Computational Complexity of Probabilistic Turing Machines

Testable and untestable classes of first-order formulae

“…Probabilistic machines seem to have been first formalized by de Leeuw et al [32], who showed that such machines cannot compute uncomputable properties under reasonable assumptions. However, they mention the possibility that probabilistic machines could be more efficient than deterministic machines, a topic which was then investigated by Gill [19]. An early example of such a result is Freivalds' [18] matrix multiplication checker.…”

Immunity and simplicity in relativizations of probabilistic complexity classes

“…BPP(A) is the class of sets probabilistically accepted by some NPi with oracle A with bounded error probability [5]; this is equivalent to saying that for each x either more than one half of the computations accept x (and x is considered accepted) or less than one-fourth the computations accept x (and x is considered rejected); R(A) is the class of sets probabilistically accepted by some NP t with oracle A with one-sided errors: for each x either more than half the computations accept or no computation accepts; co-R(A) is the class of compléments of sets in R (A); ZPP{A) is R{A) H CO-JRC4); characterizations of ZPP(A) in terms of the number of accepting or rejecting computations may be found in [5] and [11]; PSPACE(^4) is the class of sets accepted by any Turing machine in polynomial space with oracle A; we assume that the polynomial space bound holds for the query tape.…”

“…The probabilistic classes ZPP, R, BPP and PP were introduced by Gill [5]. The class R is denoted VPP there.…”

The Kahr-Moore-Wang Class Contains Untestable Properties

“…They showed that such machines cannot compute uncomputable properties under reasonable assumptions, but mention the possibility that probabilistic machines could perhaps be more efficient than deterministic machines. This topic attracted considerable attention including Gill (1977). Early examples of such results were presented by Freivalds (1977), (1979) including his matrix multiplication checker.…”