Theoretical and Experimental DNA Computation (Natural ...

3.5 Membrane Models 67
objects, and are of the form before → after, meaning “evolve every instance of
before into an instance of after.” Note that, as we are considering multi-sets,
this rule may be applied to multiple objects. Evolution rules are represented
by a pair (u, v) of strings over the alphabet V . v may be either v ′ or v ′ δ,where
δ is a special symbol not in V . v ′ is a string over {ahere,aout,ainj } where j is
a membrane identifier. ahere means “a copy of a remains in this region”, aout
means “send a copy of a through the membrane and out of this region”, and
ainj means “send a copy of a through the membrane of the region labelled
j” (i.e., place a copy of a in membrane j, noting that this is only possible if
the current region featuring this rule contains j). When the special symbol δ
is encountered, the membrane defining the current region (assuming it is not
the skin membrane) is “dissolved”, and the contents of the current region are
placed in the “parent” region (with reference to Fig. 3.8, if membrane 5 were
to be dissolved, then region 3 would contain the accessible regions 4, 6 and 7,
whereas only regions 4 and 5 are accessible with membrane 5 intact).
In order to simplify the notation, we omit the subscript “here”, as it is
largely redundant for our purposes. Thus the rule a → ab means “retain a
copy of a here and create a copy of b here”, whereas a → bδ means “transform
every instance of a into b and then dissolve the membrane.” Note that objects
on the left hand side of evolution rules are “consumed”, or removed, during
the process of evaluation.
We may also impose priorities upon rules. This is denoted by >, andmay
be read as follows, using the example (ff → f) > (f → δ): “transform ff to
f as often as possible (halving the number of occurrences of f in the process)
until no instances of ff remain, and then transform the one remaining f to
δ, dissolving the membrane.”
We assume the existence of a “clock” that synchronizes the operation of the
system. At each time step, the configuration of the system is transformed by
the application of rules in each region in a nondeterministic, maximally parallel
fashion. This means that objects are assigned to rules nondeterministically in
parallel, until no further assignment is possible (hence maximal). This series
of transitions from one configuration to another forms the computation. A
computation halts if no rules may be applied in any region (i.e., nothing can
happen). The result of a halting computation is the number of objects sent
out through the skin membrane to the outside environment.
We now give a small worked example, taken from [118] and depicted in
Fig. 3.9.
This P system calculates n 2 for any given n ≥ 0. The alphabet V =
{a, b, d, e, f}. Region 3 contains one copy each of objects a and f, andthree
evolution rules. No objects are present in regions 1 and 2, so no rules can be
applied until we reach region 3. We iterate the rules a → ab and f → ff n
times in parallel, where n ≥ 0 is the number we wish to square. This gives n
copies of b and 2 n copies of f. Wethenusea → bδ instead of a → ab, replacing
the single a with b and dissolving the membrane. This leaves n + 1 copies of b
and 2 n+1 copies of f in region 2; the rules from region 3 are “destroyed” and