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Proving SAT does not have small circuits with an application to the two queries problem

2008
*
Journal of computer and system sciences (Print)
*

We show that if

doi:10.1016/j.jcss.2007.06.017
fatcat:ocot2yals5hanhrb2kwexmcymu
*SAT**does**not**have**small**circuits*, then there must exist a*small*number of satisfiable formulas such that every*small**circuit*fails*to*compute satisfiability correctly on at least one of ... Even showing that*the*hierarchy collapsed*to*Σ p 2 remained open prior*to*this paper. ... Acknowledgments*The*authors thank Richard Chang for his comments on*an*earlier version of this paper.*The*third author thanks Alan Selman for his helpful insights and valuable suggestions. ...##
###
Proving SAT does not have small circuits with an application to the two queries problem

*
18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.
*

*Application*

*to*

*Two*

*Queries*In this section we show

*an*

*application*of our lemma

*to*

*the*

*two*

*queries*

*problem*. . ... If there are counter-examples, then some paths will guess

*the*correct counter-examples. However,

*the*existence of counter-examples

*to*& ¢ -size

*circuits*implies that

*SAT*

*does*

*not*

*have*. ...

##
###
Algorithms for Circuits and Circuits for Algorithms

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

For instance,

doi:10.1109/ccc.2014.33
dblp:conf/coco/Williams14
fatcat:t2irb2av2bcb5hzfi2dxkrsvqu
*an*algorithm determining whether a given*circuit*has*an*input that forces a true output would solve*the*NP-complete*Circuit*-*SAT**problem*. ...*Circuits*for algorithms refers*to**the*modeling of uniform algorithms*with*non-uniform*circuit*families (or*proving*such modeling is impossible). ... ACKNOWLEDGMENT*The*author acknowledges support by*the*Alfred P. Sloan foundation, Microsoft Research, and*the*NSF under grant CCF-1212372. ...##
###
Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy

1999
*
BRICS Report Series
*

Informally, a function f is said

doi:10.7146/brics.v6i46.20116
fatcat:z2tzdkuxpzbtnew6hyyboc5pgm
*to*be half-exponential if<br />f composed*with*itself is exponential ... What is*the*best*circuit*<br /> size lower bound that can be shown for*the*classes MA-TIME[f],<br />ZP-TIME^NP[f], . . . using*the*techniques currently known? ... But this*does**not*seem*to*imply (*), as*the*fact that*SAT**does**not**have*polynomial sized*circuits**does**not*imply that it has exponential sized*circuits*. ...##
###
Pseudorandomness for Approximate Counting and Sampling

2006
*
Computational Complexity
*

We also define

doi:10.1007/s00037-007-0218-9
fatcat:cfndygmw6jc2lhb6obaznpzlfu
*two*new primitives that we regard as*the*natural pseudorandom objects associated*with*approximate counting and sampling of NP-witnesses. ...*An*(n, s, )-discrepancy set is a subset T ⊆ {0, 1} n*with**the*property that for all Boolean*circuits*C of size at most s: Pr x C(x) = 1 − Pr t∈T C(t) = 1 ≤ . cc 15 (2007) Pseudorandomness for approx. counting ... Umans' research was supported by NSF grant CCF-0346991, BSF grant 2004329, and*an*Alfred P. Sloan Research Fellowship. ...##
###
Improving exhaustive search implies superpolynomial lower bounds

2010
*
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10
*

that can

doi:10.1145/1806689.1806723
dblp:conf/stoc/Williams10
fatcat:27ullhp2ajbsnhblvpr5of4gii
*query**an*NP oracle (*with**queries*of 2 O(n) length). ... Since k can be arbitrary, NTIME[2 n ]*does**not**have*polynomial size universal witness*circuits*, hence NEXP also*does**not*. By*the*contrapositive of Lemma 3.1, we conclude that NEXP ⊆ P/poly. ... Acknowledgments I am grateful*to*Russell Impagliazzo, Dick Lipton, Ken Regan, Mohan Paturi, and John Rogers for useful discussions on this work. Special thanks are due*to*V. ...##
###
Improving Exhaustive Search Implies Superpolynomial Lower Bounds

2013
*
SIAM journal on computing (Print)
*

that can

doi:10.1137/10080703x
fatcat:imqehe4hijbpfa354vudwpah5u
*query**an*NP oracle (*with**queries*of 2 O(n) length). ... Since k can be arbitrary, NTIME[2 n ]*does**not**have*polynomial size universal witness*circuits*, hence NEXP also*does**not*. By*the*contrapositive of Lemma 3.1, we conclude that NEXP ⊆ P/poly. ... Acknowledgments I am grateful*to*Russell Impagliazzo, Dick Lipton, Ken Regan, Mohan Paturi, and John Rogers for useful discussions on this work. Special thanks are due*to*V. ...##
###
Amplifying lower bounds by means of self-reducibility

