Computing in Matrix Groups without memory
Cameron, P.J. and Fairbairn, Ben and Gadoleau, M. (2014) Computing in Matrix Groups without memory. Working Paper. Birkbeck, University of London, London, UK.

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Abstract
Memoryless computation is a novel means of computing any function of a set of regis ters 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 sin gle 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 permutations 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 func tions 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.
Metadata
Item Type:  Monograph (Working Paper) 

Additional Information:  Birkbeck Pure Mathematics Preprint Series #5 
Keyword(s) / Subject(s):  memoryless computation, linear functions, matrix groups, general linear group, special linear group, generating sets, sequential updates 
School:  Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics 
Research Centre:  Applied Macroeconomics, Birkbeck Centre for 
Depositing User:  Administrator 
Date Deposited:  22 Mar 2019 13:18 
Last Modified:  26 Jul 2019 17:54 
URI:  http://eprints.bbk.ac.uk/id/eprint/26719 
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