Using automata to characterise fixed point temporal logics

This work examines propositional fixed point temporal and modal logics called mu-calculi and their relationship to automata on infinite strings and trees. We use correspondences between formulae and automata to explore definability in mu-calculi and their fragments, to provide normal forms for formulae, and to prove completeness of axiomatisations. The study of such methods for describing infinitary languages is of fundamental importance to the areas of computer science dealing with non-terminating computations, in particular to the specification and verification of concurrent and reactive systems. To emphasise the close relationship between formulae of mu-calculi and alternating automata, we introduce a new first recurrence acceptance condition for automata, checking intuitively whether the first infinitely often occurring state in a run is accepting. Alternating first recurrence automata can be identified with mu-calculus formulae, and ordinary, non-alternating first recurrence automata with formulae in a particular normal form, the strongly aconjunctive form. Automata with more traditional Büchi and Rabin acceptance conditions can be easily unwound to first recurrence automata, i.e. to mu-calculus formulae. In the other direction, we describe a powerset operation for automata that corresponds to fixpoints, allowing us to translate formulae inductively to ordinary Büchi and Rabin-automata. These translations give easy proofs of the facts that Rabin-automata, the full mu-calculus, its strongly aconjunctive fragment and the monadic second-order calculus of n successors SnS are all equiexpressive, that Büchi-automata, the fixpoint alternation class Pi_2 and the strongly aconjunctive fragment of Pi_2 are similarly related, and that the weak SnS and the fixpoint-alternation-free fragment of mu-calculus also coincide. As corollaries we obtain Rabin's complementation lemma and the powerful decidability result of SnS. We then describe a direct tableau decision method for modal and linear-time mu-calculi, based on the notion of definition trees. The tableaux can be interpreted as first recurrence automata, so the construction can also be viewed as a transformation to the strongly aconjunctive normal form. Finally, we present solutions to two open axiomatisation problems, for the linear-time mu-calculus and its extension with path quantifiers. Both completeness proofs are based on transforming formulae to normal forms inspired by automata. In extending the completeness result of the linear-time mu-calculus to the version with path quantifiers, the essential problem is capturing the limit closure property of paths in an axiomatisation. To this purpose, we introduce a new \exists\nu-induction inference rule

Algorithmic analysis of parity games

Parity games are discrete infinite games of two players with complete information. There are two main motivations to study parity games. Firstly the problem of deciding a winner in a parity game is polynomially equivalent to the modal µ-calculus model checking, and therefore is very important in the field of computer aided verification. Secondly it is the intriguing status of parity games from the point of view of complexity theory. Solving parity games is one of the few natural problems in the class NP∩co-NP (even in UP∩co-UP), and there is no known polynomial time algorithm, despite the substantial amount of effort to find one. In this thesis we add to the body of work on parity games. We start by presenting parity games and explaining the concepts behind them, giving a survey of known algorithms, and show their relationship to other problems. In the second part of the thesis we want to answer the following question: Are there classes of graphs on which we can solve parity games in polyno

Topological Complexity of Sets Defined by Automata and Formulas

In this thesis we consider languages of infinite words or trees defined by automata of various types or formulas of various logics. We ask about the highest possible position in the Borel or the projective hierarchy inhabited by sets defined in a given formalism. The answer to this question is called the topological complexity of the formalism.It is shown that the topological complexity of Monadic Second Order Logic extended with the unbounding quantifier (introduced by Bojańczyk to express some asymptotic properties) over ω-words is the whole projective hierarchy. We also give the exact topological complexities of related classes of languages recognized by nondeterministic ωB-, ωS- and ωBS-automata studied by Bojańczyk and Colcombet, and a lower complexity bound for an alternating variant of ωBS-automata.We present the series of results concerning bi-unambiguous languages of infinite trees, i.e. languages recognized by unambiguous parity tree automata whose complements are also recognized by unambiguous parity automata. We give an example of a bi-unambiguous tree language G that is analytic-complete. We present an operation σ on tree languages with the property that σ(L) is topologically harder than any language in the sigma-algebra generated by the languages continuously reducible to L. If the operation is applied to a bi-unambiguous language than the result is also bi-unambiguous. We then show that the application of the operation can be iterated to obtain harder and harder languages. We also define another operation that enables a limit step iteration. Using the operations we are able to construct a sequence of bi-unambiguous languages of increasing topological complexity, of length at least ω square.W niniejszej rozprawie rozważane są języki nieskończonych słów lub drzew definiowane poprzez automaty różnych typów lub formuły różnych logik. Pytamy o najwyższą możliwą pozycję w hierarchii borelowskiej lub rzutowej zajmowaną przez zbiory definiowane w danym formalizmie. Odpowiedź na to pytanie jest nazywana złożonością topologiczną formalizmu.Przedstawiony został dowód, że złożonością topologiczną Logiki Monadycznej Drugiego Rzędu rozszerzonej o kwantyfikator Unbounding (wprowadzony przez Bojańczyka w celu umożliwienia wyrażania własności asymptotycznych) na słowach nieskończonych jest cała hierarchia rzutowa. Obliczone zostały również złożoności topologiczne klas języków rozpoznawanych przez niedeterministyczne ωB-, ωS- i ωBS-automaty rozważane przez Bojańczyka i Colcombet'a, oraz zostało podane dolne ograniczenie złożoności wariantu alternującego ωBS-automatów.Zaprezentowane zostały wyniki dotyczące języków podwójnie jednoznacznych, tzn. języków rozpoznawanych przez jednoznaczne automaty parzystości na drzewach, których dopełnienia również są rozpoznawane przez jednoznaczne automaty parzystości. Podany został przykład podwójnie jednoznacznego języka drzew G, który jest analityczny-zupełny. Została wprowadzona operacja σ na językach drzew taka, że język σ(L) jest topologicznie bardziej złożony niż jakikolwiek język należący do sigma-algebry generowanej przez języki redukujące się w sposób ciągły do języka L. W wyniku zastosowania powyższej operacji do języka podwójnie jednoznacznego otrzymujemy język podwójnie jednoznaczny. Zostało pokazane, że kolejne iteracje aplikacji powyższej operacji dają coraz bardziej złożone języki. Została również wprowadzona druga operacja, która umożliwia krok graniczny iteracji. Używając obydwu powyższych operacji można skonstruować ciąg długości ω kwadrat złożony z języków podwójnie jednoznacznych o coraz większej złożoności

Playing with Trees and Logic

This document proposes an overview of my research sinc

New Perspectives on Games and Interaction

This volume is a collection of papers presented at the 2007 colloquium on new perspectives on games and interaction at the Royal Dutch Academy of Sciences in Amsterdam. The purpose of the colloquium was to clarify the uses of the concepts of game theory, and to identify promising new directions. This important collection testifies to the growing importance of game theory as a tool to capture the concepts of strategy, interaction, argumentation, communication, cooperation and competition. Also, it provides evidence for the richness of game theory and for its impressive and growing application