Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
The probabilistic communication complexity of set intersection
SIAM Journal on Discrete Mathematics
Algebraic methods for interactive proof systems
Journal of the ACM (JACM)
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On the distributional complexity of disjointness
Theoretical Computer Science
Making games short (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Exponential separation of quantum and classical communication complexity
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
In search of an easy witness: exponential time vs. probabilistic polynomial time
Journal of Computer and System Sciences - Complexity 2001
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
Pseudorandomness and Average-Case Complexity via Uniform Reductions
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
On the Power of Quantum Proofs
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Oracles Are Subtle But Not Malicious
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Circuit lower bounds for Merlin-Arthur classes
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On Computation and Communication with Small Bias
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
A survey of lower bounds for satisfiability and related problems
Foundations and Trends® in Theoretical Computer Science
On determinism versus non-determinism and related problems
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
On determinism versus non-determinism and related problems
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
How to generate and exchange secrets
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
An axiomatic approach to algebrization
Proceedings of the forty-first annual ACM symposium on Theory of computing
Unconditional Lower Bounds against Advice
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna
Journal of the ACM (JACM)
Streaming computations with a loquacious prover
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Competing provers protocols for circuit evaluation
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Arthur-Merlin streaming complexity
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Nonuniform ACC Circuit Lower Bounds
Journal of the ACM (JACM)
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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 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 nonrelativizing 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 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 article 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 (√nlogn) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL ≠ NP.