Universal enzymatic numerical P systems with small number of enzymatic variables

Numerical P systems (for short, NP systems) are distributed and parallel computing models inspired from the structure of living cells and economics. Enzymatic numerical P systems (for short, ENP systems) are a variant of NP systems, which have been successfully applied in designing and implementing controllers for mobile robots. Since ENP systems were proved to be Turing universal, there has been much work to simplify the universal systems, where the complexity parameters considered are the number of membranes, the degrees of polynomial production functions or the number of variables used in the systems. Yet the number of enzymatic variables, which is essential for ENP systems to reach universality, has not been investigated. Here we consider the problem of searching for the smallest number of enzymatic variables needed for universal ENP systems. We prove that for ENP systems as number acceptors working in the all-parallel or one-parallel mode, one enzymatic variable is sufficient to reach universality; while for the one-parallel ENP systems as number generators, two enzymatic variables are sufficient to reach universality. These results improve the best known results that the numbers of enzymatic variables are 13 and 52 for the all-parallel and one-parallel systems, respectively.