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202
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2658 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
Descriptive Complexity
, 1999
"... book is dedicated to Daniel and Ellie. Preface This book should be of interest to anyone who would like to understand computation from the point of view of logic. The book is designed for graduate students or advanced undergraduates in computer science or mathematics and is suitable as a textbook o ..."
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Cited by 312 (18 self)
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book is dedicated to Daniel and Ellie. Preface This book should be of interest to anyone who would like to understand computation from the point of view of logic. The book is designed for graduate students or advanced undergraduates in computer science or mathematics and is suitable as a textbook or for self study in the area of descriptive complexity. It is of particular interest to students of computational complexity, database theory, and computer aided verification. Numerous examples and exercises are included in the text, as well as a section at the end of each chapter with references and suggestions for further reading. The book provides plenty of material for a one semester course. The core of the book is contained in Chapters 1 through 7, although even here some sections can be omitted according to the taste and interests of the instructor. The remaining chapters are more independent of each other. I would strongly recommend including at least parts of Chapters 9, 10, and 12. Chapters 8 and 13 on lower bounds include some of the nicest combinatorial arguments. Chapter 11 includes a wealth of information on uniformity; to me, the lowlevel nature of translations between problems that suffice to maintain completeness is amazing and provides powerful descriptive tools for understanding complexity. I assume that most readers will want to study the applications of descriptive complexity that are introduced in Chapter 14.
Almost Optimal Lower Bounds for Small Depth Circuits
 RANDOMNESS AND COMPUTATION
, 1989
"... We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when ..."
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Cited by 269 (8 self)
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We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k1. Our main lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely.
Boundedwidth polynomialsize branching programs recognize exactly those languages
 in NC’, in “Proceedings, 18th ACM STOC
, 1986
"... We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such prog ..."
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Cited by 263 (14 self)
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We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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Cited by 220 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Learning Decision Trees using the Fourier Spectrum
, 1991
"... This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each ..."
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Cited by 202 (10 self)
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This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over GF (2)). This paper shows how to learn in polynomial time any function that can be approximated (in norm L 2 ) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function f whose L 1 norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial L 1 norm can be learned deterministically. The algorithm can a...
Modern cryptography, probabilistic proofs and pseudorandomness, volume 17 of Algorithms and Combinatorics
, 1999
"... all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that new copies bear this notice and the full citation on the first page. Abstracting with credit is permitted. IIPreface You can start by put ..."
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Cited by 130 (13 self)
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all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that new copies bear this notice and the full citation on the first page. Abstracting with credit is permitted. IIPreface You can start by putting the do not disturb sign. Cay, in Desert Hearts (1985). The interplay between randomness and computation is one of the most fascinating scientific phenomena uncovered in the last couple of decades. This interplay is at the heart of modern cryptography and plays a fundamental role in complexity theory at large. Specifically, the interplay of randomness and computation is pivotal to several intriguing notions of probabilistic proof systems and is the focal of the computational approach to randomness. This book provides an introduction to these three, somewhat interwoven domains (i.e., cryptography, proofs and randomness). Modern Cryptography. Whereas classical cryptography was confined to
Why does unsupervised pretraining help deep learning?
, 2010
"... Much recent research has been devoted to learning algorithms for deep architectures such as Deep Belief Networks and stacks of autoencoder variants with impressive results being obtained in several areas, mostly on vision and language datasets. The best results obtained on supervised learning tasks ..."
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Cited by 129 (21 self)
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Much recent research has been devoted to learning algorithms for deep architectures such as Deep Belief Networks and stacks of autoencoder variants with impressive results being obtained in several areas, mostly on vision and language datasets. The best results obtained on supervised learning tasks often involve an unsupervised learning component, usually in an unsupervised pretraining phase. The main question investigated here is the following: why does unsupervised pretraining work so well? Through extensive experimentation, we explore several possible explanations discussed in the literature including its action as a regularizer (Erhan et al., 2009b) and as an aid to optimization (Bengio et al., 2007). Our results build on the work of Erhan et al. (2009b), showing that unsupervised pretraining appears to play predominantly a regularization role in subsequent supervised training. However our results in an online setting, with a virtually unlimited data stream, point to a somewhat more nuanced interpretation of the roles of optimization and regularization in the unsupervised pretraining effect.
On the Power of SmallDepth Threshold Circuits
, 1990
"... We investigate the power of threshold circuits of small depth. In particular we give functions which require exponential size unweigted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are mone tone functions fk which can be computed in depth k and linear s ..."
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Cited by 115 (2 self)
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We investigate the power of threshold circuits of small depth. In particular we give functions which require exponential size unweigted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are mone tone functions fk which can be computed in depth k and linear size A, Vcircuits but require exponential size to compute by a depth k 1 monotone weighted threshold circuit.