# Algebrization: A New Barrier in Complexity Theory

```@article{Aaronson2009AlgebrizationAN,
title={Algebrization: A New Barrier in Complexity Theory},
author={Scott Aaronson and Avi Wigderson},
journal={ACM Trans. Comput. Theory},
year={2009},
volume={1},
pages={2:1-2:54}
}```
• Published 1 February 2009
• Mathematics, Computer Science
• ACM Trans. Comput. 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 (e.g., 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 article, we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we…
226 Citations
BARRIERS IN COMPLEXITY THEORY
Several proof methods have been successfully employed over the years to prove several important results in Complexity Theory, but no one has been able to settle the question P =?NP. This article
Formal Barriers to Proving P 6 = NP : Relativization and Natural Proofs One Anonymous
• Computer Science, Mathematics
• 2019
The theoretical computer science community of the last century has continuously been stumped by a seemingly simple conjecture: namely, P 6= NP. While most complexity theorists believe that it is true
Relativization and Interactive Proof Systems in Parameterized Complexity Theory
Here, a new and non-trivial characterization of the A-Hierarchy in terms of oracle machines is obtained, and a famous result of Baker, Gill, and Solovay (1975) is parameterized by proving that, relative to specific oracles, FPT and A can either coincide or differ.
Affine Relativization
• Mathematics
ACM Trans. Comput. Theory
• 2018
This work defines 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, and dispels the widespread misconception that the notion of oracle access is inherently tied to a computational model.
An axiomatic approach to algebrization
• Computer Science, Mathematics
STOC '09
• 2009
An axiomatic approach to "algebrization", which complements and clarifies the approaches of [For94] and [AW08], and presents logical theories formalizing the notion of algebrizing techniques in the following sense.
A Relativization Perspective on Meta-Complexity
• Computer Science
Electron. Colloquium Comput. Complex.
• 2021
It is shown that a relativization barrier applies to many important open questions in meta-complexity, and gives relativized worlds where MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved, even approximately.
PRAMs over integers do not compute maxflow efficiently
• Computer Science, Mathematics
ArXiv
• 2018
This article analyzes two proofs of complexity lower bound: Ben-Or's proof of minimal height of algebraic computational trees deciding certain problems and Mulmuley's proof that restricted Parallel Random Access Machines (prams) over integers can not decide P-complete problems efficiently.
Circuit Complexity, Proof Complexity, and Polynomial Identity Testing
• Computer Science, Mathematics
J. ACM
• 2018
A new and natural algebraic proof system, whose complexity measure is essentially the algebraic circuit size of Nullstellensatz certificates, that shows that any super-polynomial lower bound on any Boolean tautology in this proof system implies that the permanent does not have polynomial-size algebraic circuits.
Why are certain polynomials hard?: A look at non-commutative, parameterized and homomorphism polynomials
This thesis tries to answer the question why specific polynomials have no small suspected arithmetic circuits and introduces a new framework for arithmetic circuits, similar to fixed parameter tractability in the boolean setting.
Succinct hitting sets and barriers to proving algebraic circuits lower bounds
• Mathematics, Computer Science
STOC
• 2017
Following a similar result of Williams in the boolean setting, it is shown that the existence of an algebraic natural proofs barrier is equivalent to the existenceof succinct derandomization of the polynomial identity testing problem.