An Improved Zero-One Law for Algorithmically Random Sequences

Results on random oracles typically involve showing that a class {X : P(X)} has Lebesgue measure one, i.e., that some property P(X) holds for “almost every X.” A potentially more informative approach is to show that P(X) is true for every X in some explicitly defined class of random sequences or languages. In this note we consider the algorithmically random sequences originally defined by Martin-L¨of and their generalizations, the n-random sequences. Our result is an effective form of the classical zero-one law: for each n ≥ 1, if a class {X : P(X)} is closed under finite variation and has arithmetical complexity Σ0 n+1 or Π0 n+1 (roughly, the property P can be expressed with n+1 alternations of quantifiers), then either P holds for every n-random sequence or else holds for none of them. This result has been used by Book and Mayordomo to give new characterizations of complexity classes of the form ALMOST-R, the languages which can be ≤R-reduced to almost every oracle, ...

• Type: Journal
• Year: 1998
• Where: TCS
• Authors: Steven M. Kautz
• Keywords: N-random Sequence, Random Oracles, Random Sequences, TCS 1998, Theoretical Computer Science