tag:blogger.com,1999:blog-3722233Sat, 28 Nov 2015 20:48:44 +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)Blogger2330125tag:blogger.com,1999:blog-3722233.post-98517605919665296Mon, 23 Nov 2015 15:30:00 +00002015-11-23T10:30:55.573-05:00Star Trek ComputingIn the wake of Leonard Nimoy's death last February, I decided to rewatch the entire original Star Trek series, all 79 episodes. I had watched them each many times over in high school in the 70's, though the local station removed a scene or two from each episode to add commercial time and I often missed the opening segment because I didn't get home from school in time. Back in those stone ages we had no DVR or other method to record shows. I hadn't seen many episodes of the original series since high school.<br />
<br />
Now I can watch the entire episodes whenever I want in full and in order through the magic of Netflix. I finished this quest a few days ago. Some spoilers below.<br />
<br />
I could talk about the heavy sexism, the ability to predict future technologies (the flat screen TV in episode 74), the social issues in the 23rd century as viewed from the 60's, or just the lessons in leadership you can get from Kirk. Given the topic of this blog, let's talk about computing in Star Trek which they often just get so wrong, such as when Spock asks the computer to compute the last digit of π to force Jack-the-Ripper to remove his consciousness from the ship's computers.<br />
<br />
Too many episodes end with Kirk convincing a computer or robot to destroy itself. I'd like to see him try that with Siri. In one such episode "<a href="https://en.wikipedia.org/wiki/The_Ultimate_Computer">The Ultimate Computer</a>", a new computer is installed in the Enterprise that replaces most of the crew. A conversation between Kirk and McCoy sounds familiar to many we have today (<a href="http://www.chakoteya.net/StarTrek/53.htm">source</a>).<br />
<br />
MCCOY: Did you see the love light in Spock's eyes? The right computer finally came along. What's the matter, Jim?<br />
KIRK: I think that thing is wrong, and I don't know why.<br />
MCCOY: I think it's wrong, too, replacing men with mindless machines.<br />
KIRK: I don't mean that. I'm getting a Red Alert right here. (the back of his head) That thing is dangerous. I feel. (hesitates) Only a fool would stand in the way of progress, if this is progress. You have my psychological profiles. Am I afraid of losing my job to that computer?<br />
MCCOY: Jim, we've all seen the advances of mechanisation. After all, Daystrom did design the computers that run this ship.<br />
KIRK: Under human control.<br />
MCCOY: We're all sorry for the other guy when he loses his job to a machine. When it comes to your job, that's different. And it always will be different.<br />
KIRK: Am I afraid of losing command to a computer? Daystrom's right. I can do a lot of other things. Am I afraid of losing the prestige and the power that goes with being a starship captain? Is that why I'm fighting it? Am I that petty?<br />
MCCOY: Jim, if you have the awareness to ask yourself that question, you don't need me to answer it for you. Why don't you ask James T. Kirk? He's a pretty honest guy.<br />
<br />
Later in the episode the computer starts behaving badly and Kirk has to convince it to shut itself down. But what if the computer just did its job? Is that our real future: Ships that travel to stars controlled only by machine. Or are we <a href="http://news.mit.edu/2015/nasa-gives-mit-humanoid-robot-future-space-missions-1117">already there</a>?http://blog.computationalcomplexity.org/2015/11/star-trek-computing.htmlnoreply@blogger.com (Lance Fortnow)2tag:blogger.com,1999:blog-3722233.post-6052109793999482549Thu, 19 Nov 2015 13:53:00 +00002015-11-19T08:53:05.061-05:00A Silly String TheoremFirst a note on a serious theorem: Babai has posted a <a href="http://people.cs.uchicago.edu/~laci/2015-11-10talk.mp4">video</a> (mp4, 1h 40 m, 653MB) of his first talk on his Graph Isomorphism algorithm.<br />
<b><br /></b>I was giving a talk on the Kleene star operator (don't ask) and came across this cute little problem. Say a language L commutes if for all u,v in L, uv=vu.<br />
<b><br />Problem: </b>Show that L commutes if and only if L is a subset of w* for some fixed string w.<br />
<br />
Here w* is the set of strings consisting of zero or more concatenations of w with itself. The if case is easy, but I found the other direction pretty tricky and came up with an ugly proof. I found a cleaner proof in Seymour Ginsburg's 1966 textbook <i>The Mathematical Theory of Context Free Languages </i>which I present here.<br />
<b><br /></b>
<b>⇐</b> If u=w<sup>i</sup> and v=w<sup>j</sup> then uv=vu=w<sup>i+j</sup>.<br />
<br />
<b>⇒</b> Trivial if L contains at most one string. Assume L contains at least two strings.<br />
<br />
Proof by induction on the length of the shortest non-empty string v in L.<br />
<br />
Base case: |v|=1<br />
Suppose there is an x in L with x not in v*. Then x = v<sup>i</sup>bz for b in Σ-{v}. Then the i+1st character of xv is b and the i+1st character of vx is v, contradicting xv=vx. So we can let w=v.<br />
<br />
Inductive case: |v|=k>1.
<br />
<br />
Lemma 1: Let y=v<sup>r</sup>u be in L (u might be empty). Then uv=vu.<br />
Proof: Since yv=vy we have v<sup>r</sup>uv=vv<sup>r</sup>u=v<sup>r</sup>vu. So vu=uv.<br />
<br />
Let x in L be a string not in v* (if no such x exists we can let w=v). Pick j maximum such that x=v<sup>j</sup>z with z not the empty string. By Lemma 1 we have zv=vz. Note |z| < |v| otherwise v is a prefix of z, contradicting the maximality of j.<br />
<br />
Lemma 2: For all y in L we have yz=zy.<br />
Proof: As before let y = v<sup>r</sup>u. By Lemma 1, uv=vu<br />
Since yx=xy we have v<sup>r</sup>uv<sup>j</sup>z = v<sup>j</sup>zv<sup>r</sup>u<br />
v<sup>r</sup>uv<sup>j</sup>z = v<sup>r</sup>uzv<sup>j</sup> by swapping v's and z's.<br />
v<sup>j</sup>zv<sup>r</sup>u = zv<sup>r</sup>uv<sup>j</sup> by swapping v's and z's, and v's and u's.<br />
So we have v<sup>r</sup>uzv<sup>j</sup> = zv<sup>r</sup>uv<sup>j</sup><br />
or yzv<sup>j</sup> = zyu<sup>j</sup> and thus yz=zy.<br />
<br />
By Lemma 2, the set L∪{z} commutes. Since 0 < |z| < |v| by induction L∪{z} is a subset of w* for some w so L is a subset of w*.http://blog.computationalcomplexity.org/2015/11/a-silly-string-theorem.htmlnoreply@blogger.com (Lance Fortnow)4tag:blogger.com,1999:blog-3722233.post-6976871382979968922Mon, 16 Nov 2015 17:20:00 +00002015-11-16T14:15:24.782-05:00Is the word Quantum being used properly by civilians? Understood by them? You know the answer.<br />
I've seen the word `civilians' expanded in use from non-military to non-X for some X. Not sure I've ever seen `civilians' mean `people who don't do math stuff' until the title of todays post. Well, there is a first time for everything.<br />
<br />
I've blogged in the past about the use of the word Quantum (<a href="http://blog.computationalcomplexity.org/2008/12/quantum-leapquantum-of-solacewhat-does.html#comment-form">here) </a>. The phrase <i>Quantum Leap</i> means a BIG leap, where as Quantum stuff is small. Though, to be fair, the discovery (invention?) of Quantum Mechanics was a big leap. So maybe that IS proper use. The James Bond movie <i>Quantum of Solace</i> uses Quantum to mean small, the ONLY time I've seen Quantum used to mean small, so Kudos to the title of an absolutely awful movie. Commenters on my prior blog on the subject pointed out that the original meaning of<br />
<i>quantum </i>was <i>quantity or amount without regard to size of discreteness. </i>I think using it that way now would be very rare.<br />
<br />
I came across a theatre called <a href="http://www.quantumtheatre.com/about/what-we-are/">Quantum Theatre.</a> What is Quantum Theatre? The following are actual quotes from their website. <br />
<br />
<i>Quantum artists mine all kinds of non-traditional spaces for the sensory possibilities they offer when combined with creative design.</i><br />
<br />
<i> We find it meaningful to place the audience and performer together, the moving parts inside the works.</i><br />
<br />
<i> We want to move people with our experiements.</i><br />
<br />
<i>The shows run the gamut from those you thought you knew but now
experience like never before, to shows that didn’t exist until their
elements mixed in our laboratory.</i><br />
<br />
<i> </i>I came across this article in a place it didn't belong-- in the middle of an article about Google and NASA trying to build a quantum computer (see <a href="http://www.thedailybeast.com/articles/2015/09/30/google-and-nasa-team-up-on-quantum-computer.html">here.)</a> This news might be exciting but the article was full of mistakes and bad-signs so I'm not to excited about it. Plus the reference to Quantum Theatre is just odd.<br />
<br />
The BEST use of the word <i>quantum </i>that I've heard recently was in the episode <i>The Ricks must be crazy</i> of the excellent TV show <i>Ricky and Morty</i>:<br />
<br />
The car is broken<br />
<br />
Morty (a 14 year old): W-Whats wrong Rick? Is it the Quantum carburetor or something?<br />
<br />
Rick (his grandfather, a brilliant scientist): Quantum carburetor? You can't just add a Sci-fi word to a car word and hope it means something.<br />
<div style="left: -99999px; position: absolute;">
W-what's wrong, Rick? Is it the quantum carburetor or something?<br />
<br />
Read more at: http://transcripts.foreverdreaming.org/viewtopic.php?f=364&t=20185</div>
<div style="left: -99999px; position: absolute;">
W-what's wrong, Rick? Is it the quantum carburetor or something?<br />
<br />
Read more at: http://transcripts.foreverdreaming.org/viewtopic.php?f=364&t=20185</div>
