Encoding Invariance in Average Case Complexity

Abstract

When we represent a decision problem, like CIRCUIT-SAT, as a language over the binary alphabet, we usually do not specify how to encode instances by binary strings. This relies on the empirical observation that the truth of a statement of the form "CIRCUIT-SAT belongs to a complexity class C" does not depend on the encoding, provided both the encoding and the class C are "natural". In this sense most of the Complexity theory is "encoding invariant".The notion of a polynomial time computable distribution from Average Case Complexity is one of the exceptions from this rule. It might happen that a distribution over some objects, like circuits, is polynomial time computable in one encoding and is not polynomial time computable in the other encoding. In this paper we suggest an encoding invariant generalization of a notion of a polynomial time computable distribution. The completeness proofs of known distributional problems, like Bounded Halting, are simpler for the new class than for polynomial time computable distributions. This paper has no new technical contributions. All the statements are proved using the known techniques.

A remark on pseudo proof systems and hard instances of the satisfiability problem

“…We thank Yijia Chen, Jan Krajíček and Ján Pich for valuable comments during early stages of this work. We thank Edward Hirsch for pointing out reference . Finally, we are grateful to the reviewers.…”

“…Our notion of invertibility is more restrictive than the one considered in[38].C 2018 The Authors. Mathematical Logic Quarterly published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org…”