Computing in matrix groups without memory

Cameron, P. and Fairbairn, Ben and Gadouleau, M. (2014) Computing in matrix groups without memory. Chicago Journal of Theoretical Computer Science , ISSN 1073-0486.

Abstract

Memoryless computation is a novel means to compute any function of a set of registers by updating one register at a time while using no memory. This aims at emulating how computations are performed on modern cores, since they typically involve updates of single 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 inquestion 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 functions 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.

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