tag:blogger.com,1999:blog-3722233Thu, 23 Mar 2017 10:43:19 +0000typecastfocs metacommentsComputational ComplexityComputational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarchhttp://blog.computationalcomplexity.org/noreply@blogger.com (Lance Fortnow)Blogger2466125tag:blogger.com,1999:blog-3722233.post-2431524779430977178Thu, 23 Mar 2017 09:16:00 +00002017-03-23T05:17:33.150-04:00The Dagstuhl Family<div class="separator" style="clear: both; text-align: center;">
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This week I'm at the <a href="http://www.dagstuhl.de/en/program/calendar/semhp/?semnr=17121">Dagstuhl workshop on Computational Complexity of Discrete Problems</a>. As you long time readers know Dagstuhl is a German center that hosts weekly computer science workshops. I've been coming to Dagstuhl for some 25 years now but for the first time brought my family, my wife Marcy and daughter Molly, so they can see where I have spent more than half a year total of my life. Molly, currently a freshman at the University of Chicago, was the only Chicago representative, though the attendees included four Chicago PhDs, a former postdoc and a former professor.<br />
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We had a different ice breaker, where each person wrote topics they think about which ended up looking look like an interesting bipartite graph.<br />
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Molly has a few thoughts on Dagstuhl:<br />
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The coolest thing about the study of computer science is this place.<br />
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Okay, I know my dad would disagree with me (he probably thinks the coolest thing about computer science is the computer science itself). But for me, someone quite removed from the math and science and thinking, this place is by far the coolest thing about the computer science community. The point of it is isolation, as well simultaneous connection. The isolation comes in the form of a meeting center in rural Germany, separated from the world, devices which can (and do) block wifi in rooms like lecture rooms and the dining hall, resulting in a week without much interaction with the outside world. The connection stems from this very isolation -- in this highly isolated place, people are forced to connect with each other face-to-face, and to get to know each other, as well as the ideas and problems people are working on. The isolation creates a heightened sense of community, both in social and intellectual senses of the word. Forced to be so close and so interconnected, it’s no wonder so many problems get solved here.<br />
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I’m glad I got to come see why my father has been coming here for a quarter century. He is very old.http://blog.computationalcomplexity.org/2017/03/the-dagstuhl-family.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-6465740678490690424Mon, 20 Mar 2017 03:54:00 +00002017-03-19T23:54:20.997-04:00If you want to help your bad students DO NOT give an easy exam<br />
1) When I was a grad student TAing Formal Lang Theory we had a final ready to give out but noticed that one problem was too hard. So we changed it. But we made it too easy. Whoops. My thought at the time was <i>this will help the bad students.</i> I was wrong. Roughly speaking the students who got 70-80 on the midterm now got 90-100 on the final whereas the students who got 30-40 on the midterm got 35-45 on the final. So the bad students improved, but the better students improved more.<br />
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2) When I teach Discrete Math to LOTS of students we have a policy about midterm regrade requests. Rather than have them argue in person they have to:<br />
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In writing make a clear concise argument as to why it was mis-graded<br />
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If your argument displays that you really don't know the material, even when you can reflect on it, you can lose points. (True Story: We ask for an example of a Boolean Function with two satisfying assignments. They gave us a formula with only one, so they got -5. In the regrade request they try to still argue that it has two satisfying assignments. They lost 2 more points.)<br />
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In reality the policy is more preventative and we rarely remove points. However even this policy benefits the better students more than the poor ones who have a hard time even articulating why what they wrote is actually right (likely it is not).<br />
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3) Just this winter teaching a 3-week 1-credit course we were grading a problem and giving lots of 15/25 since the students were all making the same mistake. Half way through I got suspicious that maybe WE were incorrect. Looking at the exact wording of the question I realized WE were wrong, and, given the wording and what they would quite reasonably think we wanted, they were right. So we went back and upgraded many students from 15 to 25. And again, this lifted students in the 70's to 90's, but did NOTHING for the students below 50 since none of them had anything like a correct answer to any way to view the question.<br />
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Okay, so what does all of this mean? It means that an easy exam or a generous grading policy is devastating for the bad students. <br />
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However, that's just my experience- what are your experiences with this?<br />
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<br />http://blog.computationalcomplexity.org/2017/03/if-you-want-to-help-your-bad-students.htmlnoreply@blogger.com (GASARCH)19tag:blogger.com,1999:blog-3722233.post-3468358529935078919Thu, 16 Mar 2017 18:54:00 +00002017-03-16T14:54:33.872-04:00NP in ZPP implies PH in ZPPIf NP is in <a href="https://complexityzoo.uwaterloo.ca/Complexity_Zoo:Z#zpp">ZPP</a> is the entire polynomial-time hierarchy in ZPP? I saw this result used in an old <a href="http://cstheory.stackexchange.com/questions/430/does-exp-neq-zpp-imply-sub-exponential-simulation-of-bpp-or-np">TCS Stackexchange post</a> but I couldn't find a proof (comment if you know a reference). The proof that NP in BPP implies PH in BPP is <a href="http://blog.computationalcomplexity.org/2003/10/when-good-theorems-have-bad-proofs.html">harder than it looks</a> and NP in BQP implies PH is in BQP is <a href="http://blog.computationalcomplexity.org/2005/12/pulling-out-quantumness.html">still open</a> as far as I know.<br />
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I found a simple proof that NP in ZPP implies PH in ZPP and then an even simpler one.<br />
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Assume NP in ZPP. This implies NP in BPP so PH is also in BPP. So we need only show BPP in ZPP.<br />
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BPP is in ZPP<sup>NP</sup> follows directly by Lautemann's proof that BPP is in Σ<sub>2</sub><sup>P</sup> or by the fact that BPP is in MA is in <a href="https://complexityzoo.uwaterloo.ca/Complexity_Zoo:S#s2p">S<sub>2</sub><sup>P</sup></a> is in ZPP<sup>NP</sup>. By assumption, BPP in ZPP<sup>NP</sup> implies BPP in ZPP<sup>ZPP</sup> = ZPP.<br />
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And this is even simpler.<br />
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ZPP = RP∩co-RP in NP∩co-NP. Σ<sub>2</sub><sup>P</sup> = NP<sup>NP</sup> in NP<sup>ZPP</sup> (by assumption) in NP<sup>NP∩co-NP</sup> = NP in ZPP. You can get the higher levels of the hierarchy by an easy induction.http://blog.computationalcomplexity.org/2017/03/np-in-zpp-implies-ph-in-zpp.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-6936460121917102079Mon, 13 Mar 2017 15:49:00 +00002017-03-13T11:49:25.198-04:00Other fields of math don't prove barrier results- why do we?Before FLT was solved did some people prove theorems like:<br />
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FLT cannot be proven using techniques BLAH. This is important since all current proofs use BLAH.<br />
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I do not believe so.<br />
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Replace FLT with Goldbach's conjectures or others and I do not believe there were ever such papers.<br />
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I have sometimes seen a passing reference like `the techniques of this paper cannot get past BLAH but it was not dwelled on. The most striking example of this (and what got me to right this post) was the<br />
Erdos Distance Problem (see <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem">here</a>)--- when the result Omega( n^{ (48-14e)/(55-16e) - epsilon}) was shown I heard it said that this was as far as current techniques could push it. And then 11 years later the result Omega(n/log n) was proven. I asked around and YES the new paper DID use new techniques. But there was not the same kind of excitement I here when someone in TCS uses new techniques (e.g., IP=PSPACE used techniques that did not relativize!!!!!!!!)<br />
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With P vs NP and other results we in TCS DO prove theorems and have papers like that. I am NOT being critical-- I am curious WHY we do this and other fields don't. Some options<br />
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1) Bill is WRONG- other fields DO do this- see BLAH. Actually proof theory, and both the recursive math program and the reverse math program DID look into `does this theorem require this technique' but this was done for theorems that were already proven.<br />
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2) Bill is WRONG- we are not that obsessed with barrier results.<br />
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3) P vs NP is SO HARD that we are forced into considering why its hard. By contrast there has been progress on FLT and Goldbach over time. Rather than ponder that they NEED new techniques they went out and FOUND new techniques. Our inability to do that with P vs NP might be because it's a harder problem- though we'll know more about that once its solved (in the year 3000).<br />
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4) P vs NP is closer to logic so the notion of seeing techniques as an object worth studying is more natural to them.<br />
