tag:blogger.com,1999:blog-37222332018-07-17T11:11:39.367-04:00Computational ComplexityComputational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill GasarchLance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.comBlogger2599125tag:blogger.com,1999:blog-3722233.post-76084851423887525462018-07-16T12:55:00.001-04:002018-07-16T12:55:50.300-04:00The Mystical Bond Between Man and MachineYou just can't watch a movie these days without being inundated with trailers. First came <a href="https://www.youtube.com/watch?v=--8nr2kt4uk">Axl</a>, a movie about a boys love for a military robotic dog.<br />
<br />
<iframe allow="autoplay; encrypted-media" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/--8nr2kt4uk" width="560"></iframe><br />
<br />
"It's only a robot," says his father. "It's an intelligent robot" replies the kid. Then comes the generic ET-like story of the government coming for the robot.<br />
<br />
Next came a <a href="https://youtu.be/fAIX12F6958">trailer</a> for a movie that start off with Hailee Steinfeld discovering a VW bug with the background voice going "The driver don't pick the car. The car picks the driver". I'm thinking it's either a new <a href="https://www.imdb.com/title/tt0064603/">Herbie</a> movie or Transformers. Spoiler: Transformers.<br />
<br />
<iframe allow="autoplay; encrypted-media" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/fAIX12F6958" width="560"></iframe>
<br />
"There's a mystical bond between man and machine" the voice intones. Followed by action that looks just like Axl.<br />
<br />
Movie love for machines is hardly new. You can go back to <a href="https://www.imdb.com/title/tt1798709/">Her</a> or <a href="https://www.imdb.com/title/tt0091949/">Short Circuit</a> or even <a href="https://www.imdb.com/title/tt0017136/">Metropolis</a> in 1927. But in an age that <a href="http://www.chicagotribune.com/lifestyles/parenting/ct-life-parenting-alexa-rudeness-20180509-story.html">parents worry about their kids being rude to Alexa</a> perhaps this mystical bond is starting to get just a little too real.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-30536894019722984052018-07-12T01:33:00.002-04:002018-07-12T01:33:55.481-04:00The Six Degrees of VDW A long long time ago a HS student, Justin Kruskal (Clyde's son) was working with me on upper bounds on some Poly VDW numbers (see <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/polyvdwstat.pdf">here</a> for a statement of PVDW). His school required that he have an application. Here is what he ended up doing: rather than argue that PVDW had an application he argued that Ramsey Theory itself had applications, and since this was part of Ramsey Theory it had an application.<br />
<br />
How many degrees of separation were there from his work and the so called application.<br />
<br />
<ol>
<li>The best (at the time) Matrix Multiplication algorithm used 3-free sets.</li>
<li>3-free sets are used to get lower bounds on VDW numbers.</li>
<li>Lower bounds on VDW numbers are related to upper bounds on VDW numbers</li>
<li>Upper bounds on VDW are related to upper bounds on PVDW numbers.</li>
</ol>
Only 4 degrees! The people in charge of the HS projects recognized that it was good work and hence gave him a pass on the lack of real applications. Or they didn't quite notice the lack of applications. He DID end up being one of five students who got to give a talk on his project to the entire school. <br />
<br />
When you say that your work has applications is it direct? one degree off? two? Are all theorems no more than six degrees away from an application? Depends on how you define <i>degree</i> and <i>application.</i>GASARCHnoreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-29858330466226496682018-07-09T16:05:00.001-04:002018-07-10T11:52:32.357-04:00Soliciting answers for THIRD survey about P vs NP<br />
<br />
I have done two surveys for SIGACT NEWS Complextiy Column (edited by Lane Hemaspaandra)<br />
on P vs NP and related topics. Lane has asked me to do a third. I annouced it in my open problems column <a href="https://www.cs.umd.edu/users/gasarch/open/pnp.pdf">here</a> For those who don't read SIGACT news<br />
<br />
1) You should!<br />
<br />
2) Here is where to go to fill out the survey: <a href="https://www.surveymonkey.com/r/PversusNP">here</a><br />
<br />
bill g.<br />
<br />
P.S. (do people use P.S. anymore? Do young people know that it means Post Script, and that it<br />
does not refer to ps-files?)<br />
<br />A commenter requested I add what the DEADLINE for responding was. I originally thought people would read the post and immediately respond (and I HAVE had a BIG uptick in responses in the last day). I still believe this. BUT there are people who read the blog days, weeks, months, even years after I post it (though the comments we get on very old posts tend to contain clicks for good deals on Tuxedo's (I am serious. Tuxedo's? Not well targeted unless they count my Tuxedo T-shirt).<br />
<br />
Okay, enough babbling -- the point is that I should have a deadline for those who read the blog later than now.<br />
<br />
DEADLINE: Oct 1, 2018. STRICT!<br />
<br />
P.P.S - does anyone use P.P.S anymore?<br />
<br />
<br />
<br />
<br />GASARCHhttp://www.blogger.com/profile/03615736448441925334noreply@blogger.com7tag:blogger.com,1999:blog-3722233.post-41748240071887022112018-07-05T09:15:00.003-04:002018-07-05T09:15:31.343-04:00Happy 90th Juris!<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-KgCJcPhjO1o/Wz4ZxGnntbI/AAAAAAABhAM/f9n9VDzAgsgTg3pdB10tDsZpnWqIBlFGQCLcBGAs/s1600/Juris.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="639" data-original-width="450" height="200" src="https://2.bp.blogspot.com/-KgCJcPhjO1o/Wz4ZxGnntbI/AAAAAAABhAM/f9n9VDzAgsgTg3pdB10tDsZpnWqIBlFGQCLcBGAs/s200/Juris.gif" width="140" /></a></div>
<a href="https://en.wikipedia.org/wiki/Juris_Hartmanis">Juris Hartmanis</a> turns 90 today. Hartmanis with Richard Stearns received the 1993 Turing Award for their seminar work <a href="http://dx.doi.org/10.2307/1994208">On the Computational Complexity of Algorithms</a>. I've <a href="https://blog.computationalcomplexity.org/2005/02/favorite-theorems-seminal-paper.html">talked about that paper before</a>, after all it started our field and gave the name that we use for the blog. So instead I'll use this blog post to talk about how Hartmanis led me into this wondrous field.<br />
<br />
At Cornell in 1983 I was an undergraduate double major in math and CS destined at the time for grad school in math. I took the undergraduate Introduction to Theory course from Hartmanis and immediately got excited about the field. Hartmanis said the top student in the course was typically an undergrad followed by a bunch of graduate students. I was a counterexample coming in second in the course list (back then your grades were posted by ID number). I never found out who came in first.<br />
<br />
In that same semester I took Introduction to Linguistics. Chomsky and context-free languages in both classes. Seemed cool at the time.<br />
<br />
Based almost solely on that course with Hartmanis I decided to do graduate work in theoretical computer science. In the spring of 1985, while most of my fellow second-semester seniors took Introduction to Wine, I took graduate Complexity Complexity again with Hartmanis. That cemented my career as a complexity theorist. The course started with some basic computability theory (then called recursion theory) and used that as a jumping point to complexity. A large emphasis on the <a href="https://blog.computationalcomplexity.org/2003/03/berman-hartmanis-isomorphism.html">Berman-Hartmanis Isomorphism Conjecture</a>. The conjecture implies P ≠ NP and Hartmanis said anyone who could prove the conjecture, even assuming P ≠ NP, would get an automatic A. The problem remains open but I did years later have a <a href="http://doi.org/10.1137/S0097539793248305">couple</a> of <a href="http://doi.org/10.1145/276698.276737">proofs</a> giving an oracle making BH true. That should be good enough for a B.<br />
<br />
