# BPP has subexponential time simulations unlessEXPTIME has publishable proofs

@article{Babai2005BPPHS, title={BPP has subexponential time simulations unlessEXPTIME has publishable proofs}, author={L{\'a}szl{\'o} Babai and Lance Fortnow and Noam Nisan and Avi Wigderson}, journal={computational complexity}, year={2005}, volume={3}, pages={307-318} }

AbstractWe show thatBPP can be simulated in subexponential time for infinitely many input lengths unless exponential timeℴ collapses to the second level of the polynomial-time hierarchy.ℴ has polynomial-size circuits andℴ has publishable proofs (EXPTIME=MA).
We also show thatBPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we showBPP can be simulated in subexponential time for infinitely many input lengths…

## 73 Citations

### Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

- Computer Science, Mathematics2010 IEEE 25th Annual Conference on Computational Complexity
- 2010

If Arthur-Merlin protocols can be derandomized, then there is a language computable in deterministic exponential-time with access to an NP oracle that requires circuits of exponential size and the lower bound in the conclusion of the theorem suffices to construct pseudorandom generators with exponential stretch.

### Holographic proofs and derandomization

- Mathematics18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.
- 2003

We derive a stronger consequence of EXP having polynomial-size circuits than was known previously, namely that there is a simulation of P in MAPOLYLOG that fools all deterministic polynomial-time…

### If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances

- Computer Science, Mathematics20th Annual IEEE Conference on Computational Complexity (CCC'05)
- 2005

It is shown that there is a fixed distribution on instances of NP-complete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case).

### Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

- Computer Science, Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

A new and natural algebraic proof system, which has tight connections to (algebraic) circuit complexity, 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 (VNP≠VP).

### Circuit lower bounds for nondeterministic quasi-polytime: an easy witness lemma for NP and NQP

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2017

We prove that if every problem in NP has nk-size circuits for a fixed constant k, then for every NP-verifier and every yes-instance x of length n for that verifier, the verifier’s search space has an…

### On TC0 Lower Bounds for the Permanent

- Mathematics, Computer ScienceCOCOON
- 2012

This paper proves a new parameterized lower bound that includes each of the previous results as sub-cases for the permanent and implies that the permanent does not have Boolean threshold circuits of the following kinds.

### If NP Languages are Hard on the Worst-Case Then It is Easy to Find Their Hard Instances

- Computer Science, MathematicsComputational Complexity Conference
- 2005

It is shown that there is a fixed distribution on instances of NP-complete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial time algorithms (unless NP is easy in the worst-case).

### Solvable Group Isomorphism is (almost) in NP CoNP

- Mathematics
- 2004

The Group Isomorphism problem consists in decidingwhether two input groups G_1 and G_2 givenby their multiplication tables are isomorphic. Wefirst give a 2-round Arthur-Merlin protocol for theGroup…

### Nonuniform ACC Circuit Lower Bounds

- Computer ScienceJACM
- 2014

The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail these lower bounds, while the second step requires a strengthening of the author’s prior work.

### Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

- Computer Sciencecomputational complexity
- 2003

The technique combines the nonuniform hardness versus randomness tradeoff of Miltersen and Vinodchandran with “instance checking” and identifies a novel “resilience” property of hardness vs.randomness tradeoffs.

## References

SHOWING 1-10 OF 33 REFERENCES

### Multi-prover interactive proofs: how to remove intractability assumptions

- Computer Science, MathematicsSTOC '88
- 1988

It is proved that all NP languages have perfect zero-knowledge proof-systems in this model, without making any intractability assumptions, and its properties and applicability to cryptography are examined.

### Some connections between nonuniform and uniform complexity classes

- Mathematics, Computer ScienceSTOC '80
- 1980

This work aims to understand when nonuniform upper bounds can be used to obtain uniform upper bounds, and how to relate it to more common notions.

### Sparse sets in NP-P: Exptime versus nexptime

- Computer ScienceInf. Control.
- 1985

The paper exploits the recently discovered upward separation method and uses relativization techniques to determine logical possibilities, limitations of these proof techniques, and, for the first time, to exhibit structural differences between relativized NP and CoNP.

### Hiding Instances in Multioracle Queries

- Computer Science, MathematicsSTACS
- 1990

It is shown that, if f is an NP-hard function, A cannot query a single oracle B while hiding all but the size of the instance, assuming that the polynomial hierarchy does not collapse.

### Arithmetization: A new method in structural complexity theory

- Mathematicscomputational complexity
- 2005

A technique of arithmetization of the process of computation is introduced in order to obtain novel characterizations of certain complexity classes viamultivariate polynomials, demonstrating the applicability of this technique to classical complexity classes.

### On Relativized Exponential and Probabilistic Complexity Classes

- Mathematics, Computer ScienceInf. Control.
- 1986

### On the existence of pseudorandom generators

- Mathematics, Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

It is shown that if aRegular functions in which every image of a k-bit string has the same number of preimages of length k are considered, then pseudorandom generators do exist, and assuming the intractability of general factoring, it can be proved that they do exist.

### The knowledge complexity of interactive proof-systems

- Mathematics, Computer ScienceSTOC '85
- 1985

A computational complexity theory of the “knowledge” contained in a proof is developed and examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity.

### Two theorems on random polynomial time

- Computer Science19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
- 1978

Where the traditional method of polynomial reduction has been inapplicable, randomness has been used in demonstrating intractibility by Adleman and Manders, and in showing problems equivalent by Rabin, a new examination of randomness is in order.