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What do you think?http://blog.computationalcomplexity.org/2017/03/other-fields-of-math-dont-prove-barrier.htmlnoreply@blogger.com (GASARCH)13tag:blogger.com,1999:blog-3722233.post-1457580569596784380Thu, 09 Mar 2017 21:47:00 +00002017-03-09T16:47:29.072-05:00The Beauty of ComputationLisa Randall wrote a <a href="https://www.nytimes.com/2017/03/03/books/review/reality-is-now-what-it-seems-carlo-rovelli.html">New York Times book review</a> of Carlo Rovelli's <a href="https://www.amazon.com/Reality-Not-What-Seems-Journey/dp/0735213925/ref=as_li_ss_tl?ie=UTF8&qid=1489087471&sr=8-1&keywords=reality+is+not+what+it+seems&linkCode=ll1&tag=computation09-20&linkId=2023fb24a0adf6543e28ed1a6e28abb0">Reality Is Not What It Seems</a> with some <a href="http://www.math.columbia.edu/~woit/wordpress/?p=9155">interesting</a> <a href="https://www.facebook.com/Prof.Rovelli/posts/1622834574408273">responses</a>. I want to focus on a single sentence from Randall's review.<br />
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The beauty of physics lies in its precise statements, and that is what is essential to convey.</blockquote>
I can't speak for physics but I couldn't disagree more when it comes to computation. It's nice we have formal models, like the Turing machine, for that gives computation a firm mathematical foundation. But computation, particularly a computable function, transcend the model and remain the same no matter what reasonable model of computation or programming language you wish to use. This is the Church-Turing thesis, exciting exactly because it doesn't have a formality that we can prove or disprove.<br />
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Likewise the P versus NP question remains the same under any reasonable computational model. Russell Impagliazzo goes further in his <a href="https://dx.doi.org/10.1109/SCT.1995.514853">description</a> of his world Algorithmica.<br />
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Algorithmica is the world in which P = NP or some <i>moral equivalent</i>, e.g. NP in BPP [probabilistic polynomial time]. </blockquote>
In other words the notion of easily finding checkable solutions transcends even a specifically stated mathematical question.<br />
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That's why I am not a huge fan of results that are so specific to a single model, like finding the fewest number of states for a universal Turing machine. I had an email discussion recently about the busy beaver function which I think of in general terms: a mapping from some notion of program size to program output as opposed to some <a href="http://mathworld.wolfram.com/BusyBeaver.html">precise definition</a>. I find the concept incredibly interesting and important, no one should care about the exact values of the function.<br />
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We need the formal definitions to prove theorems but we really care about the conceptual meaning.<br />
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Maybe that's what separates us from the physicists. They want precise definitions to capture their conceptual ideas. We want conceptual ideas that transcend formal definitions.http://blog.computationalcomplexity.org/2017/03/the-beauty-of-computation.htmlnoreply@blogger.com (Lance Fortnow)4tag:blogger.com,1999:blog-3722233.post-5945310957520182605Mon, 06 Mar 2017 18:01:00 +00002017-03-06T13:01:07.767-05:00why are regular expressions defined the way they are<br />
BILL: The best way to prove closure properties of regular languages is to first prove the equiv of DFA's, NDFA's and Reg Expressions. Then, if you want to prove a closure property, choose the definition of regular that makes it easiest. For example, to prove Reg Langs closed under intersection I would use DFA's, NOT Reg Expressions.<br />
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STUDENT: I thought reg expressions were<br />
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a) finite sets<br />
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b) if alpha and beta are reg exp then so are alpha UNION beta, alpha INTERSECT beta, alpha CONCAT beta and alpha*<br />
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BILL: No. Regular expressions are defined just using UNION, CONCAT, and *.<br />
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STUDENT: Why? Had the defined it my way then closure under INTERSECTION would be easier. For that matter toss in COMPLIMENTATION and you're get that easily also.<br />
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BILL: First off, thats not quite right. You compliment a DFA by saying how lovely its states are. I think you mean complement. Second off, GOOD question!- Why are Reg Expressions defined the way they are. I"ll try to look that up and if I can't find anything I'll blog about it.<br />
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STUDENT: When will you blog about it?<br />
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BILL: I just did. Now, let me ask the question more directly:<br />
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The definition of Reg Exp is essentially closure under UNION, CONCAT, STAR. Why not other things? There are very broadly three possibilities:<br />
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a) Historical Accident.<br />
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b) Some good math or CS reason for it.<br />
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c) Something else I haven't thought of.<br />
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I hope its (b). Moreover, I hope one of my readers knows and can enlighten me and the other readers.<br />
<br />http://blog.computationalcomplexity.org/2017/03/why-are-regular-expressions-defined-way.htmlnoreply@blogger.com (GASARCH)11tag:blogger.com,1999:blog-3722233.post-4605708902834011128Thu, 02 Mar 2017 13:02:00 +00002017-03-02T08:02:38.362-05:00International ScienceI did some counting and the 35 academic faculty members in the Georgia Tech School of Computer Science come from 14 different countries. My co-authors come from at least 20 different nations. My 10 successful PhD students hail from 7 different countries. I have benefited immensely from global collaborations thanks to relatively open borders and communication during most of my academic career and I am hardly the only academic who has done so.<br />
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I'm old enough to remember the days of the cold war where travel between East and West was quite difficult. We had considerable duplication of effort--many important theorems were proven independently on both sides of the iron curtain but even worse ideas took a long time to permeate from one side to the other. We could not easily build on each other's work. Science progressed slower as a result. Pushing back the boundaries of science is not a zero-sum game, quite the opposite--we can only grow knowledge. We grow that knowledge much faster working together.<br />
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As the United States and other countries take on a more nationalistic point of view, we'll see fewer people travel, fewer people willing or even able to spend significant parts of their career in other countries. We will (hopefully) continue to have an open Internet so information will still flow but nothing can replace the focus of face-to-face collaboration to share ideas and create new ones.<br />
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The real loss for America will be an invisible one: the students who won't be coming here to study and later become our professors, scientists and colleagues, to make our universities, industries and society stronger. Sad.http://blog.computationalcomplexity.org/2017/03/international-science.htmlnoreply@blogger.com (Lance Fortnow)9tag:blogger.com,1999:blog-3722233.post-9008528016432895661Mon, 27 Feb 2017 03:44:00 +00002017-03-01T00:25:17.064-05:00Should we learn from the Masters or the Pupils (Sequel)<div>
A while back I had a blog entry <a href="http://blog.computationalcomplexity.org/search?q=Masters">Should we learn from the Masters of the Pupils?</a> The Masters may have more insights but he Pupils may have a better view aided by a modern viewpoint.</div>
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Sometimes the Masters are in a different language or not in the modern style but you still want to know what they did and why. As I blogged about earlier (See <a href="http://blog.computationalcomplexity.org/search?q=Hilbert">here</a>) Villarino/Gasarch/Regan have a paper which explains Hilbert's Proof of Hilbert's Irreducibility Theorem (<a href="https://arxiv.org/abs/1611.06303">see</a>) Tao has a paper on Szemeredi's Proof of Szemeredi's Theorem (on Tao's webpage: <a href="http://www.math.ucla.edu/~tao/preprints/acnt.html">here</a>). Villarino has a paper on Merten's Proof of Merten's Theorem (<a href="https://arxiv.org/abs/math/0504289">here</a>).</div>
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Mark Villarino read that blog entry (good to know someone did!) and then presented me with MANY examples where the MASTER is worth reading, which I present to you. For all of them reading a well written exposition of what the Master did would also be good (as good? better?) if such exists.</div>
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Here is his letter with a few of my comments.</div>
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I would suggest the following examples where the original teaches and illuminates more than the modern slick version:</div>
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1. Euclid's proof of the Pythagorean Theorem (and its converse). Indeed, once you understand the diagram, the proof is immediate and beautiful. See <a href="http://www.cut-the-knot.org/pythagoras/Proof1.shtml">here</a>.</div>
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2. Gauss' first proof (by induction) of quadratic reciprocity. If you REALLY read it, you see how Gauss was led to the proof by numerous specific examples and it is quite natural. It is a marvelous example of how numerical examples inspired the structure of the induction proof. (BILL COMMENT: Here is a Masters Thesis in Math that has the proof and lots of context and other proofs of QR: <a href="http://scholarworks.lib.csusb.edu/cgi/viewcontent.cgi?article=1324&context=etd">here</a>)</div>
