Monodirectional P systems

Abstract

Summary. We investigate the influence that the flow of information in membrane systems has on their computational complexity. In particular, we analyse the behaviour of P systems with active membranes where communication only happens from a membrane towards its parent, and never in the opposite direction. We prove that these "monodirectional P systems" are, when working in polynomial time and under standard complexity-theoretic assumptions, much less powerful than unrestricted ones: indeed, they characterise classes of problems defined by polynomial-time Turing machines with NP oracles, rather than the whole class PSPACE of problems solvable in polynomial space.

Shallow laconic P systems can count

“…We also assume that both M and M ′ have a binary alphabet. This allows us to reuse the same TM simulation described in [3].…”

“…To simulate a TM, we employ the same construction used for monodirectional P systems in [3]. Here, we just briefly recall the inner computing mechanism and the encoding used.…”

“…the objects encoding , with the encoding of the tape of M ′ already containing the oracle query. A copy of the initial encoding of is then sent out into the skin membrane M to simulate the TM M. For the details of how this send-out process can be performed, we refer the reader to [3], which employs a very similar method. We assume that M already has predefined positions (the size of the oracle answer to the query) named s 0 , … , s −1 that are set to zero and where M expects to find the answer to the oracle query, with s 0 the position of the least significant digit and s −1 the position of the most significant one.…”

“…Since all the information needed to perform the TM transition is now stored inside a single object, either of the form (q, 0) or (q, 1), it is now possible to simulate the application of the transition function (as specified in detail in [3]): Fig. 1 The initial membrane structure of the P system Simultaneously, the objects on the tape are primed again:…”

“…shallow, to solve problems in P # in polynomial time [1]. With monodirectional communication, only problems in P ∥ can be solved and, even if polynomial depth in the membrane structure is allowed, only P can be reached [3]. There exists, however, an entire spectrum of possibilities between monodirectional and full bidirectional communication; in particular, there are multiple ways of limiting the amount of bidirectional communication, for example by limiting the number of send-in rules that a membrane or its descendants (i.e.…”