Busy beavers gone wild

We show some incompleteness results a la Chaitin using the busy beaver functions. Then, with the help of ordinal logics, we show how to obtain a theory in which the values of the busy beaver functions can be provably established and use this to reveal a structure on the provability of the values of these functions

La théorie des cofinalités possibles et ses applications

Cardinal arithmetic, which has given birth to set theory,seemed to be until lately either simple (addition and multiplication ofinfinite cardinals are simple), or quite elastic (by forcing methods, it seemed possible to show the consistency with set theory of any reasonablebehaviour of cardinal exponentiation). Saharon Shelah has developped a rich theory with surprising applications in cardinal arithmetic, changing completely those beliefs. We present a state of the art of this theory anda certain number of its applications.L'arithmétique des cardinaux, qui est à l'origine de lathéorie des ensembles, semblait jusqu'il y a quelques années, soit simple (l'addition et la multiplication de cardinaux infinis sont simples), soit élastique (par le biais de forcing, on pensait pouvoir montrer laconsistence avec la théorie des ensembles de tout comportement raisonnablede l'exponentiation de cardinaux). Saharon Shelah a développé une théorie ayant des applications surprenantes pour l'arithmétique des cardinaux, changeant complètement cette vision des choses. Nous présentonsun état de l'art de cette théorie et un certain nombre d'applications decelle-ci

Infinite Time Cellular Automata: a Real Computation Model

International audienceWe define a new transfinite time model of computation, infinite time cellular automata. The model is shown to be as powerful than infinite time Turing machines, both on finite and infinite inputs; thus inheriting many of its properties. We then show how to simulate the canonical real computation model, BSS machines, with infinite time cellular automata in exactly ω steps

Gödel incompleteness revisited

ISBN 978-5-94057-377-7International audienceWe investigate the frontline of Gödel's incompleteness theorems' proofs and the links with computability

An algorithmic approach to characterizations of admissibles

International audienceSacks proved that every admissible countable ordinal is the first admissible ordinal relatively to a real. We give an algorithmic proof of this result for constructibly countable admissibles. Our study is completed by an algorithmic approach to a generalization of Sacks’ theorem due to Jensen, that finds a real relatively to which a countable sequence of admissibles, having a compatible structure, constitutes the sequence of the first admissibles. Our approach deeply involves infinite time Turing machines. We also present different considerations on the constructible ranks of the reals involved in coding ordinals

Model Theory and Computational Complexity

Model theory has lately become a domain of interest to computer scientists. The reason is that model theory, and in particular its restriction to finite models, has led to some new results in computational complexity (e.g. NLOGSPACE=co-NLOGSPACE). In this report, firstly we present a survey of this theory and we focus on the descriptive complexity aspects and other links between computational complexity and model theory. Secondly, we extend some results of Grandjean and Lynch. We give a more precise logical characterization of complexity classes NTIME(n^d) for some d. This leads us to show applications of this result and to give openings made possible by this result.Les informaticiens théoriques se sont récemment intéressés à la théorie des modèles. La raison est que la théorie des modèles, en particulier lorsque l'on se restreint à des modèles finis, a permis d'aboutir à de nouveaux résultats en complexité (e.g. NLOGSPACE=co-NLOGSPACE). Dans ce rapport, nous présentons premièrement une étude de cette théorie dans le but de présenter les notions de complexité descriptive et d'autres liens entre la complexité et la théorie des modèles. Deuxièmement, nous étendons des résultats de Grandjean et Lynch. Nous donnons une caractérisation logique plus précise des classes de complexité NTIME(n^d) pour un certain d. Enfin, nous montrons des implications de notre résultat et nous donnons différentes ouvertures rendues possibles grâce à ce résultat

On shift-invariant maximal filters and hormonal cellular automata

International audienceThis paper deals with the construction of shift-invariant maximal filters on ℤ and their relation to hormonal cellular automata, a generalization of the cellular automata computation model with some information about the global state shared among all the cells. We first design shift-invariant maximal filters in order to define this new model of computation. Starting from different assumptions, we show how to construct such filters, and analyze the computation power of the induced cellular automata computation model

Infinite Time Busy Beavers

International audienceIn 1962, Hungarian mathematician Tibor Radó introduced in [8] the busy beaver competition for Turing machines: in a class of machines, find one which halts after the greatest number of steps when started on the empty input. In this paper, we generalise the busy beaver competition to the infinite time Turing machines (ITTMs) introduced in [6] by Hamkins and Lewis in 2000. We introduce two busy beaver functions on ITTMs and show both theoretical and experimental results on these functions. We give in particular a comprehensive study, with champions for the busy beaver competition, of the classes of ITTMs with one or two states (in addition to the halt and limit states). The computation power of ITTMs is humongous and thus makes the experimental study of this generalisation of Radó’s competition and functions a daunting challenge. We end this paper with a characterisation of the power of those machines when the use of the tape is restricted in various ways

Calculs et infinis

Nous introduisons une hiérarchie de notions de calcul généralisé. L'idée est de regrouper en une notion tout ce que l'on pourrait qualifier de "calculabilité", de pouvoir étudier ces notions et en fin de compte d'établir des théorèmes de transfert entre elles. Ces notions correspondent certaines fois aussi à des modèles de calcul obtenues par le biais de machines concrètes. Nous avons ainsi un nouveau modèle de calcul avec les " automates cellulaires à temps infini " qui ont l'avantage sur les machines de Turing d'être plus homogènes (absence de tête). La notion de complexité de calcul (selon une certaine notion de calcul) est également généralisée et étudiée. Enfin, nous obtenons des notions de réels aléatoires plus fines que la notion classique de Martin-Löf (ou Kolmogorov) que l'on peut affiner de plus en plus. Tout ceci mène à la notion de complexité de Kolmogorov généralisée qui ouvre des perspectives intéressantes.We introduce a hierarchy of notions of generalized computation. The idea is to almagamate in one concept all that we could qualify of "computability", to study those notions and ultimately to have transfer theorems between those notions. Those notions correspond also in some cases to computation models obtained by means of concrete machines. We obtain in this way a new computation model, "infinite time cellular automata", which are more homogeneous than Turing machines (lack of head). The notion of computational complexity (according to a certain notion of computation) is also generalized and studied. Finally, we obtain notions of random reals that are finer than the classical notion of Martin-Löf (or Kolmogorov) and yet more and more refinable. All of this leads to a notion of generalized Kolmogorov complexity which opens up interesting prospects..LYON-ENS Sciences (693872304) / SudocSudocFranceF