Parallel computation for well-endowed rings and space-bounded probabilistic machines

It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S² parallel machine and therefore by a space S² deterministic machine. (Previous simulations ran in space S⁶.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+,−,×) ring operations.