LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Splicing Test Tube Systems 141
the third section we show that both variants of test tube systems considered in
this paper are equivalent and therefore have the same computational power. In
the fourth section we show that these variants of test tube systems using splicing
rules and P systems with splicing rules assigned to membranes are equivalent,
too. An outlook to related results and a short summary conclude the paper.
2 Definitions
In this section we define some notions from formal language theory and recall
the definitions of splicing schemes (H-schemes, e.g., see [2], [14]) and splicing test
tube systems (HTTS, e.g., see [2], [7]). Moreover, we define the new variant of
test tube systems with communication by splicing (HTTCS). Finally, we recall
the definition of membrane systems with splicing rules assigned to membranes
as introduced in [9].
2.1 Preliminaries
An alphabet V is a finite non-empty set of abstract symbols. GivenV ,thefree
monoid generated by V under the operation of concatenation is denoted by V ∗ ;
the empty string is denoted by λ, andV ∗ \{λ} is denoted by V + .By| x | we
denote the length of the word x over V . For more notions from the theory of
formal languages, the reader is referred to [4] and [17].
2.2 Splicing Schemes and Splicing Systems
A molecular scheme is a pair σ =(B,P), where B is a set of objects and P is a
set of productions. A production p in P in general is a partial recursive relation
⊆ B k × B m for some k, m ≥ 1, where we also demand that for all w ∈ B k the
range p (w) is finite, and moreover, there exists a recursive procedure listing all
v ∈ B m with (w, v) ∈ p. ForanytwosetsL and L ′ over B, we say that L ′ is
computable from L by a production p if and only if for some (w1, ..., wk) ∈ B k
and (v1, ..., vm) ∈ B m with (w1, ..., wk,v1, ..., vm) ∈ p we have {w1, ..., wk} ⊆L
and L ′ = L∪{v1, ..., vm} ;wealsowriteL =⇒p L ′ and L =⇒σ L ′ . A computation
in σ is a sequence L0, ..., Ln such that Li ⊆ B, 0 ≤ i ≤ n, n ≥ 0, as well as
Li =⇒σ Li+1, 0 ≤ i