Uniformity is Weaker than Semi-Uniformity for Some Membrane Systems

Abstract

We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family. Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks and tile assembly systems, and there are many others. However, in such models there are actually two distinct kinds of uniformity condition. The first is the most common and well-understood, where each input length is mapped to a single computing device (e.g. a Boolean circuit) that computes on the finite set of inputs of that length. The second, called semi-uniformity, is where each input is mapped to a computing device for that input (e.g. a circuit with the input encoded as constants). The former notion is well-known and used in Boolean circuit complexity, while the latter notion is frequently found in literature on nature-inspired computation from the past 20 years or so. Are these two notions distinct? For many models it has been found that these notions are in fact the same, in the sense that the choice of uniformity or semi-uniformity leads to characterisations of the same complexity classes. In other related work, we showed that these notions are actually distinct for certain classes of Boolean circuits. Here, we give analogous results for membrane systems by showing that certain classes of uniform membrane systems are strictly weaker than the analogous semi-uniform classes. This solves a known open problem in the theory of membrane systems. We then go on to present results towards characterising the power of these semi-uniform and uniform membrane models in terms of NL and languages reducible to the unary languages in NL, respectively.

P systems attacking hard problems beyond NP: a survey

“…Open Problem 2 Is there a relevant example of polynomially uniform families of P systems with active membranes where the classes of problems solvable by uniform and semi-uniform way are different? (Up to now, results of this kind are known for FAC 0 -uniform families with a tighter uniformity condition [37]. )…”

On the power of P systems with active membranes using weak non-elementary membrane division

“…So far, this conjecture was confirmed only in some special cases, for example, when dissolution rules are not used [11], when the division rules are symmetric [18], or when dissolution and elementary membrane division rules are allowed but both the use of other types of rules and the initial membrane structure are restricted [9,15,34]. The aim to prove Păun's conjecture initiated a research line in membrane computing where the computational power of restricted variants of polarizationless P systems is investigated (see, e.g., [19][20][21], where these P systems with no dissolution rules were studied and [8], where it is shown that these systems can simulate Turing machines efficiently using only evolution and dissolution rules). For a recent survey on exploring the boundary between P and NP in terms of membrane computing, see, e.g., [22].…”