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

Lance Fortnow, A. Pavan, Samik Sengupta
2008 Journal of computer and system sciences (Print)  
We show that if 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.  ... 
doi:10.1016/j.jcss.2007.06.017 fatcat:ocot2yals5hanhrb2kwexmcymu

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

L. Fortnow, A. Pavan, S. Sengupta
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 .  ... 
doi:10.1109/ccc.2003.1214433 dblp:conf/coco/FortnowPS03 fatcat:c6yug4vwpzcg7i5gh67w4pb6z4

Algorithms for Circuits and Circuits for Algorithms

Ryan Williams
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
For instance, 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.  ... 
doi:10.1109/ccc.2014.33 dblp:conf/coco/Williams14 fatcat:t2irb2av2bcb5hzfi2dxkrsvqu

Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy

Peter Bro Miltersen, Vinodchandran N. Variyam, Osamu Watanabe
1999 BRICS Report Series  
Informally, a function f is said 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.  ... 
doi:10.7146/brics.v6i46.20116 fatcat:z2tzdkuxpzbtnew6hyyboc5pgm

Pseudorandomness for Approximate Counting and Sampling

Ronen Shaltiel, Christopher Umans
2006 Computational Complexity  
We also define 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.  ... 
doi:10.1007/s00037-007-0218-9 fatcat:cfndygmw6jc2lhb6obaznpzlfu

Improving exhaustive search implies superpolynomial lower bounds

Ryan Williams
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
that can 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.  ... 
doi:10.1145/1806689.1806723 dblp:conf/stoc/Williams10 fatcat:27ullhp2ajbsnhblvpr5of4gii

Improving Exhaustive Search Implies Superpolynomial Lower Bounds

Ryan Williams
2013 SIAM journal on computing (Print)  
that can 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.  ... 
doi:10.1137/10080703x fatcat:imqehe4hijbpfa354vudwpah5u

Amplifying lower bounds by means of self-reducibility

Eric Allender, Michal Koucký
2010 Journal of the ACM  
We also show that 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.  ... 
doi:10.1145/1706591.1706594 fatcat:on7yqocrijckhbbamxiui3aolu

Amplifying Lower Bounds by Means of Self-Reducibility

Eric Allender, Michal Koucký
2008 2008 23rd Annual IEEE Conference on Computational Complexity  
We also show that 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.  ... 
doi:10.1109/ccc.2008.11 dblp:conf/coco/AllenderK08 fatcat:g7admhuhebhzdpne4ryyl6pczm

The Circuit-Input Game, Natural Proofs, and Testing Circuits With Data

Brynmor Chapman, Ryan Williams
2015 Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science - ITCS '15  
We 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.  ... 
doi:10.1145/2688073.2688115 dblp:conf/innovations/ChapmanW15 fatcat:dxgjg4xysnebvjhabo7uss6caa

On the Power of Randomized Reductions and the Checkability of SAT

Mohammad Mahmoody, David Xiao
2010 2010 IEEE 25th Annual Conference on Computational Complexity  
We first 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.  ... 
doi:10.1109/ccc.2010.16 dblp:conf/coco/MahmoodyX10 fatcat:aye2ebywvvfgfk67gw3egiwmu4

Which Problems Have Strongly Exponential Complexity?

Russell Impagliazzo, Ramamohan Paturi, Francis Zane
2001 Journal of computer and system sciences (Print)  
We also look at 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.  ... 
doi:10.1006/jcss.2001.1774 fatcat:iokfgohwmjgbxiiawfj5l2plwa

Probabilistically checkable proofs [chapter]

Madhu Sudan
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.  ... 
doi:10.1090/pcms/010/12 fatcat:4vhklk62eng4lacp5tpcexo4ve

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

Benny Applebaum, Boaz Barak, David Xiao
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
is allowed 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.  ... 
doi:10.1109/focs.2008.35 dblp:conf/focs/ApplebaumBX08 fatcat:44nkrplugncare53zwbzd26lwu

Probabilistically checkable proofs

Madhu Sudan
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.  ... 
doi:10.1145/1467247.1467267 fatcat:jwhq2op7yvfvtfezwxxqlj43i4
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