Random Permutations in Computational Complexity

Classical results of Bennett and Gill (1981) show that with probability 1, $P^A \neq NP^A$ relative to a random oracle $A$, and with probability 1, $P^\pi \neq NP^\pi \cap coNP^\pi$ relative to a random permutation $\pi$. Whether $P^A = NP^A \cap coNP^A$ holds relative to a random oracle $A$ remains open. While the random oracle separation has been extended to specific individually random oracles--such as Martin-L\"of random or resource-bounded random oracles--no analogous result is known for individually random permutations. We introduce a new resource-bounded measure framework for analyzing individually random permutations. We define permutation martingales and permutation betting games that characterize measure-zero sets in the space of permutations, enabling formal definitions of resource-bounded random permutations. Our main result shows that $P^\pi \neq NP^\pi \cap coNP^\pi$ for every polynomial-time betting-game random permutation $\pi$. This is the first separation result relative to individually random permutations, rather than an almost-everywhere separation. We also strengthen a quantum separation of Bennett, Bernstein, Brassard, and Vazirani (1997) by showing that $NP^\pi \cap coNP^\pi \not\subseteq BQP^\pi$ for every polynomial-space random permutation $\pi$. We investigate the relationship between random permutations and random oracles. We prove that random oracles are polynomial-time reducible from random permutations. The converse--whether every random permutation is reducible from a random oracle--remains open. We show that if $NP \cap coNP$ is not a measurable subset of $EXP$, then $P^A \neq NP^A \cap coNP^A$ holds with probability 1 relative to a random oracle $A$. Conversely, establishing this random oracle separation with time-bounded measure would imply $BPP$ is a measure 0 subset of $EXP$.