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We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons(More)
It is shown 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 $6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic(More)
It is shown that if formulas are used to compute Boolean functions in the presence of randomly occurring failures (as has been suggested by von Neumann and others), then 1) there is a limit shictly less than 1/2 to the failure probability per gate that can be tolerated, and 2) formulas that tolerate failures must be deeper (and, therefore, compute more(More)
We present upper bounds for sorting and selecting the median in a fixed number of rounds. These bounds match the known lower bounds to within logarithmic factors. They also have the merit of being "explicit modulo expansion"; that is, probabilistic arguments are used only to obtain expanding graphs, and when explicit constructions for such graphs are found,(More)