On the Non-Deterministic Communication Complexity of Regular Languages

Abstract

The notion of communication complexity was introduced by Yao in his sem-inal paper [Yao79]. In [BFS86], Babai Frankl and Simon developed a rich structure of communication complexity classes to understand the relationships between various models of communication complexity. This made it apparent that communication complexity was a self-contained mini-world within complexity theory. In this thesis, we study the place of regular languages within this mini-world. In particular, we are interested in the non-deterministic communication complexity of regular languages. We show that a regular language has either O(1) or Ω(log n) non-determi-nistic complexity. We obtain several linear lower bound results which cover a wide range of regular languages having linear non-deterministic complexity. These lower bound results also imply a result in semigroup theory: we obtain sufficient conditions for not being in the positive variety P ol(Com). To obtain our results, we use algebraic techniques. In the study of regular languages, the algebraic point of view pioneered by Eilenberg ([Eil74]) has led to many interesting results. Viewing a semigroup as a computational device that recognizes languages has proven to be prolific from both semigroup theory and formal languages perspectives. In this thesis, we provide further instances of such mutualism. i Résumé La notion de complexité de communication a d'abordétéabordété introduite par Yao [Yao79]. Les travaux fondateurs de Babai et al. [BFS86] ont dévoilé une riche structures de classes de complexité de communication qui permettent de mieux comprendre la puissance de divers modles de complexité de communication. Ces résultats ont fait de la complexité de communication une sorte de maquette petite chelle du monde de la complexit. Dans ce mémoire, nousétudionsnousétudions la place des langages réguliers dans cette maquette. Plus précisément, nous chercherons déterminer la complexité de communication non-déterministe de ces langages. Nous montrons qu'un langage régulier a une complexité de communication soit O(log n), soit Ω(log n). NousétablissonsNousétablissons de plus des bornes inférieures linéaires sur la complexité non-déterministe d'une vaste classe de langages. Celles-ci fournissentégalementfournissentégalement des conditions suffisantes pour qu'un langage donné n'appartienne pas la variété positive P ol(Com). Nos résultats se basent sur des techniques algébriques. Dans l'´ etude des langages réguliers, le point de vue algébrique, développé initialement par Eilenberg [Eil74] s'est révélé comme un outil central. En effet, on peut voir un semigroupe fini comme une machine capable de reconnaˆıtrereconnaˆıtre des langages et cette perspective a permis des avancées tant en théorie des semigroupes qu'en théorie des langages formels. Dans ce mémoire, nousétablissonsnousétablissons de nouveaux exemples de ce mutualisme. ii

An application of communication complexity, Kolmogorov complexity and extremal combinatorics to parity games

“…Apparently, the most important one is obtaining lower bounds on state complexity of non-deterministic automata (NFA) (see, e.g., the monograph [15]). CC is also applied to analysis of nondeterminism measures in finite automata [16] and for a number of other problems in FLT (e.g., see [1] for bounds on nondeterministic communication complexity of regular languages). Also, it is worth to mention that lower bounds on memory used in streaming algorithms, an another important application of CC (see the book [26]), can be viewed as lower bounds on size of probabilistic automata of specific form.…”

On fuzzy multiset regular languages

“…Thereafter, there were number of authors having contributed to this field (cf., [15,24,26,29,30,35]). In application point of view, fuzzy regular language have been shown to be useful in numerous engineering applications such as learning systems, computing with word, fuzzy discrete event systems, pattern recognition, database theory, lexical analysis in programming language compilation, user-interface translations, circulation design and text editing (cf., [1,7,16,17,25]). In recent years their application have been further extended to include parallel processing, image generation and compression, type theory for object-oriented languages, DNA computing, etc.…”