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Dimension of Projections in Boolean Functions

Ramamohan Paturi, Francis Zane
1998 SIAM Journal on Discrete Mathematics  
We study monochromatic projections in 2-colorings of an n-dimensional Boolean cube. We also study the dimension of the largest projection contained in a set specified by its density.  ...  We also prove almost tight upper and lower bounds on the dimension of monochromatic projections in arbitrary Boolean functions.  ...  We also obtain bounds on the dimension of the largest monochromatic projections in arbitrary Boolean functions.  ... 
doi:10.1137/s0895480197318313 fatcat:jrwu67wpifh73pz4sgikqmhbaa

Boolean Functions, Projection Operators, and Quantum Error Correcting Codes

Vaneet Aggarwal, A. Robert Calderbank
2008 IEEE Transactions on Information Theory  
This paper describes a fundamental correspondence between Boolean functions and projection operators in Hilbert space.  ...  The correspondence is widely applicable, and it is used in this paper to provide a common mathematical framework for the design of both additive and non-additive quantum error correcting codes.  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for many suggestions that improved this paper and for bringing the work of Danielson [12] to their attention.  ... 
doi:10.1109/tit.2008.917720 fatcat:n6ewbvhtgzbcxgr2gdtf6t6oki

Exponential lower bounds for depth 3 Boolean circuits

Ramamohan Paturi, Michael E. Saks, Francis Zane
1997 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97  
~: = ~1) of dimension at least w.  ...  We consider the class ~k of unbounded fan-in depth-3 boolean circuits, for which the bottom fan-in is limited by k and thetop gateis an OR.  ...  Acknowledgments: The authorswould like to thankJohan Hktad for pointing out an error in the example in section 2.  ... 
doi:10.1145/258533.258556 dblp:conf/stoc/PaturiSZ97 fatcat:hp7xra5sw5c7dcx447gbkndaxe

The construction and weight distributions of all projective binary linear codes [article]

Cunsheng Ding
2020 arXiv   pre-print
Boolean functions can be used to construct binary linear codes in many ways, and vice versa.  ...  The objective of this short article is to point out a connection between the weight distributions of all projective binary linear codes and the Walsh spectra of all Boolean functions.  ...  In addition, other connections between projective binary codes and Boolean functions could also be developed.  ... 
arXiv:2010.03184v2 fatcat:iepzzct7vfa3jllyk3d56gawrq

Exponential lower bounds for depth three Boolean circuits

R. Paturi, M. E. Saks, F. Zane
2000 Computational Complexity  
We consider the class Σ k 3 of unbounded fan-in depth three Boolean circuits, for which the bottom fan-in is limited by k and the top gate is an OR.  ...  1, x i = x j or x i =x j ) of dimension at least log(a/s)/log n.  ...  A version of this paper appeared previously as Paturi et al. ( 1997 ) .  ... 
doi:10.1007/pl00001598 fatcat:7d5qcgwdyjamjfzsfp7h7b7yam

The threshold order of a Boolean function

Chi Wang, A.C. Williams
1991 Discrete Applied Mathematics  
It is also shown that a Boolean function is a threshold function of order at least m if some sequence of "contractions" and "projections" of it to a (generally) lower dimensional space is a threshold function  ...  We have used such functions in the problem of extrapolating partially defined Boolean functions.  ...  Let f(x) be an n-dimensional Boolean function, and let Tf= {x E B" (f(x) = 1) be the set of sisting of a polynomial pseudo-Boolean function of degree m, determined by the vector w of dimension  ... 
doi:10.1016/0166-218x(91)90032-r fatcat:xlaly3sonbbgdcwb6hly3cqbnm

Classifying bent functions by their Cayley graphs [article]

Paul Leopardi
2018 arXiv   pre-print
In 1999 Bernasconi and Codenotti noted that the Cayley graph of a bent function is strongly regular.  ...  SageMath scripts and CoCalc worksheets are used to compute and display some of these relationships, for bent functions up to dimension 8.  ...  The algorithm uses 8 S-boxes, each of which consists of 32 Boolean bent functions in 8 dimensions, with degree 4, making 256 bent functions in total.  ... 
arXiv:1705.04507v6 fatcat:rblk4oisyfazjp56qkatxrpdvq

