LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

6 Artiom Alhazov, Carlos Martín-Vide, and Linqiang Pan
4 The Polynomial Complexity Class PMC �
Many practical problems are presumably intractable for conventional (electronic)
computers, because all known algorithms solving these problems spend exponential
time. P systems have an inherent parallelism and hence the capability to
solve hard problems in feasible (polynomial or linear) time.
To understand what it means that a problem can be solved in polynomial
time by membrane systems, it is necessary to recall some complexity measure
for P systems as described in [5].
A decision problem will be solved by a family of recognizing P systems in such
a way that given an instance of the problem it is necessary to fix the concrete P
system (with a suitable input multiset) that will process it. The next definition
(polynomial encoding) captures this idea.
Definition 41 Let L be a language, F a class of P systems with input and
Π =(Π(t))t∈N a family of P systems of type F. A polynomial encoding of L in
Π is a pair (g, h) of polynomial time computable functions, g : L →∪t∈NI Π(t)
and h : L → N, such that for every u ∈ L we have g(u) ∈ I Π(h(u)).
Now we define what it means solving a decision problem by a family of
recognizing P systems in time bounded by a given function.
Definition 42 Let F be a class of recognizing P systems, f : N → N a total
computable function, and X = (IX,θX) a decision problem. We say that X
belongs to MCF(f) if there exists a family, Π =(Π(t))t∈N, ofPsystemssuch
that:
–Πis F-consistent: that is, Π(t) ∈F for all t ∈ N.
–Πis polynomially uniform: that is, there exists a deterministic Turing machine
that constructs Π(t) in polynomial time from t ∈ N.
– There exists a polynomial encoding (g, h) from IX to Π verifying:
• Π is f-bounded, with regard to (g, h); that is, for each u ∈ IX, allcomputations
of Π(h(u)) with input g(u) halt in at most f(|u|) steps.
• Π is X-sound, with regard to (g, h); that is, for each u ∈ IX, if
there exists an accepting computation of Π(h(u)) with input g(u), then
θX(u) =1.
• Π is X-complete, with regard to (g, h); that is, for each u ∈ IX, if
θX(u) =1, then every computation of Π(h(u)) with input g(u) is an
accepting computation.
A polynomial encoding (g, h) fromIX to Π provides a size function, h, that
gives us the set of instances of X processed through the same P system, and an
input function, g, supplying the input multiset to be processed for the P system.
Note 1. In the above definition we have imposed a confluence property in
the following sense: for every input u ∈ IX, either every computation of Π(h(u))
with input g(u) is an accepting computation or every computation of Π(h(u))
with input g(u) is a rejecting computation.