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Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. In 1975, Simon proved that for unboundederror polynomial-time machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class, BPP path , and, more generally, we study the structural properties of BPP path. We prove that BPP path contains both NP BPP and P NP log] , and that BPP path is contained in P p 2 log] , BPP NP , and PP. We conclude that, unless the polynomial hierarchy collapses, bounded-error threshold computation is strictly more powerful than bounded-error probabilistic computation. We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length. We show, for both BPP and BPP path , that if there is any database for which this formalization of security di ers from the security given by oblivious database access, then P 6 = PSPACE. It follows that if any set lacking small circuits can be securely accepted, then P 6 = PSPACE.

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A semi-membership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A . A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent half-decade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semi-membership algorithms.

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