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3. Gauss' first proof of the fundamental theorem of algebra. The real and imaginary parts of the polynomial must vanish simultaneously. However the graph of each is a curve in the plane, and so the two curves must intersect at some point. Gauss explicitly finds a circle which contains the parts of the two curves which intersect in the roots of the polynomial. The proof of the existence of a point of intersection is quite clever and natural, although moderns might quibble. In an appendix he gives a numerical example (BILL COMMENT- Sketch of the first proof of FTOA that I ever saw: First show that the complex numbers C and the punctured plane C- {(0,0)} have different fundamental groups (The fund group of C is trivial, the fund group of C-{(0,0)} is Z,the integers.) Hence there can't be an X-morphism from C to C-{(0,0)} (I forget which X it is). If there is a poly p in C[x] with no roots in C then the map x --> 1/p(x) is an X-morphism. Contradiction. Slick but not clear what it has to to with polynomials. A far cry from the motivated proof by Gauss.)<br />
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4. Abel's proof, in Crelle's Journal, of the impossibility of solving a quintic equation by radicals. Abel explores the properties that a "formula" for the root any algebraic equation must have, for example that if you replace any of its radicals by a conjugate radical, the new formula must also identically satisfy the equation, in order to deduce that the formula cannot exist Yes, it has a few correctable errors, but the idea is quite natural. (BILL's COMMENT- proof- sounds easier than what I learned, and more natural. There is an exposition in English <a href="http://www.math.caltech.edu/~jimlb/abel.pdf">here</a>. I have to read this since I became a math major just to find out why there is no quintic equation.)</div>
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5. Jordan's proof of the Jordan curve theorem. His idea is to go from the theorem for polygons to then approximate the curve by a polygon and carry the proof over to the curve by a suitable limiting process. See <a href="http://mizar.org/trybulec65/4.pdf">here</a> for a paper on Jordan's proof of the Jordan Curve theorem.</div>
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6. Godel's 1948 paper on his rotating universe solution to the Einstein Field Equations. Although his universe doesn't allow the red-shift, it DOES allow time travel! The paper is elegant, easy to read, and should be read (in my opinion) by any mathematics student. (Added later- for the paper see <a href="http://fuchs-braun.com/media/91ac4f6879c27351ffff8191fffffff0.pdf">here</a>)</div>
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7. Einstein's two papers on special/general relativity. There are english translations. They are both elegantly written and are much better than the later "simplifications" by text-book writers. I was amazed at how natural his ideas are and how clearly and simply they are presented in the papers. English Translation <a href="https://archive.org/details/principleofrelat00eins">here</a></div>
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8. Lagrange's Analytical Mechanics. There is now an english translation. What can I say? It is beautiful. Available in English <a href="https://archive.org/details/springer_10.1007-978-94-015-8903-1">here</a>.</div>
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9. I add "Merten's proof of Merten's theorem" to the list of natural instructive original proofs. His strategy is quite natural and the details are analytical fireworks. (BILL COMMENT- as mentioned above there is an exposition in English of Merten's proof.)</div>
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I could go on, but these are some standouts.<br />
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BILL COMMENT: So, readers, feel free to ad to this list!<br />
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http://blog.computationalcomplexity.org/2017/02/should-we-learn-from-masters-or-pupils.htmlnoreply@blogger.com (GASARCH)6tag:blogger.com,1999:blog-3722233.post-356304281567214878Thu, 23 Feb 2017 12:07:00 +00002017-02-23T07:07:56.130-05:00Ken Arrow and Oscars VotingKenneth Arrow, the Nobel Prize winning economist known for his work on social choice and general equilibrium, <a href="https://www.nytimes.com/2017/02/21/business/economy/kenneth-arrow-dead-nobel-laureate-in-economics.html">passed away Tuesday</a> at the age of 95.<br />
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I can't cover Arrow's broad influential work in this blog post even if I were an economist but I would like to talk about Ken Arrow's perhaps best known work, his <a href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem">impossibility theorem</a> for voting schemes. If you have at least three candidates, there is no perfect voting method.<br />
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Suppose a group of voters give their full rankings of three candidates, say "La La Land", "Moonlight" and "Manchester by the Sea" and you have some mechanism that aggregates these votes and chooses a winner.<br />
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Now suppose we want a mechanism to have two fairness properties (for every pair of movies):<br />
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<li>If every voter prefers "Moonlight" to "La La Land" then the winner should not be "La La Land". </li>
<li>If the winner is "Moonlight" and some voters change their ordering between "La La Land" and "Manchester by the Sea" then "Moonlight" is still the winner (independence of irrelevant alternatives).</li>
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Here's one mechanism that fills these properties: We throw out every ballot except Emma Watson's and whichever movie she chooses wins.<br />
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Arrow shows these are the only mechanisms that fulfill the properties: There is no non-dictatorial voting system that has the fairness properties above.<br />
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Most <a href="https://en.wikipedia.org/wiki/Arrow's_impossibility_theorem#Informal_proof">proofs</a> of Arrow's theorem are combinatorial in nature. In 2002 Gil Kalai <a href="http://dx.doi.org/10.1016/S0196-8858(02)00023-4">gave a clever proof</a> based on Boolean Fourier analysis. Noam Nisan goes over this proof in a 2009 <a href="https://agtb.wordpress.com/2009/03/31/from-arrow-to-fourier/">blog post</a>.<br />
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Arrow's theorem that no system is perfect doesn't mean that some systems aren't better than others. The Oscars use a reasonably good system known as Single Transferable Voting. Here is a short version updated from a <a href="http://abcnews.go.com/Entertainment/academy-awards-voting-process-works-steps/story?id=37067797">2016 article</a>.<br />
<blockquote class="tr_bq">
For the past 83 years, the accounting firm PricewaterhouseCoopers has been responsible for tallying the votes, and again this year partners Martha Ruiz and Brian Cullinan head up the operation. The process of counting votes for Best Picture isn't as simple as one might think. According to Cullinan, each voter is asked to rank the nine nominated films 1-9, one being their top choice. After determining which film garnered the least number of votes, PWC employees take that title out of contention and look to see which movie each of those voters selected as their second favorite. That redistribution process continues until there are only two films remaining. The one with the biggest pile wins. "It doesn’t necessarily mean that who has the most number one votes from the beginning is ensured they win," he added. "It’s not necessarily the case, because going through this process of preferential voting, it could be that the one who started in the lead, doesn’t finish in the lead."</blockquote>
Another article <a href="http://www.thewrap.com/how-oscar-best-picture-winner-chosen-moonlight-la-la-land/">explicitly asks</a> about strategic voting.<br />
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<b>So if you’re a big fan of “Moonlight” but you’re scared that “La La Land” could win, you can help your cause by ranking “Moonlight” first and “La La Land” ninth, right?</b></blockquote>
<blockquote class="tr_bq">
Wrong. That won’t do a damn thing to help your cause. Once you rank “Moonlight” first, your vote will go in the “Moonlight” stack and stay there unless “Moonlight” is eliminated from contention. Nothing else on your ballot matters as long as your film still has a chance to win. There is absolutely no strategic reason to rank your film’s biggest rival last, unless you honestly think it’s the worst of the nominees.</blockquote>
Arrow's theorem says there must be a scenario where you can act strategically. It might make sense for this fan to put "Fences" as their first choice to potentially knock out "La La Land" in an early round. A similar situation <a href="https://en.wikipedia.org/wiki/Bids_for_the_2016_Summer_Olympics#Election">knocked out</a> Chicago from hosting the 2016 Olympics.<br />
<br />
Maybe the Oscars should just let Emma Watson choose the winner.http://blog.computationalcomplexity.org/2017/02/ken-arrow-and-oscars-voting.htmlnoreply@blogger.com (Lance Fortnow)5tag:blogger.com,1999:blog-3722233.post-8949775073949839802Mon, 20 Feb 2017 04:25:00 +00002017-02-19T23:25:46.911-05:00The benefits of Recreational MathematicsWhy study Recreational Mathematics?<br />
<br />
Why do recreational Mathematics?<br />
<br />
1) The line between recreational and serious mathematics is thin. Some of the problems in so-called recreational math are harder than they look.<br />
<br />
2) Inspiring. Both Lance and I were inspired by books by Martin Gardner, Ray Smullyan, Brian Hayes, and others.<br />
<br />
3) Pedagogical: Understanding Godel's Inc. theorem via the Liar's paradox (Ray S has popularized that approach) is a nice way to teach the theorem to the layperson (and even to non-laypeople).<br />
<br />
4) Rec math can be the starting point for so-called serious math. The Konigsberg bridge problem was the starting point for graph theory, The fault diagnosis problem is a generalization of the Truth Tellers and Normals Problem. See <a href="https://www.math.iupui.edu/~bleher/Papers/1983_On_a_logical_problem.pdf">here</a> for a nice paper by Blecher on the ``recreational'' problem of given N people of which over half are truth tellers and the rest are normals, asking questions of the type ``is that guy a normal'' to determine whose who. See <a href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/normals.pdf">here</a> for my writeup of the algorithm for a slightly more general problem. See William Hurwoods Thesis: <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.6597&rep=rep1&type=pdf">here</a> for a review of the Fault Diagnosis Literature which includes Blecher's paper.<br />
<br />
I am sure there are many other examples and I invite the readers to write of them in the comments.<br />
