¨ jedes Generizitatskonzept¨ die zugehorigen¨ gener- ischen Mengen eine co-magere Klasse bilden, ist die Analyse generischer Mengen ein wichtiges Hifsmittel fur¨ eine quantitative Analyse struktureller Phanomene.¨ Typischer- weise werden Generizitatskonzepte¨ mit Hilfe von Erweiterungsfunktionen definiert, wobei die Starke¨ eines Konzepts von der Komplexitat¨ der zugelassenen Erwiterungsfunktionen abhangt.¨ Hierbei erweisen sich die sog. schwachen Generizitatskonzepte,¨ bei denen nur totale Erweiterungsfunktionen berucksichtigt¨ werden, meist als wesentlich schwacher¨ als die vergleichbaren allgemeinen Konzepte, bei denen auch partielle Funktionen zugelassen sind. Weiter sind die sog. beschrankten¨ Generizitatskonzepte¨ – basierend auf Erweiterun- gen konstanter Lange¨ – besonders interessant, da hier die Klassen der zugehorigen¨ gener- ischen Mengen nicht nur co-mager sind sondern zusatzlich¨ Maß 1 haben. Generische Mengen diesen Typs sind daher typisch sowohl im topologischen wie im maßtheoretis- chen Sinn. In dieser Dissertation initiieren wir die Untersuchung von Generizitat¨ im Bereich der Theorie der Formalen Sprachen: Wir fuhren¨ finite-state-Generizitatskonzepte¨ ein und ver- wenden diese, um die Diagonalisierungsstarke¨ endlicher Automaten zu erforschen. Wir konzentrieren uns hierbei auf die beschrankte¨ finite-state-Generizitat¨ und Spezial- falle¨ hiervon, die wir durch die Beschrankung¨ auf totale Erweiterungsfunktionen bzw. auf Erweiterungen konstanter Lange¨ erhalten. Wir geben eine rein kombinatorische Charak- terisierung der beschrankt¨ finite-state-generischen Mengen: Diese sind gerade die Men- gen, deren charakteristische Folge saturiert ist, d.h. jedes Binarwort¨ als Teilwort enthalt.¨ Mit Hilfe dieser Charakterisierung bestimmen wir die Komplexitat¨ der beschrankt¨ finite- state-generischen Mengen und zeigen, dass solch eine generische Menge nicht regular¨ sein kann es aber kontext-freie Sprachen mit dieser Generizitatseigenschaft¨ gibt. Weiter un- tersuchen wir den Einfluss der Lange¨ der Erweiterungen und der Beschrankung¨ auf totale Erweiterungsfunktionen auf die Starke¨ der korrespondierenden Generizitatskonzepte.¨ Die Untersuchung von eingeschrankten¨ Erweiterungsfunktionen, deren Wert jeweils nur von der Eingabenlange¨ oder einem Endstuck¨ der Eingabe konstanter Lange¨ abhangt,¨ verdeut- licht weiter die geringe Diagonalisierungsstarke¨ endlicher Automaten. Wir beenden un- sere Untersuchung der beschrankten¨ finite-state-Generizitat¨ damit, dass wir zeigen, dass die Starke¨ dieser Konzepte dramatisch erhoht¨ wird, wenn wir Erweiterungsfunktionen zu- grundelegen, deren Eingaben Anfangstucke¨ in redundanter Darstellung sind. Auf diese Art erhalten wir beschrankt¨ finite-state-generische Mengen, die REG-bi-immun sind, d.h. deren Erkennung die Kapazitat¨ eines endlichen Automaten nicht nur unendlich oft sondern fast uberall¨ uberschreitet.¨ Die von uns betrachteten unbeschrankten¨ finite-state-Generizitatskonzepte¨ basieren auf Moore-Funktionen und auf Verallgemeinerungen dieser Funktionen. Auch hier verglei- chen wir die Starke¨ der verschiedenen korrespondierenden Generizitatskonzepte¨ und eror-¨ tern die Frage, inwieweit diese Konzepte machtiger¨ als die beschrankte¨ finite-state-Generi- zitat¨ sind. Unsere Untersuchungen der finite-state-Generizitat¨ beruhen zum Teil auf neuen Ergeb- nissen uber¨ Bi-Immunitat¨ in der Chomsky-Hierarchie, einer neuen Chomsky-Hierarchie fur¨ unendliche Folgen und einer grundlichen¨ Untersuchung der saturierten Folgen. Diese Ergebnisse – die von unabhangigem¨ Interesse sind – werden im ersten Teil der Dissertation vorgestellt. Sie konnen¨ unabhangig¨ von dem Hauptteil der Arbeit gelesen werden. Abstract Algorithmic genericity notions play a major role in computability theory and computa- tional complexity theory. These notions are closely related to important diagonalization techniques and they can be used for obtaining strong separations of complexity classes. Moreover, since for any genericity concept, the class of the correspondent generic sets is comeager, the analysis of generic sets leads to a quantitative analysis of structural phenom- ena. Typically, genericity concepts are based on partial or total extension functions, where the strength of a concept is determined by the complexity of the admissible extension func- tions, where in general weak genericity notions based only on total extension functions are much weaker than the