Computational Error Complexity Classes

The complexity classes Nearly-BPP and Med DisP had been recently proposed as limits of eecient computation Yam1 96, Schi 96]. For both classes, a polynomial time bounded algorithm with bounded probabilistic error has to compute correct outputs for at least 1 ? 1 n !(1) of inputs of length n. We generalize this notion to general error probabilities and arbitrary complexity classes. For proving the intractability of a problem it is necessary to show that it cannot be computed within a given error bound for every input length. For this, we introduce a new error complexity class, where the error is only innnitively often bounded by the error function. We identify sensible bounds for the error function and derive new diagonalizing techniques. Using these techniques we present time hierarchies of a new quality: We are able to show that there are languages computable in time T that a machine with asymptotically slower running time cannot predict within a smaller error than 1 2. Further, we investigate two classical non recursive problems: the halting problem and the Kolmogorov complexity function. We give strict lower bounds proving that any heuristic algorithm that claims to solve one of these problems makes unrecoverable errors with constant probability.