A Nondeterministic Model for Abstract Geometrical Computation

Abstract

A signal machine is an abstract geometrical model for computation, proposed as an extension to the one-dimensional cellular automata, in which discrete time and space of cellular automata is replaced with continuous time and space in signal machine. A signal machine is defined as a set of meta-signals and a set of rules. A signal machine starts from an initial configuration which is a set of moving signals. Signals are moving in space freely until a collision. Rules of signal machine specify what happens after a collision, or in other words, specify out-coming signals for each set of colliding signals. Originally signal machine is defined by its rule as a deterministic machine. In this paper, we introduce the concept of non-deterministic signal machine, which may contain more than one defined rule for each set of colliding signals. We show that for a specific class of nondeterministic signal machines, called k-restricted nondeterministic signal machine, there is a deterministic signal machine computing the same result as the nondeterministic one, on any given initial configuration. k-restricted nondeterministic signal machine is a nondeterministic signal machine which accepts an input iff produces a special accepting signal, which have at most two nondeterministic rule for each collision, and at most k collisions before any acceptance.