Results 1  10
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23
Oracles and Queries that are Sufficient for Exact Learning
 Journal of Computer and System Sciences
, 1996
"... We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected poly ..."
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Cited by 83 (5 self)
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We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected polynomial time with equivalence queries and the aid of an NPoracle. We also show that circuits are exactly learnable in deterministic polynomial time with equivalence queries and a \Sigma 3 oracle. The hypothesis class for the above learning algorithms is the class of circuits of largerbut polynomially relatedsize. Also, the algorithms can be adapted to learn the class of DNF formulas with hypothesis class consisting of depth3  formulas (by the work of Angluin [A90], this is optimal in the sense that the hypothesis class cannot be reduced to DNF formulas, i.e. depth2  formulas).
Analysis of Random Processes via AndOr Tree Evaluation
 In Proceedings of the 9th Annual ACMSIAM Symposium on Discrete Algorithms
, 1998
"... We introduce a new set of probabilistic analysis tools based on the analysis of AndOr trees with random inputs. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including random lossresilient ..."
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Cited by 73 (23 self)
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We introduce a new set of probabilistic analysis tools based on the analysis of AndOr trees with random inputs. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including random lossresilient codes, solving random kSAT formula using the pure literal rule, and the greedy algorithm for matchings in random graphs. In addition, these tools allow generalizations of these problems not previously analyzed to be analyzed in a straightforward manner. We illustrate our methodology on the three problems listed above. 1 Introduction We introduce a new set of probabilistic analysis tools related to the amplification method introduced by [12] and further developed and used in [13, 5]. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including the random lossresilient codes introduced ...
The quantum adversary method and classical formula size lower bounds
 Computational Complexity
"... We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f: S → T, where S ⊆ Σ n for some alphabet Σ and T an arbitrary set. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query com ..."
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Cited by 21 (9 self)
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We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f: S → T, where S ⊆ Σ n for some alphabet Σ and T an arbitrary set. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the socalled quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [ ˇ SS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI 2 (f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [H˚as98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is
AttributeEfficient Learning in Query and MistakeBound Models
, 1996
"... We consider the problem of attributeefficient learning in query and mistakebound models. Attributeefficient algorithms make a number of queries or mistakes that is polynomial in the number of relevant variables in the target function, but only sublinear in the number of irrelevant variables. We c ..."
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Cited by 20 (2 self)
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We consider the problem of attributeefficient learning in query and mistakebound models. Attributeefficient algorithms make a number of queries or mistakes that is polynomial in the number of relevant variables in the target function, but only sublinear in the number of irrelevant variables. We consider a variant of the membership query model in which the learning algorithm is given as input the number of relevant variables of the target function. We show that in this model, any projection and embedding closed class of functions (including parity) that can be learned in polynomial time can be learned attributeefficiently in polynomial time. We show that this does not hold in the randomized membership query model. In the mistakebound model, we consider the problem of learning attributeefficiently using hypotheses that are formulas of small depth. Our results extend the work of Blum et al. [4] and Bshouty et al. [7].
Exact Identification of Readonce Formulas Using Fixed Points of Amplification Functions
 SIAM Journal on Computing
, 1993
"... In this paper we describe a new technique for exactly identifying certain classes of readonce Boolean formulas. The method is based on sampling the inputoutput behavior of the target formula on a probability distribution which is determined by the fixed point of the formula's amplification functio ..."
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Cited by 13 (0 self)
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In this paper we describe a new technique for exactly identifying certain classes of readonce Boolean formulas. The method is based on sampling the inputoutput behavior of the target formula on a probability distribution which is determined by the fixed point of the formula's amplification function (defined as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the fixedpoint distribution, we are able to efficiently infer all structural information about any logarithmicdepth formula (with high probability). We apply our results to prove the existence of short universal identification sequences for large classes of formulas. We also describe extensions of our algorithms to handle high rates of noise, and to learn formulas of unbounded depth in Valiant's model with respect to specific distributions. Most of this research was carried out while all three aut...
Amplification and Percolation
, 1992
"... Moore and Shannon had shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone readonce formulae of size O(n ff+2 ) (where ff = log p 5\Gamma1 2 ' 3:27) that amplify (/ \Gamma 1 n ; / + ..."
