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Page 5112 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews  
911:68054 911:68054 68Q25 68Q15 Cai, Jin-Yi (1-PRIN-CS) Lower bounds for constant-depth circuits in the presence of help bits. Inform. Process. Lett. 36 (1990), no. 2, 79-83.  ...  Each help bit can be an arbitrary Boolean function of the input. We prove an exponential lower bound on the size of the circuit computing m parity functions in the presence of m — 1 help bits.  ... 

Quantum Lower Bounds for Fanout [article]

Maosen Fang, Stephen Fenner, Frederic Green, Steven Homer, Yong Zhang
2003 arXiv   pre-print
We prove several new lower bounds for constant depth quantum circuits.  ...  In the case of a non-constant number a of ancillae, we give a tradeoff between a and the required depth, that results in a non-trivial lower bound for fanout when a = n^1-o(1).  ...  This work was supported in part by the National Security Agency (NSA) and Advanced Research and Development Agency (ARDA) under Army Research Office (ARO) contract numbers DAAD 19-02-1-0058 (for M.  ... 
arXiv:quant-ph/0312208v1 fatcat:ikmw6hbxpfeq3kfyuzecvyu3lm

NEXP Does Not Have Non-uniform Quasipolynomial-Size ACC Circuits of o(loglogn) Depth [chapter]

Fengming Wang
2011 Lecture Notes in Computer Science  
We show that there exists a language in non-deterministic exponential time which can not be computed by any non-uniform family of ACC m circuits of quasipolynomial size and o(log log n) depth, where m  ...  ACC m circuits are circuits consisting of unbounded fan-in AND, OR and MOD m gates and unary NOT gates, where m is a fixed integer.  ...  Acknowledgement We thank Eric Allender, Luke Friedman and Ryan Williams for helpful discussions.  ... 
doi:10.1007/978-3-642-20877-5_17 fatcat:lxkowswvgvc2bbzj6dgnf232qu

Can large fanin circuits perform reliable computations in the presence of faults?

Rüdiger Reischuk
2000 Theoretical Computer Science  
Here, we consider the same question for unbounded fanin circuits which in the fault-free case can compute Boolean functions in sublogarithmic depth.  ...  We will show that in case of faulty and/or-circuits as well as threshold circuits an increase of fanin and size cannot be traded for a depth reduction if the error probabilities are unknown.  ...  Our analysis also gives an upper bound on the deterministic, resp. probabilistic complexity of functions that can be computed in the presence of faults in bounded depth.  ... 
doi:10.1016/s0304-3975(99)00237-6 fatcat:5muctxnjuzbtnniczg26bmnhiq

Lower Bounds on Interactive Compressibility by Constant-Depth Circuits

Arkadev Chattopadhyay, Rahul Santhanam
2012 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science  
We answer this question affirmatively for compression using constant-depth polynomial-size circuits.  ...  Dubrov and Ishai [DI06] proved a lower bound which can be interpreted in our setting as saying that there cannot be a 1-round protocol of cost O(n 1−δ ) for Parity on n bits in the AC 0 -compression game  ...  The second author would like to thank Yuval Ishai for suggesting it might be interesting to study compression games, and for preliminary discussions.  ... 
doi:10.1109/focs.2012.74 dblp:conf/focs/ChattopadhyayS12 fatcat:hezo4obwdfdxfljp4ccnuqyu7q

Quantum algorithms and approximating polynomials for composed functions with shared inputs [article]

Mark Bun, Robin Kothari, Justin Thaler
2021 arXiv   pre-print
The previous best size lower bound was Ω(n^1+4^-(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).  ...  This improves on the bound of O(Q(f) ·√(n)) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f.  ...  Acknowledgements We thank Nikhil Mande, Ronald de Wolf, and Shuchen Zhu for comments on earlier drafts of this paper. R.K. thanks Luke Schaeffer for comments on the proof of Theorem 1.  ... 
arXiv:1809.02254v3 fatcat:5k4gx2yxabaarnm5gzkptxbylq

Quantum algorithms and approximating polynomials for composed functions with shared inputs

Mark Bun, Robin Kothari, Justin Thaler
2021 Quantum  
d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits.  ...  The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).  ...  Acknowledgements We thank Nikhil Mande, Ronald de Wolf, and Shuchen Zhu for comments on earlier drafts of this paper. R.K. thanks Luke Schaeffer for comments on the proof of Theorem 1.  ... 
doi:10.22331/q-2021-09-16-543 fatcat:uspa34j5vrbslnqvrma3esj7sq

Circuit Lower Bounds, Help Functions, and the Remote Point Problem [article]