<br />
5) Rec math can be used to inspire HS students who don't quite have enough background to do so-called serious mathematics.<br />
<br />
This post is a bit odd since I cannot imagine a serious counter-argument; however, if you disagree, leave an intelligent thoughtful comment with a contrary point of view.http://blog.computationalcomplexity.org/2017/02/the-benefits-of-recreational-mathematics.htmlnoreply@blogger.com (GASARCH)5tag:blogger.com,1999:blog-3722233.post-93268597532832374Thu, 16 Feb 2017 12:51:00 +00002017-02-16T07:54:31.609-05:00Liberatus Wins at Poker<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://www.cmu.edu/news/stories/archives/2017/january/images/poker_853x480-min.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="180" src="https://www.cmu.edu/news/stories/archives/2017/january/images/poker_853x480-min.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Tuomas Sandholm (center) and Ph.D. student Noam Brown (via <a href="https://www.cmu.edu/news/stories/archives/2017/january/AI-beats-poker-pros.html">CMU</a>)</td></tr>
</tbody></table>
Congratulations to Liberatus the <a href="https://www.wired.com/2017/02/libratus/">new poker champ</a>. Liberatus, an AI program, beat several top-ranked players in heads-up (two player) no-limit Texas hold 'em.<br />
<br />
For those unfamiliar with <a href="https://en.wikipedia.org/wiki/Texas_hold_%27em">Texas hold 'em</a><br />
<blockquote class="tr_bq">
Two cards, known as the hole cards or hold cards, are dealt face down to each player, and then five community cards are dealt face up in three stages. The stages consist of a series of three cards ("the flop"), later an additional single card ("the turn") and a final card ("the river"). Each player seeks the <a href="https://en.wikipedia.org/wiki/List_of_poker_hands">best five card poker hand</a> from the combination of the community cards and their own hole cards. Players have betting options to check, call, raise or fold. Rounds of betting take place before the flop is dealt, and after each subsequent deal.</blockquote>
Unlike the computers that defeated the best humans in chess, Jeopardy and go, Liberatus comes directly from academia, from Tuomas Sandholm and his student Noam Brown at Carnegie-Mellon.<br />
<br />
Unlike chess and go, poker is a game of incomplete information in many forms.<br />
<ul>
<li>Information both players have: the community cards already played.</li>
<li>Information only one player has: the hole card</li>
<li>Information neither player has: the community cards yet to be played.</li>
</ul>
<div>
Betting in poker plays the primary role of raising the stakes but betting can also signal what hole cards you have. Players can bluff (betting large amounts without a corresponding strong hand), trying to cause other players to misread the signal. There is no perfect play in poker, just a mixed equilibrium though we still don't know how to compute the equilibrium and even if we could we might deviate from the equilibrium to gain an advantage. Deviating also make you vulnerable.</div>
<div>
<br /></div>
<div>
All of this makes poker a far more complicated game for computers to tackle. But through persistence and new tools in machine learning, Sandholm and Brown have found success.</div>
<div>
<br /></div>
<div>
If history holds up, it won't be long before we have champion-caliber poker apps on our phones. Will we see cheating like has been <a href="http://www.uschess.org/content/view/12677/763">happening in chess</a>? Will online poker sites just disappear?</div>
<div>
<br /></div>
<div>
What is the next great game to fall to computers? I'm guessing NASCAR. </div>
http://blog.computationalcomplexity.org/2017/02/liberatus-wins-at-poker.htmlnoreply@blogger.com (Lance Fortnow)6tag:blogger.com,1999:blog-3722233.post-2012957173445019786Mon, 13 Feb 2017 18:45:00 +00002017-02-18T14:19:26.067-05:00Raymond Smullyan: Logician, Recreational math writer, Philosopher, passed awayRaymond Smullyan was born on May 25 1919 and passed away recently at the age of 97. He was a logician (PhD from Princeton under Alonzo Church in 1959) who did serious, recreational, and philosophical work. I doubt he invented the truth-teller/liar/normals and knight/knave/Knormal problems, but he popularized them and (I suspect) pushed them further than anyone before him.<br />
<br />
He was active all of his life:<br />
<br />
His last book on Logic Puzzles, <i>The Magic Garden of George B and other Logic Puzzle</i>s was published in 2015. Wikipedia lists 14 books on logical puzzles, from 1978 untl 2015.<br />
<br />
His last book classified (on Wikipedia) as Philosophy/Memoir,<i> A Mixed Bag: Jokes, Riddles</i>,<i> Puzzles</i>, <i>and Memorabilia </i>was published in 2016. Wikipedia lists 8 books in this category, from 1977 until 2016.<br />
<br />
His last Academica book,<i> A beginners further guide to mathematical logic </i>was published in 2016. (It was a sequel to his 2014 <i>A beginners guide to mathematical logic</i>.) Wikipedia lists 8 books in this category, from 1961 to 2016.<br />
<br />
<br />
<b>Recreational Work:</b><br />
<br />
His recreational work was of the Knights/Knaves/Knormals/Sane/Insane/ variety.a<br />
<br />
Knights always tell the truth.<br />
<br />
Knaves always lie,<br />
<br />
Knormals may tell the truth or lie.<br />
<br />
Insane people only believe false things,<br />
<br />
Sane people only believe true things.<br />
<br />
He also added a complication: a species that says ALPHA and BETA for YES and NO but<br />
you don't know which of ALPHA, BETA means YES and which one means NO.<br />
<br />
<br />
Note that a truth teller Insane Knight will answer YES to 1+1=3.<br />
<br />
He invented (discovered?) the so called <a href="https://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever">hardest logic puzzle ever</a>. He wrote many books on recreational math. We mention four of them that show the line between recreational and serious mathematics is thin.<br />
<div>
<div>
<br /></div>
<div>
<i>To Mock a Mockingbird</i>. This book has logic puzzles based on combinatory logic. Is that really recreational?</div>
<div>
<br /></div>
<div>
<i>Forever Undecided.</i> This book introduces the layperson to Godel's theorem.</div>
<div>
<br /></div>
<div>
<i>Logical Labyrinth</i>s. This is a textbook for a serious logic course that uses puzzles to teach somewhat serious logic. It was published in 2009 when he was only 89 years old.</div>
<div>
<br /></div>
<div>
A Personal Note: I read the following, from his</div>
<div>
<i>The Lady or the Tiger, </i>when I was in high school, and I still don't know the answer!:</div>
<div>
<br /></div>
<div>
<i>My brother told me he would fool me on April Fools Day. I lay awake that night wondering how he would fool me. All day I was worried how he would fool me. At midnight I asked him <i>Hey, you said you would fool me but you didn't He replied April Fools!. To this day I don't know if I was fooled or not.</i></i></div>
</div>
<div>
<i><br />
</i></div>
<div>
<i><br />
</i></div>
<div>
<div>
<i><b>Serious Math Work</b>. </i>His serious work included the Double Recursion Theorem. (you can write two programs that know both their indices and each others indices) and other theorem in logic. (ADDED LATER: Lipton and Regan have a blog post with lots of great information about Ray S's serious math work <a href="https://rjlipton.wordpress.com/">here</a>.)</div>
<div>
<br /></div>
<div>
<b>Philosophy.</b> I'm not qualified to comment on this; however, it looks like he did incorporate his knowledge of logic.</div>
<div>
<br /></div>
<div>
Looking over his books and these points it seems odd to classify his books as the recreational books had some serious logic in them, and the academic books had some math that a layperson could understand<br />
<br />
I think its rarer now to do both recreational and serious mathematics, though I welcome intelligent debate on this point.<br />
<br />
Before he died, was he the oldest living mathematician? No- Richard Guy is 100 years old. wikipedia claims that Guy is still an active math Guy. Is he the oldest living mathematican? The oldest living active mathematician? It was hard to find out on the web so I ask you.</div>
</div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<br /></div>
<div>
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http://blog.computationalcomplexity.org/2017/02/raymond-smullyan-was-born-on-may-25.htmlnoreply@blogger.com (GASARCH)11tag:blogger.com,1999:blog-3722233.post-301310271299156009Thu, 09 Feb 2017 12:11:00 +00002017-02-09T10:51:44.800-05:00The Dichotomy ConjectureArash Rafiey, Jeff Kinne and Tomás Feder settle the Feder-Vardi dichotomy conjecture in their paper <a href="https://arxiv.org/abs/1701.02409">Dichotomy for Digraph Homomorphism Problems</a>. Jeff Kinne is my academic grandchild--how quickly they grow up.<br />
<br />
Keep in mind the usual caveat that this work has not yet been fully refereed and vetted by the community, though there is no reason to think it won't be (though some skepticism in the <a href="http://blog.computationalcomplexity.org/2017/02/the-dichotomy-conjecture.html#comments">comments</a>).<br />
<br />
A homomorphism from a graph G = (V,E) to H=(V',E') is a function f:V→V' such that if (u,v) is in E then (f(u),f(v)) is in E'. For a fixed graph H, define L(H) as the set of graphs G such that there is a homomorphism from G to H.<br />
<br />
If H is just a single edge then L(H) is the set of bipartite graphs. If H is a triangle then L(H) is the set of 3-colorable graphs. If H has a self-loop then L(H) is the set of all graphs.<br />
<br />
L(H) is always in NP by guessing the homomorphism. In 1990 Pavol Hell and Jaroslav Nešetřil <a href="http://dx.doi.org/10.1016/0095-8956(90)90132-J">showed</a> the following dichotomy result: If H is bipartite or has a self-loop then L(H) is computable in polynomial-time, otherwise L(H) is NP-complete. There are no undirected graphs H such that L(H) is not in P or NP-complete.<br />
<br />
In 1998 Tomás Feder and Moshe Vardi <a href="http://dx.doi.org/10.1137/S0097539794266766">conjectured</a> that even for all directed graphs H, L(H) is either in P or NP-complete. Rafiey, Kinney and Feder settle the conjecture by showing a polynomial-time algorithm for a certain class of digraphs H.<br />
<br />
Details in the <a href="https://arxiv.org/abs/1701.02409">paper</a>. The authors recommend first watching their <a href="https://youtu.be/PBX51Qv5wtw">videos</a> <a href="https://youtu.be/vD68km24Rh0">below</a> though I would suggest reading at least the introduction of the paper before tackling the videos.<br />
