LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

98 Rodica Ceterchi, Carlos Martín-Vide, and K.G. Subramanian
For L ∈ SSH(1, 3) the result is proved in [8]. For L ∈ SSH(2, 4) the same
proof holds, replacing prefix edge with suffix edge.
According to [7], a language L is strictly locally testable if and only if there
exists a positive integer K such that every string in V ∗ of length at least K is a
constant of L. Wehavethus:
Corollary 3. (extension of Corollary 30.2 of [8]) If L is a (1, 3) or a (2, 4)semi-simple
splicing language, then L is strictly locally testable.
5 A Hierarchy of Classes of Semi-simple Languages
To illustrate the number and variety of open problems in this area, let us note
that there is no connection formulated yet between the semi-simple languages
as introduced by Goode and Pixton in [8], and the classes introduced by Tom
Head in [10].
Recall that, inspired by the study of simple splicing languages introduced in
[12], Head introduced the families of splicing languages {SkH | k ≥−1} using the
null context splicing rules defined in [9]. A splicing rule, according to the original
definition of [9], is a sixtuple (u1,x,u2; u3,x,u4) (x is called the cross-over of the
rule) which acts on strings precisely as the (Păun type) rule (u1x, u2; u3x, u4).
(For a detailed discussion of the three types of splicing – definitions given by
Head, Păun, and Pixton – and a comparison of their generative power, we send
the reader to [1], [2], [3], [4].)
A null context splicing rule is a rule of the particular form (1,x,1; 1,x,1)
(all contexts ui are the empty string 1) and thus will be precisely a rule of type
(x, 1; x, 1) with x ∈ V ∗ . Such a rule is identified by Tom Head with the string
x ∈ V ∗ .Forafixedk ≥−1, a splicing system (V,A,R) withrules(x, 1; x, 1) ∈ R
such that |x| ≤k is called an SkH system, and languages generated by an SkH
system are the SkH languages (more appropriately, SkH(1, 3) languages).
Note that the class S1H(1, 3) is precisely the class SH of simple H systems
of [12].
For k = −1, the S−1H(1, 3) systems are precisely those for which the set of
rules R is empty. Thus the languages in the class S−1H(1, 3) are precisely the
finite languages, i.e., S−1H(1, 3) = FIN.
For k =0,theS0H(1, 3) systems have either R empty, or R = {(1, 1; 1, 1)}.
Thus infinite languages in the class S0H(1, 3) are generated by H systems of
the form S =(V,A,{(1, 1; 1, 1)}), with A �= ∅, and it can be easily shown that
L(S) =alph(A) ∗ ,wherealph(A) denotes the set of symbols appearing in strings
of A.
The union of all classes SkH is precisely the set of null context splicing languages.
It is shown in [10] that this union coincides with the family of strictly
locally testable languages (in the sense of [7]). Also from [10] we have the following
result:
Lemma 10. The sequence {SkH(1, 3) | k ≥−1} is a strictly ascending infinite
hierarchy of language families for alphabets of two or more symbols.