Summary: Algebrization: A New Barrier in Complexity Theory
Institute for Advanced Study
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 algebriza-
tion. 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 delin-
eate 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 algebrize. Second, we show that almost all of
the major open problems--including P versus NP, P versus RP, and NEXP versus P/poly--will
require non-algebrizing techniques. In some cases algebrization seems to explain exactly why