Kleene-like Theorem Research Articles

Series-parallel languages on scattered and countable posets

In this paper, we investigate the recognition by finite automata of languages of countable labelled posets. We unify and generalize several previous results from two different directions: the theory of finite or ω N -free posets, and automata over countable and scattered linear orderings. First, we establish that the smallest class of posets obtained from the empty set and the singleton and closed under finite parallel operation and sequential concatenation indexed by all linear orderings corresponds precisely to the class of scattered and countable N -free posets without infinite antichains. Next, we prove a Kleene-like theorem. We define automata and rational expressions for languages of countable, scattered, N -free labelled posets without infinite antichains, and show that both formalisms have the same expressive power.

Formal language properties of hybrid systems with strong resets

We study hybrid systems with strong resets from the perspective of formal language theory. We define a notion of hybrid regular expression and prove a Kleene-like theorem for hybrid systems. We also prove the closure of these systems under determinisation and complementation. Finally, we prove that the reachability problem is undecidable for synchronized products of hybrid systems.

A KLEENE THEOREM FOR LANGUAGES OF WORDS INDEXED BY LINEAR ORDERINGS

In a preceding paper, Bruyère and Carton introduced automata, as well as rational expressions, which allow to deal with words indexed by linear orderings. A Kleene-like theorem was proved for words indexed by countable scattered linear orderings. In this paper we extend this result to languages of words indexed by all linear orderings.

Hierarchy Among Automata on Linear Orderings

In a preceding paper, automata and rational expressions have been introduced for words indexed by linear orderings, together with a Kleene-like theorem. We here pursue this work by proposing a hierarchy among the rational sets. Each class of the hierarchy is defined by a subset of the rational operations that can be used. We then characterize any class by an appropriate class of automata, leading to a Kleene theorem inside the class. A characterization by particular classes of orderings is also given.

Several properties of array languages

The languages generated by connected and disconnected array grammars are studied. We define substitutions of array languages, and we study closure properties of classes of array languages under substitutions. A characterization by means of substitutions for connected context-free array languages and a Kleene-like theorem for regular array languages are given.