The non-recursive power of erroneous computation

Abstract

We present two new complexity classes which are based on a complexity class C and an error probability function F. The first, F-Err C, reflects the (weak) feasibility of problems that can be computed within the error bound F. As a more adequate measure to investigate lower bounds we introduce F-Errio C where the error is infinitely often bounded by the function F. These definitions generalize existing models of feasible erroneous computations and cryptographic intractability. We identify meaningful bounds for the error function and derive new diagonalizing techniques. These techniques are applied to known time hierarchies to investigate the influence of error bound. It turns out that in the limit a machine with slower running time cannot predict the diagonal language within a significantly smaller error prob. than 1/2. Further, we investigate two classical non-recursive problems: the halting problem and the Kolmogorov complexity problem. We present strict lower bounds proving that any heuristic algorithm claiming to solve one of these problems makes unrecoverable errors with constant probability. Up to now it was only known that infinitely many errors will occur. © Springer-Verlag Berlin Heidelberg 1999.