My favorite quote from Juris: "We all know that P is different from NP, we just don't know how to prove it." Still true today.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-3729153606112455212018-07-02T17:28:00.000-04:002018-07-02T17:28:01.431-04:00The BREAKTHROUGH on Chromatic Number of the Plane (guest post)(The new SIGACT News chair wnated me to post a letter he send to all SIGACT members on my blog in case you are not in SIGACT. He thinks you should be. I think so to so next year he won't ask me to do this. Here is his letter: <a href="https://dmatheorynet.blogspot.com/2018/07/new-sigact-executive-committee.html">here</a>)<br />
<br />
As many of you know there was a BREAKTHROUGH on the problem of the <i>The Chromatic Number of The Plane. </i>There have been fine blog posts on this by Gil Kalai <a href="https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/">here</a> amd Scott Aaronson <a href="https://www.scottaaronson.com/blog/?p=3697">here</a>. Rather than blog on it ourselves we have recruited and expert in the field, Alexander Soifer. He has written a book on the history and mathematics of coloring problems (see <a href="https://www.amazon.com/Mathematical-Coloring-Book-Mathematics-Colorful/dp/0387746404/ref=asap_bc?ie=UTF8">here</a> for the amazon link to the book and <a href="https://www.cs.umd.edu/users/gasarch/BLOGPAPERS/soiferrev.pdf">here</a> for my review of the book). The Chromatic Number of the Plane is his favorite problem. Much of the work on it is either by him or inspired by him. Much of the book is on it.<br />
<br />
Alexander also is the editor of a journal on problems like this called GEOCOMBINATORICS (oddly enough. Geometric Combinatorics is a different field). See <a href="http://geombina.uccs.edu/">here</a> for that journal- which I recommend!<br />
<br />
He wrote an 8-page essay, which is a concise version of an essay that will appear in a Special Issue XXVIII (1) of Geocombinatorica (July 2018, 5-17), dedicated to 5-chromatic unit distance graphs. The essay includes contributions from Aubrey D.N.J de Grey, Marijn J.H. Heule, and Geoffrey Exoo<br />
and Dan Ismailescu.<br />
<br />
What I find remarkable is that even though the result is new, the essay contains NEWER results. The modern world moves fast!<br />
<br />
Without further ado, here is that essay: <a href="https://www.cs.umd.edu/users/gasarch/BLOGPAPERS/soifer.pdf">here</a>.GASARCHhttp://www.blogger.com/profile/03615736448441925334noreply@blogger.com1tag:blogger.com,1999:blog-3722233.post-76786179425044578682018-06-28T18:31:00.001-04:002018-06-28T18:31:33.090-04:00STOC 50 Part IIOn Wednesday, <a href="http://acm-stoc.org/stoc2018/">STOC</a> had a great complexity session and the best complexity paper of the conference, Cody Murray and Ryan Williams extending Ryan’s <a href="https://blog.computationalcomplexity.org/2010/11/breakthrough-circuit-lower-bound.html">celebrated result</a> separating NEXP from ACC<sup>0</sup>. Cody and Ryan <a href="http://people.csail.mit.edu/rrw/easy-witness-nqp.pdf">show</a> there are problems in quasipolynomial time (2<sup>poly log(n)</sup>) that do not sit in ACC<sup>0</sup>, a near exponential improvement from Ryan’s original work. The key insight shows that if NP has n<sup>k</sup>-size circuits then for some j, each accepting input has a witness describable by a n<sup>j</sup>-size circuit.<br />
<br />
By the way everyone now has full access to the <a href="http://acm-stoc.org/stoc2018/toc.html">STOC Proceedings</a>. Read and enjoy.<br />
<br />
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<br />
<br />
Before the business meeting, Moses Charikar led a panel reminiscing over the 50 years of STOC. Panelists Dick Karp, Al Borodin, Ronitt Rubinfeld and Avrim Blum talked over memories of the early conferences and the good old days of transparencies and paper submissions.<br />
<br />
Highlights of the business meeting:
<br />
<ul>
<li>About 350 attendees, slightly larger than years past though the organizers hoped for a larger crowd given the 50th celebration and the theory fest.
</li>
<li>112 accepted papers of 416 submitted. There are now 29 PC members--time to rethink the standard in-person meeting.
</li>
<li><a href="https://www.irif.fr/~focs2018/">FOCS 2018</a> in Paris October 7-9. STOC 2019 as part of <a href="https://fcrc.acm.org/">FCRC</a> in Pheonix June 22-28. STOC 2020 likely in Copenhagen.
</li>
<li>New SIGACT executive committee: Samir Khuller (Chair), Eric Allender, Shuchi Chawla, Nicole Immorlica and Bobby Kleinberg.
</li>
<li>Shuchi Chawla taking over <a href="https://thmatters.wordpress.com/catcs/">CATCS</a>. Lots of goodies on the <a href="https://thmatters.wordpress.com/">website</a> for those applying for funding or looking for jobs.
</li>
<li>Sandy Irani is leading a new effort to <a href="https://www.ics.uci.edu/~irani/safetoc.html">combat harrassment</a> in the theoretical computer science community.
</li>
<li>Tracy Kimbrel gave NSF report. The NSF <a href="https://www.hpcwire.com/off-the-wire/nsf-appoints-dr-rance-cleaveland-as-division-director-for-division-of-computing-and-communication-foundations/">recently appointed</a> Rance Cleaveland as head of CCF, the division that includes algorithmic foundations. New rule: You can’t submit to both CRII and CAREER in the same year so pick your poison.
</li>
</ul>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-35770936270492686802018-06-26T18:31:00.001-04:002018-06-26T18:31:24.333-04:00STOC 50 Part IThis week I'm in Los Angeles attending the 50th Symposium on the Theory of Computing. Most attendees weren't even born before the first STOC. Many of them weren't even born when I went to my first STOC in 1986 in Berkeley.<br />
<br />
Most of the festivities come later but let me mention the best paper winners, both of whose titles give the theorem. <a href="https://arxiv.org/abs/1708.04215">A Constant-Factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem</a> by Ola Svensson, Jakub Tarnawski and László Végh won the best paper and <a href="https://arxiv.org/abs/1711.06455">An almost-linear time algorithm for uniform random spanning tree generation</a> by Aaron Schild won the Danny Lewin Best Student Paper Award. For those who don't know, Danny Lewin was an MIT grad student and co-founder of Akamai who <a href="https://www.cnn.com/2013/09/09/tech/innovation/danny-lewin-9-11-akamai/index.html">lost his life on 9/11</a>.<br />
<br />
A nice feature of the STOC theory fest, a tradition started last year, are the many invited talks. This morning we saw Stanford statistician Emmanuel Candes talk about irreproducible scientific results. The scientific method typically makes hypotheses, designs experiments to test predictions, updates the hypotheses and repeat. Today with we automatically generate hypotheses from big data using machine learning techniques which often leads to false positive correlations. Candes talked about his approach to mitigating this problem with <a href="https://statweb.stanford.edu/~candes/papers/FDR_regression.pdf">knockoff variables</a>.<br />
<br />
I also enjoy the senior junior lunch which I had today with students Rachit Nimavat, Jiyu Zhang and Zhixian Lei. Great discussions about the theory life.<br />
<br />
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<br />Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-43974376249185797782018-06-22T13:31:00.001-04:002018-07-10T13:11:57.868-04:00The Muffin ProblemI've been meaning to post on THE MUFFIN PROBLEM for at least a year. Its a project I've been working on for two years, but every time I wanted to post on it I thought.<br />
<br />
<br />
<i>I'm in the middle of a new result. I'll wait until I get it!</i><br />
<br />
However, I was sort of forced to finally post on it since Ken Regan (with my blessing) posted on it. In fact its better this way- you can goto his post for the math and I get to just tell you other stuff.<br />
<i><br /></i>
The problem was first defined by Alan Frank in a math email list in 2009.<br />
<i><br /></i>
I'll define the problem, though for more math details goto Ken's post: <a href="https://rjlipton.wordpress.com/2018/06/21/muffins-and-integers/">here</a>.<br />
<br />
You have m muffins and s students. You want to give each student m/s piece and<br />
divide the muffins to maximize the min piece. Let f(m,s) be the size of the min piece<br />
in an optimal divide-and-distribute procedure.<br />
<br />
Go and READ his post, or skim it, and then come back.<br />
<br />
Okay, you're back. Some informal comments now that you know the problem and the math<br />