<br />
<br />http://blog.computationalcomplexity.org/2015/11/is-word-quantum-being-used-properly-by.htmlnoreply@blogger.com (GASARCH)5tag:blogger.com,1999:blog-3722233.post-1680472021350980055Thu, 12 Nov 2015 13:30:00 +00002015-11-17T07:48:46.145-05:00A Primer on Graph Isomorphism<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-huVsxq73vP0/VkOIj2UgO5I/AAAAAAABVgc/KFrQJ8bJkJI/s1600/CTeqULsVAAA-Edn.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="106" src="http://1.bp.blogspot.com/-huVsxq73vP0/VkOIj2UgO5I/AAAAAAABVgc/KFrQJ8bJkJI/s400/CTeqULsVAAA-Edn.jpg" width="400" /></a></div>
<br />
I spent 14 years on the faculty at the University of Chicago. I know László Babai well, we collaborated on some of my best known work. I also know Ryerson 251, a room where I've seen hundreds of talks and given more than a few myself. So I could imagine the excitement in that room on Tuesday as Babai gave the most anticipated talk in the history of theoretical computer science, the first of <a href="http://people.cs.uchicago.edu/~laci/quasipoly.html">several talks</a> Babai is giving on his new algorithm for graph isomorphism [<a href="http://people.cs.uchicago.edu/~laci/2015-11-10talk.mp4">Video</a>]. Gabriel Gaster extensively <a href="https://storify.com/ptwiddle/babai-s-first-graph-isomorphism-talk">live tweeted</a> the event. Jeremy Kun has <a href="http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/">some details</a>.<br />
<br />
For this post instead of doing a bad job trying to overview Babai's proof, I'll explain the graph isomorphism problem and why it is important in the complexity world.<br />
<br />
Suppose we have two high schools, say HS North and HS South, each with 1000 students. Consider a diagram (or graph) containing a point for each student at HS North with lines between students who are facebook friends, and a similar diagram for HS South. Is there a 1-1 map from the students at HS North to the students at HS South so that these diagrams look identical? That's the graph isomorphism problem.<br />
<br />
To determine whether these two graphs were isomorphic you could look at all the possible mappings between students, but that's 1000! or more than 4x10<sup>2567</sup> possible maps. There has long been known how to search a smaller number of possibilities, especially if we put some restrictions on the diagrams, but always exponential in n (the number of students) in the worst case. Ideally we'd like a polynomial-time algorithm and Babai gets very close, an algorithm that runs in time n<sup>log<sup>k</sup>n</sup> time for some fixed k.<br />
<br />
Graph Isomorphism is one of those few problems, like factoring, known to be in NP but not known to be in P or NP-complete. Graph non-isomorphism is the poster child for the class AM, the problems solved by a randomized verifier asking a single question to a powerful prover. Graph non-isomorphism in AM implies that Graph Isomorphism is not likely NP-complete and that under reasonable derandomization assumptions that Graph non-isomorphism is in NP. Kobler, Schöning and Toran wrote a <a href="http://www.springer.com/us/book/9780817636807">whole book</a> on the computational complexity issues of graph isomorphism.<br />
<br />
Even small progress in graph isomorphism creates waves. At a theory conference in the late 80's a speaker caused a major stir when he casually mentioned he had a proof (he didn't) that Graph non-isomorphism was in NP. Babai's announced caused a huge tsunami and those of us who know him realize he wouldn't make such an announcement without being sure he has a proof. The talk put together a large number of ideas from combinatorics and graph theory. My impression is that those who saw the talk didn't leave convinced of the proof, but did feel Babai had found the pieces to make it work.<br />
<br />
With Babai's breakthrough algorithm, the smart money says now that graph isomorphism sits in P. It took decades to get from quasipolynomial time to polynomial time for primality testing and the same time frame may be needed to get polynomial time for graph isomorphism. But it will likely happen and the complexity of graph isomorphism then gets a whole lot simpler.<br />
<br />
A couple of thoughts: All the breakthrough results that I can remember were released as papers, ready to devour. This is the first result of this caliber I remember being announced as a talk.<br />
<br />
Also we think of theory as a young person's game, most of the big breakthroughs coming from researchers early in their careers. Babai is 65, having just won the <a href="http://www.sigact.org/Prizes/Knuth/">Knuth Prize</a> for his lifetime work on interactive proofs, group algorithms and communication complexity. Babai uses his extensive knowledge of combinatorics and group theory to get his algorithm. No young researcher could have had the knowledge base or maturity to be able to put the pieces together the way that Babai did.<br />
<br />
More on Babai and graph isomorphism from <a href="http://www.scottaaronson.com/blog/?p=2521">Scott</a>, <a href="https://lucatrevisan.wordpress.com/2015/11/03/laci-babai-and-graph-isomorphism/">Luca</a>, <a href="http://blog.computationalcomplexity.org/2015/11/looking-forward-to-gi-result.html">Bill</a>, <a href="https://plus.google.com/+TimothyGowers0/posts/JbSPELbKQED">Tim</a>, <a href="https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-graph-isomorphism/">Dick and Ken</a>, <a href="https://www.reddit.com/r/math/comments/3sdixw/babais_breakthrough_on_graph_isomorphism/">Reddit</a>, <a href="https://www.sciencenews.org/article/new-algorithm-cracks-graph-problem">Science News</a>, <a href="https://www.newscientist.com/article/dn28478-complex-problem-made-simple-sends-computer-scientists-wild">New Scientist</a> and <a href="http://news.sciencemag.org/math/2015/11/mathematician-claims-breakthrough-complexity-theory">Science</a>.http://blog.computationalcomplexity.org/2015/11/a-primer-on-graph-isomorphism.htmlnoreply@blogger.com (Lance Fortnow)4tag:blogger.com,1999:blog-3722233.post-5246892705611455093Mon, 09 Nov 2015 02:59:00 +00002015-11-08T21:59:58.656-05:00Looking forward to the GI resultAs you all know Laszlo Babai will give a talk Tuesday Nov 10 on a result: GI in quasipolynomial time (2 to a polylog). Other theory blogs have already commented on this (<a href="https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-graph-isomorphism/">GLL</a>,<a href="https://lucatrevisan.wordpress.com/2015/11/03/laci-babai-and-graph-isomorphism/">In Theory</a>,<a href="http://www.scottaaronson.com/blog/?p=2521">Shetl-Opt</a>)<br />
<br />
When I was in Graduate School (early 1980's) it looked like GI would get into P. Three key results: graphs of bounded degree, graphs of bounded genus, graphs of bounded eigenvalue multiplicity, were all shown to be in P. These results used group theory and linear algebra in serious ways so the thought was that more advanced group theory, perhaps the classification of all finite simple groups (CFSG) would be used to show GI in P. If CFSG was used in an algorithm for GI in P then the algorithm might have an enormous constant, but that's okay since the quest for GI in P is not for practical reasons (I think that all the people who want to solve it already have fast enough algorithms) . I was looking forward to the debates on <i>if an algorithm with that big a constant would count as poly time</i> (I would say yes). But no algorithm for GI in P, using CFSG or not, was found. Also note that it was possible (and is still possible) that GI is NOT in P. In my 2012 survey of P vs NP I also asked about GI. Of the 20 people who responded 14 thought GI is in P, 6 thought not. Note that GI is not known to be in co-NP or BPP. It IS in IP[2], and if GI was NPC<br />
then PH collapses, so it is unlikely that GI is NP-complete. This tells us NOTHING about whether it is in P.<br />
<br />
An analogous thing happened with PRIMES. Lots of results that got it almost in P, lots of hard number theory used, but then the progress stopped. Two differences: Most people thought PRIMES IS in P (likely because it's in coNP and BPP),and this was finally proven in 2002.<br />
<br />
Is Babai's proof likely to be correct? There are three parameters to consider when looking at that question: (1) Who is making the claim?, (2) How likely the claim is to be true?, (3) How likely the claim is to be provable with what is known today?.You can give different weights to each one in different combinations..<br />
<br />
Laszlo Babai is an expert in GI with a track record in the area. (reading that over again it undersells the case- Babai is, as the kids say, awesome!) That GI is in quasipolynomial time is quite believable. Even those 6 who think that Gi is not in P might believe that. Its harder to know if today's techniques are up to the task; however, there are no results like `group theory won't suffice' or any kind of obstacles, so certainly the claim seems provable with todays technology. <br />
<br />
Here's hoping...http://blog.computationalcomplexity.org/2015/11/looking-forward-to-gi-result.htmlnoreply@blogger.com (GASARCH)2tag:blogger.com,1999:blog-3722233.post-7684043071724994534Thu, 05 Nov 2015 16:10:00 +00002015-11-05T13:30:43.095-05:00Computation and Journalism Part 2<i>The <a href="http://www.scottaaronson.com/blog/?p=2521">theory</a> <a href="https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-graph-isomorphism/">blogosphere</a> is atwitter with an announcement of a talk next Tuesday at Chicago by László Babai <a href="https://calendar.google.com/calendar/render?eid=czNzOXNtZ2tydG00OG5obDJlZ3I3c21uY2cgYzU3c2hpY2k0NW0xN3FsMGdodmw1NmVrMzhAZw&ctz=America/Chicago">Graph Isomorphism in Quasipolynomial Time</a>. <a href="https://lucatrevisan.wordpress.com/">Luca</a> will blog from the scene and we will have a full report as details emerge. </i><br />