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Cited by 10 (3 self)
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Moore and Shannon had shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone readonce formulae of size O(n ff+2 ) (where ff = log p 5\Gamma1 2 ' 3:27) that amplify (/ \Gamma 1 n ; / + 1 n ) (where / = p 5\Gamma1 2 ' 0:62) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) and deduced as a consequence the existence of monotone formulae of the same size that compute the majority of n bits. Boppana had shown that any monotone readonce formula that amplifies (p \Gamma 1 n ; p + 1 n ) to ( 1 4 ; 3 4 ) (where 0 ! p ! 1 is constant) has size of at least\Omega\Gamma n ff ) and that any monotone, not necessarily readonce, contact network (and in particular any monotone formula) that amplifies ( 1 4 ; 3 4 ) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) has size of at least \Omega\Gamma n 2 ). We extend Boppana's results in two ways. We first show that his two lower bounds...
Shrinkage of de Morgan formulae under restriction
, 1993
"... It is shown that a random restriction leaving only a fraction " of the input variables unassigned reduces the expected de Morgan formula size of the induced function by a factor of O(" 5\Gamma p 3 2 ) = O(" 1:63 ). (A de Morgan, or unate, formula is a formula over the basis f; ; :g.) This imp ..."
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Cited by 10 (6 self)
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It is shown that a random restriction leaving only a fraction " of the input variables unassigned reduces the expected de Morgan formula size of the induced function by a factor of O(" 5\Gamma p 3 2 ) = O(" 1:63 ). (A de Morgan, or unate, formula is a formula over the basis f; ; :g.) This improves a longstanding result of O(" 1:5 ) by Subbotovskaya and a recent improvement to O(" 21\Gamma p 73 8 ) = O(" 1:55 ) by Nisan and Impagliazzo. The new exponent yields an increased lower bound of n 7\Gamma p 3 2 \Gammao(1) = \Omega\Gamma n 2:63 ) for the de Morgan formula size of a function in P defined by Andreev. This is the largest formula size lower bound known, even for functions in NP.
Optimal Carry Save Networks
"... A general theory is developed for constructing the asymptotically shallowest networks and the asymptotically smallest networks (with respect to formula size) for the carry save addition of n numbers using any given basic carry save adder as a building block. Using these optimal carry save additi ..."
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Cited by 8 (0 self)
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A general theory is developed for constructing the asymptotically shallowest networks and the asymptotically smallest networks (with respect to formula size) for the carry save addition of n numbers using any given basic carry save adder as a building block. Using these optimal carry save addition networks the shallowest known multiplication circuits and the shortest formulae for the majority function (and many other symmetric Boolean functions) are obtained. In this paper, simple basic carry save adders are described, using which multiplication circuits of depth 3:71 log n (the result of which is given as the sum of two numbers) and majority formulae of size O(n 3:21 ) are constructed. Using more complicated basic carry save adders, not described here, these results could be further improved. Our best bounds are currently 3:57 log n for depth and O(n 3:13 ) for formula size. 1. Introduction The question `How fast can we multiply?' is one of the fundamental questions...
On the Fourier Analysis of Boolean Functions
, 1996
"... We study the Fourier representation of Boolean functions. The goal is to look at the frequency domain of Boolean functions to get complexity properties. Preliminary results indicate that this might be fruitful. In addition to presenting new results, we review some of the most significant work on the ..."
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Cited by 7 (5 self)
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We study the Fourier representation of Boolean functions. The goal is to look at the frequency domain of Boolean functions to get complexity properties. Preliminary results indicate that this might be fruitful. In addition to presenting new results, we review some of the most significant work on the subject. Istituto di Matematica Computazionale, Consiglio Nazionale delle Ricerche, and Dipartimento di Informatica, Pisa (Italy). y Istituto di Matematica Computazionale, Consiglio Nazionale delle Ricerche, Pisa (Italy). email: codenotti@iei.pi.cnr.it z Department of Computer Science, The University of Chicago. Portions of this work were done while visiting IEICNR in Pisa, sponsored by a grant from CNR. 1 Introduction The Fourier transform of a Boolean function is an invertible linear mapping of the values of the function onto a set of coefficients, known as Fourier coefficients. This transformation is such that the Fourier coefficients contain information about the regularitie...