On membrane hierarchy in P systems

Abstract

We look at a restricted model of a communicating P system, called RCPS, whose environment does not contain any object initially. The system can expel objects into the environment but only expelled objects can be retrieved from the environment. Such a system is initially given an input $a_1^{i_1}\dots a_n^{i_n}$ (with each $i_j$ representing the multiplicity of distinguished object $a_i$, $1⩽i⩽n$) and is used as an acceptor. We show that RCPSs are equivalent to two-way multihead finite automata over bounded languages (i.e., subsets of $a_1^{*}\dots a_n^{*}$, for some distinct symbols $a_1,\dots ,a_n$). We then show that there is an infinite hierarchy of RCPS's in terms of the number of membranes: For every $r$, there is an $s>r$ and a unary language $L$ accepted by an RCPS with $s$ membranes that cannot be accepted by an RCPS with $r$ membranes. This provides an answer to an open problem in (Membrane Computing: An Introduction, Springer, Berlin, 2002) which asks whether there is a nonuniversal model of a membrane computing system which induces an infinite hierarchy on the number of membranes. We also consider variants/generalizations of RCPSs, e.g, acceptors of languages; models that allow a “polynomial bounded” supply of objects in the environment initially; models with tentacles, etc. We show that they also form an infinite hierarchy with respect to the number of membranes (or tentacles). The proof techniques can be used to obtain similar results for other restricted models of P systems, like symport/antiport systems.