  Memoryless computation is a new 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.
