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On Structural Similarities of Finite Automata and Turing Machine Enumerability Classes On Structural Similarities of Finite Automata and Turing Machine Enumerability Classes Till Tantau A A¯ × A A A¯ A¯ × × On Structural Similarities of Finite Automata and Turing Machine Enumerability Classes vorgelegt von Diplom-Informatiker und Diplom-Mathematiker Till Tantau von der Fakult¨at iv { Elektrotechnik und Informatik der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften { Dr. rer. nat. { genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Bernd Mahr Berichter: Prof. Dr. Dirk Siefkes Berichter: Prof. Dr. Johannes K¨obler Tag der wissenschaftlichen Aussprache: 17. Februar 2003 Berlin 2003 D 83 Preface This dissertation presents the lion's share of the research I did since I started working at the Technical University of Berlin at the end of 1999. Unfortunately, since I failed to focus my research on a single subject during the last three years, the results of several interesting research projects could not be included. If I had included them, the title would have had to be `On a Bunch of Interesting, but Unrelated Theorems of Theoretical Computer Science'. When I went to my advisor Dirk Siefkes at the beginning of 2002, we pondered on which papers would be fit to form the basis of a dissertation. There were two alternatives: either two papers on a new concept, namely enumerability by finite automata; or different technical reports on reducibility to selective languages, prover-verifier protocols, and reachability problems. Reducibility classes of selective languages would have been more `en vogue' and the results applicable in standard complexity theory and in practice (like a constant parallel time reachability algorithm for tournaments). In the end, the mathematical beauty of results that are presented in the following won over. I have presented the central theorems of this dissertation at two con- ferences in 2002, namely at the 19th Symposium on Theoretical Aspects of Computer Science in Antibes{Juan les Pins, France, and at the 27th Sym- posium on Mathematical Foundations of Computer Science in Warsaw, Poland. This dissertation also includes results that were presented at other conferences and in technical reports, but these are not at the core of my topic. Compared to the two main conference papers, this dissertation fo- cusses on demonstrating how the main results are part of broader contexts. This text is best read front to back, but each chapter is as self-contained as possible. Especially the fifth chapter, which is a `tutorial' on a new diagonalisation method, can be read independently of the other chapters. Notations and special terminology are explained directly preceding their first use and they are also explained in the list of notations. There are many people whom I have to thank, since without them this dissertation would have been much worse. My `being indebted' relation on them forms a partial ordering, but I decided to linearise this ordering to simplify its presentation, see theorems 2.38 and 2.39 for more details on lin- earisations. So, in alphabetical ordering I am deeply grateful to Sebastian Bab, Lane Hemaspaandra, Chris Homan, Johannes K¨obler,Carsten Lenz, Arfst Nickelsen, Margit Russ, Birgit Schelm, Dirk Siefkes, Gerda Tantau, Karl Tantau, Leen Torenvliet, and Tux. Till Tantau Berlin, autumn of 2002 1 Abstract There are different ways of measuring the complexity of functions that map words to words. Well-known measures are time and space complexity. Enu- merability is another possible measure. It is used in recursion theory, where it plays a key r^olein bounded query theory, but also in resource-bounded complexity theory, especially in connection with nonuniform computations. This dissertation transfers enumerability to automata theory. It is shown that enumerability behaves similarly in recursion theory and in automata theory, but differently in complexity theory. The enumerability of a function f is the smallest m such that there exists an m-enumerator for f. An m-enumerator is a machine that produces, for every input word w, a set of up to m possibilities for f(w). By varying the parameter m and the class of allowed enumerators, different enumerability classes can be defined. In recursion theory, one allows arbitrary Turing machines as enumerators; in automata theory, only finite automata. A deep structural result that holds both for finite automata and for Turing machine enumerability is the following cross product theorem: if f g is (n + m)-enumerable, then either f is n-enumerable or g is m-enumerable.