Tissue P Systems with Small Cell Volume

Traditionally, P systems allow their membranes or cells to grow exponentially (or even more) in volume with respect to the size of the multiset of objects they contain in the initial configuration. This behaviour is, in general, biologically unrealistic, since large cells tend to divide in order to maintain a suitably large surface-area-to-volume ratio. On the other hand, it is usually the number of cells that needs to grow exponentially with time by binary division in order to solve NP-complete problems in polynomial time. In this paper we investigate families of tissue P systems with cell division where each cell has a small volume (i.e., sub-polynomial with respect to the input size), assuming that each bit of information contained in the cell, including both those needed to represent the multiset of objects and the cell label, occupies a unit of volume. We show that even a constant volume bound allows us to reach computational universality for families of tissue P systems with cell division, if we employ an exponential-time uniformity condition on the families. Furthermore, we also show that a sub-polynomial volume does not suffice to solve NP-complete problems in polynomial time, unless the satisfiability problem for Boolean formulae can be solved in sub-exponential time, and that solving an NP-complete problem in polynomial time with logarithmic cell volume implies P = NP.