Computing in matrix groups without memory

TL;DR

This paper explores memoryless computation for linear functions over finite fields, analyzing complexity, instruction sets, and group properties to demonstrate efficiency and minimal instruction requirements in a memoryless model.

Contribution

It provides the first detailed analysis of linear permutation complexity, instruction minimality, and group computability within the memoryless computation framework.

Findings

Maximum complexity of linear functions with linear instructions determined

Special linear group is internally computable but not fast

Small instruction sets suffice to generate linear groups

Abstract

Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.