Algebrization: A New Barrier in Complexity Theory

Any proof of P = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization--both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP P/poly --do indeed...

• Type: Journal
• Year: 2008
• Where: ECCC
• Authors: Scott Aaronson, Avi Wigderson
• Keywords: Complexity Theory, ECCC 2008, Lower Bound, Open Problems