### Volume 2014

#### Computing in Matrix Groups Without Memory

Peter J. Cameron
School of Mathematics and Statistics
University of St Andrews
Fife KY16 9AJ, United Kingdom
pjc20 AT st-andrews DOT ac DOT uk

Ben Fairbairn
Department of Economics, Mathematics and Statistics
Birbeck, University of London
London, UK
b DOT fairbairn AT bbk DOT ac DOT uk

and

School of Engineering and Computer Science
Durham University
Durham, UK
m DOT r DOT gadouleau AT durham DOT ac DOT uk

November 2, 2014

#### 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 permutation 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.

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