<br />
More from the Comp and Journalism conference<br />
<br />
0) (ADDED WRITE BEFORE POSTING) The conference was about how journalists do or should use technology. Part of this is how news gets to people. Twitter and Blogs are one method, as the above note about Babai's talk shows. <br />
<br />
1) There was a panel discussion on context When one reads an article it depends on where one is coming from. Here is a strange thing that technology has allowed journalists to do: An article can have slight differences depending on where its being read. For example if someone from New York is reading it may refer to Broadway, whereas if someone in LA is reading it it might refer to Hollywood. The tech also exists to have the reader (online) have a choice- so you can read it as if you are in place X. This scares me a bit- what if an article is can be tinkered with to appeal to lefthanded latino lesbians who work on the local<br />
Lovasz Lemma. Isn't our country fragmented enough? STILL, the point that context matters came through nicely.<br />
<br />
<br />
2) There was a panel of three ACTUAL JOURNALISTS (I think that all of the other speakers were academics) who used BIG DATA or MACHINE LEARNING or FILL IN BUZZWORD to write articles. Propublica, a nonprofit newspaper (aren't they all?) did an article about rating doctors<a href="http://projects.propublica.org/graphics/investigating-doctors"> (here ) </a>that seems to be very well checked (example- if a patient has to come back again because of a complication, is it the surgeons fault? hard to answer). However, some doctors (those that got low ratings?) complained that there article was not peer reviewed. This brings up a question--- journalists run things by fact checkers and proofreaders, but do not have peer-review. Academics do. Which is the better system? Another journalist did a study of the Supreme court and found that there is a small set of lawyers that are well connected (one used to play darts with Alito) who are involved in over half of the cases the court sees. What interested me is--- what if you have an idea for an article and you do ALL of the number crunching, bring in ML people to help and find out in the end OH, the Supreme court actually is doing a fair job or OH, Doctors are pretty good. No story! They said this happens about 1/3 of the time where either you don't have enough data for a story or the story is not really a story.<br />
<br />
3) There was a paper about YELP ratings of doctors. It seems that the actual<br />
doctor-stuff is rarely mentioned, usually it was how long the wait time was, <br />
how boring the magazines in the office were, etc. Also, most doctors got <br />
either 1 star or 5 stars. One thing they didn't mention but I will- Walter Palmer, the Dentist who killed Lion in Africa, got lots of very negative YELP reviews from people who were never patients of his. So YELP ratings can be skewed by things independent of what is supposed to be measured. That is, of course, and extreme case.<br />
<br />
4) Key Note by Chris Wiggins who is an academic who, on his Sabbatical, worked for the NYT on issues of <i>How many copies of the paper should we print</i>? and other Operations Research type questions. Print subscriptions still out number online subscriptions, the online is doing pretty well and bringing in money, but they still NEED to find a better business model.I am reminded that when Jeff Bezos bought the Wash Post Stephen Colbert said<i> there are more people buying newspapers than there are people buying newspapers.</i><br />
(My spellchecker things online is not a word. Everyone else does.)<i><br /></i><br />
<br />
5) There was a panel on Education. From there point of view very pessimistic---<br />
most schools don't offer courses in how to use Technology in journalism, no real<br />
books, no agreement on what's important. And I suspect they will have a hard time updating courses once they have them. I wonder if the early days of Comp Science were like that.<br />
<br />
6) The issue of jobs in journalism going away was not quite ignored but also not<br />
quite confronted either. Some people told me that Journalism schools are not being honest with their students about the dismal job market. Then again, they are journalism students- they should investigate!<br />
<br />
7) A journalism student told me of a case going to the supreme court that may be<br />
very important for privacy: Spokeo vs Robins:<a href="https://www.epic.org/amicus/spokeo/"> here</a><br />
<br />
8) UPSHOT- I am glad I went, I learned some things outside of what I usually think about. Unlike the RATLOCC (Ramsey Theory and Logic) I was able to tell my mom what I learned. I didn't bring my laptop nor had TV access I was able to work on some math and had a minor breakthrough. But that,s for a later blog <br />
(My spellcheck thinks<i> blog </i>is not a word. It also thinks spellcheck is not a word.)http://blog.computationalcomplexity.org/2015/11/computation-and-journalism-part-2.htmlnoreply@blogger.com (GASARCH)0tag:blogger.com,1999:blog-3722233.post-4523514760544332060Tue, 03 Nov 2015 05:22:00 +00002015-11-03T00:22:01.907-05:00Conference on Computation and Journalism (Part I) (In honor of Boole's Birthday yesterday see <a href="http://calculushumor.tumblr.com/post/65825942666/boole-orders-lunch">this.)</a><br />
<br />
I recently (Oct 2 and 3 2015) went to a conference on Computation and Journalism (see <a href="http://cj2015.brown.columbia.edu/index.html#schedule-anchor">here</a> for their schedule). I went since I co-blog and have a Wikipedia page (<a href="https://en.wikipedia.org/wiki/William_Gasarch">here</a>) hence I"ve gotten interested in issues of technology and information (not information-theory, but real world information).<br />
<br />
I describe some of what I saw in two posts, today and thursday. Many of the articles I mention can be gotten from the schedule page above.<br />
<br />
1) I understood all of the talks. This is very different from STOC, FOCS, CCC, etc; however, old timers tell me that back in STOC 1980 or so one could still understand all of the talks.<br />
<br />
2) Lada Adamic gave a keynote on Facebook about whether or not Facebook causes an echo chamber (e.g., liberals only having liberal friends) Her conclusion was NO- about 25% of the friends of an L are a C and vice versa. This makes sense since people friend people who are workers and family, not based on political affiliation (Some of my in-laws are farther to the right than... most of my readers.) She also noted that 10% of the articles passed on by L's are Conservative- though this might not be quite indicative since they may often be `look at this crap that FOX news is saying now' variety. Note that Lada works for Facebook. Talking to people over lunch about that the consensus was that (1) the study was valid, but (2) if the conclusion had been that Facebook is tearing apart our country THEN would they have been allowed to publish it? I leave that as a question.<br />
<br />
3) There was a Panel on Comments. The NYT has 14 people whose sole job is the soul-crushing job of reading peoples comments on articles and deciding what to let through. Gee, complexity blog just needs Lance and me. I found out that the best way to ensure intelligent comments and to get rid of trolls is (1) people must register, give their real name, and even have a picture of themselves, and (2) engage with the commenters (like Scott A does, though they did not use him as an example). Lance and I do neither of these things. Oh well.(3)COMPUTERS- can we have a program that can tell if an comment is worth posting? Interesting research question. One basic technique is to not allow comments with curse words in them- NOT out of a prudish impulse, but because curse words are an excellent indicator of articles that will not add to the discourse. <br />
<br />
The most interesting question raised was <i>Do Trolls know the are trolls</i>? YES- they just like destroying things for no reason. Picture Bart Simpson using a hammer on Mustard packets just cause.<br />
<br />
4) There was two sessions of papers on Bias in the news.<br />
<br />
a) R<i>anking in the Age of Algorithms and Curated News </i>By Suman Deb Rob. How should news articles be ranked? If articles are ranked by popularity then you end up covering Donald Trump too much. If articles are ranked by what the editors think thats too much of THE MAN telling me whats important. Also, there are articles that are important for a longer period of time but never really top-ten important. Nate Silver has written that the most important story of the 2016 election is that the number of serious (defined: Prior Senators of Govs) who are running for the Rep nomination is far larger than ever before. But thats not quite a story.<br />
<br />
b) <i>The Gamma: Programming tools for transparent data journalism</i> by Tomas Petricek. So there are data journalists that you can see right through! Wow! Actually he had a tool so that you could, using World Bank Data, get maps that show stuff via colors. His example was the story that China produces more carbon emission than the USA, and a nice color graphic for it, but then the user or the journalist can easily get a nice color graphic of carbon-emms-per-person in which case the USA is still NUMBER ONE! (Yeah!) Later he had a Demo session. I tried to look at % of people who go to college by country, but it didn't have info on some obscure countries like Canada. Canada? The project was Limited by the data we have available.<br />
