A Relativization Perspective on Meta-Complexity.

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where: 1. MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC’16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof. 2. The complexities of MCSP[2] and MCSP[2] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not “robust” to the choice of the size function. 3. Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2 + ) in polynomial time, for any > 0. 4. Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS’20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets. 5. DistNP does not reduce to GapMINKT by a family of “robust” reductions. This presents a technical barrier for solving a question of Hirahara [FOCS’20]. ∗rhl16@mails.tsinghua.edu.cn †rahul.santhanam@cs.ox.ac.uk ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 89 (2021)