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Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates [article]

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor Oliveira
2020 arXiv   pre-print
We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^1.99]∘G, for classes G of functions with low communication complexity.  ...  The class FORMULA[s] ∘G consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class G.  ...  We also thank Mahdi Cheraghchi for several discussions on the analysis of Boolean circuits with a bottom layer of parity gates.  ... 
arXiv:2002.08533v1 fatcat:qfiearqugbaklfftdb2ku7flsa

Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor C. Oliveira, Shubhangi Saraf
2020 Computational Complexity Conference  
We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]∘𝒢, for classes 𝒢 of functions with low communication complexity.  ...  As a corollary, we get an average-case lower bound for 𝖦𝖨𝖯^k_n against 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^{1.99}]∘𝖯𝖳𝖥^{k-1}, i.e., sub-quadratic-size de Morgan formulas with degree-(k-1) PTF (polynomial threshold  ...  The size of a de Morgan formula is the number of its leaf gates. In this work, we denote by FORMULA[s] the class of Boolean functions computable by size-s de Morgan formulas.  ... 
doi:10.4230/lipics.ccc.2020.15 dblp:conf/coco/KabanetsKLMO20 fatcat:jzht3ie3ezg7di7goqaak6z32e

Super-cubic lower bound for generalized Karchmer-Wigderson games [article]

Artur Ignatiev, Ivan Mihajlin, Alexander Smal
2022 Electronic colloquium on computational complexity  
, and hence for proving new lower bounds on De Morgan formula size.  ...  Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic.  ...  Let D(f ) denotes the minimal depth of De Morgan formula for function f .  ... 
dblp:journals/eccc/IgnatievMS22 fatcat:c2rqoh6ojbbsxaewmskwwqiygq

Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)

Manindra Agrawal, Thomas Thierauf, Christopher Umans, Marc Herbstritt
2013 Dagstuhl Reports  
We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques.  ...  This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings.  ...  For de Morgan formulas, seed length s 1/3+o(1) ; 2. For formulas over an arbitrary basis, seed length s 1/2+o(1) ; 3. For read-once de Morgan formulas, seed length s 0.234... ; 4.  ... 
doi:10.4230/dagrep.2.10.60 dblp:journals/dagstuhl-reports/AgrawalTU12 fatcat:dg7ithf6xfgadkzwkxjtjhy7ge

Multiparty Karchmer - Wigderson Games and Threshold Circuits

Alexander Kozachinskiy, Vladimir Podolskii, Shubhangi Saraf
2020 Computational Complexity Conference  
In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables.  ...  We suggest a generalization of Karchmer - Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates.  ...  Acknowledgements The authors are grateful to Alexander Shen for suggesting to generalize our initial results.  ... 
doi:10.4230/lipics.ccc.2020.24 dblp:conf/coco/KozachinskiyP20 fatcat:mtevyuog4nfulgnbob3leylmxa

Multiparty Karchmer-Wigderson Games and Threshold Circuits [article]

Alexander Kozachinskiy, Vladimir Podolskii
2020 arXiv   pre-print
In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables.  ...  We suggest a generalization of Karchmer-Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates.  ...  The authors are grateful to Alexander Shen for suggesting to generalize our initial results.  ... 
arXiv:2002.07444v1 fatcat:osvrwznibbgytg23icgxj2eate

Depth Lower Bounds against Circuits with Sparse Orientation [article]

Sajin Koroth, Jayalal Sarma
2015 arXiv   pre-print
However, using specific properties of the Clique function and the Karchmer-Wigderson framework (Karchmer and Wigderson, 1988), we go beyond the limitations and obtain lower bounds when the weight restrictions  ...  We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d × w must be Ω(n).  ...  And the best known depth lower bound for an explicit function against general circuits of bounded fan-in is (derived from formula size lower bound due to Håstad [19] ) less than 3 log n.  ... 
arXiv:1404.7443v2 fatcat:lrp3oauqxfgcjmzg4i5zghpvpy

Depth Lower Bounds against Circuits with Sparse Orientation [chapter]

Sajin Koroth, Jayalal Sarma
2014 Lecture Notes in Computer Science  
However, using specific properties of the CLIQUE function (used in [5] ) and the Karchmer-Wigderson framework [12], we go beyond the limitations and obtain lower bounds when the weight restrictions are  ...  We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d×w must be Ω(n).  ...  And the best known depth lower bound for an explicit function against general circuits of bounded fan-in is (derived from formula size lower bound due to Håstad [19] ) less than 3 log n.  ... 
doi:10.1007/978-3-319-08783-2_51 fatcat:xz5rpndedrbuzpr6g47dz25pfq

