In 1978, Courcelle asked for a complete set of axioms and rules for the equational theory of (discrete regular) words equipped with the operations of product, omega power and omega-op power. In this paper we find a simple set of equations and prove they are complete. Moreover, we show that the equational theory is decidable in polynomial time.

We exhibit a new class of grammars with the help of weightfunctions. They are characterized by decreasing the weight during the derivation process. A decision algorithm for the emptiness problem is developed. This class contains non-contextfree grammars. The corresponding language class is identical to the class of ultralinear languages.

We present a description of asymptotic behaviour of languages of bi-infinite words obtained by iterating morphisms defined on free monoids.

We provide alternative proofs and algorithms for results proved by Sénizergues on rational and recognizable free group languages. We consider two different approaches to the basic problem of deciding recognizability for rational free group languages following two fully independent paths: the symmetrification method (using techniques inspired by the study of inverse automata and inverse monoids) and the right stabilizer method (a general approach generalizable to other classes of groups). Several different algorithmic characterizations of recognizability are obtained, as well as other decidability results.

This paper establishes computational equivalence of two seemingly unrelated concepts: linear conjunctive grammars and trellis automata. Trellis automata, also studied under the name of one-way real-time cellular automata, have been known since early 1980s as a purely abstract model of parallel computers, while linear conjunctive grammars, introduced a few years ago, are linear context-free grammars extended with an explicit intersection operation. Their equivalence implies the equivalence of several other formal systems, including a certain restricted class of Turing machines and a certain type of language equations, thus giving further evidence for the importance of the language family they all generate.