Computing in Permutation Groups Without Memory

Cameron, Peter; Fairbairn, Ben; Gadouleau, Maximilien

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.

Citation

Cameron, P., Fairbairn, B., & Gadouleau, M. (2014). Computing in Permutation Groups Without Memory. Chicago journal of theoretical computer science, 2014(7), 1-20. https://doi.org/10.4086/cjtcs.2014.007

• 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
• Electronic ISSN: 1073-0486
• Publisher: Massachusetts Institute of Technology Press
• Peer Reviewed: Peer Reviewed
• Volume: 2014
• Issue: 7
• Article Number: 7
• Pages: 1-20
• DOI: https://doi.org/10.4086/cjtcs.2014.007
• Keywords: Memoryless computation, Permutation groups, Symmetric group, Alternating group, Generating sets, Boolean networks, Sequential updates.
• Public URL: https://durham-repository.worktribe.com/output/1429520