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<br />http://blog.computationalcomplexity.org/2017/02/the-dichotomy-conjecture.htmlnoreply@blogger.com (Lance Fortnow)17tag:blogger.com,1999:blog-3722233.post-6017135610692729306Mon, 06 Feb 2017 04:44:00 +00002017-02-06T09:25:28.790-05:00The Hardness of Reals Hierarchy In my last post (<a href="http://blog.computationalcomplexity.org/2017/01/what-was-first-result-in-complexity.html">here</a>) I defined the following hierarchy (which I am sure is not original- if someone has a source please leave a comment on it)<br />
<br />
Z_d[x] is the set of polys of degree d over Z (the integers)<br />
<br />
ALG_d is the set of roots of these polys.<br />
<br />
ALG_1 = Q (The rationals)<br />
<br />
Given a real alpha I think of its complexity as being the least d such that alpha is in ALG_d. This is perhaps a hierarchy of hardness of reals (though there are an uncountable number of reals that are NOT in any ALG_d.)<br />
<br />
I then said something like<br />
<br />
Clearly<br />
<br />
ALG_1 PROPER SUBSET ALG_2 PROPER SUBSET ALG_3 etc.<br />
<br />
But is that obvious? <br />
<br />
``Clearly'' 2^{1/d} is not in ALG_{d-1}. And this is not a trick- YES 2^{1/d} is NOT in ALG_{d-1} But how would you prove that. My first thought was `I'll just use Galois theory' And I am sure I could dust off my Galois theory notes (I had the course in 1978 so the notes are VERY dusty) and prove it. But is there an elementary proof. A proof a High School Student could understand.<br />
<br />
How to find out? Ask a bright high school student to prove it! Actually I asked a Freshman Math Major who is very good, Erik Metz. I thought he would find a proof, I would post about it asking if it was known, and I would get comments telling me that OF COURSE its known but not give me a reference (have I become cynical from years of blogging. Yes.)<br />
<br />
But that's not what happened. Erik had a book <i>Problems from the Book </i>by Dospinescu and Andreescu that has in it a lovely elementary proof (it uses Linear Algebra) that 2^{1/d} is NOT in ALG_{d-1}.<br />
<br />
Hence the hardness of reals hierarchy is proper. For a write up of just the proof that<br />
7^{1/3} is not in ALG_2 (which has most of the ideas) see <a href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/hardreals.pdf">this writeup</a>http://blog.computationalcomplexity.org/2017/02/the-hardness-of-reals-hierarchy.htmlnoreply@blogger.com (GASARCH)7tag:blogger.com,1999:blog-3722233.post-5654741233125813331Thu, 02 Feb 2017 19:15:00 +00002017-02-02T14:15:34.851-05:00We Are All Iranians<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/--xxGCHPAfmg/WJOEdzGG-BI/AAAAAAABZWY/Oui3E65mVQEBnO3JV_nBtagFpdOO50ObwCKgB/s1600/00001IMG_00001_BURST20170202114143.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://2.bp.blogspot.com/--xxGCHPAfmg/WJOEdzGG-BI/AAAAAAABZWY/Oui3E65mVQEBnO3JV_nBtagFpdOO50ObwCKgB/s400/00001IMG_00001_BURST20170202114143.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A <a href="https://www.facebook.com/events/615927695273045/?active_tab=discussion">solidarity rally</a> held at Georgia Tech today</td></tr>
</tbody></table>
There are ten Iranian members of my department, the School of Computer Science at Georgia Tech, all of whom face a very uncertain future in America. Luckily none of them were outside the US when the executive order was signed last Friday.<br />
<br />
We have nine Iranian Ph.D. students. It was already difficult for them to leave the US and return and with the new executive order essentially impossible, even for family emergencies. One expressed disappointment “Why did we even bother to come to the United States to study?”<br />
<br />
We also have a young Iranian professor, a very successful computer architect and my first hire as chair, in the final stage before getting his green card now on hold. If things don’t change he and his wife may be forced to leave the country they now call home. That would be a huge loss for Georgia Tech and the United States.<br />
<br />
This is not the America I believe in.http://blog.computationalcomplexity.org/2017/02/we-are-all-iranians.htmlnoreply@blogger.com (Lance Fortnow)9tag:blogger.com,1999:blog-3722233.post-8074027353717692435Mon, 30 Jan 2017 04:09:00 +00002017-01-29T23:09:11.036-05:00What was the first result in complexity theory?Let Z_d[x] be the set of polynomials of degree d over the integers.<br />
<br />
Let ALG_d be the set of roots of polys in Z_d.<br />
<br />
One can easily show that<br />
<br />
ALG_1 is a proper subset ALG_2is a proper subset ...<br />
<br />
and that there are numbers not in any of the ALG_i (by a countability argument).<br />
<br />
I began my ugrad automata theory course with this theorem (also a good review of countability- I found out, NOT to my surprise, that they never really understood it as freshman taking Discrete Math, even the good students.)<br />
<br />
I presented this as the<i> first theorem in complexity.</i><br />
<i><br /></i>
But is it? I suspect that the question <i>What was the first result in complexity?</i><br />
<i><br /></i>
has no real answer, but there are thoughts:<br />
<br />
Complexity theory is about proving that you can't do BLAH with resource bound BLAHBLAH.. We will distinguish it from Computability theory by insisting that the things we want to compute are computable; however, if someone else wants to argue that (say)<i> HALT is undecidable </i>is the first result in complexity, I would not agree but I would not argue against it.<br />
<br />
Here are other candidates:<br />
<br />
1) sqrt(2) is irrational. This could be considered a result in descriptive complexity theory.<br />
<br />
2) The number of primes is infinite. If you view `finite' as a complexity class then this takes PRIMES out of that class.<br />
<br />
3) You cannot, with ruler and compass, trisect an angle, square a circle, or double a cube. This seems very close to the mark--- one can view ruler-and-compass as a well defined model of computation and these are lower bounds in that model.<br />
<br />
4) There is no quintic equation. Also close to the mark as this is a well defined lower bound.<br />
<br />
5) In the early 60s the definition of P (Cobram) and of DTIME, etc (Hartmanis-Stearns). The result I would point to is the time hiearchy theorem. While these results are much later than those above, they are also far closer to our current notion of complexity.<br />
<br />
6) I'm not sure which paper to point to, but Knuth's observation that one can analyse algorithms without running them. This is more algorithms than complexity, but at this level the distinction may be minor.<br />
<br />
7) Cook-Levin Theorem. Probably not the first theorem in Complexity, but certainly a big turning point.<br />
<br />
I urge you to comment with other candidates!<br />
<br />
<br />http://blog.computationalcomplexity.org/2017/01/what-was-first-result-in-complexity.htmlnoreply@blogger.com (GASARCH)9tag:blogger.com,1999:blog-3722233.post-7533256557125815169Fri, 27 Jan 2017 01:47:00 +00002017-01-26T20:47:26.893-05:0060 years of Eric and Mike<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-_eVxHDeClSA/WIqmJdBx1BI/AAAAAAABZTM/rCeoAt6gSvsgiL08Q3gl8CrLPz8NEbFFQCLcB/s1600/mike.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-_eVxHDeClSA/WIqmJdBx1BI/AAAAAAABZTM/rCeoAt6gSvsgiL08Q3gl8CrLPz8NEbFFQCLcB/s200/mike.jpg" width="163" /></a></div>
<a href="https://1.bp.blogspot.com/-lf0ZbUa9Wic/WIqmGCpw08I/AAAAAAABZTI/WelazE_HdYweCEpJvo953teOmcXcuczuQCLcB/s1600/eric.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://1.bp.blogspot.com/-lf0ZbUa9Wic/WIqmGCpw08I/AAAAAAABZTI/WelazE_HdYweCEpJvo953teOmcXcuczuQCLcB/s200/eric.jpg" width="136" /></a>As I checked in at the Holiday Inn in New Brunswick Wednesday night, they asked me if I had stayed there before. I said it has been a while and they looked in their computer: October 2007. I was last at DIMACS for the <a href="http://dimacs.rutgers.edu/Workshops/EconomicTheory/">Workshop on the Boundary between Economic Theory and Computer Science</a>, the closing workshop of the <a href="http://dimacs.rutgers.edu/SpecialYears/2004_CSEC/">Special Focus on Computation and the Socio-Economic Sciences</a> which Rakesh Vohra and I had organized. DIMACS, the Center for Discrete Math and Computer Science at Rutgers, started as an NSF center and I went often, even serving on the executive committee when I worked at NEC in the early 2000's. So an odd feeling to be back after ten years.<br />
<br />
But back for a good reason, the DIMACS <a href="http://dimacs.rutgers.edu/Workshops/EMC/">Workshop on E+M=C<sup>2</sup></a>, the joint 60th birthday celebration for Rutgers Professors Eric Allender and Michael Saks. A great turnout of Eric and Mike's former colleagues and students. I've known both Eric and Mike for many years but it is Eric I've had the closest connections to. Eric and I share many research interests, especially in Kolmogorov complexity. I took over from Eric as chair of the Computational Complexity conference and Eric took over from me as editor-in-chief of ACM Transactions on Computation Theory. Combined we've attended 61 of the 31 complexity conferences so far (I missed 2012 in Porto) and many Dagstuhl workshops and other meetings together on four continents. But we've oddly enough never co-authored.<br />
<br />
Congrats to Eric and Mike and their careers worth savoring.http://blog.computationalcomplexity.org/2017/01/60-years-of-eric-and-mike.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-3194218073277459202Mon, 23 Jan 2017 01:54:00 +00002017-01-22T20:54:43.919-05:00My once-every-four-years Presidential Quiz/how should quizes work in the e-era?Every four years I post a PRESIDENTIAL QUIZ which I must update based on new information since we have a new prez and veep. The questions are here:<a href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/prezquiz.pdf">here</a>.<br />
<br />
I will post a link to the answers next week. The answers will also contain more information if interest beyond the question, so the answers are worth reading whether or not you get it right.<br />
<br />
The quiz is 43 questions (hmmm- that is long for a quiz)<br />
Start with 14 points, and the questions are 2 points each. No penalty for wrong answers.<br />
<br />
Why take a quiz when you can find most (maybe all) answers on the web?<br />
<br />
Well- there are two ways to view this:<br />
<br />
1) Take the quiz without the aid of the web and give yourself 3 hours. You are on your scouts honor. Doesn't matter much since you truly are just testing yourself.<br />
<br />
2) USE the web. This is NOT cheating. But TIME how long it takes you. Stop when you want but your score is (120 times the number you got right)/(number of minutes it took you) + 14. So if using the web you get them all right in an hour, you get a 100. There may be other ways to do this.<br />