<br />
1) I saw the problem in a pamplet at the 12th Gathering for Gardner. Yada Yada Yada I have 8 co-authors and 200 pages, and a paper in FUN with algorihtms You never know when a problem will strike you and be worth working on!<br />
(The 8 coauthors are Guangiqi Cui, John Dickerson, Naveen Dursula, William Gasarch, Erik Metz,<br />
Jacob Prinze, Naveen Raman, Daniel Smolyak, Sung Hyun Yoo. The 200 pages are <a href="https://arxiv.org/abs/1709.02452">here</a>. The FUN with algorithms paper, only 20 pages, is <a href="http://drops.dagstuhl.de/opus/volltexte/2018/8806/pdf/LIPIcs-FUN-2018-15.pdf">here</a>)<br />
<br />
2) Many of the co-authors are HS students. The problem needs very little advanced mathematics (though Ken thinks it might connect to some advanced math later). This is a PRO (HS students can work on it, people can understand it) and a CON (maybe harder math would get us more unified results)<br />
<br />
3) The way the research had gone is a microcosm for math and science progress:<br />
<br />
We proved f(m,s) = (m/s)f(s,m) (already known in 2009) by Erich Friedman in that math email list)<br />
<br />
Hence we need only look at m>s.<br />
<br />
First theorem: we got a simple function FC such that<br />
<br />
f(m,s) ≤ FC(m,s)<br />
<br />
for MANY m,s we got f(m,s) = FC(m,s).<br />
<br />
GREAT! - conjecture f(m,s) = FC(m,s)<br />
<br />
We found some exceptions, and a way to get better upper bounds called INT.<br />
<br />
GREAT! - conjecture f(m,s) = MIN(FC(m,s),INT(m,s))<br />
<br />
We found some exceptions, and a way to get better upper bounds called ... We now have<br />
<br />
conjecture<br />
<br />
f(m,s) = MIN(FC(m,s), INT(m,s), ERIK(m,s), JACOB(m,s), ERIKPLUS(m,s), BILL(m,s))<br />
<br />
and it looks like we still have a few exceptions.<br />
<br />
This is how science and math works- you make conjectures which are false but the refutations lead<br />
to better and better results.<br />
<br />
Also, we have over time mechanized the theorems, a project called:<br />
<br />
<i>Making Erik Obsolete</i><br />
<br />
since Erik is very clever at these problems, but we would like to not have to rely on that.<br />
<br />
4) I have worked hard on this problem as is clear from this: <a href="https://www.cs.umd.edu/users/gasarch/BLOGPAPERS/billmuffins.JPG">picture</a><br />
<br />
<br />GASARCHnoreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-11930882487122276932018-06-17T22:31:00.001-04:002018-06-18T11:07:40.883-04:00Its good to be mii<div class="separator" style="clear: both; text-align: center;">
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When I taught ugrad Elementary Theory of Computation (Reg, CFL, P, NP, Dec, c.e.) I made 5% of the grade be MEET THE PROF- come to my office, in groups of 3-6 (though sometimes I got 10) just to talk to me. I ask them what there plans are, why they took the course, and they get to ask ME what they want.<br />
<br />
The most common question I got:<br />
<br />
Why is your mii the example of a mii on the Wikipedia entry on mii: <a href="https://en.wikipedia.org/wiki/Mii">here</a> (<a href="https://en.wikipedia.org/w/index.php?title=Mii&oldid=845863614">permalink</a>)<br />
<br />
I was originally hesitant to blog about this because (1) it may go away soon, and (2) I don't know.<br />
<br />
But (1) its been there and stable for a while, and (2) I can speculate:<br />
<br />
1) A prof in my department (not me) made and posted a Wikipedia page about me.<br />
<br />
2) That page used the mii.<br />
<br />
3) The powers that be at Wikipedia took down the mii (along with the list of my grad students who got PhD's). This post is not a rant about this, but I will note that I think they should have allowed the mii since it looks like me. whenever I am going to meet someone at an airport I email them the mii and it always works.<br />
<br />
4) Speculation: since it was on my Wikipedia page this mii was in the public domain and they could easily access it. Hence they used it. Is Wikipedia this arbitrary? Yes.<br />
<br />
5) My darling thinks is unfair that the mii page can use my mii but my page can't. I just think its odd.GASARCHnoreply@blogger.com4tag:blogger.com,1999:blog-3722233.post-84025247698314201812018-06-14T15:44:00.000-04:002018-06-14T15:44:07.053-04:00Hoteling<div class="" style="clear: both; text-align: left;">
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Thanks to Grigory Yaroslavtsev for taking over the <a href="https://docs.google.com/spreadsheets/d/1P5okKjeNlvkEEFMzX3l8VcL4SHpWBKNFET-5mPTWfDQ/edit?usp=sharing">Theory Jobs Spreadsheet</a>. Details on <a href="http://grigory.us/blog/theory-jobs-2018/">Grigory's blog</a>. Check out who is going where next year.</div>
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<br />
<br />
My office has an awesome view of Midtown Atlanta. Midtown has seen considerable construction over the last decade and I get to see the skyline change before my eyes. <a href="https://en.wikipedia.org/wiki/NCR_Corporation">NCR</a> opened an impressive glass building for their new world headquarters just a few months ago not coincidentally a short walk from Georgia Tech. A few weeks ago I got a tour of this facility.<br />
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<br /></div>
<div class="" style="clear: both; text-align: left;">
Most of the employees in the building do not get their own offices or desks. They have an open floor plan with hoteling. They use an app to reserve a desk up to a few weeks ahead of time. Each desk has a keyboard and two screens that they stick their laptops into. Their cell phones become their main phones. There are many conference rooms of different sizes, even some tiny ones meant for a single person to have a phone call or escape to some quietness.<br />
<br />
Would hoteling work in the academic world? Never, you learn quickly that professors will never give up two things: their office or their parking. When we do talk about hoteling, we talk about a shared office for people who have a main office in a different building.<br />
<br />
With faculty who often travel at far too many conference, or on some days working out of home or coffee shops and rarely using the books in their office, if they have them at all, why do we stick to the old fashioned offices?<br />
<br />
The best academic research require collaboration, between faculty, students, postdocs and other researchers. The best times I've had as a professor is when a student or colleague walks into my office with a question, an idea or sometimes a proof. Better if I have some place to be found and not just a location in an app.<br />
<br />
I'd also miss the view. </div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com4tag:blogger.com,1999:blog-3722233.post-60216048273592770982018-06-11T00:45:00.002-04:002018-06-11T00:46:54.305-04:00How the Villarino-Gasarch-Regan paper came about(This post overlaps a prior one <a href="http://blog.computationalcomplexity.org/2016/12/the-very-first-ramseyian-theorem.html">here</a>. The paper I am blogging about was also blogged about by Lipton <a href="https://rjlipton.wordpress.com/2018/06/08/hilberts-irreducibility-theorem/">here</a>. The paper itself is on arxiv <a href="https://arxiv.org/abs/1611.06303">here</a>. My slides for a talk I will be giving on this material are <a href="https://www.cs.umd.edu/users/gasarch/hilbertalk.pdf">here</a>)'<br />
<br />
<br />
<br />
Todays post is on how the paper came about. A later post will be about why someone else didn't do it earlier.<br />
<br />
How this paper came about:<br />
<br />
Many years ago Bill noticed that while several books on Ramsey theory (see my prior post for quotes) state that the HCL was the first Ramseyian theorem. I think one source mentioned in passing that Hilbert used it to prove the Hilbert Irreducibility theorem (HIT). Bill could not find a modern English exposition of the proof. So he asked Ken Regan (who not only knows German but can recite The Lewis Carol Poem J<i>abberwocky</i> in German!) to translate it, and then Bill would put it in modern language, and there would be an exposition. Bill got bogged down in some of the math, and they both got bogged down with other things (For Ken catching chess-cheaters, for Bill mentoring HS students, for both of them, blogging.) Many years passed.<br />
