Determinising Parity Automata

Parity word automata and their determinisation play an important role in automata and game theory. We discuss a determinisation procedure for nondeterministic parity automata through deterministic Rabin to deterministic parity automata. We prove that the intermediate determinisation to Rabin automata is optimal. We show that the resulting determinisation to parity automata is optimal up to a small constant. Moreover, the lower bound refers to the more liberal Streett acceptance. We thus show that determinisation to Streett would not lead to better bounds than determinisation to parity. As a side-result, this optimality extends to the determinisation of B\"uchi automata

Parity and generalised Büchi automata - determinisation and complementation

In this thesis, we study the problems of determinisation and complementation of finite automata on infinite words. We focus on two classes of automata that occur naturally: generalised Büchi automata and nondeterministic parity automata. Generalised Büchi and parity automata occur naturally in model-checking, realisability checking and synthesis procedures. We first review a tight determinisation procedure for Büchi automata, which uses a simplification of Safra trees called history trees. As Büchi automata are special types of both generalised Büchi and parity automata, we adjust the data structure to arrive at suitably tight determinisation constructions for both generalised Büchi and parity automata. As the parity condition describes combinations of Büchi and CoBüchi conditions, instead of immediately modifying the data structure to handle parity automata, we arrive at a suitable data structure by first looking at a special case, Rabin automata with one accepting pair. One pair Rabin automata correspond to parity automata with three priorities and serve as a starting point to modify the structures that result from Büchi determinisation: we then nest these structures to reflect the standard parity condition and describe a direct determinisation construction. The generalised Büchi condition is characterised by an accepting family with 'k' accepting sets. It is easy to extend classic determinisation constructions to handle generalised Büchi automata by incorporating the degeneralization algorithm in the determinisation construction. We extend the tight Büchi construction to do exactly this. Our determinisation constructions go to deterministic Rabin automata. It is known that one can determinise to the more convenient parity condition by incorporating the standard Latest Appearance Record construction in the determinisation procedure. We determinise to parity automata using this technique. We prove lower bounds on these constructions. In the case of determinisation to Rabin automata, our constructions are tight to the state. In the case of determinisation to parity, there is a constant factor ≤ 1.5 between upper and lower bounds reducing to optimal(to the state) in the case of Büchi and 1-pair Rabin. We also reconnect tight determinisation and complementation and provide constructions for complementing generalised Büchi and parity automata by starting withour data structure for determinisation. We introduce suitable data structures for the complementation procedures based on the data structure used for determinisation. We prove lower bounds for both constructions that are tight upto an O(n) factor where 'n' is the number of states of the nondeterministic automaton that is complemented

Programmation par contraintes sur les flux de données

We study the generalization of constraint programming on variables finite domains with variable flow. On the one hand, the flow of concepts, infinite sequences and infinite words have been the subject of numerous studies, and a goal is to achieve a state of the art covering language theory, classical and temporal logics as well as many related formalisms. The reconciliation performed with temporal logics is a first step towards unification formalisms on flows and temporal logics being themselves many, we establish a classification of these will allow the extrapolation of contributions to other contexts. The second objective is to identify the elements of these formalisms that allow the processing of satisfaction problems with the techniques of constraint programming on finite domain variables. Compared to the expressiveness of temporal logic, that of our formalism is more limited. This is due to the fact that constraint programming allows only the conjunction of constraints and requires integrating the disjunction in the notion of constraint propagator. Our formalism allows a gain in conciseness and reuse of the concept of propagator. The issue of generalization to more expressive logics is left open.Nous étudions la généralisation de la programmation par contraintes sur les variables à domaines finis aux variables flux. D'une part, les concepts de flux, de séquences infinies et de mots infinis ont fait l'objet de nombreux travaux, et un objectif consiste à réaliser un état de l'art qui couvre la théorie des langages, les logiques classiques et temporelles, ainsi que les nombreux formalismes apparentés. Le rapprochement effectué avec les logiques temporelles est un premier pas vers l'unification des formalismes sur les flux, et les logiques temporelles étant elles-même nombreuses, nous établissons une classification de celles-ci qui permettra l'extrapolation des contributions à d'autres contextes. Le second objectif consiste à identifier les éléments de ces formalismes qui permettent le traitement des problèmes de satisfaction avec les techniques de la programmation par contraintes sur les variables à domaines finis. Comparée à l'expressivité des logiques temporelles, celle de notre formalisme est plus limitée. Ceci est dû au fait que la programmation par contraintes ne permet que la conjonction de contraintes, et impose d'intégrer la disjonction dans la notion de propagateur de contraintes. Notre formalisme permet un gain en concision et la réutilisation de la notion de propagateur. La question de la généralisation à des logiques plus expressives est laissée ouverte

IST Austria Technical Report

Computing the winning set for Büchi objectives in alternating games on graphs is a central problem in computer aided verification with a large number of applications. The long standing best known upper bound for solving the problem is ̃O(n·m), where n is the number of vertices and m is the number of edges in the graph. We are the first to break the ̃O(n·m) boundary by presenting a new technique that reduces the running time to O(n2). This bound also leads to O(n2) time algorithms for computing the set of almost-sure winning vertices for Büchi objectives (1) in alternating games with probabilistic transitions (improving an earlier bound of O(n·m)), (2) in concurrent graph games with constant actions (improving an earlier bound of O(n3)), and (3) in Markov decision processes (improving for m > n4/3 an earlier bound of O(min(m1.5, m·n2/3)). We also show that the same technique can be used to compute the maximal end-component decomposition of a graph in time O(n2), which is an improvement over earlier bounds for m > n4/3. Finally, we show how to maintain the winning set for Büchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in O(n) amortized time per operation. This is the first dynamic algorithm for this problem

Automates cellulaires non-uniformes

This thesis introduces a new dynamical system which generalizes the cellular automata (CA) : non-uniform cellular automata (nuCA). The nuCAs are obtained by relaxing the spatial uniformity of the local rule of CAs. The global rule is now given by a distribution of local rules. Several classes of nuCAs are defined with respect to restrictions on the distributions. Firstly, nuCAs are used to modelize structural perturbations on CAs. This allows to study the resiliency of CAs in this field. Secondly, the new model is studied for itself. In particular, a characterization of rules distributions inducing some properties (number conserving, surjectivity, injectivity, equicontinuity, ...) are given.Cette thèse introduit un nouveau système dynamique qui généralise les automates cellulaires classiques (CA) : les automates cellulaires non-uniformes (nuCA). Les nuCA sont obtenus en relaxant l'uniformité spatiale de la règle local d'un automate cellulaire, la règle globale est alors obtenu par une distribution de règles locales. Plusieurs classes de nuCA sont identifiées en fonction de restrictions sur les distributions. Dans un premier temps, les nuCA sont utilisés pour modéliser des perturbations structurelles d'un CA. Cette approche permet d'étudier la résilience des CA dans ce cadre. Dans un deuxième temps, le nouveau modèle est étudié pour lui-même. En particulier, des caractérisations sont données sur les distributions de règles qui engendrent certaines propriétés (conservation du nombre, surjectivité, injectivité, équicontinuité, ...)