It has been shown in previous recent work that
multiplicity automata are predictable from multiplicity
and equivalence queries. In this paper we generalize
related notions in a matrix representation
and obtain a basis for the solution
of a number of open problems in learnability theory.
Membership queries are generalized ... more >>>

In this paper, we consider the question of determining whether
a function $f$ has property $P$ or is $\e$-far from any
function with property $P$.
The property testing algorithm is given a sample of the value
of $f$ on instances drawn according to some distribution.
In some cases,
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In a variety of PAC learning models, a tradeoff between time and
information seems to exist: with unlimited time, a small amount of
information suffices, but with time restrictions, more information
sometimes seems to be required.
In addition, it has long been known that there are
concept classes ... more >>>

This is a revised version of work which has appeared
in preliminary form in the 36th FOCS, 1995.

Given a function $f$ mapping $n$-variate inputs from a finite field
$F$ into $F$,
we consider the task of reconstructing a list of all $n$-variate
degree $d$ polynomials which agree with $f$
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We show that the class of monotone $2^{O(\sqrt{\log n})}$-term DNF
formulae can be PAC learned in polynomial time under the uniform
distribution. This is an exponential improvement over previous
algorithms in this model, which could learn monotone
$o(\log^2 n)$-term DNF, and is the first efficient algorithm
for ... more >>>

A theory, in this context, is a Boolean formula; it is
used to classify instances, or truth assignments. Theories
can model real-world phenomena, and can do so more or less
correctly.
The theory revision, or concept revision, problem is to
correct a given, roughly correct concept.
This problem is ... more >>>

We give an algorithm that with high probability properly learns random monotone t(n)-term
DNF under the uniform distribution on the Boolean cube {0, 1}^n. For any polynomially bounded function t(n) <= poly(n) the algorithm runs in time poly(n, 1/eps) and with high probability outputs an eps accurate monotone DNF ... more >>>

We initiate the study of \emph{inverse} problems in approximate uniform generation, focusing on uniform generation of satisfying assignments of various types of Boolean functions. In such an inverse problem, the algorithm is given uniform random satisfying assignments of an unknown function $f$ belonging to a class $\C$ of Boolean functions ... more >>>

The threshold degree of a Boolean function $f$ is the minimum degree of
a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv\mathrm{sgn}\; p(x)$. In a seminal 1969
monograph, Minsky and Papert constructed a polynomial-size constant-depth
$\{\wedge,\vee\}$-circuit in $n$ variables with threshold degree $\Omega(n^{1/3}).$ This bound underlies ... more >>>

For every fixed constant $\alpha > 0$, we design an algorithm for computing the $k$-sparse Walsh-Hadamard transform of an $N$-dimensional vector $x \in \mathbb{R}^N$ in time $k^{1+\alpha} (\log N)^{O(1)}$. Specifically, the algorithm is given query access to $x$ and computes a $k$-sparse $\tilde{x} \in \mathbb{R}^N$ satisfying $\|\tilde{x} - \hat{x}\|_1 \leq ... more >>>

The threshold degree of a Boolean function $f$ is the minimum degree of
a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv\mathrm{sgn}\; p(x)$. Introduced
in the seminal work of Minsky and Papert (1969), this notion is central to
some of the strongest algorithmic and complexity-theoretic results for
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