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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]

2003
*
arXiv
*
pre-print

We prove several new

arXiv:quant-ph/0312208v1
fatcat:ikmw6hbxpfeq3kfyuzecvyu3lm
*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. ...##
###
NEXP Does Not Have Non-uniform Quasipolynomial-Size ACC Circuits of o(loglogn) Depth
[chapter]

2011
*
Lecture Notes in Computer Science
*

We show that there exists a language

doi:10.1007/978-3-642-20877-5_17
fatcat:lxkowswvgvc2bbzj6dgnf232qu
*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. ...##
###
Can large fanin circuits perform reliable computations in the presence of faults?

2000
*
Theoretical Computer Science
*

Here, we consider

doi:10.1016/s0304-3975(99)00237-6
fatcat:5muctxnjuzbtnniczg26bmnhiq
*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*. ...##
###
Lower Bounds on Interactive Compressibility by Constant-Depth Circuits

2012
*
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
*

We answer this question affirmatively

doi:10.1109/focs.2012.74
dblp:conf/focs/ChattopadhyayS12
fatcat:hezo4obwdfdxfljp4ccnuqyu7q
*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. ...##
###
Quantum algorithms and approximating polynomials for composed functions with shared inputs
[article]

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. ...

##
###
Quantum algorithms and approximating polynomials for composed functions with shared inputs

2021
*
Quantum
*

d) upper

doi:10.22331/q-2021-09-16-543
fatcat:uspa34j5vrbslnqvrma3esj7sq
*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. ...##
###
Circuit Lower Bounds, Help Functions, and the Remote Point Problem
[article]

2009
*
arXiv
*
pre-print

More precisely, proving

arXiv:0911.4337v1
fatcat:xm4i6qxryfclnaws5nt6apmvgi
*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. ...##
###
Depth Lower Bounds against Circuits with Sparse Orientation
[article]

2015
*
arXiv
*
pre-print

We study

arXiv:1404.7443v2
fatcat:lrp3oauqxfgcjmzg4i5zghpvpy
*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. ...##
###
Depth Lower Bounds against Circuits with Sparse Orientation
[chapter]

2014
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-319-08783-2_51
fatcat:xz5rpndedrbuzpr6g47dz25pfq
*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. ...##
###
New Bounds for Energy Complexity of Boolean Functions
[chapter]

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. ...

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

1998
*
Theoretical Computer Science
*

This is

doi:10.1016/s0304-3975(97)00227-2
fatcat:urnpqd5gbzggjkrzueaduhms6a
*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. ...##
###
On the complexity of powering in finite fields

2011
*
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
*

Our proof revisits

doi:10.1145/1993636.1993702
dblp:conf/stoc/Kopparty11
fatcat:isepqqscnrarrohsxg3efhjbcq
*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. ...##
###
Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

2021
*
Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
*

We also show similar unconditional unprovability results

doi:10.1145/3406325.3451117
fatcat:zwtwvpqcfbcvtlkxx75w7jkf5i
*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. ...##
###
Parallel random access machines with bounded memory wordsize

1991
*
Information and Computation
*

Finally, an efficient simulation by boolean

doi:10.1016/0890-5401(91)90069-e
fatcat:hshtr2dlrzbvtlzlj55hfuzrx4
*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. ...
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