<br />
<br />
REQUEST: Do not post answers to quiz questions since that may deprive others the joy.<br />
You CAN (and I would like this) post how well you did using either criteria (1) or (2).<br />
<br />
Is this the future of trivia contests? Rather than ban the use of the web embrace it!<br />
<br />
bill g.<br />
<a href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/prezquiz.pdf"></a>http://blog.computationalcomplexity.org/2017/01/my-once-every-four-years-presidential.htmlnoreply@blogger.com (GASARCH)0tag:blogger.com,1999:blog-3722233.post-8508515779454734672Thu, 19 Jan 2017 13:25:00 +00002017-01-19T08:25:17.985-05:00Infinite Series and Markov ChainsThere's a wonderful new series of math videos <a href="https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g/">PBS Infinite Series</a> hosted by Cornell Math Phd student Kelsey Houston-Edwards. Check out this latest <a href="https://youtu.be/63HHmjlh794">video</a> on Markov Chains.<br />
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She gives an amazingly clear description of why, on a random walk on an undirected graphs, the stationary distribution puts probability on each node proportional to its degree.</div>
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Houston-Edwards also relies without proof on the following fact: The expected time to start and return to a vertex v is 1/p where p is the probability given to v in the stationary distribution. Why should that be true?</div>
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Didn't seem so obvious to me, so I <a href="http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-MCII.pdf">looked it up</a>. Here is an informal proof:</div>
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Let V1 be the random variable that is the expected length of time to get from v back to v. Let's take that walk again and let V2 be the expected length of time to get from v back to v the second time, and so on. Let An=V1+V2+..+Vn, the random variable representing the number of transitions taken to return to v n times. In a Markov chain each Vi is distributed the same so E(An) = n E(V1). </div>
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As n gets large the Markov chain approaches the stationary distribution and will be in state v about a p fraction of the time. After E(An) transitions we should return to v about p E(An) times, where we actually return n times. So we have p E(An) approaches n, or p n E(V1) approaches n and the only way this could happen is if E(V1)=1/p.</div>
http://blog.computationalcomplexity.org/2017/01/infinite-series-and-markov-chains.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-1536965085783366798Mon, 16 Jan 2017 03:24:00 +00002017-01-15T22:25:55.708-05:00My REU program/REU's in general/Flyers? Why do some Transcripts...I run an REU program (Research Experience for Undergraduates) and I would normally urge you to urge undergrads who would benefit to apply to it and present both this link: <a href="http://www.cs.umd.edu/projects/reucaar/index.html">here</a> and this flyer: <a href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/caarflyers2017.pdf">here</a>.<br />
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I just did that. But I want to talk about REU programs, not just mine. A few points<br />
which can be construed as advise- though its more questions and what I do.<br />
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1) Do flyers still have any effect? If there is a bulletin board in a dept that is known for where to look for announcements of summer opps, then YES. Otherwise--- not sure. when I asked this question to a professor I emailed the flyer to she said that at her school they stick flyers in the bathroom stalls where they are sure to get a captive audience.<br />
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2) Should you visit schools directly? I have done this; however, most of the applicants find out about it from the NSF REU website.<br />
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3) Should students pick projects ahead of time or in the first week? When students apply they list a set of projects they want to work on (unranked but the Statement of Purpose can say which one(s) they REALLY want) so we can start on Day 1. Some of the mentors are in contact with students ahead of time. Other programs have them decide the first week. There are PROS and CONS to both.<br />
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<br />
4) How to do admissions? I do them myself since there are few enough of them (about 180 for 10 slots- though see next item) and since there are so many criteria to balance I'd rather not get into arguments with other committee members. I will sometimes (for example) say<br />
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``John, here are two people who want to work on your project. What do you think of them''<br />
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5) Of the 180 applicants about 50 do not have a statement of purpose. For me this is a deal-breaker. Either they were in the middle of applying and got another offer and took it- which is fine, but no reason for me to even look at the application, OR they have a hard time completing things and again, not worth my time. A prof once asked me `But what if they are really good''-- there are plenty of really good applicants who do fill out the statement of purpose.<br />
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6) The main activity is research but we have some social activities as well.<br />
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7) How big should teams be? We try to avoid teams of 1. We usually have teams of 2 or 3.<br />
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8) What about students from your own school? My REU grant tends to urge them to go elsewhere since the NSF likes to have people from NON-research schools, and because I personally think its better for broadening to go elsewhere. Other programs are set up differently.<br />
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9) Why do some transcript not have the name of the school on them. Drives me nuts.<br />
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In the Summer of 2017 I will be running this program for the 5th time. Feel free to leave questions about how to run one, OR your own experiences, in the comments.<br />
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http://blog.computationalcomplexity.org/2017/01/my-reu-programreus-in-generalflyers-why.htmlnoreply@blogger.com (GASARCH)2tag:blogger.com,1999:blog-3722233.post-2677209759857653054Thu, 12 Jan 2017 17:33:00 +00002017-01-12T12:33:19.617-05:00Guest Post about the first Women in Computational Topology (WinCompTop) WorkshopThe first Women in Computational Topology <a href="https://www.ima.umn.edu/2015-2016/SW8.15-19.16">WinCompTop</a> workshop was held in August at the Institute for Mathematics and its Applications (IMA) in Minneapolis, MN. In total, 27 women participated, ranging from undergraduates to full professors; in addition, five future topologists (children of the participants) attended the various social events scattered throughout the week.<br />
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The central goal of this workshop was to establish research collaborations among both junior and senior women in the field, as well as to provide an opportunity for mentoring at all levels. There were four working groups, each led by a more senior member of the community:<br />
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<a href="https://dentistry.sitecore.ualberta.ca/AboutUs/FacultyStaff/FacultyStaffCollection/GiseonHeo.aspx">Giseon Heo</a> (University of Alberta) outlined a project that extends one dimensional scale-space persistent homology (a fundamental tool in computational topology) to a pseudo-multidimensional persistence tool that can be applied to a variety of applications.<br />
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<a href="http://web.cs.ucdavis.edu/~amenta/">Nina Amenta</a> (University of California, Davis) posed a problem of producing an explicit representation of a surface S from an input point cloud P drawn from a distribution over S (possibly with some noise).<br />
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<a href="http://web.cse.ohio-state.edu/~yusu/">Yusu Wang</a> (The Ohio State University) discussed a new method of persistence-based profiles to compare metric graphs and outlined that further exploration of what information is captured by persistence-based profiles and understanding their discriminative power would be the focus of their working group.<br />
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<a href="http://www.cs.tulane.edu/~carola/">Carola Wenk</a> (Tulane University) and <a href="https://www.cs.montana.edu/brittany/">Brittany Terese Fay</a> investigated the use of topology in map construction and comparison, particularly understanding directed graphs with multiple lanes and overpasses.<br />
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The workshop began with each of the senior researchers presenting an overview of their working group’s topic. After the overview of each project, working groups began to explore their topic; over the course of the week, substantial progress was made in each group. Each working group will contribute an article to a special AWM/IMA Springer journal, co-edited by the organizers of WinCompTop 2016 (Erin Chambers, Brittany Terese Fasy, and Lori Ziegelmeier). In addition, many of the participants who attended WinCompTop will meet once again at a special session of the AWM Research Symposium in April (see <a href="https://sites.google.com/site/awmmath/home/RS17">here</a>).<br />
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The workshop also had several outings and other social events, including a poster session where over a dozen posters were presented, a panel on work-life balance, an open problem session, and several receptions or banquets. These events let participants come together as a group, establish future collaborations, and connect with one another. In addition to formally scheduled outings, informal activities such as a marshmallow roast one evening, group dinners, and many other gatherings happened throughout the week.<br />
<br />
<i>What we (the organizers) have learned, and some questions for the community:</i><br />
