Abstract
Memoryless computation is a novel means of computing any function of a set of registers 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 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 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 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.
Highlights
Amongst the results derived in the literature is the non-trivial fact that any function of n-bit input and n-bit ouput can be computed using memoryless computation
Memoryless computation has the potential to speed up computations by avoiding timeconsuming communication with the memory and by effectively combining the values contained in registers
This indicates that memoryless computation can be viewed as an analogue in computing to network coding [1, 17], an alternative to routing on networks
Summary
Swapping the contents of two variables x and y requires a buffer t, and proceeds as follows (using pseudo-code): t←x x←y y ← t. Memoryless computation has the potential to speed up computations by avoiding timeconsuming communication with the memory and by effectively combining the values contained in registers. This indicates that memoryless computation can be viewed as an analogue in computing to network coding [1, 17], an alternative to routing on networks. It is shown in [12] that for certain manipulations of registers, memoryless computation uses arbitrarily fewer updates than traditional, “black-box” computing. Memoryless computation is a unique area of theoretical computer science, which brings new insights and possible applications to some well-known results in algebra
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