Against an adaptive adversary, we show that the power of randomization in online algorithms is severely limited! We prove the existence of an efficient "simulation" of randomized online algorithms by deterministic ones, which is best possible in general. The proof of the upper bound is existential. We deal with the issue of computing the efficient deterministic algorithm, and show that this is possible in very general cases. 1 Introduction and Overview of Results Beginning with the work of Sleator and Tarjan [17], there has recently been a development of what might be called a Theory of Online Algorithms. The particular algorithmic problems analyzed in the Sleator and Tarjan paper are "list searching" and "paging", both well studied problems. But the novelty of their paper lies in a new measure of performance, later to be called the "competitive ratio", for online algorithms. This new approach, called "competitive analysis" in Karlin, Manasse, Rudolph and Sleator [11], seems to have...

We consider Borel sets of the form A ` ! (with usual topology) where cardinality of is less than some uncountable regular cardinal . We obtain a "normal form" of A, by finding a Borel set\Omega\Gamma ff) such that A and\Omega\Gamma ff) continuously reduce to each other. We do so by defining Borel operations which are homomorphic to the first Veblen ordinal functions of base required to compute the Wadge degree of the set A: the ordinal ff.

This thesis presents research done by the author on competitive analysis of on-line problems. An on-line problem is a problem that is given and solved one piece at a time. An on-line strategy for solving such a problem must give the solution to each piece knowing only the current piece and preceding pieces, in ignorance of the pieces to be given in the future. We consider on-line strategies that are competitive (guaranteeing solutions whose costs are within a constant factor of optimal) for several combinatorial optimization problems: paging, weighted caching, the k-server problem, and weighted matching. We introduce variations on the standard model of competitive analysis for paging: allowing randomization, allowing resource-bounded lookahead, and loose competitiveness, in which performance over a range of fast memory sizes is considered and noncompetitiveness is allowed provided the fault rate is insignificant. Each variation leads to substantially better competitive ratios. We prese...

Abstract. We study infinite games played by arbitrarily many players on a directed graph. Equilibrium states capture rational behaviour in these games. Instead of the well-known notion of a Nash equilibrium, we focus on the notion of a subgame perfect equilibrium. We argue that the latter one is more appropriate for the kind of games we study, and we show the existence of a subgame perfect equilibrium in any infinite game with ω-regular winning conditions. As, in general, equilibria are not unique, it is appealing to compute one with a maximal payoff. This problem corresponds naturally to the problem of deciding given a game and two payoff vectors whether the game has an equilibrium with a payoff in between the given thresholds. We show that this problem is decidable for games with ω-regular winning conditions played on a finite graph and analyse its complexity. Moreover, we establish that any subgame perfect equilibrium of a game with ω-regular winning conditions played on a finite graph can be implemented by finite-state strategies. Finally, we consider logical definability. We state that if we fix the number of players together with an ω-regular winning condition for each of them and two payoff vectors the property that a game has a subgame perfect equilibrium with a payoff in between the given thresholds is definable in the modal µ-calculus. 1

Given well ordered countable sets of the form , we consider Borel mappings from ! with countable image inside the ordinals. The ordinals and ! are respectively equipped with the discrete topology and the product of the discrete topology on . The Steel well-ordering on such mappings is dened by i there exists a continuous function f such that (x) (x) holds for any x 2 ! . It induces a hierachy of mappings which we give a complete description of. We provide, for each ordinal , a mapping T() whose rank is precisely in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by . These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy. 1

Many temporal logics were suggested as branching time specification formalisms during the last 20 years. These logics were compared against each other for their expressive power, model checking complexity and succinctness. Yet, unlike the case for linear time logics, no canonical temporal logic of branching time was agreed upon. We offer an explanation for the multiplicity of temporal logics over branching time and provide an objective quantified `yardstick' to measure these logics. We define

We consider finite-state games as a model of nonterminating reactive computations. A natural type of specification is given by games with Streett winning condition (corresponding to automata accepting by conjunctions of fairness conditions). We present an algorithm which solves the problem of program synthesis for these specifications. We proceed in two steps: First, we give a reduction of Streett automata to automata with the Rabin chain (or parity) acceptance condition. Secondly, we develop an inductive strategy construction over Rabin chain automata which yields finite automata that realize winning strategies. For the step from Rabin chain games to winning strategies examples are discussed, based on an implementation of the algorithm. 1 Introduction In recent years, methods of automatic verification for finite--state programs have been applied successfully, which have clearly reached the level of practical use. For the existing automata theoretic results on finite--state program sy...

This paper presents a game semantics for LP, Artemov’s Logic of Proofs. The language of LP extends that of propositional logic by adding formula-labeling terms, permitting us to take a term t and an LP formula A and form the new formula t:A. We define a game semantics for this logic that interprets terms as winning strategies on the formulas they label, so t:A may be read as “t is a winning strategy on A. ” LP may thus be seen as a logic containing in-language descriptions of winning strategies on its own formulas. We apply our semantics to show how winnable instances of certain extensive games with perfect information may be embedded into LP. This allows us to use LP to derive a winning strategy on the embedding, from which we can extract a winning strategy on the original, non-embedded game. As a concrete illustration of this method, we compute a winning strategy for a winnable instance of the well-known game Nim. 1

. We discuss the extent to which game semantics is implicit in the basic concepts of linear logic. Introduction The purpose of this paper is to show that a version of game semantics for linear logic is implicit in the logic itself and the basic intuitions underlying the logic. Like the talk at CSL'93 on which it is based, the body of this paper is intended to be accessible to people with little or no previous knowledge of linear logic or game semantics. Comments that do presuppose such prior knowledge have been relegated to a series of notes at the end of the paper. Propositions as Types The relevance of various constructive propositional logics, including linear logic, to computation and particularly to type theory is largely based on the propositionsas -types paradigm, also often called the Curry-Howard isomorphism [8, 9, 13]. In its simplest form, this paradigm involves a correspondence between the constructive logic of implication and simple typed combinatory logic. Constructive...

Abstract. We study the complexity of Nash equilibria in infinite (turnbased, qualitative) multiplayer games. Chatterjee & al. showed the existence of a Nash equilibrium in any such game with ω-regular winning conditions, and they devised an algorithm for computing one. We argue that in applications it is often insufficient to compute just some Nash equilibrium. Instead, we enrich the problem by allowing to put (qualitative) constraints on the payoff of the desired equilibrium. Our main result is that the resulting decision problem is NP-complete for games with co-Büchi, parity or Streett winning conditions but fixed-parameter tractable for many natural restricted classes of games with parity winning conditions. For games with Büchi winning conditions we show that the problem is, in fact, decidable in polynomial time. We also analyse the complexity of strategies realising a Nash equilibrium. In particular, we show that pure finite-state strategies as opposed to arbitrary mixed strategies suffice to realise any Nash equilibrium of a game with ω-regular winning conditions with a qualitative constraint on the payoff. 1