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Computing in Permutation Groups Without Memory

Cameron, Peter; Fairbairn, Ben; Gadouleau, Maximilien

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Peter Cameron

Ben Fairbairn


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.


Cameron, P., Fairbairn, B., & Gadouleau, M. (2014). Computing in Permutation Groups Without Memory. Chicago journal of theoretical computer science, 2014(7), 1-20.

Journal Article Type Article
Acceptance Date Sep 27, 2014
Publication Date Nov 2, 2014
Deposit Date Oct 14, 2015
Publicly Available Date Oct 21, 2015
Journal Chicago journal of theoretical computer science
Publisher Massachusetts Institute of Technology Press
Peer Reviewed Peer Reviewed
Volume 2014
Issue 7
Article Number 7
Pages 1-20
Keywords Memoryless computation, Permutation groups, Symmetric group, Alternating group, Generating sets, Boolean networks, Sequential updates.


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