A Nondeterministic Model for Abstract Geometrical Computation

TL;DR

This paper introduces a nondeterministic extension to signal machines, an abstract geometrical computation model, and proves equivalence with deterministic machines for a specific class of these nondeterministic models.

Contribution

It defines k-restricted nondeterministic signal machines and demonstrates their computational equivalence to deterministic signal machines for certain classes.

Findings

k-restricted nondeterministic signal machines can be simulated by deterministic ones

The model extends the classical deterministic signal machine framework

Equivalence holds for machines with at most two nondeterministic rules per collision

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.