In 1978, Courcelle asked for a completeset 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 equationsand 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-infinitewords obtained by iterating morphisms defined on free monoids.

We provide alternative proofs and algorithms for resultsproved by Sénizergues on rational and recognizable freegroup languages. We consider two different approaches to the basicproblem of deciding recognizability for rational free group languagesfollowing two fully independent paths: the symmetrificationmethod (using techniques inspired by the study ofinverse automata and inverse monoids) andthe right stabilizer method (a general approach generalizable to otherclasses of groups). Several different algorithmic characterizations ofrecognizability 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, whilelinear conjunctive grammars, introduced a few years ago, are linear context-free grammars extendedwith 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, thusgiving further evidence for the importance of the language family they all generate.