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c) <i>The quest to automate Fact-Checking</i> by Hassan, Adair, Hamilton, Li, Tremayne, Yang, Yu. We are pretty far from being able to do this, however, they had a program that could identify whether or not something uttered IS checkable . <i>When I am president I will lower taxes</i> is not checkable, whereas Bill Mahr's <a href="http://www.theglobeandmail.com/life/celebrity-news/my-dad-was-not-an-orangutan-donald-trump-says-in-lawsuit/article8467052/">claim </a> that <i>Donald Trumps father is an Orangutan</i>g is checkable.</div>
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d) <i>Consumer and supplies: Attention Asymmetries</i> by Abbar, An, Kwak, Messaoui, Borge-Holthofer. They had 2 million comments posted by 90,000 unique readers and analyzed this data to see if there is an asymtery between what readers want and what they are getting. Answer: YES<br />
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<br />http://blog.computationalcomplexity.org/2015/11/conference-on-computation-and.htmlnoreply@blogger.com (GASARCH)0tag:blogger.com,1999:blog-3722233.post-532636741821976893Mon, 02 Nov 2015 10:52:00 +00002015-11-02T05:52:30.492-05:00George Boole (1815-1864)<div class="separator" style="clear: both; text-align: center;">
<a href="http://ichef-1.bbci.co.uk/news/660/media/images/79578000/jpg/_79578978_53009793.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://ichef-1.bbci.co.uk/news/660/media/images/79578000/jpg/_79578978_53009793.jpg" height="200" width="200" /></a></div>
<a href="http://www-history.mcs.st-and.ac.uk/Biographies/Boole.html">George Boole</a>, the father of symbolic logic, was born two hundred years ago today. His 1854 treatise, <i><a href="http://www.gutenberg.org/ebooks/15114">An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities</a>, </i>gave us what we now call Boolean algebra, variables that take on just two values, TRUE and FALSE, and the basic operations AND, OR and NOT. Boole could not have imagined that a century later this logic would form the foundation for digital computers.<br />
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Where would computational complexity be without Boolean algebra? We can use AND, OR, and NOT gates in place of Turing machines to capture computation: A polynomial-sized circuit of these gates can simulate any polynomial-time computable algorithm.<br />
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Stephen Cook analyzed the complexity of determining whether a formula φ of Boolean variables connected by AND, OR and NOTs was a tautology, a mathematical law. Cook <a href="http://dx.doi.org/10.1145/800157.805047">showed</a> that every nondeterministic polynomial-time computable problem reduced to checking if φ is not a tautology, or equivalently that (NOT φ) is satisfiable. That paper, as you well know, gave us the P v NP problem.<br />
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So here's to George Boole, whose desire to give mathematics a firm foundation produced simple tools powerful enough to describe everything.
http://blog.computationalcomplexity.org/2015/11/george-boole-1815-1864.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-99090894343850375Thu, 29 Oct 2015 11:38:00 +00002015-10-29T07:47:28.930-04:00S. Barry Cooper (1943-2015)<div class="separator" style="clear: both; text-align: center;">
<a href="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Barry_Cooper.jpg/220px-Barry_Cooper.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Barry_Cooper.jpg/220px-Barry_Cooper.jpg" width="146" /></a></div>
<a href="https://en.wikipedia.org/wiki/S._Barry_Cooper">Barry Cooper</a>, a computability theorist and professor at the University of Leeds, passed away on Monday after a brief illness. Cooper was a big proponent of computability and Alan Turing in particular. He chaired the advisor committee for the <a href="http://www.mathcomp.leeds.ac.uk/turing2012/">Turing Year</a> celebrations in 2012, and organized a number of events in Cambridge around the centennial of Turing's birth on June 23.<br />
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For the Turing year, Cooper and Jan van Leeuwen co-edited a massive volume <a href="http://www.amazon.com/Alan-Turing-His-Work-Impact/dp/0123869803">Alan Turing: His Work and Impact</a> which includes all of Turing's publications and a number of commentaries on the legacy of that work. I wrote a piece on <a href="https://www.dropbox.com/s/wtbthhylhlzrgku/Turing-Dots.pdf?dl=0">Turing's Dots</a> for the volume. Alan Turing: His Work and Impact received the <a href="https://proseawards.com/winners/2013-award-winners/#body">R. R. Hawkins Award</a>, the top award for professional and scholarly publishing across all the arts and sciences.<br />
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Barry Cooper was also the driving force behind <a href="http://www.computability.org.uk/">Computability in Europe</a>, an eclectic meeting to discuss all things related to computation.<br />
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I hope Barry now gets to meet his hero Turing but we'll miss him greatly down here.http://blog.computationalcomplexity.org/2015/10/s-barry-cooper-1943-2015.htmlnoreply@blogger.com (Lance Fortnow)5tag:blogger.com,1999:blog-3722233.post-4122561839268621412Mon, 26 Oct 2015 16:51:00 +00002015-10-26T12:51:25.972-04:00A human-readable proof that every planar graph is 4.5-colorableAbout 20 years ago I went to a talk by <a href="http://faculty.uml.edu/jpropp/">Jim Propp</a> on the fractional chromatic number of a graph. (<a href="http://www.ams.jhu.edu/~ers/fgt/index.html">Here</a> is a link to a free legal copy of the book on fractional graph theory by <a href="http://www.ams.jhu.edu/~ers/">Ed Scheinerman</a> and <a href="http://home.gwu.edu/~dullman/">Dan Ullman</a>.)<br />
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Let c be a natural number. A graph G=(V,E) is c-colorable if there is a mapping of the vertices of G to the vertices of K<sub>c</sub> such that (x,y) ∈ E --> (x,y) is an edge in K<sub>c</sub><br />
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Let K<sub>t:k</sub> be the <a href="https://en.wikipedia.org/wiki/Kneser_graph">Knesser graph</a>: the vertices are all k-element subsets of {1,...,t}, (A,B) is an edge if A∩B=∅. A graph is t/k-colorable if there is a mapping of the vertices of G to the vertices of K<sub>t:k</sub> such that (x,y) ∈ E --> (x,y) is an edge in K<sub>t:k</sub>. (That is not quite right as that might depend on if you use, say, 15/10 or 3/2 but is close. See page 40-42 in the book linked to above for the formal definition.)<br />
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One can also show that this is equivalent to other definitions of fractional chromatic number (one involves a relaxation of an LP problem).<br />
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The big open problem in the field and perhaps the motivation for it was this:<br />
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Known and the proof is human-readable: Every Planar Graph is 5-colorable.<br />
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Known and the current proof is NOT human-readable: Every Planar Graph is 4-colorable.<br />
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OPEN: Is there some number 4<c<5 such that every planar graph is c-colorable and the proof is human-readable?<br />
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I had not thought about this problem for a long time when I was invited by <a href="http://www.people.vcu.edu/~dcranston/">Dan Cranston</a> to give a talk at a Discrete Math Seminar in Virginia Commonwealth University. Looking over the prior <a href="http://www.people.vcu.edu/~dcranston/DM-seminar">talks in the seminar</a> I noticed a talk on <a href="http://arxiv.org/abs/1501.01647">Fractional Chromatic Number of the Plane</a> by Dan Cranston. I inquired:<br />
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BILL: Fractional colorings! Has there been progress on finding a number less than 5 so that the proof that every planar graph is c colorable is reasonable?<br />
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DAN: Yes. By myself and <a href="http://asu.academia.edu/LandonRabern">Landon Rabern</a>. We have shown that every planar graph is 4.5-colorable in <a href="http://arxiv.org/abs/1410.7233">this paper!</a> I've given talks on it, <a href="http://www.people.vcu.edu/~dcranston/slides/planar-fractional-SFU.pdf">here are my slides.</a><br />
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BILL: This is big news! I am surprised I haven't heard about it.<br />
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DAN: The paper is on arXiv but not published yet. Also, while you and me think its a great result, since its already known that all planar graphs are 4-colorable, some people are not interested.<br />
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BILL: Too bad you didn't prove it in 1975 (the four-color theorem was proven in 1976).<br />
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DAN: I was in kindergarten. <br />
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BILL: Were you good at math?<br />
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DAN: My finger paintings were fractional colorings of the plane and I never used more than 4.5 colors.<br />