<br />
Sometime before 2015 Larry Washington showed me a nice proof that (ignoring mult constants)<br />
<br />
∑ 1/p ≤ ln(ln(n)) + O(1) (the sum is over all primes p ≤n )<br />
<br />
Read that carefully. There are many proofs in the web that the sum isat least ≥ ln(lg n) but I could not find any that the sum was ≤ ln(ln n). Larry Washington told me that the result and<br />
the proof were not new. I told him that, even so, it doesn't seem to be out there. So we agreed to write and and post to arXiv but not publish in a journal. It's <a href="https://arxiv.org/abs/1511.01823">here</a>.<br />
<br />
This arXiv paper caught the attention of Mark since he had an exposition of Merten's proof see <a href="https://arxiv.org/abs/math/0504289">here</a> that that sum diverges. Mertens proof had explicit bounds which are missing from modern proofs.<br />
<br />
I got into an email discussion with Mark about Math and History and I casually mentioned that Ken and I had worked on-and-off on HRL and HIT. A few weeks later he emailed me a translation of the paper. WOW! We worked together on polishing it, combining it with what Ken had done, and later brought Ken back into the project. I say without embarasment that we NEEDED Mark to resolve some of the issues we had and push the paper out the door. A Win-Win-Win.<br />
<br />
And a lesson here--- Larry Washington was reluctant to publish on arXiv a paper on stuff that was already known. I should have told him<br />
<br />
<i>But Larry, if we do that I might find someone to help me finish the Hilbert paper</i><br />
<br />
In a word: Serendipity.GASARCHnoreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-74755238938063752862018-06-06T19:40:00.001-04:002018-06-06T19:40:47.448-04:00I tell my class that P is important because... but is that really true?When teaching P vs NP the questions arises (and if not then I bring it up) what if you have algorithm in P that takes n^{100} time?. Or even n^{5} time which might be too long.<br />
<br />
I have given the following answers for years; however, I post this and will use your comments as a sanity check. In fact, I suspect I am wrong on some points.<br />
<br />
1) IF SAT was in P then something very clever was done. Even if its n^{100} it was very clever. That cleverness will almost surely result in other better algorithms. They may only be better in practice by not in theory (it works in practice- if only we can get it to work in theory :-) ), but it will still work and be a great advance.<br />
<br />
2) IF SAT was in P then perhaps we could USE SAT in P to get a better algorithm for SAT in P. This was the theme of a chapter in Lance Fortnow's book <a href="https://www.amazon.com/Golden-Ticket-NP-Search-Impossible/dp/0691175780/ref=sr_1_1?ie=UTF8&qid=1491666026&sr=8-1&keywords=Lance+Fortnow">The Golden Ticket: P, NP. and the search for the impossible</a> and is also likely behind something I once heard (I think it was from Scott but I can't find it on his blog) <i>If P=NP then math would be easy and we would have already found a proof that P=NP. Hence P ≠ NP</i><br />
<br />
The two above really can't be WRONG or even RIGHT or even NOT EVEN WRONG since they are speculations. The next few is where I need my sanity checked<br />
<br />
3) Whenever a real world natural problem is in P it has a low degree poly algorithm OR there are ideas to make it work in practice, if not in theory. When Lance read a prior version of this post he pointed out that `real world natural problem' is not a well defined term. True. Not sure what to do about that. Even so, is this true? roughly true? Something theorists say to be able to sleep at night?<br />
<br />
4) In the real world people say ``OH- the problem is NP-complete. Better look for approx solutions instead'' Or similar things. While this sounds true since I have never worked in the real world I really don't know. Is that what people say? do?<br />
<br />
5) If Lance proves P=NP next week then the consequences are enormous for real people working on real computing problems. But what if Lance proves P NE NP? Would that affect people working on real computing problems? A long time Carl Smith told me <i>If P NE NP was proven then the proof would have great insight into computing which would have a real affect on real people working on real computing problems</i>. My Darling (who is a Software Engineer) is skeptical of that. What do you think?GASARCHnoreply@blogger.com18tag:blogger.com,1999:blog-3722233.post-44829780107826753342018-06-01T14:09:00.002-04:002018-06-03T11:14:50.186-04:00BQP not in the Polynomial-Time Hierarchy in Relativized WorldsThe quantum complexity world is a rocking with the paper released yesterday by Ran Raz and Avishay Tal, <a href="https://eccc.weizmann.ac.il/report/2018/107/">Oracle Separation of BQP and PH</a>, resolving a question open since quantum complexity got its start over two decades ago. Scott Aaronson's <a href="https://www.scottaaronson.com/papers/bqpph.pdf">2010 paper</a> that sets some of the groundwork for the Raz-Tal result gives a nice overview of the question.<br />
<br />
All of you readers should be familiar with the P v NP question. P is the set of problems efficiently solvable by a deterministic computer. NP are the problems for which there exists a witness that we can verify quickly. The polynomial-time hierarchy (PH) is the constant alternation generalization of NP that has played a <a href="https://blog.computationalcomplexity.org/2005/06/favorite-theorems-polynomial-time.html">major role in computational complexity</a> So why don't we talk about the P vs PH question? That's just equivalent to P vs NP.<br />
<br />
BQP is the the class of problem efficiently solved by a quantum computer. Since P sits in BQP which sits in PSPACE we can't prove outright any separations for BQP without separating P from PSPACE. We can though get an understanding of complexity by allowing the machines access to the same oracle and seeing what we can separate. We already knew BQP doesn't sit in NP relative to an oracle. Same was true for BPP (efficient probabilistic computation) but BPP is in PH for all oracles as are constant round interactive proofs. So we might expect we can have BQP in PH, BQP solvable with constant alternations, relative to all oracles. Raz and Tal refute that hypothesis.<br />
<br />
Raz and Tal's proof builds on the Forrelation concept developed by <a href="https://www.scottaaronson.com/papers/bqpph.pdf">Aaronson</a> (under the name Fourier Checking) and extended by <a href="https://arxiv.org/abs/1411.5729">Aaronson and Ambainis</a>. The oracle isn't complicated: Pick n numbers x<sub>1</sub>,...,x<sub>n</sub> each x<sub>i</sub> chosen from a normal distribution with mean 0 and variance a small ε > 0. Let y=y<sub>1</sub>,...,y<sub>n</sub> be the <a href="https://en.wikipedia.org/wiki/Hadamard_transform">Hadamard transform</a> applied to x<sub>1</sub>,...,x<sub>n</sub>. You then probabilistically round the x's and y's to 1 or -1. A quantum machine can distinguish this distribution from uniform using the fact that quantum machines can do Hadamards easily and Raz and Tal use Fourier show that low-depth alternating circuits (that capture PH) can't distinguish so easily. The <a href="https://eccc.weizmann.ac.il/report/2018/107">paper</a> is well-written if you want details.<br />
<br />
A few questions for further directions:<br />
<ul>
<li>Can you show a relativized world where P = NP (=PH) but P ≠ BQP? I'd suspect it will be a messy but not too hard generalization of this paper.</li>
<li>How far can you push the Raz-Tal result, i.e., for what function f(n) can we show that BQP cannot be solved by f(n)-alternating polynomial-time Turing machines. Can you show f(n) = n<sup>Ω(1)</sup>?</li>
</ul>
<div>
Also see Scott's <a href="https://www.scottaaronson.com/blog/?p=3827">personal take</a> on this new result.</div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com2tag:blogger.com,1999:blog-3722233.post-17614356622899058532018-05-31T11:09:00.001-04:002018-05-31T14:13:25.293-04:00Seventh Grade Math ContestI stumbled upon an old <a href="https://www.lesswrong.com/posts/EdFDwjsLNpgtTMJAp/great-mathematicians-on-math-competitions-and-genius">blog post</a> on the Lesswrong weblog that quotes several famous mathematicians on the connections, or lack thereof, between mathematics competitions and mathematics research. Let me tell you how a seventh grade math contest altered the course of my life.<br />
