Finite-state Strategies in Delay Games (full version)

What is a finite-state strategy in a delay game? We answer this surprisingly non-trivial question by presenting a very general framework that allows to remove delay: finite-state strategies exist for all winning conditions where the resulting delay-free game admits a finite-state strategy. The framework is applicable to games whose winning condition is recognized by an automaton with an acceptance condition that satisfies a certain aggregation property. Our framework also yields upper bounds on the complexity of determining the winner of such delay games and upper bounds on the necessary lookahead to win the game. In particular, we cover all previous results of that kind as special cases of our uniform approach

Polishness of some topologies related to word or tree automata

We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the proceedings of the 26th EACSL Annual Conference on Computer Science and Logic, CSL 2017. The main addition with regard to the conference paper consists in the study of the B\"uchi topology and of the Muller topology in the case of a space of trees, which now forms Section

Decidability and Universality in Symbolic Dynamical Systems

Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a model-checking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2: minor orthographic changes v3: section 5.2 (collatz functions) mathematically improved v4: orthographic corrections, one reference added v5:27 pages. Important modifications. The formalism is strengthened: temporal logic replaced by finite automata. New results. Submitte

Borel Ranks and Wadge Degrees of Context Free Omega Languages

We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power than Turing machines equipped with a B\"uchi acceptance condition. In particular, for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free languages accepted by 1-counter B\"uchi automata, and the supremum of the set of Borel ranks of context free omega languages is the ordinal gamma^1_2 which is strictly greater than the first non recursive ordinal. This very surprising result gives answers to questions of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", LNCS 803, Springer, 1994, p. 583-621]

The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter B\"uchi automaton and a 2-tape B\"uchi automaton such that the \omega-language of the first and the infinitary rational relation of the second in one model of ZFC are \pi_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by B\"uchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author