How many women in Field X does it take to justify creating a “Women in X” network? (Or, more generally, an <insert-subpopulation> in X network?) This question was brought to our attention by Bill Gasarch (thanks for letting us post on your blog, BTW). We started this community as a listserv over two years ago (by the way, visit here if you’d like to join: <a href="https://groups.google.com/forum/#!forum/wincomptop/join">Here</a>. Today, we have over 100 subscribers, several of whom are not women. Regularly, opportunities are posted through this listserv, and lively discussions sometimes ensue (for example, we recently had a lengthy thread listing all of the researchers under whom one might be able to pursue a Ph.D. in computational topology). This network was started by just a handful of us who decided that there needed to be a more formal way for junior researchers to seek advice and for organizers of events to find diverse speakers. So, perhaps the answer to Bill’s question is: just a handful of people, and you’ll be surprised how quickly things grow.</insert-subpopulation><br />
<insert-subpopulation><br /></insert-subpopulation>
Acknowledgements<br />
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Finally, we want to end this post with a note of gratitude. We thank NSF,<br />
which funded the travel and local expenses for most of the participants (NSF DMS grant #1619908). We thank Microsoft Research for providing a generous donation, which funded the social events and travel for one of our international group leaders. Thanks also to AWM, which provided logistical support and advice for the event, and financial support for the upcoming follow-up events. Most enthusiastically, we thank the IMA for all of their support of both time and resources. The IMA donated the meeting spaces, breakfasts, as well as poster-printing, all in-kind. Last but not least, we thank every participant of WinCompTop 2016. We’ll see you again in 2018!<br />
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If you have any questions or comments about our experience organizing WinCompTop, we encourage you to contact us:<br />
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Erin Chambers erin.chambers@gmail.com<br />
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Brittany Terese Fasy brittany@fasy.us<br />
<insert-subpopulation></insert-subpopulation><br />
Lori Ziegelmeier lziegel1@macalester.edu<br />
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http://blog.computationalcomplexity.org/2017/01/guest-post-about-first-women-in.htmlnoreply@blogger.com (GASARCH)0tag:blogger.com,1999:blog-3722233.post-2740454628817287885Tue, 10 Jan 2017 19:27:00 +00002017-01-10T14:37:54.211-05:00Babai Strikes Back<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-pWBQD_8C61I/WHTYVOsZeWI/AAAAAAABZNw/leOuDhK9pas3R9AprOwq_27BppfapGNpgCPcB/s1600/IMG_20170109_172658.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://2.bp.blogspot.com/-pWBQD_8C61I/WHTYVOsZeWI/AAAAAAABZNw/leOuDhK9pas3R9AprOwq_27BppfapGNpgCPcB/s320/IMG_20170109_172658.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">"So it is quasipolynomial time again"</td></tr>
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Short Version: Babai <a href="http://people.cs.uchicago.edu/~laci/update.html">fixed his proof</a>.<br />
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In November 2015 László Babai announced a talk "Graph Isomorphism in Quasipolynomial Time" at the University of Chicago, an incredible breakthrough for this important problem. The <a href="https://arxiv.org/abs/1512.03547">algorithm</a> is a tour-de-force masterpiece combining combinatorics and group theory. The result sent shock waves through the community, we <a href="http://blog.computationalcomplexity.org/2015/11/a-primer-on-graph-isomorphism.html">covered</a> <a href="http://blog.computationalcomplexity.org/2015/11/looking-forward-to-gi-result.html">the result</a> as did most everyone else, despite Babai's disclaimer that the result was not yet peer reviewed.<br />
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Robin Thomas at Georgia Tech immediately suggested we get Babai down to Atlanta to talk on the paper. Babai's dance card filled up quickly but he eventually agreed to give two talks at Georgia Tech. January 9-10, 2017, right after the Joint Mathematics Meeting in Atlanta. Robin scheduled the <a href="http://aco25.gatech.edu/">25th anniversary celebration</a> of our Algorithms, Combinatorics and Optimization PhD program around Babai's talks.<br />
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Around 4 AM last Wednesday we got a lengthy email from Babai with the subject "new development" and the first line "The GI algorithm does not run in quasipolynomial time." The email explained the new running time, still a huge breakthrough being the first subexponential time algorithm for graph isomorphism, but not as strong as before. Harald Helfgott had <a href="https://valuevar.wordpress.com/2017/01/04/graph-isomorphism-in-subexponential-time/">discovered the problem</a> while going through the proof carefully preparing for a Bourbaki seminar talk on the result. The email asked if we still wanted him to give the talk (of course we did) and instructions for removing "quasipolynomial" from his <a href="http://aco25.gatech.edu/abstracts#babai1">talk abstracts</a>. That email must have been very hard for Babai to write.<br />
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Babai posted an <a href="http://people.cs.uchicago.edu/~laci/update.html">update</a> on his home page which I too quickly <a href="https://twitter.com/fortnow/status/816628539608932356">tweeted</a>. News spread quickly in <a href="https://rjlipton.wordpress.com/2017/01/04/babais-result-still-a-breakthrough/">blog</a> <a href="https://windowsontheory.org/2017/01/05/on-expexpsqrtlog-n-algorithms/">posts</a> and by Thursday an online article in Quanta titled <a href="https://www.quantamagazine.org/20170105-graph-isomorphism-retraction/">Complexity Theory Strikes Back.</a> Scott Aaronson picked a lousy day to <a href="http://www.scottaaronson.com/blog/?p=3095">release</a> his new book-length survey on the P v NP problem.<br />
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On top of all this snow in Atlanta threatened to cancel the Monday talk. But the snow never came and Babai started his talk at 4:30 PM to a packed room. Without telling any of us beforehand, an hour earlier he had posted an <a href="http://people.cs.uchicago.edu/~laci/update.html">update</a> to his update and announced it in his talk.<br />
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<a href="https://1.bp.blogspot.com/-GoBOzMmjBZ8/WHTYVAdPtsI/AAAAAAABZNw/uBVV61KR5FwGbG0STGD_yfVg9C8pI7CcwCPcB/s1600/IMG_20170109_164434.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="201" src="https://1.bp.blogspot.com/-GoBOzMmjBZ8/WHTYVAdPtsI/AAAAAAABZNw/uBVV61KR5FwGbG0STGD_yfVg9C8pI7CcwCPcB/s320/IMG_20170109_164434.jpg" width="320" /></a></div>
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We were watching history. From the talk I <a href="https://twitter.com/fortnow/status/818575040048463872">tweeted</a> the new news though Bill Cook, also in the audience, <a href="https://twitter.com/wjcook/status/818574848016478208">beat me to the punch</a>. Babai went on to describe the issue, an error in the analysis of the running time in the recursion, and the fix, basically a way to avoid that recursive step, but I can't do it justice here. At the end he proclaimed "So it is quasipolynomial time again". And so it was.<br />
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Today Babai talked about the group theory behind the algorithm as though there was never an issue in the first place.http://blog.computationalcomplexity.org/2017/01/babai-strikes-back.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-5146589104526336803Thu, 05 Jan 2017 15:09:00 +00002017-01-05T10:09:14.000-05:00Learning About Learning<div>
Yesterday Scott Aaronson released his sweeping new <a href="http://www.scottaaronson.com/papers/pnp.pdf">P v NP survey</a>. Babai gave an <a href="http://people.cs.uchicago.edu/~laci/update.html">update on graph isomorphism</a>, in short while he still has the first subexponential time algorithm for GI, he no longer claims quasipolynomial-time. We'll have more on graph isomorphism next week.<br />
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Having seen the power of neural networks for learning, how do they actually work? Over the break I decided to catch myself up. I worked through Michael Nielsen's online book <a href="http://neuralnetworksanddeeplearning.com/">Neural Networks and Deep Learning</a> chosen mainly because he co-authored one of the <a href="http://amzn.to/2iAikk5">great books on quantum computing</a> followed by the beginning of the <a href="https://www.tensorflow.org/tutorials/">TensorFlow tutorial</a>. To try out code I downloaded <a href="https://www.python.org/downloads/">Python</a> and <a href="https://www.tensorflow.org/get_started/">TensorFlow</a> and used <a href="https://www.jetbrains.com/pycharm/">PyCharm</a> as my IDE (free for academics). I should really learn <a href="https://github.com/">GitHub</a> since that holds all the code samples. Where were all these cool tools when I was a kid?</div>
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So what did I learn? Here is my high level understanding, but read the above materials to get a fuller picture. A neural network is just a weighted threshold circuit, though instead of precise threshold functions you use more continuous and differentiable functions like <a href="https://en.wikipedia.org/wiki/Sigmoid_function">sigmoid</a> or a <a href="https://en.wikipedia.org/wiki/Rectifier_(neural_networks)">rectifier</a>. If you fix the weights and biases (the negation of the threshold value), the network computes a function from input to output, for example from images of digits to the digits themselves. Typically you have a sample of labelled data and you can create an error function to see how well your network computes the solution. Fix the labelled data and now you have a new function from the weights and biases to the error.</div>
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You want to choose weights and biases to minimize the error. The general approach is gradient descent, improve a given solution by moving a small distance in the opposite direction in the gradient and repeat. I had naively thought you estimated the gradient numerically but in fact the gradient is computed symbolically using a dynamic programming-like algorithm called backpropagation based on the chain rule for partial derivatives.</div>
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Convolution nets has a special first layer that captures features of pieces of the image. Recurrent neural networks allow feedback loops.</div>