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DAN: By the way, the paper I spoke about in seminar was about fractional colorings of the plane. The graph is: all points in the plane are vertices, and two points have an edge if they are an inch apart. The ordinary chromatic number is known to be between 4 and 7. Tighter bounds are known for the fractional chromatic number, see the paper linked to above when you first mention that paper.<br />
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Some thoughts:<br />
<br />
1) I would have thought that they would first get something like 4.997 and then whittle it down. No, it went from 5 right to 4.5. Reducing it any further looks hard and they proved that it cannot be a tweak of their proof.<br />
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2) The paper is readable. Its very clever and doesn't really use anything not known in 1975. But the problem wasn't known then and of course the right combination of ideas would have been hard to find back the. In particular, the method of discharging is better understood now.<br />
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3) When I told this to Clyde Kruskal he wanted to know if there was a rigorous definition of ``human-readable'' since the open question asked for a human-readable proof. I doubt there is or that there can be, but this paper clearly qualifies. Perhaps a paper is human-readable if 2 humans have actually read it and understood it. Or perhaps you can parameterize is and have n-human-readable for n humans.<br />
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4) Why does my spell checker thing <i>colorable</i> and <i>colorings</i> and <i>parameterize</i> are not words?<br />
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5) The serendipity: I'm very happy to know the result. I came upon it by complete accident. I'm bothered that there may be other results I want to know about but don't. How does one keep up? One way is to check arXiv's once-a-week or on some regular basis. But is that a good use of your time? I ask non-rhetorical. http://blog.computationalcomplexity.org/2015/10/a-human-readable-proof-that-every.htmlnoreply@blogger.com (GASARCH)17tag:blogger.com,1999:blog-3722233.post-54499659883146446Thu, 22 Oct 2015 11:55:00 +00002015-10-22T07:55:27.112-04:00Complexity Blast from the Past<i>My friend Marty in grad school mysteriously disappeared in 1985 and showed up yesterday looking exactly the same as I remember him. He said he traveled into my time and asked me how we finally proved P different than NP. I didn't give him the answer he seeked but I let him look around at what complexity has done in the past thirty years. He read through the night and agreed to share his thoughts on this blog.</i><br />
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When I left in '85 Yao and Håstad had just showed that Parity require exponential-size circuits and Razborov showed clique does not have poly-size monotone circuits. Looks like showing some NP problem didn't have poly-size circuits was right around corner so I'm a bit disappointed that didn't happen. I would have thought you would have proven NP is not computable in log space (nope), NP not computable by Boolean formulae (nope) or NP not computable by constant depth circuits with threshold gates (nope). All I see are excuses like "natural proofs". You did show lower bounds for constant depth circuits with mod<sub>m</sub> gates for m prime but that happened back just a year after I left. For m composite you need to go all the way to NEXP and even that took another 25 years.<br />
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Back in 1984 we had a 3n lower circuit bound for an explicit function and surely you have much better bounds now, at least quadratic. I almost didn't find any improvement until I saw a <a href="http://eccc.hpi-web.de/report/2015/166/">paper written last week</a> that improves this lower bound to a whopping 3.011n. You can't make this stuff up.<br />
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I shouldn't be that hard on you complexity theorists. All that work on interactive proofs, PCPs, approximations and unique games is pretty cool. Some real impressive stuff on pseudorandom generators and error-correcting codes. Who would have thunk that NL = co-NL or that the entire polynomial-time hierarchy reduces to counting? Nice to see you finally proved that Primes in P and undirected graph connectivity in log-space.<br />
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Oh no! Biff stole Lance's copy of Arora-Barak and is taking it back in time. This could change computational complexity forever. I'd better go find Doc Brown and make things right.<br />
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[Lance's Note: Thanks to Mostafa Ammar for inspiring this post.]http://blog.computationalcomplexity.org/2015/10/complexity-blast-from-past.htmlnoreply@blogger.com (Lance Fortnow)4tag:blogger.com,1999:blog-3722233.post-5862285161843940695Sun, 18 Oct 2015 22:13:00 +00002015-10-18T18:13:47.960-04:00Amazon going after fake reviews but mine is still postedI have always wondered why YELP and Amazon Reviews and other review sites work as well as they do since a company COULD flood one with false reviews (high marks for their company or low marks for their competitors). From what I've seen this does not seem to be a big problem.<br />
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Even so, it is A problem, and amazon is taking action on it. See <a href="http://www.theguardian.com/technology/2015/oct/18/amazon-sues-1000-fake-reviewers">here </a>for details.<br />
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A while back I posted a review that is... not sure its false, but I am surprised they allowed it and still allow it. What happened: I bought a copy of my own book <a href="http://www.amazon.com/Bounded-Queries-Recursion-Progress-Computer/dp/1461268486">Bounded Queries in Recursion Theory (Gasarch and Martin)</a> since my dept wanted a copy to put in one of those glass cases of faculty books. I bought it used for about $10.00. I got email from amazon asking if I wanted to review it, so I thought YES, I'll review my book! I wrote:<br />
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This is a great book. I should know, I wrote it.<br />
I am surprised Amazon ASKED me for my opinion.<br />
Seriously- I also have a survey which says whats in the book<br />
better, email me if you want it, gasarch@cs.umd.edu<br />
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That was in November of 2014. Its still posted. Is it a false review? Not clear since everything I say in it is true and I am honestly saying who I am.<br />
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If Amazon removed it I would be surprised they noticed.<br />
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If Amazon does not remove it I'm surprised they allow it.<br />
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Is it possible to be surprised both ways?http://blog.computationalcomplexity.org/2015/10/amazon-going-after-fake-reviews-but.htmlnoreply@blogger.com (GASARCH)3tag:blogger.com,1999:blog-3722233.post-2196224000699687478Thu, 15 Oct 2015 12:12:00 +00002015-10-15T08:12:12.067-04:002015 Fall Jobs PostWhen the 2014 fall jobs post is our most popular post, you know it is time for the 2015 jobs post. This year instead of (or in addition to) posting your jobs in the comments, post your theory jobs to <a href="http://cstheory-jobs.org/">Theoretical Computer Science Jobs</a> site coordinated by Boaz Barak.<br />
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For job searchers the standard official sites for CS faculty jobs are the the <a href="http://www.cra.org/ads/">CRA</a> and the <a href="http://jobs.acm.org/c/search_results.cfm?max=25&site_id=1603&vnet=0&keywords=faculty+or+postdoc&search=Search">ACM</a>. Check out postdocs and other opportunities on the <a href="http://cstheory-jobs.org/">TCS Jobs</a> site mentioned above and <a href="http://dmatheorynet.blogspot.com/">Theory Announcements</a>. It never hurts to check out the webpages of departments or to contact people to see if positions are available. Even if theory is not listed as a specific hiring priority you may want to apply anyway since some departments may hire theorists when other opportunities to hire dry up.<br />
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Many if not most computer science department should be trying to expand again this year to keep up with ever growing enrollments. Hiring in theoretical computer science is harder to predict, not likely to be a priority in many departments. You can make yourself more valuable by showing a willingness to participate and teach beyond core theory. Machine learning, data science and information security are areas of great need where theorists can play a large role.<br />
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Don't ignore postdoc and lecturer positions that will give you some extra training and a stronger CV for future searchers. Think global--there are growing theory groups around the world.<br />
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Good luck out there and I look forward to seeing your names on the Spring 2016 jobs post.http://blog.computationalcomplexity.org/2015/10/2015-fall-jobs-post.htmlnoreply@blogger.com (Lance Fortnow)15tag:blogger.com,1999:blog-3722233.post-1944588051277695048Mon, 12 Oct 2015 03:52:00 +00002015-10-11T23:52:31.627-04:00Moore's Law of Code OptimizationAn early version of Moore's law is as follows:<br />
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<blockquote>
HARDWARE: The number of transistors on an integrated circuits doubles every 18 MONTHS.</blockquote>
