Theoretical and Experimental DNA Computation (Natural ...

3.5 Membrane Models 65
while the set contains ≥ two elements do begin
select two elements, compare them and remove the smaller
end
In order to represent such programs, Banâtre and Le Métayer developed the
GAMMA language: the General Abstract Model for Multiset Manipulation
[20, 21]. In GAMMA, the above statement would be expressed thus:
maxset(s) =Γ ((R, A))(s) where
R(x, y) =x ≤ y
A(x, y) ={y}
The function R specifies the property to be satisfied by selected elements x
and y; inthiscase,x should be less than or equal to y. These elements are then
replaced by applying function A. There is no implied order of evaluation, if
several disjoint pairs of elements satisfy R, they can be performed in parallel.
Banâtre and Le Métayer recognized that GAMMA programs could almost
be viewed as chemical reactions, with the set being the chemical solution, R
(the reaction condition) a property to be satisfied by reacting elements, and
A (the action) the resulting product of the reaction. The computation ends
when no reactions can occur, and a stable state is reached.
Before we conclude our treatment of GAMMA, let us examine one more
program, prime number generation, again taken from [21].
The goal is to produce all prime numbers less than a given N, with
N > 2. The GAMMA solution removed multiple elements from the multiset
{2,...,N}. We assume the existence of a function multiple(x, y), which
returns true if x is a multiple of y, andfalse otherwise. The resulting multiset
contains all primes less than N:
primes(N) =Γ ((R, A))({2 ...N}) where
R(x, y) =multiple(x, y)
A(x, y) ={y}
One possible “trace” through the computation for N =10isdepictedin
Fig. 3.7. The similarities between this approach and the parallel filtering model
described earlier are apparent – in the latter model, no ordering is implied
when the remove operation is performed.
2
10 8 5
7
3
9 4 6
3
7
9
2
4
5
6
2
3
7
5
6
Fig. 3.7. Trace of primes(10)
2
7
3
The GAMMA model was extended by Berry and Boudol with the development
of the CHAM [30]. The CHAM formalism provides a syntax for molecules as
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