corresponding genericity notions allowing partial extension func- tions too. Moreover, so called bounded genericity concepts based on extensions of con- stant length are of particular interest since the classes of the corresponding generic sets are not only comeager but also have measure 1. So generic sets of these types are abundant in the topological and the measure theoretic sense. In this thesis we initiate the investigation of genericity in the setting of formal lan- guage theory: We introduce finite-state genericity notions, i.e., genericity notions related to the lowest class in the Chomsky hierarchy and we apply these concepts to explore the diagonalization strength of finite automata. We focus on bounded finite-state genericity and some special cases hereof allowing only total extensions and extensions of fixed length. We give a purely combinatorial char- acterization of bounded finite-state genericity by showing that a set A is bounded finite-state generic if and only if its characteristic sequence is saturated, i.e., contains any binary string as a subword. We use this characterization for determining the complexity of bounded finite-state generic sets. In particular we show that no bounded finite-state generic lan- guage is regular but that there are such languages which are context-free. Moreover, we explore the impact of the length of the admissible extensions and of the question whether we allow partial or only total extension functions. We further illustrate the limitations of the diagonalization strength of finite automata by considering some restricted types of extension strategies, namely length invariant and oblivious extensions. We complete our investigation of bounded finite-state genericity by showing that the strength of these con- cepts can be dramatically increased if we work with more redundant representations of initial segments: This way we obtain bounded finite-state generic sets which are REG-bi- immune, i.e., sets which exceed the capacity of finite automata not only infinitely often but almost everywhere. The unbounded finite-state genericity concepts which we consider are based on Moore functions and various generalizations of these functions. Again we compare the strength of different concepts and discuss the question in which respect these concepts are more powerful than bounded finite-state genericity. Our analysis of finite-state genericity is based in part on new results on bi-immunity in the Chomsky hierarchy, on a Chomsky hierarchy of sequences, and a thorrough analysis of saturated sequences. These results – which are of independent interest – are presented in the first part of the thesis and can be read independently. Contents 1 Introduction 1 2 Formal Languages and Infinite Sequences 11 2.1 Notation and Basic Concepts . 13 2.2 Grammars and Automata . 15 2.2.1 Chomsky Grammars and the Chomsky Hierarchy . 15 2.2.2 Regular Languages and Finite Automata . 18 2.2.3 Context Free Languages and Push Down Automata . 21 2.2.4 Turing Machines and Complexity . 23 2.3 Regular Functions . 29 2.4 Strong Separations in the Chomsky Hierarchy . 37 2.4.1 Bi-Immunity and Almost Everywhere Complexity . 38 2.4.2 Definitions and Basic Facts . 39 2.4.3 Immunity to the Class of Regular Languages . 42 2.4.4 Immunity to the Classes of Linear and Context-Free Lan- guages . 43 2.4.5 Immunity to the Higher Classes of the Chomsky Hierarchy 44 2.4.6 Summary . 45 2.5 A Chomsky Hierarchy For Sequences . 47 2.5.1 Definitions and Basic Facts . 47 2.5.2 Regular and Context-Free Sequences . 48 2.5.3 Context-Sensitive and Recursive Sequences . 51 2.5.4 The Chomsky Hierarchy Theorem For Sequences . 55 2.5.5 Prediction Machines . 56 2.6 Saturated Sequences . 64 2.6.1 Definitions and Basic Facts . 64 2.6.2 Closure Properties and Some Technical Properties . 66 2.6.3 Saturated Sequences and Languages and the Chomsky Hi- erarchy . 70 2.6.4 Saturation and Predictability . 73 2.6.5 Computational Complexity of Saturated Sequences and Lan- guages . 76 2.6.6 Saturated and Disjunctive Languages . 78 2.6.7 Partial Saturation . 80 II CONTENTS 3 Baire Category, Forcing, Genericity 83 3.1 Baire Category and the Cantor Space . 85 3.2 Extension Functions . 87 3.3 Baire Category and Lebesgue Measure . 92 3.4 Finite Extension Arguments . 96 3.5 Generic Sets . 104 4 Bounded Finite-State Genericity 109 4.1 Bounded reg-Genericity . 112 4.1.1 Definitions
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