Reversible parallel computation: An evolving space-model

Abstract

We study the totality of the possible evolution “laws” of “colored spaces”, i.e. Euclidean spaces whose points have time-variable colors possibly representing microphenomena. Such spaces obey some physical principles meaningful in computer science: a limit on the speed of information transmission, microscopic reversibility and some further restrictions, which however make possible a mathematical analysis of the problem. We suppose that the set of the phenomena occuring inside a computer may be schematized as the evolution of a colored space according to one of the laws first sketched. In such a case we further specify new axioms that set bounds to the compressibility of information (which is justified in computer science by the repeatability and reliability required for computation). This allows us to rigorously define insurmountable bounds for the time spent in computing functions and, therefore, a hierarchy (subdivision into classes of the set of functions computable in polynomial time) based on such limitations. Regarding these concepts, we show various results; in particular we show, under very general assumptions, the inability for any real computer to compute Turing non-polynomial functions in polynomial time. The way in which “laws” specify the evolution of a colored space is similar to that of a cellular automaton. However, it is by no means certain that the two concepts are equivalent with respect to the time needed to compute functions. The continuity concept introduced allows us to solve (in a fairly intuitive fashion) an open problem about cellular automata: the characterization of the class of all reversible cellular automata.