LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

368 Sergey Verlan
The number of test tubes used in [1] to generate a language was dependent on
the language itself. It was also shown that only one tube generates the class of
regular languages.
A series of papers then showed how to create test tube systems able to
generate any recursively enumearble language using fewer and fewer test tubes.
In [2] this result is obtained with 9 test tubes, in [11] with 6, and in [15] with 3
tubes.
In [1], it is shown that test tube systems with two test tubes can generate
non regular languages, but the problem whether or not we can generate all
recursively enumerable languages with two test tubes is still open. Nevertheless,
in [3] R. and F. Freund showed that two test tubes are able to generate all
recursively enumerable languages if the filtering process is changed. In the variant
they propose, a filter is composed by a finite union of sets depending upon the
simulated system. In [4] P. Frisco and C. Zandron propose a variant which uses
two symbols filter, which is a set of single symbols and couples of symbols. A
string can pass the filter if it is made from single symbols or it contains both
elements from a couple. In the case of this variant two test tubes suffice to
generate any recursively enumerable language.
In this paper we propose a new variant of test tube systems which differs from
the original definition by the filtering process. Each filter is replaced by a tuple
of filters, each of them being an alphabet (i.e., like in the original definition),
and the current filter i (the one which shall be used) is replaced by the next
element of the tuple (i + 1) after its usage. When the last filter in the tuple is
reached the first filter is taken again and so on.
We show that systems with two tubes are enough to generate any recursively
enumerable language. We present different solutions depending on the number
of elements in the filter tuple. In Section 3 we describe a system having tuples
of two filters. Additionally both filters of the first tube coincide. In Section 4 we
present a system having tuples of four filters, but which contains no rules in the
second tube. In this case, all the work is done in the first tube and the second
one acts like a garbage collector. In the same section it is shown how to reduce
the number of filters to three. Finally, in Section 5 we present a system with two
filters which contains no rules in the second tube and for which the two filters
in a tuple differ only in one letter.
2 Basic Definitions
2.1 Grammars and Turing Machines
For definitions on Chomsky grammars and Turing machines we shall refer to [7].
In what follows we shall fix the notations that we use. We consider nonstationary
deterministic Turing machines, i.e., at each step the head shall move
totheleftorright.WedenotesuchmachinesbyM =(Q, T, a0,q0,F,δ), where Q
is the set of states and T isthetapealphabet,a0 ∈ T is the blank symbol, q0 ∈ Q
is the initial state, F ⊆ Q is the set of final (halting) states. By δ we denote the