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Moore's law is often used to refer to the following:<br />
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<blockquote>
SPEED: The speed of computers due to hardware doubles every 18 MONTHS.</blockquote>
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There are other versions as well. I've heard that some versions of it are no longer working (it couldn't go on forever). But what about the gains made NOT by hardware? Is there a Moore's Law of Code Optimization?<br />
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There is! Its called <a href="http://www.cs.virginia.edu/~techrep/CS-2001-12.pdf">Proebstring's law</a><br />
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<blockquote>
SPEED: The speed of computers due to code opt doubles every 18 YEARS. </blockquote>
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The paper pointed to gives evidence for this law.<br />
So is Code Opt worth it? I've heard the following:<br />
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1) Some of the Code Opts eventually go into the hardware, so its not quite fair to say that Code Opts improve speed that slowly.<br />
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2) Any improvement is worth having.<br />
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3) Being forced to think about such issues leads to other benefits.http://blog.computationalcomplexity.org/2015/10/moores-law-of-code-optimization.htmlnoreply@blogger.com (GASARCH)3tag:blogger.com,1999:blog-3722233.post-8347866383318862104Wed, 07 Oct 2015 12:53:00 +00002015-10-08T08:45:24.576-04:00Randomness by ComplexityLet n be the following number<br />
135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563 (<a href="https://en.wikipedia.org/wiki/RSA_numbers#RSA-1024">RSA 1024</a>)<br />
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What is the probability that the fifth least significant bit of the smallest prime factor of n is a one? This bit is fully defined--the probability is either one or zero. But if gave me better than 50/50 odds one way or the other I would take the bet, unless you worked for RSA Labs.<br />
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How does this jibe with a Frequentist or Bayesian view of probability? No matter how often you factor the number you will always get the same answer. No matter what randomized process was used to generate the original primes, conditioned on n being the number above, the bit is determined.<br />
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Whether we flip a coin, shuffle cards, choose lottery balls or use a PRG, we are not creating truly random bits, just a complex process who unpredictability is,we hope, indistinguishable from true randomness. We know from <a href="http://dx.doi.org/10.1145/258533.258590">Impagliazzo-Wigderson</a>, and its many predecessors and successors, that any sufficiently complex process can be converted to a PRG indistinguishable from true randomness. Kolmogorov complexity tells us we can treat a single string with no short description as a "random string".<br />
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That's how we ground randomness in complexity: Randomness is not some distribution over something not determined, just something we cannot determine.http://blog.computationalcomplexity.org/2015/10/randomness-by-complexity.htmlnoreply@blogger.com (Lance Fortnow)6tag:blogger.com,1999:blog-3722233.post-1811916031439740547Mon, 05 Oct 2015 21:48:00 +00002015-10-05T17:48:55.699-04:00Is Kim Davis also against Nonconstrutive proofs?<br />
Recall that Kim Davis is the Kentucky clerk who refused to issue marriage licenses for same-sex couples and was cheered on by Mike Huckabee and other Republican candidates for Prez. Had she refused to issue marriage license for couples that had been previously divorced than neither Mike Huckabee would not be supporting her, and the Pope wouldn't have a private 15 minute meeting with her telling her to stay strong (NOTE- I wrote this post in that tiny window of time when it was believed she did have such a meeting with the Pope, which is not true. The Pope DID meet with a former student of his who is gay, and that studen'ts partner.) Had she refused to issue marriage licenses to inter-racial couples (this did happen in the years after the Supreme court said that states could not ban interracial marriage, <a href="https://en.wikipedia.org/wiki/Loving_v._Virginia">Loving vs Virginia, 1967</a> ) then ... hmm, I'm curious what would happen. Suffie to say that Mike H and the others would prob not be supporting her.<br />
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David Hilbert solved Gordon's problem using methods that were nonconstructive in (I think) the 1890's. This as considered controversial and Gordon famously said <i>this is not math, this is theology.</i> Had someone else solved this problem in 1990 then the fact that the proof is non-constructive might be noted, and the desire for a constructive proof might have been stated, but nobody would think the proof was merely theology.<br />
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I don't think the Prob Method was ever controversial; however, it was originally not used much and a paper might highlight its use in the title or abstract. Now its used so often that it would be unusual to point to it as a novel part of a paper. If I maintained a website of <i>Uses of the Prob Method in Computer Science</i> then it would be very hard to maintain since papers use it without commentary.<br />
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The same is becoming true of Ramsey Theory. I DO maintain a website of apps of Ramsey Theory to CS (see <a href="http://www.cs.umd.edu/~gasarch/TOPICS/ramsey/ramsey.html">here</a>) and its geting harder to maintain since using Ramsey Theory is not quite so novel as to be worth a mention.<br />
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SO- when does a math technique (e.g., prob method) or a social attuitude (e.g., acceptance of same-sex marriage) cross a threshold where its no longer controversial? Or no longer novel? How can you tell? Is it sudden or gradual? Comment on other examples from Math! CS! http://blog.computationalcomplexity.org/2015/10/is-kim-davis-also-against.htmlnoreply@blogger.com (GASARCH)3tag:blogger.com,1999:blog-3722233.post-4214365156853800764Thu, 01 Oct 2015 16:48:00 +00002015-10-06T12:09:24.755-04:00Cancer Sucks<div>
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<a href="http://www.cc.gatech.edu/sites/default/files/images/people/karsten-schwan_1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://www.cc.gatech.edu/sites/default/files/images/people/karsten-schwan_1.jpg" height="200" width="160" /></a></div>
Karsten Schwan said the title quote when we were gathered as a faculty two years ago mourning the Georgia Tech School of Computer Science faculty member Mary Jean Harrold who <a href="http://cacm.acm.org/news/168025-in-memoriam-mary-jean-harrold-1947-2013/fulltext">died from the disease</a>. Karsten just <a href="http://www.scs.gatech.edu/content/college-computing-mourns-loss-regents%E2%80%99-professor-karsten-schwan">lost his own battle with cancer</a> monday morning and my department is now mourning another great faculty member.<br />
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Just a few months ago, Alberto Apostolico, an algorithms professor at Georgia Tech, also <a href="http://www.cc.gatech.edu/news/427841/college-computing-remembers-alberto-apostolico">passed away from cancer</a>. </div>
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I went back through the obituaries in the blog and we lost quite a few to cancer, often way too young, including <a href="http://xn--mihai%20ptracu-ovb98l/">Mihai Pătraşcu</a>, <a href="http://blog.computationalcomplexity.org/2010/10/benoit-mandelbrot-1924-2010.html">Benoît Mandelbrot</a>, <a href="http://blog.computationalcomplexity.org/2010/10/partha-niyogi-1967-2010.html">Partha Niyogi</a>, <a href="http://blog.computationalcomplexity.org/2008/12/ingo-wegener-1950-2008.html">Ingo Wegener</a>, <a href="http://blog.computationalcomplexity.org/2005/04/clemens-lautemann.html">Clemens Lautemann</a> and <a href="http://blog.computationalcomplexity.org/2004/07/carl-smith-1950-2004.html">Carl Smith</a>. I just taught Lautemann's proof that BPP is in Σ<sup>p</sup><sub>2</sub>∩Π<sup>p</sup><sub>2</sub> in class yesterday.<br />
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With apologies to Einstein, God does play dice with people's lives, taking them at random in this cruel way. Maybe someday we'll find a way to cure or mitigate this terrible disease but for now all I can say is Cancer Sucks. </div>
http://blog.computationalcomplexity.org/2015/10/cancer-sucks.htmlnoreply@blogger.com (Lance Fortnow)2tag:blogger.com,1999:blog-3722233.post-339437391065343906Mon, 28 Sep 2015 17:14:00 +00002015-09-28T13:14:41.926-04:00Venn Diagrams are used (yeah) but abused (boo)<br />
In a <a href="http://blog.computationalcomplexity.org/search?q=Venn+diagrams">prior blog entry</a> I speculated about which math phrases will enter the English Language and will they be used correctly. I thought <i>Prisoner's Dilemma</i> would enter and be used correctly, but <i>Turing Test</i> and <i>Venn Diagram</i> would not enter.<br />
<br />
Since then I've seen Turing Test used, but only because of the movie <i>The</i> <i>Imitation Game</i>. I don't think it will enter the human language until a computer actually passes it for real, which might not be for a while. See <a href="http://www.scottaaronson.com/blog/?p=1858">this</a> excellent post by Scott Aaronson about a recent bogus claim that a machine passed the Turing Test.<br />
