Diagonalization Without Relativization A Closer Look at the Baker-Gill-Solovay Theorem

TL;DR

This paper introduces a new approach called semi-relativization that distinguishes certain complexity class separations from relativization, providing insights into longstanding barriers like Baker-Gill-Solovay and natural proofs.

Contribution

It defines semi-relativization as a novel method to differentiate complexity classes without relativization, and applies it to show separations like R and RE, and P and NP, evade known barriers.

Findings

Semi-relativization separates R and RE classes.

Polynomial acceptance problem reduces to CIRCUIT-SAT and 3-CNF-SAT.

Semi-relativization evades relativization, algebrization, and natural proofs barriers.

Abstract

We already know that several problems like the inequivalence of P and EXP as well as the undecidability of the acceptance problem and halting problem relativize. However, relativization is a limited tool which cannot separate other complexity classes. What has not been proven explicitly is whether the Turing-recognizability of the acceptance problem relativizes. We will consider an oracle for which R and RE are equivalent; RA = REA, where A is an oracle for the equivalence problem in the class ALL, but not in RE nor co-RE. We will then differentiate between relativization and what we will call "semi-relativization", i.e., separating classes using only the acceptance problem oracle. We argue the separation of R and RE is a fact that only "semi-relativization" proves. We will then "scale down" to the polynomial analog of R and RE, to evade the Baker-Gill-Solovay barrier using "semi-relativized" diagonalization, noting this subtle distinction between diagonalization and relativization. This "polynomial acceptance problem" is then reducible to CIRCUIT-SAT and 3-CNF-SAT proving that these problems are undecidable in polynomial time yet verifiable in polynomial time. "Semi-relativization" does not employ arithmetization to evade the relativization barrier, and so itself evades the algebrization barrier of Aaronson and Wigderson. Finally, since semi-relativization is a non-constructive technique, the natural proofs barrier of Razborov and Rudich is evaded. Thus the separation of R and RE as well as P and NP both do not relativize but do "semi-relativize", evading all three barriers.