13th International Conference on Membrane Computing - MTA Sztaki

P. Sosík
1. The family Π is polynomially uniform by Turing machines, that is, there
exists a deterministic Turing machine working in polynomial time which
constructs the system Π(n) from n ∈ N.
2. There exists a pair (cod, s) of polynomial-time computable functions over I X
such that:
(a) for each instance u ∈ I X , s(u) is a natural number and cod(u) is an
input multiset of the system Π(s(u));
(b) for each n ∈ N, s −1 (n) is a finite set;
(c) the family Π is polynomially bounded with regard to (X, cod, s), that is,
there exists a polynomial function p, such that for each u ∈ I X every
computation of Π(s(u)) with input cod(u) is halting and it performs at
most p(|u|) steps;
(d) the family Π is sound with regard to (X, cod, s), that is, for each u ∈ I X ,
if there exists an accepting computation of Π(s(u)) with input cod(u),
then θ X (u) = 1;
(e) the family Π is complete with regard to (X, cod, s), that is, for each
u ∈ I X , if θ X (u) = 1, then every computation of Π(s(u)) with input
cod(u) is an accepting one.
From the soundness and completeness conditions above we deduce that every
P system Π(n) is confluent, in the following sense: every computation of a system
with the same input multiset must always give the same answer.
Let R be a class of recognizer tissue P systems. We denote by PMC R the
set of all decision problems which can be solved in a uniform way and polynomial
time by means of families of systems from R. The following results have been
proved:
Theorem 1 ([6]). P = PMC T DC(1)
Theorem 2 ([15]). NP ∪ co-NP ⊆ PMC T DC(3)
As a consequence, both NP and co-NP are contained in the class PMC T DC .
In this paper we impose an upper bound on PMC T DC .
3 Simulation of Tissue P Systems with Cell Division in
Polynomial Space
In this section we demonstrate that any computation of a recognizer tissue P
system with cell division can be simulated in space polynomial to its initial
size and the number of steps. Instead of simulating a computation of a P system
from its initial configuration onwards (which would require exponential space for
storing configurations), we create a recursive function which computes content
of any cell h after a given number of steps. Thus we do not need to store content
of cells interacting with h but we calculate it recursively whenever needed.
Simulated P systems are confluent, hence possibly nondeterministic, but the
simulation will be performed in a deterministic way: only one possible sequence
of configurations of the P system is traced. This corresponds to a weak priority
relation between rules:
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