TR08-005 | 15th January 2008 00:00

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 the 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 MA-EXP not in 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 progress stopped where it did: for example, why we

have superlinear circuit lower bounds for PromiseMA but not for NP.

Our second set of results follows from lower bounds in a new model of

algebraic query complexity, which we introduce in this paper and which

is interesting in its own right. Some of our lower bounds use direct

combinatorial and algebraic arguments, while others stem from a

surprising connection between our model and communication complexity.

Using this connection, we are also able to give an MA-protocol for the

Inner Product function with O(sqrt(n) log(n)) communication (essentially

matching a lower bound of Klauck), as well as a communication complexity

conjecture whose truth would imply NL!=NP.