Chicago Journal of Theoretical Computer Science

Volume 2014

Article 7

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

Computing in Permutation 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
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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


Memoryless computation is a modern technique to compute any function of a set of registers by updating one register at a time while using no memory. Its aim is to emulate how computations are performed in modern cores, since they typically involve updates of single registers. The memoryless computation model can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we consider how efficiently permutations can be computed without memory. We determine the minimum number of basic updates required to compute any permutation, or any even permutation. The small number of required instructions shows that very small instruction sets could be encoded on cores to perform memoryless computation. We then start looking at a possible compromise between the size of the instruction set and the length of the resulting programs. We consider updates only involving a limited number of registers. In particular, we show that binary instructions are not enough to compute all permutations without memory when the alphabet size is even. These results, though expressed as properties of special generating sets of the symmetric or alternating groups, provide guidelines on the implementation of memoryless computation.

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

DOI: 10.4086/cjtcs.2014.007

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