Marie-Pierre Béal

Symbolic dynamics and finite automata

Symbolic dynamics is a field which was born with the work in topology of Marston Morse at the beginning of the 1920s [44]. It is, according to Morse, an “algebra and geometry of recurrence”. The idea is the following. Divide a surface into regions named by certain symbols. We then study the sequences of symbols obtained by scanning the successive regions while following a trajectory starting from a given point. A further paper by Morse and Hedlund [45] gave the basic results of this theory. Later, the theory was developed by many authors as a branch of ergodic theory (see for example the collected works in [59] or [12]). One of the main directions of research has been the problem of the isomorphism of shifts of finite type (see below the definition of these terms). This problem is not yet completely solved although the latest results of Kim and Roush [35] indicate a counterexample to a long-standing conjecture formulated by F. Williams [61].

Squaring transducers: an efficient procedure for deciding functionality and sequentiality

We describe here a construction on transducers that give a new conceptual proof for two classical decidability results on transducers: it is decidable whether a finite transducer realizes a functional relation, and whether a finite transducer realizes a sequential relation. A better complexity follows then for the two decision procedures.

A QUADRATIC UPPER BOUND ON THE SIZE OF A SYNCHRONIZING WORD IN ONE-CLUSTER AUTOMATA

Cernýs conjecture asserts the existence of a synchronizing word of length at most (n - 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p·ar = q·as for some integers r, s (for a state p and a word w, we denote by p·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n2). This applies in particular to Huffman codes.

Conjugacy and equivalence of weighted automata and functional transducers

We show that two equivalent

Minimal Forbidden Words and Symbolic Dynamics

\mathbb{K}

On the equivalence of Z -automata

-automata are conjugate to a third one, when

Determinization of transducers over finite and infinite words

\mathbb{K}

A quadratic algorithm for road coloring

is equal to

On the bound of the synchronization delay of a local automaton

\mathbb{B, N, Z}

A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata

, or any (skew) field and that the same holds true for functional tranducers as well.