2010
*
Journal of the ACM
*

We also show that

doi:10.1145/1706591.1706594
fatcat:on7yqocrijckhbbamxiui3aolu
*problems**with**small*uniform constant-depth*circuits**have*algorithms that simultaneously*have**small*space and time bounds. ... As*an*example of what this observation yields, consider*the*Boolean Formula Evaluation*problem*(BFE), which is complete for NC 1 . ... We thank Ryan Williams for calling our attention*to**the*work of Srinivasan [19] . We thank*the*program committee (and their referees) for helpful suggestions. ...##
###
Amplifying Lower Bounds by Means of Self-Reducibility

2008
*
2008 23rd Annual IEEE Conference on Computational Complexity
*

We also show that

doi:10.1109/ccc.2008.11
dblp:conf/coco/AllenderK08
fatcat:g7admhuhebhzdpne4ryyl6pczm
*problems**with**small*uniform constant-depth*circuits**have*algorithms that simultaneously*have**small*space and time bounds. ... As*an*example of what this observation yields, consider*the*Boolean Formula Evaluation*problem*(BFE), which is complete for NC 1 . ... We thank Ryan Williams for calling our attention*to**the*work of Srinivasan [19] . We thank*the*program committee (and their referees) for helpful suggestions. ...##
###
The Circuit-Input Game, Natural Proofs, and Testing Circuits With Data

2015
*
Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science - ITCS '15
*

We

doi:10.1145/2688073.2688115
dblp:conf/innovations/ChapmanW15
fatcat:dxgjg4xysnebvjhabo7uss6caa
*prove*that designing*small*test suites for f is equivalent*to**proving**circuit*lower bounds on f :*the*data complexity of testing f is "*small*" if and only if*the**circuit*complexity of f is "large." ... We give*two*new*applications*of these classical results*to**circuit*complexity: Natural properties useful against self-checking*circuits*are equivalent*to**circuit*lower bounds. ... Acknowledgements We thank*the*ITCS reviewers for their helpful comments. In particular, one reviewer improved*an*earlier version of Theorem 2.3. ...##
###
On the Power of Randomized Reductions and the Checkability of SAT

2010
*
2010 IEEE 25th Annual Conference on Computational Complexity
*

We first

doi:10.1109/ccc.2010.16
dblp:conf/coco/MahmoodyX10
fatcat:aye2ebywvvfgfk67gw3egiwmu4
*prove*that BPP PrSZK ⊆ AM ∩ coAM, where PrSZK is*the*class of promise*problems**having*statistical zero knowledge proofs. ... Then we investigate*the*power of randomized oracle reductions*with*relation*to**the*notion of instance checking (Blum and Kannan, J. ACM '95). ... We thank Salil Vadhan for sharing his insight into zero knowledge and promise*problems*, as well as giving pointers*to**the*literature on promise*problems*. ...##
###
Which Problems Have Strongly Exponential Complexity?

2001
*
Journal of computer and system sciences (Print)
*

We also look at

doi:10.1006/jcss.2001.1774
fatcat:iokfgohwmjgbxiiawfj5l2plwa
*the*issue of*proving*strongly exponential lower bounds for AC 0 , that is, bounds of*the*form 2 W(n) . This*problem*is even open for depth-3*circuits*. ... We show that*Circuit*-*SAT*is SERF-complete for all NP-search*problems*, and that for any fixed k \ 3, k-*SAT*, k-Colorability, k-Set Cover, Independent Set, Clique, and Vertex Cover, are SERF-complete for ... We also thank*the*referees for*the*useful comments. ...##
###
Probabilistically checkable proofs
[chapter]

2004
*
Computational Complexity Theory
*

*An*instance of

*Circuit*

*Sat*consists of a Boolean

*circuit*C comprising of

*Not*gates and And, Xor gates of fan-in

*two*, and

*the*goal is

*to*decide if C has a satisfying input. ...

*To*

*prove*NP-hardness, we reduce from

*Circuit*

*Sat*. ...

##
###
On Basing Lower-Bounds for Learning on Worst-Case Assumptions

2008
*
2008 49th Annual IEEE Symposium on Foundations of Computer Science
*

is allowed

doi:10.1109/focs.2008.35
dblp:conf/focs/ApplebaumBX08
fatcat:44nkrplugncare53zwbzd26lwu
*to*output any efficient hypothesis approximating*the*concept, including*an*"improper" hypothesis that is*not*itself in*the*concept class. ... In particular, we*prove*that if a language L reduces*to**the*task of improper learning of*circuits*, then, depending on*the*type of*the*reduction in use, either (1) L has a statistical zero-knowledge argument ... Indeed,*the*only efficiency guarantee we*have*is that f z has a*small**circuit*for every fixed z. ...##
###
Probabilistically checkable proofs

2009
*
Communications of the ACM
*

*An*instance of

*Circuit*

*Sat*consists of a Boolean

*circuit*C comprising of

*Not*gates and And, Xor gates of fan-in

*two*, and

*the*goal is

*to*decide if C has a satisfying input. ...

*To*

*prove*NP-hardness, we reduce from

*Circuit*

*Sat*. ...

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