Generic Oracles, Uniform Machines, and Codes

Abstract The basic properties of generic oracles are reviewed, and proofs given that they separate P and NP and are weakly incompressible. A new notion of generic oracle, called t-generic, is defined. It is shown that t-generic oracles do not exist, and consequently a nondeterministic oracle machine which for any oracle X accepts the tautologies relativized to X when running with oracle X does not run in polynomial time at any oracle. A weak form of t-generic oracle, called r-generic, is shown to exist, and it is shown that if there exists an r-generic oracle X at which the r-query relativized tautologies are not in co NP X then NP ≠ co NP . The notion of a code for the Boolean functions is defined, and it is shown that generic oracles do not have short codes in any code. Universal circuits of size O(n log4 n) are shown to exist, and it is shown that increasing the number of ⋏, ⋎ gates from g to 2g + 1 allows the computation of new Boolean functions.