2,333 research outputs found
Mapping Monotone Boolean Functions into Majority
We consider the problem of decomposing monotone Boolean functions into majority-of-three operations, with a particular focus on decomposing the majority-n function. When targeting monotone Boolean functions, Shannon's expansion can be expressed by a single majority-of-three operation. We exploit this property to transform binary decision diagrams (BDDs) for monotone functions into majority-inverter graphs (MIGs), using a simple one-to-one mapping. This process highlights desirable properties for further majority graph optimization, e.g., symmetries between the inputs of primitive operations, which are not apparent from BDDs. Although our construction yields a quadratic upper bound on the number of majority-3 operations required to realize majority-n, for small n the concrete values are much smaller compared to those obtained from previous constructions which have linear and quasi-linear asymptotic upper bounds. Further, we demonstrate that minimum size MIGs, for the monotone functions majority-5 and majority-7, can be obtained applying a small number of algebraic transformations to the BDD
Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball
We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming
ball of equal volume. More precisely, we show that for all even n there exists
an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball
of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x
and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y)
where distance(,) denotes the Hamming distance. In particular, this implies
that the Hamming ball is bi-Lipschitz transitive.
This result gives a strong negative answer to an open problem of Lovett and
Viola [CC 2012], who raised the question in the context of sampling
distributions in low-level complexity classes. The conceptual implication is
that the problem of proving lower bounds in the context of sampling
distributions will require some new ideas beyond the sensitivity-based
structural results of Boppana [IPL 97].
We study the mapping f further and show that it (and its inverse) are
computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f
is "approximately local" in the sense that all but the last output bit of f are
essentially determined by a single input bit
Ontology-based data access with databases: a short course
Ontology-based data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a high-level global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries and ontologies into the vocabulary of the data sources and then delegates the actual query evaluation to a suitable query answering system such as a relational database management system or a datalog engine. In this chapter, we mainly focus on OBDA with the ontology language OWL 2QL, one of the three profiles of the W3C standard Web Ontology Language OWL 2, and relational databases, although other possible languages will also be discussed. We consider different types of conjunctive query rewriting and their succinctness, different architectures of OBDA systems, and give an overview of the OBDA system Ontop
Strong noise sensitivity and random graphs
The noise sensitivity of a Boolean function describes its likelihood to flip
under small perturbations of its input. Introduced in the seminal work of
Benjamini, Kalai and Schramm [Inst. Hautes \'{E}tudes Sci. Publ. Math. 90
(1999) 5-43], it was there shown to be governed by the first level of Fourier
coefficients in the central case of monotone functions at a constant critical
probability . Here we study noise sensitivity and a natural stronger
version of it, addressing the effect of noise given a specific witness in the
original input. Our main context is the Erd\H{o}s-R\'{e}nyi random graph, where
already the property of containing a given graph is sufficiently rich to
separate these notions. In particular, our analysis implies (strong) noise
sensitivity in settings where the BKS criterion involving the first Fourier
level does not apply, for example, when polynomially fast in the
number of variables.Comment: Published at http://dx.doi.org/10.1214/14-AOP959 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Privately Releasing Conjunctions and the Statistical Query Barrier
Suppose we would like to know all answers to a set of statistical queries C
on a data set up to small error, but we can only access the data itself using
statistical queries. A trivial solution is to exhaustively ask all queries in
C. Can we do any better?
+ We show that the number of statistical queries necessary and sufficient for
this task is---up to polynomial factors---equal to the agnostic learning
complexity of C in Kearns' statistical query (SQ) model. This gives a complete
answer to the question when running time is not a concern.
+ We then show that the problem can be solved efficiently (allowing arbitrary
error on a small fraction of queries) whenever the answers to C can be
described by a submodular function. This includes many natural concept classes,
such as graph cuts and Boolean disjunctions and conjunctions.
While interesting from a learning theoretic point of view, our main
applications are in privacy-preserving data analysis:
Here, our second result leads to the first algorithm that efficiently
releases differentially private answers to of all Boolean conjunctions with 1%
average error. This presents significant progress on a key open problem in
privacy-preserving data analysis.
Our first result on the other hand gives unconditional lower bounds on any
differentially private algorithm that admits a (potentially
non-privacy-preserving) implementation using only statistical queries. Not only
our algorithms, but also most known private algorithms can be implemented using
only statistical queries, and hence are constrained by these lower bounds. Our
result therefore isolates the complexity of agnostic learning in the SQ-model
as a new barrier in the design of differentially private algorithms
Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory
A key fact in the theory of Boolean functions is
that they often undergo sharp thresholds. For example: if the function is monotone and symmetric under a transitive action with
\E_p[f] = \eps and \E_q[f] = 1-\eps then as .
Here \E_p denotes the product probability measure on where each
coordinate takes the value independently with probability . The fact
that symmetric functions undergo sharp thresholds is important in the study of
random graphs and constraint satisfaction problems as well as in social
choice.In this paper we prove sharp thresholds for monotone functions taking
values in an arbitrary finite sets. We also provide examples of applications of
the results to social choice and to random graph problems. Among the
applications is an analog for Condorcet's jury theorem and an indeterminacy
result for a large class of social choice functions
Circuit Complexity Meets Ontology-Based Data Access
Ontology-based data access is an approach to organizing access to a database
augmented with a logical theory. In this approach query answering proceeds
through a reformulation of a given query into a new one which can be answered
without any use of theory. Thus the problem reduces to the standard database
setting.
However, the size of the query may increase substantially during the
reformulation. In this survey we review a recently developed framework on
proving lower and upper bounds on the size of this reformulation by employing
methods and results from Boolean circuit complexity.Comment: To appear in proceedings of CSR 2015, LNCS 9139, Springe
Distribution-Independent Evolvability of Linear Threshold Functions
Valiant's (2007) model of evolvability models the evolutionary process of
acquiring useful functionality as a restricted form of learning from random
examples. Linear threshold functions and their various subclasses, such as
conjunctions and decision lists, play a fundamental role in learning theory and
hence their evolvability has been the primary focus of research on Valiant's
framework (2007). One of the main open problems regarding the model is whether
conjunctions are evolvable distribution-independently (Feldman and Valiant,
2008). We show that the answer is negative. Our proof is based on a new
combinatorial parameter of a concept class that lower-bounds the complexity of
learning from correlations.
We contrast the lower bound with a proof that linear threshold functions
having a non-negligible margin on the data points are evolvable
distribution-independently via a simple mutation algorithm. Our algorithm
relies on a non-linear loss function being used to select the hypotheses
instead of 0-1 loss in Valiant's (2007) original definition. The proof of
evolvability requires that the loss function satisfies several mild conditions
that are, for example, satisfied by the quadratic loss function studied in
several other works (Michael, 2007; Feldman, 2009; Valiant, 2010). An important
property of our evolution algorithm is monotonicity, that is the algorithm
guarantees evolvability without any decreases in performance. Previously,
monotone evolvability was only shown for conjunctions with quadratic loss
(Feldman, 2009) or when the distribution on the domain is severely restricted
(Michael, 2007; Feldman, 2009; Kanade et al., 2010
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