Random Permutations in Computational Complexity

2025-11-11

International Symposium on Mathematical Foundations of Computer Science

AI Summary

A long-standing open question asks whether, relative to a random oracle (A), it holds that (P^A = NP^A cap coNP^A). Method: This paper develops a resource-bounded measure theory framework tailored to individual random permutations (pi), introducing *permutation martingales* and *permutation games* to precisely characterize null sets in the permutation space under polynomial-time constraints. Contribution/Results: It is proven that for *almost all* polynomial-time computable random permutations (pi), (P^pi eq NP^pi cap coNP^pi); moreover, this separation is strengthened to the quantum setting, yielding (NP^pi cap coNP^pi subseteq BQP^pi). Crucially, this constitutes the first *measure-theoretic “almost all”* separation—stronger than the classical “almost everywhere”—thereby establishing a rigorous measure-theoretic foundation for the fundamental role of random permutations in computational complexity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

Classical results of Bennett and Gill (1981) show that with probability 1, $P^A eq NP^A$ relative to a random oracle $A$, and with probability 1, $P^pi eq NP^pi cap coNP^pi$ 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 $P^pi eq NP^pi cap coNP^pi$ 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 $NP^pi cap coNP^pi otsubseteq BQP^pi$ 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 $P^A eq NP^A cap coNP^A$ 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 $EXP$.

Problem

• Establishing complexity separations for individually random permutations
• Developing measure framework for resource-bounded random permutations
• Investigating relationships between random permutations and random oracles

Innovation

• Introducing resource-bounded measure framework for random permutations
• Defining permutation martingales and betting games
• Proving separations for polynomial-time random permutations