LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

86 Rodica Ceterchi, Carlos Martín-Vide, and K.G. Subramanian
systems can generate non-regular circular languages (see [17],[15]). On the other
hand, there are regular circular languages (for instance the language ˆ(aa) ∗ b)
which cannot be generated by any finite circular splicing system (with Păun
type of rules) (see [2], [3]). The fact that the behavior in the circular case is
by no means predictable from the behavior in the linear case is made apparent
by the example of the ˆ(aa) ∗ circular language, which is both a regular and a
splicing language (see [2], [3]).
This has motivated the beginning of the study of simple circular H systems
in the paper [5]. We recall briefly in Section 6 some results of [5], which are
completed here by some closure properties.
In Section 7 we introduce the semi-simple circular splicing systems and first
steps in the study of such systems are made by proving the incomparability of
classes SSH ◦ (1, 3) and SSH ◦ (2, 4), a situation which is totally dissimilar to the
simple circular case, where more classes colapse than in the simple linear case.
The following table summarizes the notations used for the different types of
H systems and the respective classes of languages considered throughout the
paper, and the main references for the types already studied in the literature.
Table 1. Types of splicing systems and languages
linear case, V ∗ circular case, V ◦
simple SH [12] SH ◦
[5]
semi-simple SSH [8] SSH ◦
k-simple SkH [10]
k-semi-simple SSkH
2 Preliminaries and Notations
For a finite alphabet V we denote by V ∗ the free monoid over V ,by1theempty
string, and we let V + = V ∗ \{1}. Forastringw ∈ V ∗ we denote by |w| its
length, and if a ∈ V is a letter, we denote by |w|a the number of occurrences of
a in w. Weletmi : V ∗ −→ V ∗ denote the mirror image function, and we will
denote still by mi its restriction to subsets of V ∗ . For a word w ∈ V ∗ ,wecall
u ∈ V ∗ a factor of w, ifthereexistx, y ∈ V ∗ such that w = xuy. Wecallu ∈ V ∗
a factor of a language L, ifu is a factor of some word w ∈ L.
On V ∗ we consider the equivalence relation given by xy ∼ yx for x, y ∈ V ∗
(the equivalence given by the circular permutations of the letters of a word).
A circular string over V will be an equivalence class w.r.t. the relation above.
We denote by ˆw the class of the string w ∈ V ∗ .WedenotebyV ◦ the set of
all circular strings over V , i.e., V ◦ = V ∗ / ∼. The empty circular string will be
denoted ˆ1, and we let V ⊕ = V ◦ \{ˆ1}. Any subset of V ◦ will be called a circular
language.
For circular words, the notions of length and number of occurrences of a
letter, can be immediately defined using representatives: |ˆw| = |w|, and|ˆw|a =