Circular splicing has been very recently introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we restrict our study to the relationship between regular circular languages and languages generated by finite circular splicing systems and provide some results towards a characterization of the intersection between these two classes. We consider the class of languages X*, called here star languages, which are closed under conjugacy relation and with X being a regular language. Using automata theory and combinatorial techniques on words, we show that for a subclass of star languages the corresponding circular languages are (Paun) circular splicing languages. For example, star languages belong to this subclass when X* is a free monoid or X is a finite set. We also prove that each (Paun) circular splicing language L over a one-letter alphabet has the form L = X+ ∪ Y, with X,Y finite sets satisfying particular hypotheses. Cyclic and weak cyclic languages, which will be introduced in this paper, show that this result does not hold when we increase the size of alphabets, even if we restrict ourselves to regular languages.

We give a linear time algorithm which, given a simply connected figure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed flow lines which match inputs and outputs according to the colors, in such a way that each edge of any cell is crossed by at most one line. The main tool is the notion of height function, previously introduced for tilings. It appears as an extension of the notion of potential of a flow in a planar graph.

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

We introduce the notion of a differentiation function of a context-free grammar which gives the number of terminal words that can be derived in a certain number of steps. A grammar is called narrow (or k-narrow) iff its differentiation function is bounded by a constant (by k). We present the basic properties of differentiation functions, especially we relate them to structure function of context-free languages and narrow grammars to slender languages. We discuss the decidability of the equivalence of grammars with respect to the differentiation function and structure function and prove the decidability of the k-narrowness of context-free grammars. Furthermore, we introduce languages representing the graph of the differentiation and structure function and relate these languages to those of the Chomsky hierarchy.

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny et al. (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny et al. (2004) use it.