Center for Communications Research

San Diego, CA, United States

Center for Communications Research

San Diego, CA, United States
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Jain S.,National University of Singapore | Stephan F.,National University of Singapore | Teutsch J.,Center for Communications Research
Journal of Computer and System Sciences | Year: 2011

This paper studies the Turing degrees of various properties defined for universal numberings, that is, for numberings which list all partial-recursive functions. In particular properties relating to the domain of the corresponding functions are investigated like the set DEQ of all pairs of indices of functions with the same domain, the set DMIN of all minimal indices of sets and DMIN * of all indices which are minimal with respect to equality of the domain modulo finitely many differences. A partial solution to a question of Schaefer is obtained by showing that for every universal numbering with the Kolmogorov property, the set DMIN* is Turing equivalent to the double jump of the halting problem. Furthermore, it is shown that the join of DEQ and the halting problem is Turing equivalent to the jump of the halting problem and that there are numberings for which DEQ itself has 1-generic Turing degree. © 2010 Elsevier Inc. All rights reserved.


News Article | January 25, 2016
Site: motherboard.vice.com

Go is an old game where, broadly speaking, players take turns placing white and black pieces on a board in an attempt to surround the most empty territory on the playing field. The earliest known reference to it dates back to China in the 5th century B.C., which means people were playing it along the shores of the Yangtze before the Parthenon cast its shadow over Athens. At once simple and complex, it's known for the high degree of versatility its gridded playing surface offers, and for years—centuries, even—some players assumed the number of legal positions must be infinite on the game's standard 19x19 boards. But that's not exactly true, as computer scientist John Tromp revealed last week. There's just a friggin' lot of them. Specifically, Tromp discovered this is the number of legal ways you can use the board's 361 points with the black and white playing pieces and empty spaces: That's hardly the kind of thing you can figure out with pen and paper. As one Reddit user pointed out in reference to Tromp's earlier work, that's bigger than the total number of observable atoms in the universe. Tromp started the calculations back in March 6 of last year, with the help of staff and big servers at both the Institute for Advanced Study's School of Natural Sciences and the IDA's Center for Communications Research in Princeton, New Jersey, along with some help from Hewlett-Packard's Helion Cloud. "The software was developed mostly in 2005, so from then we had the ability to attempt it, but not the required hardware resources," Tromp tells me in an e-mail. "By 2007, we (me and Michal Koucký) were just able to compute the number of legal 17x17 positions, and that exhausted the resources available at the Dutch Centre for Mathematics and Computer Science where I worked." Want to double-check his work? Tromp provides the software he used on his github repository, but he cautions that you'll need "a beefy server with 15 TB of fast scratch diskspace, 8 to 16 cores, and 192 GB of RAM." Indeed, for years the available technology held him back, limiting him to calculating boards beneath 19x19 grid standard. The leap from 17x17 to 19x19 may not seem like much, but Tromp emphasizes that each increase in the board's dimensions demands a fivefold increase in the memory, time, and diskspace required. Tromp unsuccessfully tried to get "dozens" of companies, people, and institutions like Amazon and Google to sponsor his quest to borrow the hardware to calculate the results for a 19x19 board, but the winning ticket dropped in 2014 when fellow Dutchman Piet Hut, an astronomer at the Institute for Advanced Study, came through with an offer. After Tromp and friends achieved some notoriety for figuring out the number of legal positions on an 18x18 board, the right folks jumped on board for the final push. So after solving a mystery that's been unsolved for 2,500 years, what's next? Tromp tells me he plans to continue work on his "Cuckoo Cycle" proof-of-work system and solve large-scale Connect Four problems, but he's especially interested in improving his similar work on chess. Chess, though, is a different beast. Which leaves the question of why? "Having the ability to determine the state complexity of the greatest abstract board game, and not doing it, that just doesn't sit right with me," Tromp says. "Or, as the great mathematician Hilbert proclaimed, 'We must know — we will know!"


Howe E.W.,Center for Communications Research
Finite Fields and their Applications | Year: 2017

The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil–Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at manypoints.org. © 2017 Elsevier Inc.


Fradkin A.,Center for Communications Research | Seymour P.,Princeton University
Journal of Combinatorial Theory. Series B | Year: 2015

A digraph H is infused in a digraph G if the vertices of H are mapped to vertices of G (not necessarily distinct), and the edges of H are mapped to edge-disjoint directed paths of G joining the corresponding pairs of vertices of G. The algorithmic problem of determining whether a fixed graph H can be infused in an input graph G is polynomial-time solvable for all graphs H (using paths instead of directed paths). However, the analogous problem in digraphs is NP-complete for most digraphs H. We provide a polynomial-time algorithm to solve a rooted version of the problem, for all digraphs H, in digraphs with independence number bounded by a fixed integer α. The problem that we solve is a generalization of the k edge-disjoint directed paths problem (for fixed k). © 2014 Elsevier Inc.


