A polynomial alternative to unbounded environment for tissue P systems with cell division

Abstract

The standard definition of tissue P systems includes a special alphabet whose elements are assumed to appear in the initial configuration of the system in an arbitrarily large number of copies. These objects reside in a distinguished place of the system, called the environment. Such potentially infinite supply of objects seems an unfair tool when designing efficient solutions to computationally hard problems in the framework of membrane computing, by performing a space-time trade-off. This paper deals with computational aspects of tissue P systems with cell division where there is no environment having the property mentioned above. Specifically, we prove that the polynomial complexity classes associated with tissue P systems with cell division and with or without environment are actually identical. As a consequence, we conclude that it is not necessary to have infinitely many copies of some objects in the initial configuration in order to solve NP-complete problems in an efficient way. Keywords: membrane computing; tissue P systems; cell division; environment of a tissue; computational complexityfor each instance u ∈ I X . The polynomial function r(n) fulfils the property required.Corollary 3.8 Let Π = { (n) : n ∈ N} be a family of recognizer tissue P systems with cell division solving a decision problem X = (I X , θ X ) in polynomial time. Let (cod, s) be a polynomial encoding associated with that solution. Then, there exists a polynomial function r(n) such that for each instance u ∈ I X , 2 r(|u|) is an upper bound of the number of objects from E which are moved from the environment to all cells of the system (s(u)) + cod(u) along any computation.Proof It suffices to note that from Proposition 3.7, there exists a polynomial function r(n) such that for each instance u ∈ I X , 2 r(|u|) is an upper bound of the number of objects in all cells of the system (s(u)) + cod(u).

Efficient solutions to hard computational problems by P systems with symport/antiport rules and membrane division

“…(d) The environment is not relevant for tissue P systems with symport/antiport rules and cell division from a complexity point of view Pérez-Jiménez et al (2013). What about the efficiency of recognizer P systems from CDC(k) when the alphabet of the environment is an empty set?…”

The P versus NP Problem from the Membrane Computing View

“…In Ref. 25, it is shown that every family Π of tissue P systems with cell division solving a decision problem can be simulated by a family Π ′ of tissue P systems with cell division and without environment, in an efficient way. Moreover, the upper bound of the length of communication rules of the systems from Π is equal to the maximum length of communication rules of the system from Π ′.…”

Membrane fission versus cell division: When membrane proliferation is not enough

“…In that paper, the definition of tissue P systems with symport/antiport rules is combined with the one of P systems with active membranes, yielding tissue P systems with cell division. One of the latest studies on their computational power can be found in [22]. In tissue-like systems, the membrane fission phenomenon has been considered together with symport/antiport rules [17] by means of cell separation rules.…”