Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Derandomizing Isolation in Space-Bounded Settings

We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits. A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Dagstuhl News January - December 2011

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Automates à contraintes semilinéaires = Automata with a semilinear constraint

Cette thèse présente une étude dans divers domaines de l'informatique théorique de modèles de calculs combinant automates finis et contraintes arithmétiques. Nous nous intéressons aux questions de décidabilité, d'expressivité et de clôture, tout en ouvrant l'étude à la complexité, la logique, l'algèbre et aux applications. Cette étude est présentée au travers de quatre articles de recherche. Le premier article, Affine Parikh Automata, poursuit l'étude de Klaedtke et Ruess des automates de Parikh et en définit des généralisations et restrictions. L'automate de Parikh est un point de départ de cette thèse; nous montrons que ce modèle de calcul est équivalent à l'automate contraint que nous définissons comme un automate qui n'accepte un mot que si le nombre de fois que chaque transition est empruntée répond à une contrainte arithmétique. Ce modèle est naturellement étendu à l'automate de Parikh affine qui effectue une opération affine sur un ensemble de registres lors du franchissement d'une transition. Nous étudions aussi l'automate de Parikh sur lettres: un automate qui n'accepte un mot que si le nombre de fois que chaque lettre y apparaît répond à une contrainte arithmétique. Le deuxième article, Bounded Parikh Automata, étudie les langages bornés des automates de Parikh. Un langage est borné s'il existe des mots w_1, w_2, ..., w_k tels que chaque mot du langage peut s'écrire w_1...w_1w_2...w_2...w_k...w_k. Ces langages sont importants dans des domaines applicatifs et présentent usuellement de bonnes propriétés théoriques. Nous montrons que dans le contexte des langages bornés, le déterminisme n'influence pas l'expressivité des automates de Parikh. Le troisième article, Unambiguous Constrained Automata, introduit les automates contraints non ambigus, c'est-à-dire pour lesquels il n'existe qu'un chemin acceptant par mot reconnu par l'automate. Nous montrons qu'il s'agit d'un modèle combinant une meilleure expressivité et de meilleures propriétés de clôture que l'automate contraint déterministe. Le problème de déterminer si le langage d'un automate contraint non ambigu est régulier est montré décidable. Le quatrième article, Algebra and Complexity Meet Contrained Automata, présente une étude des représentations algébriques qu'admettent les automates contraints et les automates de Parikh affines. Nous déduisons de ces caractérisations des résultats d'expressivité et de complexité. Nous montrons aussi que certaines hypothèses classiques en complexité computationelle sont reliées à des résultats de séparation et de non clôture dans les automates de Parikh affines. La thèse est conclue par une ouverture à un possible approfondissement, au travers d'un certain nombre de problèmes ouverts.This thesis presents a study from the theoretical computer science perspective of computing models combining finite automata and arithmetic constraints. We focus on decidability questions, expressiveness, and closure properties, while opening the study to complexity, logic, algebra, and applications. This thesis is presented through four research articles. The first article, Affine Parikh Automata, continues the study of Klaedtke and Ruess on Parikh automata and defines generalizations and restrictions of this model. The Parikh automaton is one of the starting points of this thesis. We show that this model of computation is equivalent to the constrained automaton that we define as an automaton which accepts a word only if the number of times each transition is taken satisfies a given arithmetic constraint. This model is naturally extended to affine Parikh automata, in which an affine transformation is applied to a set of registers on taking a transition. We also study the Parikh automaton on letters, that is, an automaton which accepts a word only if the number of times each letter appears in the word verifies an arithmetic constraint. The second article, Bounded Parikh Automata, focuses on the bounded languages of Parikh automata. A language is bounded if there are words w_1, w_2, ..., w_k such that every word in the language can be written as w_1...w_1w_2...w_2 ... w_k...w_k. These languages are important in applications and usually display good theoretical properties. We show that, over the bounded languages, determinism does not influence the expressiveness of Parikh automata. The third article, Unambiguous Constrained Automata, introduces the concept of unambiguity in constrained automata. An automaton is unambiguous if there is only one accepting path per word of its language. We show that the unambiguous constrained automaton is an appealing model of computation which combines a better expressiveness and better closure properties than the deterministic constrained automaton. We show that it is decidable whether the language of an unambiguous constrained automaton is regular. The fourth article, Algebra and Complexity Meet Constrained Automata, presents a study of algebraic representations of constrained automata and affine Parikh automata. We deduce expressiveness and complexity results from these characterizations. We also study how classical computational complexity hypotheses help in showing separations and nonclosure properties in affine Parikh automata. The thesis is concluded by a presentation of possible future avenues of research, through several open problems

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum