Affine Relativization

We strengthen existing evidence for the so-called “algebrization barrier.” Algebrization—short for algebraic relativization—was introduced by Aaronson and Wigderson (AW) (STOC 2008) to characterize proofs involving arithmetization, simulation, and other “current techniques.” However, unlike relativization, eligible statements under this notion do not seem to have basic closure properties, making it conceivable to take two proofs, both with algebrizing conclusions, and combine them to get a proof without. Further, the notion is undefined for most types of statements and does not seem to yield a general criterion by which we can tell, given a proof, whether it algebrizes. In fact, the very notion of an algebrizing proof is never made explicit, and casual attempts to define it are problematic. All these issues raise the question of what evidence, if any, is obtained by knowing whether some statement does or does not algebrize. We give a reformulation of algebrization without these shortcomings. First, we define what it means for any statement/proof to hold relative to any language, with no need to refer to devices like a Turing machine with an oracle tape. Our approach dispels the widespread misconception that the notion of oracle access is inherently tied to a computational model. We also connect relativizing statements to proofs, by showing that every proof that some statement relativizes is essentially a relativizing proof of that statement. We then define a statement/proof as relativizing affinely if it holds relative to every affine oracle—here an affine oracle is the result of a particular error correcting code applied to the characteristic string of a language. We show that every statement that AW declare as algebrizing does relativize affinely, in fact, has a proof that relativizes affinely, and that no such proof exists for any of the statements shown not-algebrizing by AW in the classical computation model. Our work complements, and goes beyond, the subsequent work by Impagliazzo, Kabanets, and Kolokolova (STOC 2009), which also proposes a reformulation of algebrization, but falls short of recovering some key results of AW, most notably regarding the NEXP versus P/poly question. Using our definitions, we obtain new streamlined proofs of several classic results in complexity, including PSPACE ⊂ IP and NEXP ⊂ MIP. This may be of separate interest.