<br />
In 1975 I attended seventh grade in a middle school in upstate New Jersey. The school divided the students into three tracks, honors, standard and remedial, and I was an unexceptional student in the standard track. We all took a math pretest to determine who would represent the school in a state-wide competition. To everyone's surprise, especially my own, I killed on the test scoring twice as many points as anyone else. The school adjusted my course schedule but because of my not-so-great English skills I became the only student taking honors math and science and standard English and History (with a racist history teacher but that's another story). I think I came in 12th in the state competition.<br />
<br />
I never did well in math contests after that, doing okay but not particularly strong in high school competitions. But the experience drove me to what we now call STEM. I got involved in computers and math in high school which led me to study engineering, <a href="https://blog.computationalcomplexity.org/2005/12/how-i-became-theorist.html">briefly</a>, at Cornell. I did <a href="https://blog.computationalcomplexity.org/2005/12/first-saturday-in-december.html">make the Putnam team</a> as a freshman at Cornell, but scored a zero which pretty much ended my career in math competitions.<br />
<br />
<br />Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-20276579834162546842018-05-30T00:22:00.001-04:002018-05-30T08:00:37.194-04:00Why is someone emailing me an offer to apply to be chair?I recently go the following email (I blocked out identifying information of who send it and what college is involved. I doubt I needed to legally but... you never know.) I include my comments on the email in cap letters.<br />
<br />
I am NOT flattered by the email. I have LESS regard for those who send it.<br />
<br />
I will say that the school is respectable and I had heard of them. I'm just surprised they heard of me.<br />
<br />
(The letter was typeset fine- if there are problems with typesetting below thats on me.)<br />
<br />
---------------------------------------------------------------<br />
Dear Dr. Gasarch,<br />
<br />
XXX Inc. has been retained to assist XXX University in the search for the<br />
next Chair of the Computer Science and Engineering Department.<br />
<br />
ADMIN EXPERIENCE: RUNNING AN REU PROGRAM. NOT NEARLY ENOUGH.<br />
<br />
Your expertise and accomplishments have come to our attention, and we would appreciate the opportunity to speak with you about this position.<br />
<br />
IF THEY MEAN ADMIN-EXPERTISE, THATS JUST SILLY.<br />
<br />
IF THEY MEAN PAPERS AND SUCH... LETS JUST SAY I AM ON THE OBSCURE END OF THEORY AND WONDER WHAT ACCOMPLISHMENTS THEY MEAN. MY MUFFIN PAPER?<br />
<br />
EXPERTISE: MENTORING HS AND UGRAD STUDENTS. BLOGGING? EXPERT OR NOT I"VE DONE IT FOR A WHILE. NOT THE STUFF OF CHAIR'S.<br />
<br />
Could you suggest some times over the next week that<br />
might work for your schedule? We will be very respectful of your time.<br />
<br />
<br />
XXX is located in XXX, XX where the city’s diverse and growing population puts it in the<br />
Top 10 U.S. cities in population and in the Top 5 fastest growing cities. XXX has an endowment of $1.5B and recently completed a $1.3B campaign. The Computer Science and Engineering Department has been targeted for significant advancement and new faculty hires. The next Department Chair will lead and direct this enhancement.<br />
<br />
<br />
The XXX Metroplex is a dynamic region with leading high-technology companies in the aerospace,<br />
defense, energy, information technology, life sciences, semiconductors, telecommunications,<br />
transportation, and biomedical industries. Some of the top companies and research institutes with a strong presence in the XXX area include LONG LIST OF COMPANIES I HAVE HEARD OF.<br />
<br />
THESE LAST TWO PARAGRAPHS WOULD BE A LOT MORE IMPRESSIVE IF I DIDN"T KNOW THEIR WAY TO PICK CANDIDATES TO ASK TO APPLY WAS SO TERRIBLE.<br />
<br />
Please click here to view a two-minute video in which the President, Provost, and Dean of the<br />
XXX School of Engineering at XXX discuss some of the possibilities for research and growth<br />
that the Chair of Computer Science and Engineering will have the opportunity to develop.<br />
Cybersecurity, particularly, is an identified area for growth and hiring to directly complement the XXX Institute for Cyber Security. A full listing of the qualifications and duties of the position can be found in the profile under “Current Searches” at XXX<br />
<br />
CYBERSECURITY- I HAVE TAUGHT A 1-CREDIT MINI-COURSE IN CRYPTO. I HAVE A SURVEY AND A PAPER ON PRIVATE INFO RETRIEVAL. I DOUBT THAT'S WHY THEY CONTACTED ME.<br />
<br />
<br />
Best,<br />
<br />
XXX<br />
-------------------------------------------------<br />
<br />
So why did they email me? Speculation<br />
<br />
1) They meant to email Lance and got confused.<br />
<br />
2) They cast a VERY wide net. The UPSIDE of doing that so is that they might find someone<br />
they would have overlooked. The DOWNSIDE of doing that is... thats the problem with<br />
spam- there is absolutely no downside.<br />
<br />
3) They emailed every computer science professor who has a wikipedia page? a blog? a book?<br />
Some criteria which I happened to fit.<br />
<div>
<br /></div>
<br />
<br />GASARCHnoreply@blogger.com5tag:blogger.com,1999:blog-3722233.post-65509338086867286062018-05-24T11:15:00.000-04:002018-05-24T11:18:12.595-04:00Kolmogorov Complexity and Causation<div class="tr_bq" style="font-family: sans-serif; font-size: 13px;">
<br /></div>
<div style="font-family: sans-serif; font-size: 13px;">
I got an interesting email question.</div>
<blockquote style="font-family: sans-serif; font-size: 13px;">
Suppose I give you a set of points S of the form (x,y). He suggested ideally they would be pairs of a real numbers. Supposing there is a causal relationship between x and y of some kind, we want to know know if it is more likely that the x value causes the y value or the y value causes the x value. One plausible way to decide what the answer should be is by answering the question <b>is the length of the shortest program which maps the x values to their y values shorter than the length of the shortest program which maps the y values to their x values.</b></blockquote>
<blockquote style="font-family: sans-serif; font-size: 13px;">
So, my intuition says that this is clearly undecidable. I'm actually having a hard time thinking of a proof, so do you happen to know of one or if this problem might actually be decidable?</blockquote>
<blockquote style="font-family: sans-serif; font-size: 13px;">
On a related note, since I'm already writing you about this question, do you happen to know about the complexity of any related questions which involve circuit size instead of program length? </blockquote>
Let's use notation from Kolmogorov complexity, letting C(x|y) be the size of the smallest program that takes y as input and outputs x. Now suppose it is decidable to determine whether C(x|y) > C(y|x). Then find an x of length n such that for all y of length n/3, C(x|y)>C(y|x). Such x exist: For any random x, C(x|y)>= 2n/3 and C(y|x) <= n/3.<br />
<br />
Now I claim C(x)>=n/4 for the x you found. If not we have C(x|y)<=C(x)<n/4 but for some y, C(y|x)>=n/3 since there aren't enough shorter programs to cover all the y's.<br />
<br />
Since there is no computable procedure to find x such that C(x)>=n/4, there can't be decidable procedure to determine whether C(x|y) > C(y|x).<br />
<br />
But does this question relate to causality. Pick a random x from strings of length n and y at random from strings of length n/3. We have C(x|y) > C(y|x) even though there is no causality.<br />
<br />
Instead you could look at the information of y in x, how many bit of x does y help describe, defined by I(y|x) = C(x)-C(x|y). This measure correlation since I(y|x)=0 iff x and y are independent but symmetry of information gives I(y|x)=I(x|y) so no hope for causation.<br />
<br />
In short, Kolmogorov complexity won't give you much on causation--you can't avoid the controlled experiments.<br />
<br />
