Looking for a theory of communication complexity for P systems, we consider
here so-called evolution-communication (EC for short) P systems, where objects
evolve by multiset rewriting rules without target commands and pass through membranes
by means of symport/antiport rules. (Actually, in most cases below we use only
symport rules.) We first propose a way to measure the communication costs by means
of “quanta of energy” (produced by evolution rules and) consumed by communication
rules. EC P systems with such costs are proved to be Turing complete in all three cases
with respect to the relation between evolution and communication operations: priority
of communication, mixing the rules without priority for any type, priority of evolution
(with the cost of communication increasing in this ordering in the universality proofs).
More appropriate measures of communication complexity are then defined, as dynamical
parameters, counting the communication steps or the number (and the weight)
of communication rules used during a computation. Such parameters can be used in
three ways: as properties of P systems (considering the families of sets of numbers generated
by systems with a given communication complexity), as conditions to be imposed
on computations (accepting only those computations with a communication complexity
bounded by a given threshold), and as standard complexity measures (defining the class
of problems which can be solved by P systems with a bounded complexity). Because
we ignore the evolution steps, in all three cases it makes sense to consider hierarchies
starting with finite complexity thresholds. We only give some preliminary results about
these hierarchies (for instance, proving that already their lower levels contain complex –
e.g., non-semilinear – sets), and we leave open many problems and research issues.Junta de Andalucía P08 – TIC 0420
