Measure One Results in Computational Complexity Theory1

Starting with Bennet and Gill’s seminal paper [13] a whole new research line in complexity theory was opened: the examination of relativized complexity theoretic statements which hold for a measure one set of oracles in the measure defined by putting each string into the oracle with probability 12 independent of all other strings (a formal definition is given below). Bennet and Gill were concerned with the subtlety of the P ? = NP question pointed out in a paper by Baker, Gill, and Solovay [8] where oracles A and B were exhibited such that PA = NPA but PB 6= NPB. However since these two oracles are of a very “intentional nature” [59], Bennet and Gill wanted to examine the relation between P and NP relative to an oracle which is produced “at random.” They showed that PA 6= NPA for a measure one set of oracles A. In their own words: “Relative to a random oracle A, PA 6= NPA 6= coNPA with probability 1” [13]. Since then results of this form have been called “random oracle results” in complexity theory. What we want to point out here is that this term does not refer to algorithmically random languages in the sense of Martin-Löf [65]. To avoid confusion, results as the one by Bennet and Gill should better be referred to as “measure one results”. This however does not deny that there are certain relations among measure one oracle sets and the set of algorithmically random oracles as will be pointed out in the next section. Though there are a lot of beautiful and compelling results along this line the status of measure one oracle sets, especially in connection with the so called Random Oracle Hypothesis, is still unclear. This hypothesis, as stated by Bennett and Gill, claims that every statement holding for a measure one set of oracles also holds in the unrelativized world. In the meantime however it turned out that this does not hold. What we will show is that even a much more restricted formulation of the hypothesis is false.