#

### Volume 2014

#### Article 8

Published by the Department of Computer Science, The University of Chicago.

#### 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

Maxmilien Gadouleau

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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Submitted November 5, 2013, revised August 27, 2014 and in final form September 29, 2014, published November 2, 2014.

Article 7
Article 9

Volume 2014
Published articles