A New Abstract Combinatorial Dimension for Exact Learning via Queries

José L. Balcázar, Jorge Castro, David Guijarro
2002 Journal of computer and system sciences (Print)  
We present a characterization of those Boolean function classes learnable in this abstract model, in terms of a new combinatorial notion that we introduce, the abstract identi cation dimension.  ...  currently known to characterize learning in these models, such as strong consistency dimension, extended teaching dimension, and certi cate size.  ...  A Boolean function of arity n is a function from f0; 1g n ! f0; 1g. The set of all Boolean functions is denoted by B n . An element x of f0; 1g n is called an assignment.  ... 
doi:10.1006/jcss.2001.1794 fatcat:3maos6rua5gfnaolgxylnfth7i

Characterization and Lower Bounds for Branching Program Size using Projective Dimension [article]

Krishnamoorthy Dinesh and Sajin Koroth and Jayalal Sarma
2017 arXiv   pre-print
Motivated by the argument in (Pudlák, Rödl 1992), we define two variants of projective dimension - projective dimension with intersection dimension 1 (denoted by upd(G)) and bitwise decomposable projective  ...  We show that there exist a Boolean function f (on n bits) for which the gap between the projective dimension and size of the optimal branching program computing f (denoted by bpsize(f)), is 2^Ω(n).  ...  5.1 follows from Remark 1.3 in [8] .  ... 
arXiv:1604.07200v2 fatcat:56n3dlpjpbggbdhvnv7zyjxiuq

Software system for solving multi-scale optimization problems [chapter]

E. Semenkin, K. Abramovich
1996 System Modelling and Optimization  
The objective F(B, D, X) is some quality index, representing the ability of the system to perform its functions. B E Bn is the vector of Boolean variables representing the system structure.  ...  X E Rk is the vector of continuous variables representing the physical characteristics of the subsystems (mass, power consumption, etc.). 2) Planning of optimal sequence of an industrial projects' implementation  ...  leads to the loss of useful properties of the objective function (monotonicity, convexity, etc.), -considerable growth of problem dimension in comparison with the dimension of an initial problem, -existence  ... 
doi:10.1007/978-0-387-34897-1_60 fatcat:zzk2hwbzcnbhfed4wvb6wutpmm

Separability is not the best goal for machine learning [article]

Wlodzislaw Duch
2018 arXiv   pre-print
For classification of difficult Boolean problems, such as the parity problem, linear projection combined with is sufficient and provides a powerful new target for learning.  ...  Simple problems are 2-separable, but problems with inherent complex logic may be solved in a simple way by k-separable projections.  ...  Direct search in 5-dimensional space for each of these functions is already prohibitively expensive.  ... 
arXiv:1807.02873v1 fatcat:mnorx7vfkrew3bxqw4v6eludia

Power Laws Variance Scaling of Boolean Random Varieties

Dominique Jeulin
2015 Methodology and Computing in Applied Probability  
The exponent is equal to n k n for Boolean varieties with dimension k in the space R n : = 2 3 for Boolean ...bers in 3D, and = 1 3 for Boolean strata in 3D.  ...  When working in 2D, the scaling exponent of Boolean ...bers is equal to 1 2 .  ...  Figure 1 : 1 Simulation of a 2D Boolean model built on isotropic Poisson lines. expressed as a function of the length l of the projection over the lines D !  ... 
doi:10.1007/s11009-015-9464-5 fatcat:jyrasvpr2zfsthvtzw6yhla5jq

Page 7300 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
In the special case of the latter when R = C;(U,T) is the ring of all continuous functions with compact support of a locally compact Boolean space U into a discrete indecomposable unitary ring 7, the authors  ...  rings of projective generators.  ... 

Page 326 of Mathematical Reviews Vol. 1, Issue 11 [page]

1940 Mathematical Reviews  
It is proved that the complemented modular lattices of finite dimensions which are not direct products coincide with projective geometries in the sense of Veblen and Young.  ...  The terms quotient, prime quotient, transpose, projective and inde- pendent are defined. The Jordan chain condition for modu- lar lattices of finite dimensions is proved.  ... 

Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann–Schwinger Equation for Two-Particle Scattering

Zeki C. Kuruoğlu
2014 Few-body systems  
Boolean combination of projections is explored as a means of going beyond tensor-product projection.  ...  Our calculations indicate that radial function bases are very promising in the context of multivariable integral equations.  ...  Tanatar on the computing facility of his research group in the Department of Physics at Bilkent University.  ... 
doi:10.1007/s00601-014-0887-2 fatcat:afctspzmbjhiloi6bv223y654i
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