4 research outputs found
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Non-Deterministic Communication Complexity of Regular Languages
In this thesis, we study the place of regular languages within the
communication complexity setting. 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 Omega(log n)
non-deterministic 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 Pol(Com).
To obtain our results, we use algebraic techniques. In the study of regular
languages, the algebraic point of view pioneered by Eilenberg (\cite{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.Comment: Master's thesis, 93 page