Circular Interval-valued Computers and Simulation of (Red-green) Turing Machines

Abstract

Interval-valued computing is a kind of massively parallel computing. It operates on specific subsets of the interval [0,1) - unions of subintervals. They serve as basic data units and are called interval-values. It was established in [9], by a rather simple observation, that interval-valued computing, as a digital computing model, has computing power equivalent to Turing machines. However, this equivalence involves an unlimited number of interval-valued variables. In [14], the equivalence with Turing machines is established using a simulation that uses only a fixed number of interval-valued variables and this number depends only on the number of states of the Turing machine - in a logarithmic way. The simulation given there allows us to extend interval-valued computations into infinite length to capture the computing power of redgreen Turing machines. In this extension of [14], based on the quasi-periodic techniques used in the simulations in that paper, a reformulation of the interval-valued computations is given, named circular interval-valued computers. This reformulation enforces the finiteness of the number of used interval-valued variables by building the finiteness into the syntax rules.