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Parallel computation for well-endowed rings and space-bounded probabilistic machines. (English) Zbl 0598.68043
We show that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time $$S^ 2$$ parallel machine and therefore by a space $$S^ 2$$ deterministic machine. (Previous simulations ran in space $$S^ 6)$$. 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 $$(+,-,\times)$$.

##### MSC:
 68Q05 Models of computation (Turing machines, etc.) (MSC2010) 68Q25 Analysis of algorithms and problem complexity 16W99 Associative rings and algebras with additional structure 03D10 Turing machines and related notions 03D15 Complexity of computation (including implicit computational complexity) 15B51 Stochastic matrices
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