Vikraman Arvind, Srikanth Srinivasan
2009 arXiv   pre-print
More precisely, proving lower bounds for ABPs with help polynomials is related to the Remote Point Problem w.r.t. the rank metric, and for constant-depth circuits with help functions it is related to the  ...  We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions.  ...  We are grateful to Jaikumar Radhakrishnan for discussions. We also thank the anonymous referee for useful comments and suggestions.  ... 
arXiv:0911.4337v1 fatcat:xm4i6qxryfclnaws5nt6apmvgi

Depth Lower Bounds against Circuits with Sparse Orientation [article]

Sajin Koroth, Jayalal Sarma
2015 arXiv   pre-print
We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function f is the characteristic vector of the minimum sized set of negated  ...  We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting), separates NP from NC.  ...  The difficulty in extending the above lower bound to more general lower bounds is the potential presence of gates computing "densely" oriented functions.  ... 
arXiv:1404.7443v2 fatcat:lrp3oauqxfgcjmzg4i5zghpvpy

Depth Lower Bounds against Circuits with Sparse Orientation [chapter]

Sajin Koroth, Jayalal Sarma
2014 Lecture Notes in Computer Science  
We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation 1 of a function f is the characteristic vector of the minimum sized set of  ...  We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting), separates NP from NC.  ...  The difficulty in extending the above lower bound to more general lower bounds is the potential presence of gates computing "densely" oriented functions.  ... 
doi:10.1007/978-3-319-08783-2_51 fatcat:xz5rpndedrbuzpr6g47dz25pfq

New Bounds for Energy Complexity of Boolean Functions [chapter]

Krishnamoorthy Dinesh, Samir Otiv, Jayalal Sarma
2018 Lecture Notes in Computer Science  
For a Boolean function f : {0, 1} n → {0, 1} computed by a circuit C over a finite basis B, the energy complexity of C (denoted by EC B (C)) is the maximum over all inputs {0, 1} n the numbers of gates  ...  of the circuit C (excluding the inputs) that output a one.  ...  Acknowledgments The authors would like to thank the anonymous reviewers for their constructive comments.  ... 
doi:10.1007/978-3-319-94776-1_61 fatcat:5qjhgjwgi5ahfet5gez3256sai

Non-commutative arithmetic circuits: depth reduction and size lower bounds

Eric Allender, Jia Jiao, Meena Mahajan, V. Vinay
1998 Theoretical Computer Science  
This is the first depth-reduction result for arithmetic circuits over a non-commutative semiring, and it complements the lower bounds of Kosaraju and Nisan showing that depth reduction cannot be done in  ...  Finally, we characterize the languages generated by efficient circuits over the semiring (2r*, union, concat) in terms of simple one-way machines, and we investigate and extend earlier lower bounds on  ...  Acknowledgements 85 The first author thanks Shiyu Zhou and David Zuckerman for helpful discussions.  ... 
doi:10.1016/s0304-3975(97)00227-2 fatcat:urnpqd5gbzggjkrzueaduhms6a

On the complexity of powering in finite fields

Swastik Kopparty
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
Our proof revisits the classical Razborov-Smolensky method for circuit lower bounds, and executes an analogue of it in the land of univariate polynomials over F2n .  ...  We study the complexity of computing the k th -power of an element of F2n by constant depth arithmetic circuits over F2 (also known as AC 0 (⊕)).  ...  The main result here is that for constant p, powering has arithmetic circuits of bounded fan-in with size poly(n) and depth O(log(n)). For circuits with bounded fan-in, this depth is clearly optimal.  ... 
doi:10.1145/1993636.1993702 dblp:conf/stoc/Kopparty11 fatcat:isepqqscnrarrohsxg3efhjbcq

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Ján Pich, Rahul Santhanam
2021 Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing  
We also show similar unconditional unprovability results in APC 1 for the conjecture of Rudich about the existence of super-bits.  ...  circuits of sub-exponential size.  ...  Acknowledgements We would like to thank Jan Krajíček for helpful discussions, for comments on the draft of the paper and for pointing out Theorem 4 to us.  ... 
doi:10.1145/3406325.3451117 fatcat:zwtwvpqcfbcvtlkxx75w7jkf5i

Parallel random access machines with bounded memory wordsize

Stephen J. Bellantoni
1991 Information and Computation  
Finally, an efficient simulation by boolean circuits is used to give a simple new proof of the tight Q((logn)/(log log n)) time bound for PARITY on smallwordsize machines.  ...  The PRAM model of parallel computation is examined with respect to wordsize, the number of bits which can be held in each global memory cell.  ...  Using Hastad's 2(1/lo)n""'T' size lower bound for circuits computing PARITY in depth T [Hastad, 19861 and the lemma above, T satisfies ~'2'~p > 2(1'10)"1"cr', where c is some constant greater than 1.  ... 
doi:10.1016/0890-5401(91)90069-e fatcat:hshtr2dlrzbvtlzlj55hfuzrx4
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