Wieslaw Zielonka

Infinite games on finitely coloured graphs with applications to automata on infinite trees

We examine a class of infinite two-person games on finitely coloured graphs. The main aim is to construct finite memory winning strategies for both players. This problem is motivated by applications to finite automata on infinite trees. A special attention is given to the exact amount of memory needed by the players for their winning strategies. Based on a previous work of Gurevich and Harrington and on subsequent improvements of McNaughton we propose a unique framework that allows to reestablish and to improve various results concerning memoryless strategies due to Emerson and Jutla, Mostowski, Klarlund.

The Power and the Limitations of Local Computations on Graphs

This paper is a contribution to understanding the power and the limitations of asynchronous local computations on graphs and networks. We use local computations to define a notion of graph recognition, in particular our model enables a simulation of finite automata on words and on trees. We introduce the notion of k-covering to examine limitations of such systems. For example we prove that we cannot recognize the families of series parallel graphs and planar graphs by means of local computations.

An extension of Kleene's and Ochman´ski's theorems to infinite traces

Abstract As was noted by Mazurkiewicz, traces constitute a convenient tool for describing finite behaviour of concurrent systems. Extending in a natural way Mazurkiewiczs original definition, infinite traces have recently been introduced enabling one to deal with infinite behaviour of nonterminating concurrent systems. In this paper we examine the basic families of recognizable sets and of rational sets of infinite traces. The seminal Kleene characterization of recognizable subsets of the free monoid and its subsequent extensions to infinite words due to Buchi and to finite traces due to Ochmanski are the cornerstones of the corresponding theories. The main result of our paper is an extension of these characterizations to the domain of infinite traces. Using recognizing and weakly recognizing morphisms, as well as a generalization of the Schutzenberger product of monoids, we prove various closure properties of recognizable trace languages. Moreover, we establish normal-form representations for recognizable and rational sets of infinite traces.

A Kleene Theorem for Infinite Trace Languages

Kleenes theorem is considered as one of the cornerstones of theoretical computer science. It ensures that, for languages of finite words, the family of recognizable languages is equal to the family of rational languages. It has been generalized in various ways, for instance, to formal power series by Schutzenberger, to infinite words by Buchi and to finite traces by Ochmanski. Finite traces have been introduced by Mazurkiewicz in order to modelize the behaviours of distributed systems. The family of recognizable trace languages is not closed by Kleenes star but by a concurrent version of this iteration. This leads to the natural definition of co-rational languages obtained as the rational one by simply replacing the Kleenes iteration by the concurrent iteration. Cori, Perrin and Metivier proved, in substance, that any co-rational trace language is recognizable. Independently,Ochmanski generalized Kleenes theorem showing that the recognizable trace languages are exactly the co-rational languages. Besides, infinite traces have been recently introduced as a natural extension of both finite traces and infinite words. In this paper we generalize Kleenes theorem to languages of infinite traces proving that the recognizable languages of finite or infinite traces are exactly the co-rational languages.

Limits of Multi-Discounted Markov Decision Processes

Markov decision processes (MDPs) are controllable discrete event systems with stochastic transitions. The payoff received by the controller can be evaluated in different ways, depending on the payoff function the MDP is equipped with. For example a mean-payoff function evaluates average performance, whereas a discounted payoff function gives more weights to earlier performance by means of a discount factor. Another well-known example is the parity payoff function which is used to encode logical specifications. Surprisingly, parity and mean-payoff MDPs share two non-trivial properties: they both have pure stationary optimal strategies and they both are approximable by discounted MDPs with multiple discount factors (multi- discounted MDPs). In this paper we unify and generalize these results. We introduce a new class of payoff functions called the priority weighted payoff functions, which are generalization of both parity and mean-payoff functions. We prove that priority weighted MDPs admit optimal strategies that are pure and stationary, and that the priority weighted value of an MDP is the limit of the multi-discounted value when discount factors tend to 0 simultaneously at various speeds.

Controlled Timed Automata

We examine some extensions of the basic model, due to Alur and Dill, of real-time automata (RTA). Our model, controlled real-time automata, is a parameterized family of real-time automata with some additional features like clock stopping, variable clock velocities and periodic tests. We illustrate the power of controlled automata by presenting some languages that can be recognized deterministically by such automata, but cannot be recognized non-deterministically by any other previously introduced class of timed automata (even with ɛ-transitions). On the other hand, due to carefully chosen restrictions, controlled automata conserve basic properties of RTA: the emptiness problem is decidable and for each fixed parameter the family of recognized real-time languages is closed under boolean operations.

On the values of repeated games with signals

We study the existence of different notions of value in two-person zero-sum repeated games where the state evolves and players receive signals. We provide some examples showing that the limsup value (and the uniform value) may not exist in general. Then we show the existence of the value for any Borel payoff function if the players observe a public signal including the actions played. We also prove two other positive results without assumptions on the signaling structure: the existence of the

Semi-Commutations and Rational Expressions

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Blackwell-Optimal Strategies in Priority Mean-Payoff Games

value in any game and the existence of the uniform value in recursive games with nonnegative payoffs.

Pure and Stationary Optimal Strategies in Perfect-Information Stochastic Games

We extend the notion of concurrent iteration defined on trace languages by E.Ochmanski to languages closed under a semi-commutation relation which is the non-symmetric version of partial commutation relation: we iterate strongly connected components of words. For a given semi-commutation relation, this leads to the definition of a new family of rational languages closed under the semi-commutation. We give a necessary and sufficient condition for a language closed under a semi-commutation relation to be a rational language and we proved that the equality between the families of rational languages and recognizable languages closed under a semi-commutation relation is true if and only if the semi-commutation is symmetric (i.e a partial commutation).