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Algebrization: A New Barrier in Complexity Theory Scott Aaronson #

Summary: Algebrization: A New Barrier in Complexity Theory
Scott Aaronson #
Avi Wigderson +
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 linear­size 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 low­degree 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 non­relativizing
results based on arithmetization---both inclusions such as IP = PSPACE and MIP = NEXP, and
separations such as MA EXP ## 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 non­algebrizing techniques. In some cases algebrization seems to explain exactly why


Source: Aaronson, Scott - Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT)


Collections: Physics; Computer Technologies and Information Sciences