<br />
Venn Diagrams seem to be used more (Yeah!) but incorrectly (Boo!)<br />
<br />
1) In <a href="http://www.thedailybeast.com/articles/2015/09/10/will-republicans-move-to-unseat-boehner.html">this article</a> (which inspired this post), about who might replace John Boehner as speaker of the house, there is the following passage:<br />
<br />
<i>Option 3: An acceptable and respected conservative like Jeb Hensarling or<br />Tom Price emerges as speaker. Why these two? First, Paul Ryan doesn’t seem<br />to want the gig, so that leaves us with only a few options for someone who<br />fits in the Venn diagram of being enough of an outsider, well liked, and<br />sufficiently conservative</i>.<br />
<br />
Is this correct use? They really mean the intersection of oustider, well-liked,<br />
and suff conservative. One can picture it and it sort of makes sense, but its not<br />
quite correct mathematically.<br />
<br />
2) In <a href="http://popchartlab.tumblr.com/post/93786896999/in-celebration-of-john-venns-180th-birthday">this ad</a> for Venn Beer (in celebration of John Venn's 180th birthday!) they really mean union, not intersection.<br />
<br />
3) <a href="http://latenightseth.tumblr.com/post/93793705536/happy-180th-birthday-to-john-venn-creator-of-the">This Venn Diagram</a> about Vladmir Putin's and your Aunt's record collection doesn't really make sense but I know what they mean and its funny.<br />
<br />
4) <a href="http://thefaultinourstarsmovie.com/post/87845256595/and-you-thought-women-were-complicated-turns-out">This Venn Diagram</a> about how to woo women is incorrect, not funny, not mathematically meaningful. <br />
<br />
5) <a href="http://imgur.com/IT5FP?tags">This Venn Diagram</a> involved Doctors, Prostitutes, and TSA agents. At first it is funny and seems to make sense. But then you realize that the intersection of Doctors and Prostitutes is NOT People who make more per hours than you make all day, its actually prostitutes with medical degrees. Its still funny and I see what they are getting at.<br />
<br />
6) This <a href="http://fivethirtyeight.com/datalab/donald-trump-is-the-nickelback-of-gop-candidates/">Venn Diagram</a> (it's later in the article) of Republican Candidates for the 2016 nomination for Prez is correct for the math and informative, though one may disagree with some of it (Is Trump really Tea-Party or should he be in his own category of Trumpness?)<br />
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<br />http://blog.computationalcomplexity.org/2015/09/venn-diagrams-are-used-yeah-but-abused.htmlnoreply@blogger.com (GASARCH)4tag:blogger.com,1999:blog-3722233.post-3357661575827123993Thu, 24 Sep 2015 12:23:00 +00002015-09-24T08:25:00.767-04:0021st Century ProblemsMy youngest daughter, Molly, a high school senior talking colleges with a woman about ten years her senior. The woman remembered all her friends watching the clock so they could go home to check their emails to see if they were accepted. Molly said "Sheesh, you had to go home to check email?"<br />
<br />
My other daughter Annie, a college junior, went on an overnight last Thursday to a place without cell phone reception. She spent Friday night with her friends in her class catching up on emails, texts and Facebook messages.<br />
<br />
Now back in my day (that being the early 80's) we got our college acceptances and rejections by postal mail, where that one crucial bit of information could be determined by the thickness of the envelope. Some of my friends had their mail held by the post office so they could find out a few hours earlier.<br />
<br />
In college I did have email access, in fact I <a href="http://blog.computationalcomplexity.org/2011/06/creating-email-system-at-cornell.html">wrote an email system for Cornell</a>. But most students didn't use email so we resorted to other means. Student organizations could hire a service that put posters on key locations throughout campus. Chalk on sidewalks worked particularly well. The personals section of the Cornell Daily Sun served as a campus bulletin board. In my freshman dorm we had phones in our rooms but no answering machines. We did put little whiteboards on our doors so people could leave us messages. We had a lounge on our floor where you could find most people and talk to them in person. You young people should try that more often.<br />
<br />
We had to coordinate activities and meeting places ahead of time, if someone was late you waited for them. On the other hand I never had to spend my Friday nights catching up on emails and texts.http://blog.computationalcomplexity.org/2015/09/21st-century-problems.htmlnoreply@blogger.com (Lance Fortnow)1tag:blogger.com,1999:blog-3722233.post-6876765130667206397Mon, 21 Sep 2015 12:55:00 +00002015-09-21T08:55:54.787-04:00When did Mathematicians realize that Fermat did not have a proof of FLT?I recently came across the following passage which is about Fermat's Last Theorem (FLT).<br />
<br />
<i>Pierre de Fermat had found a proof, but he did not bother to write it down. This is perhaps the most frustrating note in the history of mathematics, particularly as Fermat took his secret to the grave.</i><br />
<br />
<br />
AH- so at one time people thought Fermat DID have a proof of FLT. That is, a proof using just the math of his time, likely a very clever proof. I doubt anyone thinks that Fermat had a proof in this day and age. Actually it has been in fiction: in a 2010 episode Dr. Who episode <i>The eleventh hour, </i>the doctor has to prove to some very smart people that they should take his advice. He does this by showing them Fermat's proof of FLT. Good Science Fiction but highly unlikely as Science Fact. In an episode of ST-TNG (Title: The Royale. Year: 1989) it is claimed that FLT is still open. Whoops. But in an episode of ST-DSN (Title: Facets. Year: 1995) they refer to `Wiles proof of FLT'.<br />
<br />
Wikipedia states:<i> It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely.</i> I think that understates the case.<br />
<br />
<br />
So here is a question for all you math historians out there: When did the math community realize that FLT was really hard?<br />
<br />
We have one clue- the quote I began with. Its from... 2013. Whoops. The book is <i><a href="http://www.amazon.com/The-Simpsons-Their-Mathematical-Secrets/dp/1620402785/ref=pd_sim_14_6?ie=UTF8&refRID=0BWFXMJ1Z2DTNXWFQDKN&dpID=51woMcLhaRL&dpSrc=sims&preST=_AC_UL160_SR107%2C160_">The Simpsons and their mathematical secrets </a></i>by Simon Singh (Author of <a href="http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/product-reviews/0385493622/ref=cm_cr_pr_hist_2/190-0701399-4452552?ie=UTF8&filterBy=addTwoStar&showViewpoints=0&sortBy=helpful&reviewerType=all_reviews&formatType=all_formats&filterByStar=two_star&pageNumber=1">Fermat's Enigma</a> which is about the quest to proof FLT). I've read the passages about FLT in the Simpsons book over again to make sure he doesn't someplace say that Fermat prob didn't have a proof. No--- he seems to really say the Fermat had a proof. So whats going on here? Possibilities:<br />
<br />
1) I'm wrong. There are serious credible people who think Fermat had a proof and he talked to them; perhaps while working Fermat's Enigma. I find this unlikely- I have never, literally never, heard of anyone, not even math cranks, who think Fermat had a simple proof. Some cranks think THEY have a simple proof, though even that seems far less common after FLT was proven.<br />
<br />
2) I'm right. He didn't have anyone who was both serious and credible check his book. I find this unlikely. He has written Fermat's 't Enigma so surely he is in contact with people that are both credible and serious.<br />
<br />
3) He did have someone check his book but thought the story was better the way he told it. (This was common on the TV show <i>Numb3rs </i>which never let a mathematical truth get in the way of a good story.)<br />
I find this unlikely since a better way to say it is <i>we'll never know if Fermat had a proof!</i> <br />
<br />
One problem with such mistakes is that it destroys his credibility on other math things he writes of. That info about Dr. Who I got from the book, but I suspect its correct. And the stuff about the math that appears in the Simpsons is checkable and seems correct. I give one example: in one episode you see in the background<br />
<br />
3987<sup>12</sup> + 4365<sup>12</sup>= 4472<sup>12</sup><br />
<br />
which is correct on a calculator with only 10 digits of precision. Cute! But I have stopped reading any of the math history in the book for fear that I will get incorrect notions in my head.<br />
<br />
However, back to my original question: Was there a time when people thought Fermat really had a proof?Was there a time when people thought there was an easy proof? When did that change?<br />
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http://blog.computationalcomplexity.org/2015/09/when-did-mathematicians-realize-that.htmlnoreply@blogger.com (GASARCH)9tag:blogger.com,1999:blog-3722233.post-9003988703313356477Thu, 17 Sep 2015 07:49:00 +00002015-09-17T03:49:25.754-04:00The Theorems ConferenceAll too often theoretical computer scientists get more obsessed by proofs than the theorems themselves. I suggest a theorems conference. Here's how it would work:<br />
<br />
Authors would submit two versions of a paper. One has the statement of the theorem and why that theorem matters, but no proofs. The second version includes the proofs.<br />
<br />
