Sigma^0_n-Complete Properties of Programs and Martin-Löf Randomness

Abstract G.J. Chaitin proved that the binary representation of the probability that a randomly selected program halts is Martin-Lof random. In the present paper, we introduce a notion called “Σ0n-completeness with constant-overhead reductions” and show that if a property of programs is Σ0n-complete with constant-overhead reductions then the binary representation of the probability that a randomly selected program has the property is Martin-Lof random. For almost all known Σ0n-complete properties of programs, their proofs of Σ0n-completeness can be easily modified to proofs of Σ0n-completeness in this new sense.