2,073 research outputs found

    Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

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    Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1f_1, f2f_2 and f3:F2k{0,1}f_3: \mathbb{F}_2^k \to \{0, 1\} are said to be triangle free if there is no x,yF2kx, y \in \mathbb{F}_2^k such that f1(x)=f2(y)=f3(x+y)=1f_1(x) = f_2(y) = f_3(x + y) = 1. This property is known to be strongly testable (Green 2005), but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1 / \epsislon, where \epsislon is the distance between the tested function triple and triangle-freeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of (1/ϵ)2.423(1 / \epsilon)^{2.423} for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to (1/ϵ)6.619(1 / \epsilon)^{6.619}. Interestingly, we prove this by way of a combinatorial construction called \emph{uniquely solvable puzzles} that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm

    New Lower Bounds for van der Waerden Numbers Using Distributed Computing

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    This paper provides new lower bounds for van der Waerden numbers. The number W(k,r)W(k,r) is defined to be the smallest integer nn for which any rr-coloring of the integers 0,n10 \ldots, n-1 admits monochromatic arithmetic progression of length kk; its existence is implied by van der Waerden's Theorem. We exhibit rr-colorings of 0n10\ldots n-1 that do not contain monochromatic arithmetic progressions of length kk to prove that W(k,r)>nW(k, r)>n. These colorings are constructed using existing techniques. Rabung's method, given a prime pp and a primitive root ρ\rho, applies a color given by the discrete logarithm base ρ\rho mod rr and concatenates k1k-1 copies. We also used Herwig et al's Cyclic Zipper Method, which doubles or quadruples the length of a coloring, with the faster check of Rabung and Lotts. We were able to check larger primes than previous results, employing around 2 teraflops of computing power for 12 months through distributed computing by over 500 volunteers. This allowed us to check all primes through 950 million, compared to 10 million by Rabung and Lotts. Our lower bounds appear to grow roughly exponentially in kk. Given that these constructions produce tight lower bounds for known van der Waerden numbers, this data suggests that exact van der Waerden Numbers grow exponentially in kk with ratio rr asymptotically, which is a new conjecture, according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader comment

    Asymptotic entanglement transformation between W and GHZ states

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    We investigate entanglement transformations with stochastic local operations and classical communication (SLOCC) in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increase and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen we also compute the conversion rate from W to GHZ states.Comment: 11 page
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