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