For your last question, there is a notion of Kolmogorov complexity that roughly corresponds to circuit size, KT(x|y) defined as the sum of the program size and running time minimized over all programs p that take y as an input and output x. I'm guessing it's hard to determine if KT(x|y) < KT(y|x) and you could probably show it under some assumption like secure psuedorandom generators. Also symmetry of information isn't believed to hold for KT complexity so maybe there is something there. Interesting questions.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com6tag:blogger.com,1999:blog-3722233.post-13821018067996487602018-05-20T22:44:00.000-04:002018-05-20T22:44:17.759-04:00COMPUTER PROOF vs computer proof- Quadratic VDW theorem<b>Quad VDW Theorem: </b>For all c there exists W=W(c) such that for all c-colorings of {1,...,W} there exists a,d such that a and a+d<sup>2</sup> are the same color.<br />
<br />
What is known about W(c)?<br />
<br />
The first proof of Quad VDW was nonconstructive.<br />
<br />
The second proof was constructive but used VDW's theorem and gave terrible bounds, even for W(2).<br />
<br />
EASY: Show W(2)=5<br />
<br />
ON A HS MATH COMP: Show that for all 3-colorings of {1,...,2003} there exists two numbers a square apart that are the same color.<br />
<br />
One can get a bound less than 100.<br />
<br />
With a lot more detailed work one can get W(3)=29.<br />
<br />
None of above was done with a computer.<br />
<br />
SO- what about W(4)? There are three proofs that W(4) exists:<br />
<br />
a) Prove the QUAD VDW theorem. This gives terrible bounds.<br />
<br />
b) I had some students use SAT solvers and they obtained W(4)=58. I am sure they are right and I am glad to know it (especially since it confirms the general mantra that the bounds from proofs in Ramsey theory tend to be much bigger than the reality) but its somehow unsatisfying. I want a <i>HS proof. </i>I wanted to put it on the MD Math Comp for 2017 but wiser people prevailed.<br />
<br />
c) I gave the problem to a Grad Student Zach Price and he came up with a HS proof-- sort of. Look at the following graph: <a href="https://www.cs.umd.edu/users/gasarch/COURSES/858/S18/qvdw4.pdf">here</a>. It shows that in any 4-coloring of N which does not have two numbers a square apart the same color:<br />
<br />
COL(x) = COL(x+290085289)<br />
<br />
from this one can obtain<br />
<br />
W(4) ≤ 290085289<sup>2</sup> (which is MUCH better than the bound from the proof of Quad VDW) and hence a proof of QVDW that does not need a program, except that to FIND this gadget used a program. I doubt a HS student could do this on an exam.<br />
<br />
Is Zach's proof the HS proof I am looking for?<br />
<br />
PRO- once you see it its easy to verify and does not need a program.<br />
<br />
CON- coming up with it needed a program.<br />
<br />
More to the point: which proof do you like better:<br />
<br />
1) The SAT Solver proof. Not a proof you can see or feel, but gives exact bounds.<br />
<br />
or<br />
<br />
2) Zach's gadget proof. Much worse bound but you can see and feel the proof, and verify it after you know the gadget.<br />
<br />
I prefer Zach's proof.<br />
<br />
<br />GASARCHnoreply@blogger.com6tag:blogger.com,1999:blog-3722233.post-27861197150835856902018-05-17T08:37:00.000-04:002018-05-17T08:37:05.089-04:00The Complexity of the FirmIn 1937, a year after Turing had his <a href="https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf">seminal paper</a>, Ronald Coase published a paper <a href="http://dx.doi.org/10.2307/2626876">The Nature of the Firm</a> to give a framework to why we have companies and how large they become. In a perfect market economy we shouldn't need a firm at all, everyone is just an independent contractor and market pricing will drive efficient use of labor. Coase notes there are costs to creating contracts and one can gain efficiencies by avoiding these contracts by having both parties inside the same organization.<br />
<br />
Much of those costs come from complexity, it is impossible to create a contract to cover all possible outcomes and thus contracts are incomplete leading to loopholes and lawsuits.<br />
<br />
In the other direction, having the central organization of a firm has its own costs, from inefficiencies from not using markets to balance supply and demand, to the complexity of the organization processes themselves. In Coase's model, a firm grows to a size that balances the organization and contract costs at the margin.<br />
<br />
So what happens to the sizes of firms when we reduce complexity, as happens with modern computing, from better communication, optimization and data analytics. We see relatively new companies like Google and Amazon getting larger. We also see a number of startups and small companies with very few workers, sometimes just one.<br />
<br />
Computing drops both the cost of central organization and the cost of contracts so the size of a firm depends on circumstance. One can have a tech startup with a small number of workers since they can write apps that run on other people's platforms (like web browsers and on phones) and have the processing done on the cloud where they can scale or not scale as needed. Meanwhile a large company can more easily coordinate and connect using modern technology making it cost efficient to expand.<br />
<br />
As we enter a new age of machine learning and automation, how will that affect the very nature of companies in the future? How about universities, a special kind of firm in itself? How can we harness the tools and ideas from computational complexity to help understand these and other societal changes.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com1tag:blogger.com,1999:blog-3722233.post-56383662097322163862018-05-14T23:47:00.001-04:002018-05-14T23:47:31.696-04:00What does it mean for a student to guess an answer?On my final in Aut Theory I wanted to ask a TRUE/FALSE/UNKNOWN TO SCIENCE<br />
question but did not want them to guess. Hence I had +4 for a right answer, -3 for a wrong answer.<br />
Here is the question:<br />
<br />
------------------------------------------------------------------------------------------<br />
For each of the following say if its TRUE, FALSE, or UNKNOWN TO SCIENCE. No Proof Required BUT you get +4 for every right answer and -3 for every wrong answer and 0 for an answer left blank. So<br />
<br />
DO NOT GUESS!!!!!!!!!!!!!!!!!!!!!!!!!!!<br />
<br />
a) If A is regular and F is a finite set then A UNION F is regular.<br />
<br />
b) If A is in P and F is a finite set then A UNION F is in P.<br />
<br />
c) If A is in NP and F is a finite set then A UNION F is in NP.<br />
<br />
d) If A is decidable and F is a finite set then A UNION F is decidable.<br />
<br />
e) If A is undecidable and F is a finite set then A UNION F is undecidable<br />
-------------------------------------------------------------------------------------------------<br />
<br />
Note that they are all TRUE. A student who (as many did) answered T-T-T-T-F had the following conversation with me:<br />
<br />
BILL: You guessed! I told you DO NOT GUESS!!!!!!!!!!!!<br />
<br />
STUDENT: No. I REASONED that you would not make them all T, so by this reasoning the last one had to be F. I now see that my reasoning is wrong--- you would make them all T, but it was not guessing, it was reasoning.<br />
<br />
I claim the student was guessing, he claims he was not. What do you think?<br />
<br />
Having said that, the following IS a rational strategy:<br />
<br />
If I don't know the answer but it has nothing to with P vs NP then it has to be T or F. In this case guess since exp val is positive. If the answer has to do with P vs NP then do not guess.<br />
<br />
This also raises a question- if they honestly thought (say) e was F I want to just give them 0, whereas if they are guessing or using reasoning about `Bill wouldn't ...' I want to give them -3. But alas, we cannot read their minds or souls.<br />
<br />
<br />
<br />
<div>
<br /></div>
