13th International Conference on Membrane Computing - MTA Sztaki

Fast hardware implementations of P systems
The V ariant(n, Π, C, max) function can be defined as the n-th element in the
lexicographical ordering of elements of Appl(Π, C, max) and it has the following
formula
V ariant(n, Π, C, max) = r N−n−1+k1
1 r n−1+k2
2
We remark that the above formula can also be computed in constant time using
FPGA.
We could obtain the NBV ariants formula using formal power series. In order
to do this we observe that the language ∪ N>0 L N , where L N = {r p 1 rq 2 | p+q = N}
is regular. Moreover, it holds that L N = r1 ∗r∗ 2 ∩ AN , with A being the alphabet
{r 1 , r 2 }. Below we give the automaton A 1 for the language r1 ∗r∗ 2
r 1
q 0
r 2
r 2
( ) 1 1
The transfer matrix of this automaton is and the final state characteristic
vector is [1, 1] t . Using Equation (2) this yields the generating function
0 1
for L N : q 0 = 1
(1−x)
. It is easy to verify that [x n ]q 2 0 = n + 1.
We modify the previous example by considering weighted rules.
Example 3. Consider a P system Π 1 having two rules r 1 : a ka b k b1
→ u and
r 2 : b k b2
c kc → v. These rules have the following dependency graph:
q 1
r 1 r 2
k a
7 ❄ 7 ❄❄❄❄❄❄
k b1 k b2 ❄ 8 ❄❄❄❄❄❄❄
k c
7 7777 8 8888 a
b
c
Let N a , N b and N c be the number of objects a, b and c in a configuration C.
We define
N 1 = min([N a /k a ], [N b /k b1 ])
N 2 = min([N b /k b2 ], [N c /k c ])
N = min(N 1 , N 2 )
Suppose that Π evolves in a maximally parallel derivation mode. Let A 2 be
the automaton recognizing the language (r1 kb1 ) ∗ (r2 kb2 ) ∗
r k b1
1
q 0
r k b2
r k b2
2
2
Let L ′ N = A 2 ∩ A N (A = {r 1 , r 2 }). Then it is clear that
⋃ { }
Appl(Π, C, max) =
r p+k1
1 r q+k2
2 ,
pk b1 +qk b2 =N
q 1
439