× In contrast, this theorem does not hold for polynomial-time enumerability. Enumerability can be used to quantify the difficulty of a language A by asking how difficult it is to enumerate its n-fold characteristic func- n n n tion χA and cardinality function #A. A language is (m; n)-verbose if χA is m-enumerable. The inclusion structures of Turing machine and of fi- nite automata verboseness classes are identical: all (m; n)-Turing-verbose languages are (h; k)-Turing-verbose iff all (m; n)-finite-automata-verbose languages are (h; k)-finite-automata-verbose. The structure of polynomial- time verboseness classes is different. n The enumerability of #A has been studied in detail in recursion the- n ory. Kummer's cardinality theorem states that if #A is n-enumerable by a Turing machine, then A must be recursive. Evidence is gathered that this theorem also holds for finite automata: it is shown that the nonspeedup theorem, the cardinality theorem for two words, and the restricted cardi- nality theorem all hold for finite automata. The cardinality theorem does not hold for polynomial-time computations. The central proofs rely on two proof techniques that promise to be appli- cable in other situations as well: generic proofs and branch diagonalisation. Generic proofs use elementary definitions, a concept from logic, to define enumerators in terms of other enumerators. They can be instantiated for all computational models that are closed under elementary definitions. Ex- amples of such models are finite automata, but also Presburger arithmetic and ordinal number arithmetic. The second technique is a new diagonal- 2 isation method, where machines are tricked on codes of diagonalisation decision sequences, rather than on codes of machines. Branch diagonalisa- tion is not applicable universally, but where it is applicable, it can be used to diagonalise against Turing machines, using only finite automata. Results on enumerability classes have applications in unrelated areas, like finite automata protocol testing, classification problems where exam- ples are provided, and separability. An intriguing example of such an ap- plication is the following theorem: if there exist regular supersets of A A, A A¯, and A¯ A¯ whose intersection is empty, then A is regular. × × × 3 Zusammenfassung Die Komplexit¨atvon Funktionen, die Worte auf Worte abbilden, kann auf verschiedene Arten gemessen werden. Bekannte Maße sind Zeit- und Platz- komplexit¨at. Aufz¨ahlbarkeit ist ein weiteres Komplexit¨atsmaß.Sie wird in der Rekursionstheorie eingesetzt, wo sie eine zentrale Rolle in der Theorie fragenbeschr¨ankterReduktionen spielt, sowie in der ressourcenbeschr¨ank- ten Komplexit¨atstheorie,insbesondere in Verbindung mit nichtuniformen Berechnungen. In dieser Arbeit wird das Konzept der Aufz¨ahlbarkeit auf endliche Automaten ¨ubertragen. Es wird gezeigt, dass sich Aufz¨ahlbarkeit in der Rekursionstheorie und in der Automatentheorie gleichartig verh¨alt, in der Komplexit¨atstheoriehingegen andersartig. Die Aufz¨ahlbarkeit einer Funktion f ist die kleinste Zahl m, f¨urdie ein m-Aufz¨ahlerf¨ur f existiert. Ein m-Aufz¨ahler ist eine Maschine, die bei Ein- gabe eines Wortes w eine Menge von h¨ochstens m M¨oglichkeiten f¨ur f(w) ausgibt. Verschiedene Aufz¨ahlbarkeitsklassen k¨onnendurch Ver¨anderung des Parameters m und der Art der erlaubten Maschinen definiert wer- den. In der Rekursionstheorie erlaubt man beliebige Turingmaschinen als Aufz¨ahler,in der Automatentheorie lediglich endliche Automaten. Ein tiefliegendes strukturelles Resultat, das sowohl f¨urTuringmaschinen als auch f¨urendliche Automaten gilt, ist der folgende Kreuzproduktsatz: Ist f g eine (n + m)-aufz¨ahlbareFunktion, so ist f eine n-aufz¨ahlbareoder g eine× m-aufz¨ahlbareFunktion. Dieser Satz gilt nicht im Polynomialzeitfall. Aufz¨ahlbarkeit kann auch benutzt werden, um die Komplexit¨atvon Sprachen zu quantifizieren. Dazu wird gefragt, wie schwierig es ist, die n- n fache charakteristische Funktion χA und die n-fache Kardinalit¨atsfunktion n n #A einer Sprache A aufzuz¨ahlen.Eine Sprache A ist (m; n)-verbose, falls χA m-aufz¨ahlbarist. Die Inklusionsstrukturen der Verbosenessklassen von Turingmaschinen und der von endlichen Automaten sind gleich: alle (m; n)- Turing-verbosen Sprachen sind genau dann (h; k)-Turing-verbose, wenn alle in Bezug auf endliche Automaten (m; n)-verbosen Sprachen (h; k)-verbose sind. Dies gilt nicht im Polynomialzeitfall. n Die Aufz¨ahlbarkeit von #A ist in der Rekursionstheorie wohluntersucht. n Kummers Kardinalit¨atssatzbesagt, dass A rekursiv ist, falls #A von einer Turingmaschine n-aufgez¨ahltwerden kann. Vermutlich gilt dieser Satz auch f¨urendliche Automaten: Zumindest der Nonspeedupsatz, der Kardi- nalit¨atssatzf¨urzwei Worte und der eingeschr¨ankteKardinalit¨atssatzgelten f¨urendliche Automaten. Der Kardinalit¨atssatzgilt nicht im Polynomial- zeitfall. Die Hauptbeweise in dieser Arbeit benutzen zwei Techniken, deren Ein- satz auch in anderen Gebieten vielversprechend erscheint:
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