Fradkin A.,Center for Communications Research | Seymour P.,Princeton University
Journal of Combinatorial Theory. Series B | Year: 2013

We prove that if a tournament has pathwidth ≥4θ2+7θ then it has θ vertices that are pairwise θ-connected. As a consequence of this and previous results, we obtain that for every set S of tournaments the following are equivalent:•there exists k such that every member of S has pathwidth at most k,•there is a digraph H such that no subdivision of H is a subdigraph of any member of S,•there exists k such that for each T∈S, there do not exist k vertices of T that are pairwise k-connected.As a further consequence, we obtain a polynomial-time algorithm to test whether a tournament contains a subdivision of a fixed digraph H as a subdigraph. © 2013 Elsevier Inc.


de Launey W.,Center for Communications Research | Gordon D.M.,Center for Communications Research
Cryptography and Communications | Year: 2010

Let S(x) be the number of n ≤ x for which a Hadamard matrix of order n exists. Hadamard's conjecture states that S(x) is about x/4. From Paley's constructions of Hadamard matrices, we have that S(x)=Ω(x/log x). In a recent paper, the first author suggested that counting the products of orders of Paley matrices would result in a greater density. In this paper we use results of Kevin Ford to show that it does:, where C = 0. 8178.... This bound is surprisingly hard to improve upon. We show that taking into account all the other major known construction methods for Hadamard matrices does not shift the bound. Our arguments use the notion of a (multiplicative) monoid of natural numbers. We prove some initial results concerning these objects. Our techniques may be useful when assessing the status of other existence questions in design theory. © 2010 Springer Science + Business Media, LLC.


Chow T.Y.,Center for Communications Research
Journal of Computer and System Sciences | Year: 2011

Razborov and Rudich have proved that, under a widely-believed hypothesis about pseudorandom number generators, there do not exist P/poly-computable Boolean function properties with density greater than 2-poly(n) that exclude P/poly. This famous result is widely regarded as a serious barrier to proving strong lower bounds in circuit complexity theory, because virtually all Boolean function properties used in existing lower bound proofs have the stated complexity and density. In this paper, we show that under the same pseudorandomness hypothesis, there do exist nearly-linear-time-computable Boolean function properties with only slightly lower density (namely, 2-q(n) for a quasi-polynomial function q) that not only exclude P/poly, but even separate NP from P/poly. Indeed, we introduce a simple, explicit property called discrimination that does so. We also prove unconditionally that there exist non-uniformly nearly-linear-time-computable Boolean function properties with this same density that exclude P/poly. Along the way we also note that by slightly strengthening Razborov and Rudich s argument, one can show that their "naturalization barrier" is actually a barrier to proving superquadratic circuit lower bounds, not just P/poly circuit lower bounds. It remains open whether there is a naturalization barrier to proving superlinear circuit lower bounds. © 2010 Elsevier Inc. All rights reserved.


Dougherty R.,Center for Communications Research | Freiling C.,California State University, San Bernardino | Zeger K.,University of California at San Diego
Proceedings of the IEEE | Year: 2011

Networks derived from matroids have played a fundamental role in proving theoretical results about the limits of network coding. In this tutorial paper, we review many connections between matroids and network coding theory, with specific emphasis on network solvability, admissible network alphabet sizes, linear coding, and network capacity. © 2006 IEEE.


Houde R.A.,Center for Communications Research
Proceedings of Meetings on Acoustics | Year: 2013

Our understanding of the neural mechanisms underlying the very fine auditory frequency discrimination exhibited by listeners remains far from complete. To investigate this question we developed a functional model of the cochlear process in sufficient detail to allow the simulation of the principal characteristics of the cochlea's response to multi-tone and noise stimuli over a wide range of input levels. The model simulates level-dependent changes in frequency selectivity, combination-tone distortion, tone-on-tone suppression and masking, adaptation, and critical-band masking. The model is structured as 3000 channels, each consisting of a basilar membrane bandpass filter and inner-hair cell assembly. Input to each channel is the stapes displacement signal, and the output consists of ten independent stochastic point processes that are transmitted to the CNS on auditory-nerve fibers (ANFs). Our main purpose is to address these questions: (1) What narrowband spectrum information is available in the cochlea output? (2) How is this information encoded on the ANFs? (3) How might it be decoded in the CNS? An analysis of ensemble coding of the cochlear output showed that the precision (signal-to-noise ratio) of the information decoded by the CNS frequency analysis is directly related to the bandwidth of the basilar membrane filters. © 2013 Acoustical Society of America.


Gordon D.M.,Center for Communications Research | Miller V.S.,Center for Communications Research | Ostapenko P.,Center for Communications Research
IEEE Transactions on Information Theory | Year: 2010

One way to find near-matches in large datasets is to use hash functions. In recent years locality-sensitive hash functions for various metrics have been given; for the Hamming metric projecting onto k bits is simple hash function that performs well. In this paper, we investigate alternatives to projection. For various parameters hash functions given by complete decoding algorithms for error-correcting codes work better, and asymptotically random codes perform better than projection. © 2006 IEEE.

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