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Nielsen shows how from scratch to build and train a neural net. In TensorFlow you just design the network and then it automatically computes the gradient and has a highly optimized algorithm that can be run on GPUs or specialized hardware to minimize the error.</div>
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What caught my eye is how much of an art machine learning still is. How many nodes should you have in your network? How many levels? Too many may take too long to train and could cause overfitting. Too few and you don't have enough parameters to create the function you need.</div>
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What threshold function do you use and how to you aggregate results? Lots of various tricks to avoid overfitting and improve speed of optimization. There's no fixed procedure to choose the parameters, though you can test how well you do. So lots of trial and error to learning and experience (which I don't have) certainly helps.</div>
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So we have amazingly powerful learning tools but for now we still need humans to learn. For now.</div>
http://blog.computationalcomplexity.org/2017/01/learning-about-learning.htmlnoreply@blogger.com (Lance Fortnow)9tag:blogger.com,1999:blog-3722233.post-1335949062942238647Mon, 02 Jan 2017 19:32:00 +00002017-01-02T14:32:14.467-05:00Predictions for 2017Lance's Complexity Year in Review, posted the last week of the year (lets hope that P vs NP is not resolved on Dec 31) is a tradition that goes back to 2002. Bill posting predictions for the coming year is a tradition that goes back to 2016. Here is the last one: <a href="http://blog.computationalcomplexity.org/2016/01/predictions-of-new-year.html">here.</a> My predictions were not very good- all of those that came true were obvious (P vs NP will not be resolved, CS enrollment will go up). My biggest goof- that Hillary Clinton would beat .. Ted Cruz, by accusing him of being born in a foreign country. <br />
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However, being wrong never stopped me before, so here are my predictions for 2017<br />
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1) Some people think Trump be more presidential once he takes office. Some point to Checks and Balances in the System - but the Congress is ruled by his party. Some point to the media as a watchdog. e.g: <a href="http://www.thedailybeast.com/articles/2016/12/31/our-murrow-moment.html">here.</a> A more accurate picture is due to John Oliver <a href="https://www.youtube.com/watch?v=bq2_wSsDwkQ">here</a>. Some say Trump is a closet liberal. But I doubt his true beliefs, if he has any, matter. Its going to be bad. I second Scott Aaronson's call to not normalize this: <a href="http://www.scottaaronson.com/blog/?p=2969">here</a>.<br />
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2) Not a prediction but a thought: Prior arguments against Trump were countered with `Well Hillary is just as bad' They can't use this anymore. The following absurd conversation might happen:<br />
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NEWS: Today some democrats and John McCain questioned Donald Trumps nominee, Rex Tillerson for Sec of State, about his past dealings with Russia and Putin.<br />
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FOX NEWS: But Hillary was the worst Sec of State ever! Bengazi!<br />
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3) I will be asked to review at least one paper claiming P=NP (or P NE NP or it won't be clear what they are saying) and at least one paper claiming to determine a Ramsey or VDW number. They will be garbage. Cranks are getting into more sophisticated areas so others may be asked to look at ``solutions'' to open problems in harder areas of math. The Navier-Stokes equations (A Millennium problem, see h<a href="http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation">ere</a>) might be a good area for cranks since they might get out some numbers and think they've solved it. I'm glad I'm not in that area.<br />
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4) Our Popular Posts links (which is determined by a program, not by us) will continue to have some of our most recent posts AND my very old post on Russell and Whitehead using 300 pages to prove 1+1=2. Why is that post seen as being so popular? I doubt it IS that popular. So--- whats going on?<br />
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5) Recall that Josh Alman and Ryan Williams showed that one method for lower bounds prob won't work <a href="http://blog.computationalcomplexity.org/2016/11/a-few-interesting-computer-science.html">her</a>e . There will be more results that rule out techniques.<br />
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6) n-person envy-free cake cutting can now be done with number-of-cuts TOW(n).<br />
There will be better upper bounds or some lower bounds on this proven this year.<br />
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7) There will be a big breakthrough on derandomization- possibly L=RL.<br />
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8) There will be a big data breach.<br />
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9) Some minor celebrity will die the last week of the year and hence not make either the<br />
`who died in 2017' lists, nor the `who died in 2018' lists. In 2016 this happened to <a href="https://en.wikipedia.org/wiki/William_Christopher">William Christopher</a>. Why do people make the `end of the year lists' before the year ends?<br />
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10) Fake News will become worse and worse. After Pizzagate there was NO apology or regret from the people who spread the false news.<br />
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11) Fake Journals, Schools, and accreditation agencies will continue to grow.http://blog.computationalcomplexity.org/2017/01/predictions-for-2017.htmlnoreply@blogger.com (GASARCH)5tag:blogger.com,1999:blog-3722233.post-5526704316780913018Thu, 29 Dec 2016 14:05:00 +00002016-12-29T09:05:11.342-05:00Complexity Year in Review 2016Paper of the year goes to <a href="http://ieee-focs.org/FOCS-2016-Papers/3933a416.pdf">A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents</a> by Haris Aziz and Simon Mackenzie. Might not seem like a traditional complexity result but cake cutting is a computational process with a desired set of properties and this papers settles a long standing open question. Bill <a href="http://blog.computationalcomplexity.org/2016/06/there-is-now-bounded-discrete-envy-free_0.html">posted about the paper</a> back in June.<br />
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Some other nice papers from 2016: <a href="https://arxiv.org/abs/1611.05558">Hadamard not so rigid after all</a> and <a href="http://blog.computationalcomplexity.org/2016/11/a-few-interesting-computer-science.html">not usable</a> for Valiant's lower-bound program, <a href="http://eccc.hpi-web.de/report/2015/119/">2-source extractors from low-entropy sources</a> which get <a href="http://cacm.acm.org/magazines/2017/1/211100-pure-randomness-extracted-from-two-poor-sources/fulltext">near perfect randomness from two weak sources</a>, <a href="https://arxiv.org/abs/1506.04719">query complexity</a> <a href="https://arxiv.org/abs/1511.01937">separations</a>, deterministic, randomized and quantum, in two wonderful constructions <a href="http://blog.computationalcomplexity.org/2016/06/stoc-2016.html">beautifully presented at STOC</a>, <a href="https://arxiv.org/abs/1612.02788">space-efficient subset-sum</a>,and <a href="https://www.cs.sfu.ca/~kabanets/Research/natural-learning.html">learning algorithms from natural proofs</a>.<br />
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2016 will go down as a watershed year for machine learning with <a href="http://blog.computationalcomplexity.org/2016/02/go-google-go.html">AlphaGo</a>, <a href="http://www.nytimes.com/2016/12/14/magazine/the-great-ai-awakening.html">huge advances in translation</a>, <a href="https://www.tesla.com/autopilot">self-driving cars</a> and with tools like TensorFlow out in the open we'll truly see learning everywhere. Prediction didn't have such a great year with bad calls on Trump's nomination, Brexit and Trump's election. Machines learning humans still has a way to go.<br />
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We thank our guest posters <a href="http://blog.computationalcomplexity.org/2016/09/the-letter.html">Molly Fortnow</a>, <a href="http://blog.computationalcomplexity.org/2016/03/mohammadtaghi-hajiaghayi-on-david.html">MohammadTaghi HajiAghayi</a>, <a href="http://blog.computationalcomplexity.org/2016/12/guest-post-by-samir-khuller-about.html">Samir</a> <a href="http://blog.computationalcomplexity.org/2016/12/guest-post-by-samir-khuller-on-humans.html">Khuller</a> and <a href="http://blog.computationalcomplexity.org/2016/06/the-relevance-of-tcs.html">Avi Wigderson</a>.<br />
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We remember <a href="http://www.nytimes.com/2016/11/30/technology/erich-bloch-who-helped-develop-ibm-mainframe-dies-at-91.html">Erich Bloch</a>, <a href="http://www.tfs.tu-berlin.de/menue/home/team/ehrig_hartmut_prof/">Hartmut Ehrig</a>, <a href="http://www.nytimes.com/2016/12/27/movies/carrie-fisher-dead-star-wars-princess-leia.html">Carrie Fisher</a>, <a href="http://blog.computationalcomplexity.org/2016/01/rusins-freivalds-1942-2016.html">Rūsiņš Freivalds</a>, <a href="http://www.nytimes.com/2016/12/08/us/john-glenn-dies.html">John Glenn</a>, <a href="http://blog.computationalcomplexity.org/2016/03/david-johnson-1945-2016.html">David Johnson</a>, <a href="http://www.nytimes.com/2016/01/26/business/marvin-minsky-pioneer-in-artificial-intelligence-dies-at-88.html">Marvin Minsky</a>, <a href="http://cacm.acm.org/news/196208-in-memoriam-peter-naur-1928-2016/fulltext">Peter Naur</a>, <a href="http://blog.computationalcomplexity.org/2016/08/seymour-papert-1928-2016.html">Seymour Papert</a>, <a href="http://blog.computationalcomplexity.org/2016/03/hillary-putnam-passed-away-on-march-13.html">Hillary Putnam</a>, <a href="http://blog.computationalcomplexity.org/2016/03/the-value-of-shapley.html">Lloyd Shapley</a> and <a href="http://blog.computationalcomplexity.org/2016/09/boris-trakhtenbrot-1921-2016.html">Boris Trakhtenbrot</a>.<br />
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As the surprises of 2016 lead us into an uncertain 2017 let me leave with my usual advice: When in a complex world, best to keep it simple.http://blog.computationalcomplexity.org/2016/12/complexity-year-in-review-2016.htmlnoreply@blogger.com (Lance Fortnow)0