on feasible numbers (preliminary version)
We study complexity problems associated with generation and manipulation of integers. We show that many such questions may be formulated as problems about random access machines (RAMs), and recent results about RAMs with special instructions sets can be used to give estimates about the complexity of the manipulations. Finally we settle an open problem about RAMs (which becomes more interesting in this setting), namely the power of unrestricted vector machines, studied in [5] [6] [2]. We prove that allowing arbitrary shifts (instead of bounded shifts) gives at most a polynomial reduction in the time necessary to recognize languages by vector machines, thus solving the main problem left unanswered by the papers of Pratt and Stockmeyer [5], [6] and Hartmanis and Simon [2], [3]. Equivalently, it is possible to compare two integers, each obtained by a sequence of at most n operations of shift, sum, and boolean vector operations, within tape polynomial in n.
version complexity association generation manipulation random access instruction power vectorization polynomial reduction timing language paper sequence operability