GASARCHnoreply@blogger.com10tag:blogger.com,1999:blog-3722233.post-79069429538860756962018-05-11T05:29:00.000-04:002018-05-11T05:29:12.343-04:00Richard Feynman (1918-1988)<a href="https://3.bp.blogspot.com/-e3h0ABkxT64/WvMxBvu2JmI/AAAAAAABgL4/v969shFo2CUD1Amf-kWoE00uYAuM_WB5QCLcBGAs/s1600/51jL19me4oL._SX330_BO1%252C204%252C203%252C200_.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="499" data-original-width="332" height="320" src="https://3.bp.blogspot.com/-e3h0ABkxT64/WvMxBvu2JmI/AAAAAAABgL4/v969shFo2CUD1Amf-kWoE00uYAuM_WB5QCLcBGAs/s320/51jL19me4oL._SX330_BO1%252C204%252C203%252C200_.jpg" width="211" /></a><br />
When I took cryptography from Manuel Blum, he handed out copies of the chapter "Safecracker Meets Safecracker" from Richard Feynman's book <a href="https://amzn.to/2KPg97V">Surely You're Joking Mr. Feynman</a>. Feynman, the Nobel Prize winning physicist who was born a hundred years ago today, wrote this book not about physics but just a series of stories from different times in his life. This chapter described how Feynman learned how to open locked safes in Los Alamos during the Manhattan Project.<br />
<br />
We all have interesting stories to tell but Feynman finds a way to keep things compelling in a way most scientists could not--even if he sometimes comes off being a bit of a jerk, hence the title. This book inspired me to tell my own stories which occasionally show up in this blog.<br />
<br />
His most important stories form <a href="http://www.feynmanlectures.caltech.edu/">The Feynman Lectures on Physics</a> (free to read online), an amazing explanation of deep physical concepts.<br />
<br />
Richard Feynman's biggest contribution to theoretical computer science comes from a <a href="http://doc.cat-v.org/feynman/simulating-physics/simulating-physics-with-computers.pdf">1981 keynote address</a>.<br />
<blockquote class="tr_bq">
What kind of computer are we going to use to simulate physics? The full description of quantum mechanics for a large system cannot be simulated with a normal computer. And therefore, the problem is, how can we simulate the quantum mechanics? Can you do it with a new kind of computer—a quantum computer?</blockquote>
which begat a <a href="http://dx.doi.org/10.1098/rspa.1992.0167">proof-of-concept paper</a> by David Deutsch and Richard Jozsa which begat Daniel Simon's <a href="https://dx.doi.org/10.1137/S0097539796298637">exponential separation</a> which begat Peter Shor's <a href="https://dx.doi.org/10.1109/SFCS.1994.365700">factoring algorithm</a> which begat billions of research dollars and considerable expectations, real and imagined, for Feynman's vision.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com1tag:blogger.com,1999:blog-3722233.post-83988655952463966902018-05-09T08:25:00.000-04:002018-05-09T08:25:15.994-04:00Second of N posts on G4G13. Maybe(Don't forget to vote for SIGACT posistions:<a href="http://www.acm.org/elections/sigs/sigact-election-bios">here</a> 9th workshop on Flexible network design, May 22-25 at College Park, <a href="http://www.cs.umd.edu/fnd2018">here</a>.)<br />
<br />
My first poston G4G13 is arguably <a href="https://blog.computationalcomplexity.org/2018/04/gahering-for-gardner-13-first-or-third.html">here</a>. To see why its debatable, see that post.<br />
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FOXTROT HALF-EMPTY, HALF-FULL PROBLEM, INCLUDING 13 by Thomas Francic Banchoff<br />
<br />
In a Foxtrot cartoon (see <a href="https://www.google.com/search?q=Foxtrot+half+empty&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwj17IunyvjaAhVptlkKHWBrDOgQ7AkINg&biw=1266&bih=867#imgrc=-Fbl5q_9hwo0RM:">here</a>) Foxtrot has a glass which looks like it is half-full (or half-empty)<br />
and asks people if its half-full or half empty. But the jokes on them!<br />
The class is slightly angled so its actually 5/12 full (or 7/12 empty)<br />
Given the area of the top and bottom What is the angle?<br />
Generalize to other dimensions.<br />
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HEX PRIMES by Spandan Bandyopadhyay<br />
<br />
Here is an alternative definition of primes that<br />
lends itself to a generalization.<br />
<br />
A number x is PRIME if when there is a rectangle with<br />
integer sides and area x, one of the sides is 1.<br />
<br />
Lets generalize this!<br />
<br />
A number x is TRIPRIME if when there is a triangle with<br />
integer sides of area x, one of the sides is 1.<br />
<br />
Rather than use these prefixes we will go with<br />
<br />
A number x is n-PRIME if when there is a convex n-gon with<br />
integer sides and area x, one of the sides is 1.<br />
<br />
HEXPRIMES are of course 6-primes.<br />
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The problem with 5-minute talks (maybe it should have been 6 minutes)</div>
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is that the concept is intersting but I didn't get to hear much</div>
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about them. And I could not find a paper on line. Note that this</div>
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conference has many non-academics for whome PAPERS are not the</div>
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basic currency so things are more informal. This is GOOD in that</div>
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its more of a free-for-all, but bad for follow up.</div>
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ALL of G4G12's are on You-Tube, so when that happens for G4G13,</div>
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I can follow up on this.</div>
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One thing I did manage to write down- 7 is the first HEX-composite.</div>
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GASARCHnoreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-70433082879602612382018-05-03T07:12:00.000-04:002018-05-03T07:12:46.101-04:00Broader Impacts Redefined<i>The ACM Future of Computing Academy <a href="https://acm-fca.org/2018/03/29/negativeimpacts/" target="_blank">suggests</a> that "peer reviewers should require that papers and proposals rigorously consider all reasonable broader impacts, both positive and negative." Here is the broader impacts section of a future imagined grant proposal.</i><br />
<i><br /></i>My latest cryptocurrency paper will allow people to sell all sorts of paraphernalia, illegal, immoral and fattening, while avoiding paying taxes. Dr. Evil and his minions can move money around the world anonymously to more easily implement their dastardly plans. With this new work bitcoin will increase in value, causing far more mining setups that will deplete the energy resources of the world.<br />
<br />
Since I already own bitcoin, I will get filthy rich, increasing the gap between the one-percenters and the middle-class. I'll buy a fancy car and reprogram the auto-pilot to run over people who get in my way. And all those pesky bicyclists clogging the roads. That would be a positive impact.<br />
<br />
For the rest of humanity my new quantum gravitation algorithm will put all those people out of work. But their misery will be short lived because if anyone tries to run the algorithm, it may cause a small black hole that devours the earth.<br />
<br />
My proposed generic oracle separation would seem to have no impact whatsoever. But what happens if an alien race threatens to destroy the planet unless we find such a separation. I will have saved the earth, a very positive broader impact. Of course a different alien race could visit the earth and deem it mostly harmless until they discover my oracle, realized we have advanced too far and then blow us up before we become a threat.<br />
<br />
In other broader impacts, I will train graduate student to be just like me. Whether this is a positive or negative impact I leave to the reviewers.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com3tag:blogger.com,1999:blog-3722233.post-24710167149475527042018-04-30T09:03:00.000-04:002018-05-02T21:51:16.013-04:00Gathering for Gardner 13 - the first or third of many postsI attended G4G13 (Gathering for Gardner- meeting 13). Martin Gardner was the Scientific American Mathematical Recreations columnist from 1956 until 1981. He had a great influence on many math-people of my generation.<br />
<br />
Most of the talks were 5 minutes so they could tell you a problem or thought of interest but not much more. This is GOOD in that I UNDERSTOOD most of the talks! There were also some talks on Magic and on science literacy (or perhaps science illiteracy) which were also interests of Martin Gardner.<br />
