Search results for " Computational"
showing 10 items of 661 documents
On the Inner Product Predicate and a Generalization of Matching Vector Families
2018
Motivated by cryptographic applications such as predicate encryption, we consider the problem of representing an arbitrary predicate as the inner product predicate on two vectors. Concretely, fix a Boolean function $P$ and some modulus $q$. We are interested in encoding $x$ to $\vec x$ and $y$ to $\vec y$ so that $$P(x,y) = 1 \Longleftrightarrow \langle\vec x,\vec y\rangle= 0 \bmod q,$$ where the vectors should be as short as possible. This problem can also be viewed as a generalization of matching vector families, which corresponds to the equality predicate. Matching vector families have been used in the constructions of Ramsey graphs, private information retrieval (PIR) protocols, and mor…
Combinatorial proofs of two theorems of Lutz and Stull
2021
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatori…
Adaptive Lower Bound for Testing Monotonicity on the Line
2018
In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of $\epsilon$-testing monotonicity of a function $f\colon [n]\to[r]$. All our lower bounds are for adaptive two-sided testers. * We prove a nearly tight lower bound for this problem in terms of $r$. The bound is $\Omega(\frac{\log r}{\log \log r})$ when $\epsilon = 1/2$. No previous satisfactory lower bound in terms of $r$ was known. * We completely characterise query complexity of this probl…
Testing convexity of functions over finite domains
2019
We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. 1. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the upper bound $O(\frac{\log(\epsilon n)}{\epsilon})$ in the usual uniform model, and prove an $O(\frac{\log n}{\epsilon})$ upper bound in the distribution-free setting. 2. We show a tight lower bound of $\Omega(\frac{\log(\epsilon n)}{\epsilon})$ queries for testing convexity of functions $f: [n] \rightarrow \mathbb{R}$ on the line. This lower bound applies to both adaptive and non-adaptive algorithms, and matches the upper bound from item 1, showing that adap…
Sensitivity versus Certificate Complexity of Boolean Functions
2015
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been notoriously hard to obtain even exponential bounds. Since block sensitivity is known to be polynomially related to certificate complexity, an equivalent of proving this conjecture would be showing that certificate complexity is polynomially related to sensitivity. Previously, it has been shown that $bs(f) \leq C(f) \leq 2^{s(f)-1} s(f) - (s(f)-1)$. In this work, we give a better upper bound of $bs(f) \leq C(f) \leq \max\left(2^{s(f)-1}\left(s(f)-\frac 1 3\righ…
Probabilistic verifiers for asymmetric debates
2012
We examine the power of silent constant-space probabilistic verifiers that watch asymmetric debates (where one side is unable to see some of the messages of the other) between two deterministic provers, and try to determine who is right. We prove that probabilistic verifiers outperform their deterministic counterparts as asymmetric debate checkers. It is shown that the membership problem for every language in NSPACE(s(n)) has a 2^{s(n)}-time debate where one prover is completely blind to the other one, for polynomially bounded space constructible s(n). When partial information is allowed to be seen by the handicapped prover, the class of languages debatable in 2^{s(n)} time contains TIME(2^…
Tighter Relations Between Sensitivity and Other Complexity Measures
2014
Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is polynomially related to other major complexity measures. Despite much attention to the problem and major advances in analysis of Boolean functions in the past decade, the problem remains wide open with no positive result toward the conjecture since the work of Kenyon and Kutin from 2004. In this work, we present new upper bounds for various complexity measures in terms of sensitivity improving the bounds provided by Kenyon and Kutin. Specifically, we show tha…
Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma
2014
We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture. In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if $S\subseteq\{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be con…
Fast Matrix Multiplication: Limitations of the Laser Method
2014
Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time $O(n^{2.3755})$. Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time $O(n^{2.3729})$. These algorithms are obtained by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd. We show that this exact approach cannot result in an algorithm with running time $O(n^{2.3725})$, and identify a wide class of variants of this approach which cannot result in an algorithm with running time $O(n^{2.3078})$; in particular, this approach cannot prove the conjecture that f…
Alternating, private alternating, and quantum alternating realtime automata
2014
We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs o…