Genericity, Randomness, and Polynomial-Time Approximations

Polynomial-time safe and unsafe approximations for intractable sets were introduced by Meyer and Paterson [Technical Report TM-126, Laboratory for Computer Science, MIT, Cambridge, MA, 1979] and Yesha [SIAM J. Comput., 12 (1983), pp. 411–425], respectively. The question of which sets have optimal safe and unsafe approximations has been investigated extensively. Duris and Rolim [Lecture Notes in Comput. Sci. 841, Springer-Verlag, Berlin, New York, 1994, pp. 38–51] and Ambos-Spies [Proc. 22nd ICALP, Springer-Verlag, Berlin, New York, 1995, pp. 384–392] showed that the existence of optimal polynomial-time approximations for the safe and unsafe cases is independent. Using the law of the iterated logarithm for p-random sequences (which has been recently proven in [Proc. 11th Conf. Computational Complexity, IEEE Computer Society Press, Piscataway, NJ, 1996, pp. 180–189]), we extend this observation by showing that both the class of polynomial-time ∆-levelable sets and the class of sets which have optimal polynomial-time unsafe approximations have p-measure 0. Hence typical sets in E (in the sense of p-measure) do not have optimal polynomial-time unsafe approximations. We will also establish the relationship between resource bounded genericity concepts and the polynomial-time safe and unsafe approximation concepts.