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A Relativization Perspective on Meta-Complexity

Authors Hanlin Ren , Rahul Santhanam

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LIPIcs.STACS.2022.54.pdf
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ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Circuit complexity
  • Theory of computation → Problems, reductions and completeness
Keywords
  • meta-complexity
  • relativization
  • minimum circuit size problem

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Abstract

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^{n/2}] and MCSP[2^{n/4}] 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+ε) in polynomial time, for any ε > 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].

Cite As Get BibTex

Hanlin Ren and Rahul Santhanam. A Relativization Perspective on Meta-Complexity. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.54

Author Details

Hanlin Ren
  • University of Oxford, UK
Rahul Santhanam
  • University of Oxford, UK

References

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