The complexity of symmetric boolean functions [chapter]

Ingo Wegener
1987 Lecture Notes in Computer Science  
The dual function f d of f is ¬f(x 1 x n ) . By the rules of de Morgan we obtain a monotone circuit for f d by replacing in a monotone circuit for f ∧-gates by ∨-gates and vice versa.  ...  For all i j ∈ {1 n} the number of x i -and y j -leaves in the formula for g 2 is bounded by s − 1 , since the formula for g 1 includes at least one y j -leaf.  ...  This lower bound method has been introduced by Wegener (84 c) for the proof of lower bounds on the BP1-complexity of clique functions cl n k (see Def. 11.1, Ch. 6).  ... 
doi:10.1007/3-540-18170-9_185 fatcat:tmwjsublb5bohpc56dzb5zmiey

On derandomizing algorithms that err extremely rarely

Oded Goldreich, Avi Widgerson
2014 Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14  
We also obtain results for other classes of computational devices including log-space algorithms and Arithmetic circuits.  ...  In relation to the above question, we put forward the following quantified derandomization challenge: For a class of circuits C (e.g., P/poly or AC 0 , AC 0 [2]) and a bounding function B : N → N (e.g.  ...  O.G. was partially supported by the Minerva Foundation with funds from the Federal German Ministry for Education and Research.  ... 
doi:10.1145/2591796.2591808 dblp:conf/stoc/GoldreichW14 fatcat:g25g7hytyrfidh4cjipix6owfy

On Derandomized Composition of Boolean Functions [article]

Or Meir
2017 Electronic colloquium on computational complexity  
has been used several times in the past for amplifying different hardness measures of f and g.  ...  We show that this idea can be realized in the particular setting of the composition of functions and universal relations [KRW95, GMWW17].  ...  We would also like to thank anonymous referees for comments that improved the presentation of this work.  ... 
dblp:journals/eccc/Meir17b fatcat:fzjt2dr4dvea7doxpuc3o4ll6y

Integrated logic synthesis using simulated annealing

Petra Färm, Elena Dubrova, Andreas Kuehlmann
2011 Proceedings of the 21st edition of the great lakes symposium on Great lakes symposium on VLSI - GLSVLSI '11  
KTH School of Information and Communication Technology SE-164 40 Kista SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie  ...  Dynamic weighting reflects the sensitivity of the local graph structures with respect to the actual technology parameters such as gate sizes, delays, and power levels.  ...  In Lemma 1, a formula for the upper bound on the size of a two-level CAEN implementation is defined. The formula is assuming that both the function and its complement is realized.  ... 
doi:10.1145/1973009.1973095 dblp:conf/glvlsi/FarmDK11 fatcat:dlju72re25ae7jgninv5kuntia

Analysis of Boolean Functions [article]

Ryan O'Donnell
2021 arXiv   pre-print
The book can be used as a reference for working researchers or as the basis of a one-semester graduate-level course.  ...  In both years most of Chapters 1-5 and 7 were covered, along with parts of Chapters 6, 8, 9, and 11, and some additional material on additive combinatorics.  ...  Here we use the definition of general (De Morgan) circuits from Exercise 4.13.  ... 
arXiv:2105.10386v1 fatcat:h3tobmrcjbhh3euva6ogwiubya

An Approach to Distribution of Object-Oriented Applications in Loosely Coupled Networks

Sandeep Purao, Hemant K. Jain, Derek L. Nazareth
2002 Journal of Management Information Systems  
Considering the convention of treating non-leaf classes as abstract classes [67], and the low values for the depth of inheritance metric observed in prac- tice (DIT [depth of inheritance tree] median of  ...  A number of commercial systems such as GAMS [33], and solvers such as ZOOM [33], employ- ing the branch-and-bound algorithm, are available for solving the model.  ... 
doi:10.1080/07421222.2002.11045689 fatcat:bmo57pqnvzb2zgcbsxn25j5nlm

A trans-disciplinary review of deep learning research and its relevance for water resources scientists

Chaopeng Shen
2018 Water Resources Research  
DL is especially suited for information extraction from image-like data and sequential data. Techniques and experiences presented in other disciplines are of high relevance to water research.  ...  I argue that DL can help address several major new and old challenges facing research in water sciences such as inter-disciplinarity, data discoverability, hydrologic scaling, equifinality, and needs for  ...  Michigan State and three anonymous reviewers whose suggestions and comments have helped to improve this paper greatly. This review paper is theoretical and does not contain any dataset to be shared.  ... 
doi:10.1029/2018wr022643 fatcat:ruopsnchg5eg5hsiccyadinf54
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