36 research outputs found
On the Power of Conditional Samples in Distribution Testing
In this paper we define and examine the power of the {\em
conditional-sampling} oracle in the context of distribution-property testing.
The conditional-sampling oracle for a discrete distribution takes as
input a subset of the domain, and outputs a random sample drawn according to , conditioned on (and independently of all
prior samples). The conditional-sampling oracle is a natural generalization of
the ordinary sampling oracle in which always equals .
We show that with the conditional-sampling oracle, testing uniformity,
testing identity to a known distribution, and testing any label-invariant
property of distributions is easier than with the ordinary sampling oracle. On
the other hand, we also show that for some distribution properties the
sample-complexity remains near-maximal even with conditional sampling
Underapproximation for model-checking based on universal circuits
AbstractFor two naturals m,n such that m<n, we show how to construct a circuit C with m inputs and n outputs, that has the following property: for some 0⩽k⩽m, the circuit defines a k-universal function. This means, informally, that for every subset K of k outputs, every possible valuation of the variables in K is reachable.Now consider a circuit M with n inputs that we wish to model-check. Connecting the inputs of M to the outputs of C gives us a new circuit M′ with m inputs, that its original inputs have freedom defined by k. This is a very attractive feature for underapproximation in model-checking: on one hand the combined circuit has a smaller number of inputs, and on the other hand it is expected to find an error state fast if there is one.We show a random construction of a k-universal circuit that guarantees that k is very close to m, with an arbitrarily high probability. We also present a deterministic construction of such a circuit, but here the value of k is smaller with respect to a fixed value of m. We report initial experimental results with bounded model checking of industrial designs (the method is equally applicable to unbounded model checking and to simulation), which shows mixed results. An interesting observation, however, is that in 13 out of 17 designs, setting m to be n/5 is sufficient to detect the bug. This is in contrast to other underapproximation techniques that are based on reducing the number of inputs, which in most cases cannot detect the bug even with m=n/2
Hardness and Algorithms for Rainbow Connectivity
An edge-colored graph G is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connectivity
of a connected graph G, denoted rc(G), is the smallest number of colors that
are needed in order to make G rainbow connected. In addition to being a natural
combinatorial problem, the rainbow connectivity problem is motivated by
applications in cellular networks. In this paper we give the first proof that
computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to
decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given
edge-colored (with an unbounded number of colors) graph is rainbow connected.
On the positive side, we prove that for every > 0, a connected graph
with minimum degree at least has bounded rainbow connectivity,
where the bound depends only on , and the corresponding coloring can
be constructed in polynomial time. Additional non-trivial upper bounds, as well
as open problems and conjectures are also pre sented
New Results on Quantum Property Testing
We present several new examples of speed-ups obtainable by quantum algorithms
in the context of property testing. First, motivated by sampling algorithms, we
consider probability distributions given in the form of an oracle
. Here the probability \PP_f(j) of an outcome is the
fraction of its domain that maps to . We give quantum algorithms for
testing whether two such distributions are identical or -far in
-norm. Recently, Bravyi, Hassidim, and Harrow \cite{BHH10} showed that if
\PP_f and \PP_g are both unknown (i.e., given by oracles and ), then
this testing can be done in roughly quantum queries to the
functions. We consider the case where the second distribution is known, and
show that testing can be done with roughly quantum queries, which we
prove to be essentially optimal. In contrast, it is known that classical
testing algorithms need about queries in the unknown-unknown case and
about queries in the known-unknown case. Based on this result, we
also reduce the query complexity of graph isomorphism testers with quantum
oracle access. While those examples provide polynomial quantum speed-ups, our
third example gives a much larger improvement (constant quantum queries vs
polynomial classical queries) for the problem of testing periodicity, based on
Shor's algorithm and a modification of a classical lower bound by Lachish and
Newman \cite{lachish&newman:periodicity}. This provides an alternative to a
recent constant-vs-polynomial speed-up due to Aaronson \cite{aaronson:bqpph}.Comment: 2nd version: updated some references, in particular to Aaronson's
Fourier checking proble
Learning Parities in the Mistake-Bound model
We study the problem of learning parity functions that depend on at most variables (-parities) attribute-efficiently in the mistake-bound model.
We design a simple, deterministic, polynomial-time algorithm for learning -parities with mistake bound , for any constant . This is the first polynomial-time algorithms that learns -parities in the mistake-bound model with mistake bound .
Using the standard conversion techniques from the mistake-bound model to the PAC model, our algorithm can also be used for learning -parities in the PAC model. In particular, this implies a slight improvement on the results of Klivans and Servedio
cite{rocco} for learning -parities in the PAC model.
We also show that the time algorithm from cite{rocco} that PAC-learns -parities with optimal sample complexity can be extended to the mistake-bound model
IC3-Guided Abstraction
Abstract-Localization is a powerful automated abstraction-refinement technique to reduce the complexity of property checking. This process is often guided by SATbased bounded model checking, using counterexamples obtained on the abstract model, proofs obtained on the original model, or a combination of both to select irrelevant logic. In this paper, we propose the use of bounded invariants obtained during an incomplete IC3 run to derive higher-quality abstractions for complex problems. Experiments confirm that this approach yields significantly smaller abstractions in many cases, and that the resulting abstract models are often easier to verify
Testing Graph Isomorphism
Two graphs G and H on n vertices are ɛ-far from being isomorphic if at least ɛ � � n 2 edges must be added or removed from E(G) in order to make G and H isomorphic. In this paper we deal with the question of how many queries are required to distinguish between the case that two graphs are isomorphic, and the case that they are ɛ-far from being isomorphic. A query is defined as probing the adjacency matrix of any one of the two graphs, i.e. asking if a pair of vertices forms an edge of the graph or not. We investigate both one-sided error and two-sided error testers under two possible settings: The first setting is where both graphs need to be queried; and the second setting is where one of the graphs is fully known to the algorithm in advance. We prove that the query complexity of the best one-sided error testing algorithm is � Θ(n3/2) if both graphs need to be queried, and that it is � Θ(n) if one of the graphs is known in advance (where the � Θ notation hides polylogarithmic factors in the upper bounds). For two-sided error testers, we prove that the query complexity of the best tester is � Θ ( √ n) when one of the graphs is known in advance, and we show that the query complexity lies between Ω(n) and � O(n5/4) if both G and H need to be queried. All of our algorithms are additionally non-adaptive, while all of our lower bounds apply for adaptive testers as well as non-adaptive ones