<br />
I posted on one <a href="https://blog.computationalcomplexity.org/2018/04/find-8-digits-number-such-that.html">problem</a> and its <a href="https://blog.computationalcomplexity.org/2018/04/the-8-digit-number-i-asked-for.html">solution</a> so this is either my first or third post on G4G13.<br />
<br />
Withouth further ado, here are my descriptions of some of the talks. OR you could just go <a href="http://www.gathering4gardner.org/g4g13-abstracts.pdf">here</a><br />
<br />
<br />
My descriptions are long- I may have lots of posts on G4G13!<br />
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EXTENDING THE 10958 PROBLEM by Bill Ames.<br />
<br />
(paper on arxiv <a href="https://arxiv.org/abs/1302.1479">here</a>)<br />
<br />
In 2014 Taneja proposed the following math problem:<br />
<br />
Take the digits 1,2,3,4,5,6,7,8,9.<br />
<br />
You can put any of +, -, *, /, and powers you want between the digits, or erase the comma to get (for example) 34. Use parenthesis freely. Which natural numbers can you produce?<br />
<br />
Here are some examples:<br />
<br />
1 = 1 + (2 - 3) - (4 - 5) - (6 - 7) + (8 - 9)<br />
<br />
30 = 1 + 2^3 - 4*5 + 6*7 + (8-9)<br />
<br />
Which numbers can you get?<br />
<br />
Taneja got all the numbers between 0 and 11,111 EXCEPT 10958.<br />
<br />
(Is 10958 possible? The talk didn't say.)<br />
<br />
Using other operations- square roots and factorials- you can get 10958. The talk was about adding more operations and getting all numbers between 0 and 111,111<br />
<br />
THIRTEEN BOUNCES by Gary Antonick.<br />
<br />
Since this was G4G13 there were several talks about the number 13. The hotel did not have a 13th floor, and no room was of the form X13 (I was in room 515 and sometimes overshot my room since I assumed it would go 511-513-515, not 511-515). Martin Gardner would have OBJECTED to the lack of a 13th floor and would have condemned triskaidekaphobia (fear of the number 13 and the longest word I ever spelled correctly) as irrational. And he would be right. But I am preaching to the converted (also known as preaching to the choir).<br />
<br />
The talk was about firing a cannoball at a perfect reflector. If the cannon is positioned just right the ball will go back into the cannon. Can you make it do this after taking 13 bounces? What if you an only use a compass and straightedge for your calculations?<br />
<br />
A GRAMMATICAL APPROACH TO THE CURLING NUMBER CONJECTURE by Duane Bailey.<br />
<br />
(For more on the Curling Conjecture Google "Curling Number Conjecture"-- I was going to give a list of papers but there are just too many. There is no wikipedia entry on it, though I did learn about the sport of Curling!)<br />
<br />
Take any finite sequence of integers (we will just use natural numbers) for example<br />
<br />
2 3 2 3<br />
<br />
Write it as X Y Y Y ... Y with as many Y's as possible.<br />
<br />
for example<br />
<br />
(2 3)<sup>2</sup><br />
<br />
The number 2 is the Curling Number of the sequence. Append it to the sequence<br />
<br />
2 3 2 3 2<br />
<br />
Now take the Curling number of this sequence. It is 2 since the sequence is<br />
<br />
2 (3 2)<sup>2</sup><br />
<br />
Append it to the sequence:<br />
<br />
2 3 2 3 2 2<br />
<br />
Now the Curlng number of this sequence is 1 and we stop.<br />
<br />
The conjecture is that if you start with any sequence you will eventually get to 1.<br />
<br />
The talk relates the conjecture to aperiodic grammars.<br />
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<br />GASARCHnoreply@blogger.com5tag:blogger.com,1999:blog-3722233.post-281663660312524032018-04-26T10:11:00.001-04:002018-04-26T10:11:26.264-04:00The 8 digit number I asked for(On June 29th, co-located with STOC, there will be a workshop to celebrate Vijay Vazarani's 60th birthday. See <a href="https://www.cs.umd.edu/users/samir/stoc2018/">here</a>. As computer scientists shouldn't we use 64 as the milestone?)<br />
<br />
<br />
<br />
At Gathering for Gardner 13 Peter Winkler gave a great talk entitled<br />
<br />
Problems that Solve Themselves.<br />
<br />
(The title kind-of gives away how to solve it. And its just the kind of thing Peter Winkler would talk about. Hence I omitted the title and author when I first posted about it)<br />
<br />
I blogged about one of them, asking for the answer. I repeat the problem and answer it:<br />
<br />
Find an 8-digit number<br />
<br />
d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0<br />
<br />
such that<br />
<br />
d_0 is the number of 0's in the number<br />
<br />
d_1 is the number of 1's in the number<br />
<br />
d_2 is the number of 2's in the number<br />
<br />
.....<br />
<br />
d_6 is the number of 6's in the number<br />
<br />
but<br />
<br />
d_7 is NOT necc the number of 7's<br />
<br />
d_7 is the number of DISTINCT DIGITS in d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0<br />
<br />
WRONG APPROACH: Work out algebra for the digits, try to force it Actually, I had thought this was the wrong approach but some of the comments on my last blog DID do this.<br />
<br />
RIGHT APPROACH: Take an aribtrary number like<br />
<br />
1 1 1 1 1 1 1 1<br />
<br />
NOW- look at it and count the number of 0's, 1's, ... 6'ls and the number of distinct numbers<br />
For the new number let<br />
<br />
d_0 be the number of 0's in the old number<br />
<br />
....<br />
<br />
d_6 be the number of 6's in the old number<br />
<br />
d_7 be the number of distinct digits<br />
<br />
To get<br />
<br />
1 0 0 0 0 0 8 0<br />
<br />
iterate the process to get<br />
<br />
3 0 0 0 0 0 0 6<br />
<br />
3 1 0 0 0 0 0 6<br />
<br />
4 1 0 0 1 0 1 5<br />
<br />
4 0 1 1 0 0 2 3<br />
<br />
5 0 0 1 1 1 2 2<br />
<br />
4 0 1 0 0 2 3 2<br />
<br />
5 0 0 1 1 2 1 2<br />
<br />
4 0 1 0 0 2 3 2<br />
<br />
5 0 0 1 1 2 1 3<br />
<br />
5 0 1 0 1 1 3 2<br />
<br />
5 0 1 0 1 1 3 2<br />
<br />
AH- this number works!<br />
<br />
Peter Winkle claims that if you start with any 8 digits number this process will converge to a solution within at most 15 steps. I do not think he proved this. But the key here is that by NOT being clever you can easily find the answer. The problem, as the title of the talk says, solves itself!<br />
<br />
How many such numbers are there? Proving the process works? Getting statistics on how long it will take on average? I leave all of this to the reader. (I do not know the answers.) And one can ask all of this for other bases and other condidtions on the digits (though calling them digits is odd since that smacks of base 10- what to use instead of ``digits'' when in another base?)<br />
<br />
<br />
<br />
<br />
<br />
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<br />GASARCHnoreply@blogger.com1tag:blogger.com,1999:blog-3722233.post-68825711694280389922018-04-23T11:56:00.003-04:002018-04-23T11:56:45.674-04:00Find an 8-digits number such that ....(June 29th, co-located with STOC, will be a workshop to celebrate Vijay Vazarani's 60th birthday. As computer scientists shouldn't we use 64 as the milestone? See <a href="https://www.cs.umd.edu/users/samir/stoc2018/">here</a> for details about the workshop, not about 60 vs 64.)<br />
<br />
I will likely post a lot about Gathering For Gardner 13, but for now I will just give one problem I saw there. I will not say the speaker or the title of the talk as that might be a clue. I'll give the answer in my next post which will be this week<br />
<br />
<br />
Find an 8-digit number<br />
<br />
d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0<br />
<br />
such that<br />
<br />
d_0 is the number of 0's in the number<br />
<br />
d_1 is the number of 1's in the number<br />
<br />
d_2 is the number of 2's in the number<br />
<br />
.....<br />
<br />
d_6 is the number of 6's in the number<br />
<br />
but<br />
<br />
d_7 is NOT necc the number of 7's<br />
<br />
d_7 is the number of DISTINCT DIGITS in d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0<br />
<br />
You could prob find such a number by computer search- but don't.<br />
<br />
Feel free to comment. I will not block comments (except those we usually block- usually spam) but by the same token, if you don't want hints, don't read the comments.<br />
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<br />GASARCHnoreply@blogger.com7