Olivier Carton

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.

Automata on Linear Orderings

We consider words indexed by linear orderings. These extend finite, (bi-)infinite words and words on ordinals. We introduce automata and rational expressions for words on linear orderings. We prove that for countable scattered linear orderings they are equivalent. This result extends Kleenes theorem. The proofs are effective.

On the complexity of hopcroft’s state minimization algorithm

Hopcroft’s algorithm for minimizing a deterministic automaton has complexity O(n log n). We show that this complexity bound is tight. More precisely, we provide a family of automata of size n = 2k on which the algorithm runs in time k2k. These automata have a very simple structure and are built over a one-letter alphabet. Their sets of final states are defined by de Bruijn words.

An Eilenberg Theorem for Words on Countable Ordinals

We present in this paper an algebraic approach to the theory of languages of words on countable ordinals. The algebraic structure used, called an Ω1-semigroup, is an adaptation of the one used in the theory of regular languages of Ω-words. We show that finite Ω1-semigroups are equivalent to automata. In particular, the proof gives a new algorithm for determinizing automata on countable ordinals. As in the cases of finite and Ω-words, a syntactic Ω1-semigroup can effectively be associated with any regular language of words on countable ordinals. This result is used to prove an Eilenberg type theorem. There is a one-to-one correspondence between varieties of Ω1-languages and pseudo-varieties of Ω1-semigroups.

Chains and Superchains for ω-Rational Sets, Automata and Semigroups

We introduce several equivalent notions that generalize ones introduced by Klaus Wagner for finite Muller automata under the name of chains and superchains. We define such objects in relation to ω-rational sets, Muller automata or also ω-semigroups. We prove their equivalence and derive some basic properties of these objects. In a subsequent paper, we show how these concepts allow us to derive a new presentation of the hierarchy due to K. Wagner and W. Wadge.

Logic and Rational Languages of Words Indexed by Linear Orderings

We prove that every rational language of words indexed by linear orderings is definable in monadic second-order logic. We also show that the converse is true for the class of languages indexed by countable scattered linear orderings, but false in the general case. As a corollary we prove that the inclusion problem for rational languages of words indexed by countable linear orderings is decidable.

Determinization of transducers over finite and infinite words

We study the determinization of transducers over finite and infinite words. The first part of the paper is devoted to finite words. We recall the characterization of subsequential functions due to Choffrut. We describe here a known algorithm to determinize a transducer.In the case of infinite words, we consider transducers with all their states final. We give an effective characterization of sequential functions over infinite words. We describe an algorithm to determinize transducers over infinite words. This part contains the main novel results of the paper.

Computing the Rabin index of a parity automaton

The Rabin index of a rational language of infinite words given by a parity automaton with n states is computable in time O(n(2)c) where c is the cardinality of the alphabet. The number of values used by a parity acceptance condition is always greater than the Rabin index and conversely, the acceptance condition of a parity automaton can always be replaced by an equivalent acceptance condition whose number of used values is exactly the Rabin index. This new acceptance condition can also be computed in time O(n(2)C).

Decision problems among the main subfamilies of rational relations

We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the families, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the general case. In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable. Also we exhibit a single exponential algorithm for determining if a synchronous relation is recognizable.

Regular languages of words over countable linear orderings

We develop an algebraic model for recognizing languages of words indexed by countable linear orderings. This notion of recognizability is effectively equivalent to definability in monadic second-order (MSO) logic. The proofs also imply the first known collapse result for MSO logic over countable linear orderings.