The program committee first makes tentative decisions based on the first version of the paper. If tentatively accepted the PC then looks at the second version. The PC can reject the paper if the the proofs have significant flaws, gaps or readability issues. The PC cannot reject the paper for any other aspect of the proof such as length or lack of technical depth or originality.<br />
<br />
This way we truly judge papers based on what they prove--what the results add to the knowledge base of the field.<br />
<br />
Of course my plan has many flaws. Some papers with their proofs may have already been posted on archive sites which the PC members could have seen. More likely, the PC will guess the difficulty of the proof and judge the paper based on this perceived difficulty, and not on the theorem itself.<br />
<br />
We need a culture shift, away from an emphasis on proofs. That's the only way we can judge our results for the results themselves.http://blog.computationalcomplexity.org/2015/09/the-theorems-conference.htmlnoreply@blogger.com (Lance Fortnow)13tag:blogger.com,1999:blog-3722233.post-8678647584142438200Mon, 14 Sep 2015 04:00:00 +00002015-09-14T00:00:59.908-04:00An Open(?) Question about Prime Producting Polynomials Part II in 3-D!(Apologies- No, this post is not in 3-D)<br />
<br />
I posted last week about <a href="http://blog.computationalcomplexity.org/2015/09/an-open-question-about-prime-producing.html">An Open(?) Question about Prime Producing Polynomails</a><br />
<br />
I got several emails about the post with more information, inspiring this post!<br />
All the math in this post can be found in my writeup <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/polyprimes.pdf">Polynomials and Primes,</a><br />
unless otherwise specified. NONE of the results are mine. <br />
<br />
You can read this post without reading the prior one.<br />
<br />
1) KNOWN: Let f(x) ∈ Z[x] be a poly of degree d. Then there exists a non prime in f(x)'s image (actually there are an infinite number of non primes, in fact there are an infinite number of composites). If f(1) is prime then of f(1+mf(1)) as m=0,1,2,... at most 2d-2 of them are prime.<br />
<br />
2) Algorithm to find an x such that f(x) is not prime: compute f(1), f(1+f(1)),...,f(1+(2d-2)f(1)) until you find one that is not prime. This takes 2d-1 evals. OPEN(?): Is there a better deterministic algorithm where we measure complexity by number of evals? Since this is a simple model of computation lower bounds might be possible.<br />
<br />
3) There is the following randomized algorithm: Eval f(1)- if its not prime you're done. If f(1) is prime then pick a random m where 0≤ m ≤ (2d-2)<sup>2</sup> and eval f(1+mf(1)). This is non prime with prob 1- (1/(2d-1)).<br />
<br />
4) What is it about Z that makes this theorem true? In my write up I show that if D is an integral domain with only a finite number of units (numbers that have mult inverses) then any poly in D[x] has to produce non-primes infinitely often. (A prime in D is a number a such that if a=bc then either b or c is a unit.)<br />
<br />
5) What about if D has an infinite number of units? See <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/polyprimesamm.pdf">this</a> paper for examples of polynomials over integral domains D such that the poly only takes on only prime or unit values.<br />
<br />
6) What about polys over Q? over R? over C? In my write up I prove similar theorems for Q and then use that to get theorems for C.<br />
<br />
7) Looking at polys in Z[x,y] is much harder, see<a href="http://www.jstor.org/stable/2975080?seq=1#page_scan_tab_contents"> this survey. </a><br />
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8) If f(x)∈Z[x] is a poly then does there exist a prime in f(x)'s image? An infinite number of primes? Easy but stupid answer is no: f(x)=2x. Better question: assume that f(x)'s coefficients are rel prime.<br />
<br />
Dirichlet's theorem: if GCD(a,b)=1 then ax+b is prime infinitely often.<br />
<br />
Open: is x<sup>2</sup> + 1 prime infinitely often?<br />
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<br />http://blog.computationalcomplexity.org/2015/09/an-open-question-about-prime-producting.htmlnoreply@blogger.com (GASARCH)1tag:blogger.com,1999:blog-3722233.post-5586490118604985932Thu, 10 Sep 2015 14:23:00 +00002015-09-10T10:23:47.541-04:00Designer ChipsA computer architecture researcher talked to me about a role theoretical computer science can play for them: creating a new kind of computer processor. Microprocessors stopped getting faster a decade ago due to energy challenges so computer architects look for new ways to improve performance, moving away from the general-purpose CPU towards processors that handle more specific functions. The GPU, Graphics Processor Unit, has long been around to handle the graphics-intensive needs of modern computers and many have used GPUs for other purposes such as machine learning. These days we can program chips using FPGAs (Field-programming gate arrays) and are nearly at the point of cheaply compiling directly to hardware. How does this change the theory we do?<br />
<br />
What kind of specialized chips would speed up our algorithms? If we want to find matchings on graphs, for example, is there some routine one could put in a chip that would lead to a much more efficient algorithm?<br />
<br />
On the complexity side, how do we model a computer where we can program the hardware as well as the software? What are the right resource bounds and tradeoffs?<br />
<br />
In general our notions of computers are changing, now with multi-core, cloud computing and designer chips. Not only should we focus on applying theory to these new models of computing, but we should think about what future changes in computing could yield more efficient algorithms. Theorists should be involved in planning the future of computing and we're not even doing a great job reacting to changes around us.http://blog.computationalcomplexity.org/2015/09/designer-chips.htmlnoreply@blogger.com (Lance Fortnow)2tag:blogger.com,1999:blog-3722233.post-5240612111408130205Mon, 07 Sep 2015 05:20:00 +00002015-09-09T11:53:34.862-04:00An Open(?) question about prime-producing-polynomials<br />
Known Theorem: If f(x)∈ Z[x] is prime for all nat number inputs then f(x) is a constant.<br />
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NOTE- Recall that if p is a prime then so is -p.<br />
<br />
Known Proof: Assume f(x) has degree d. f(1) IS prime. Let f(1)=p. Look at<br />
<br />
f(1+p), f(1+2p),...,f(1+(2d+1)p).<br />
<br />
One can easily show that p divides all of these. Hence if they are all primes then they must all be p or -p. Since there are 2d+1 of them, at least d+1 of them are the same, say p. Hence f is the constant p.<br />
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END of known proof.<br />
<br />
Note that the proof gives the following theorem:<br />
<br />
Let f(x)∈ Z[x] of degree d. We assume f(1)≥ 0. Least a st f(a) is NOT prime is ≤ 1+(2d+1)p.<br />
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(This can prob be improved a bit with some cases, but its good enough for now.)<br />
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Recall Euler's poly x<sup>2</sup>-x+41 produces primes for x=0,...,40. But at 41 you get a composite. This is much smaller than the upper bound 1+(2d+1)p = 1+5*41=206.<br />
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Wolfram MathWorld has a <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">page</a> of other polys in Z[x] that produces lots of primes initially, but NONE come close to the bound.<br />
<br />
QUESTIONS:<br />
<br />
Proof a better upper bound.<br />
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Proof a better lower bound (Fix d and produce and infinite seq of polys of degree d...)<br />
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Close the gap!<br />
<br />
If this is already known, then let me know please.<br />
<br />
Can also ask for polys in Q[x], R[x], C[x]. For Q[x] and R[x] same theorem is true- no poly can produce all primes. I suspect also true for C[x] but I haven't seen it stated anywhere. (ADDED LATER- Proof for C[x] is easy. First proof<br />
for Q[x] and then by Lagrange interpoloation if a poly has inf many times<br />
where f(integer)=integer, poly is in Q[x].)<br />
<br />
You can also NOT include negative primes and see how that changes things.<br />
<br />http://blog.computationalcomplexity.org/2015/09/an-open-question-about-prime-producing.htmlnoreply@blogger.com (GASARCH)0tag:blogger.com,1999:blog-3722233.post-1199023809395005544Thu, 03 Sep 2015 20:01:00 +00002015-09-03T16:01:08.556-04:00WhiplashedI recently watched the movie <a href="http://www.imdb.com/title/tt2582802/">Whiplash</a>, about a college jazz band director, Fletcher played by J.K. Simmons, who torments his musicians to force them to be their best. The movie focuses on a drummer, Andrew, which makes for a great audio/video feast but in its essentials Whiplash is a story of a professor and his student.<br />
<br />
I can imagine playing the role, “Do you think your proof is correct? Yes or No? Are you applying Toda’s theorem correctly or are you using the same crazy logic your dad used when he left your mom?” OK, maybe not.<br />
<br />
Nevertheless Fletcher has a point. Too often I’m seeing graduate student doing just enough to get a paper into a conference instead of pushing themselves, trying to do great work and still not being satisfied. Fletcher says the two most dangerous words in the English language are “good job”. While that might be a little cruel, we do need to push our students and ourselves to take risks in research and be okay in failing. To roughly quote John Shedd and Grace Murray Hopper, "the safest place for a ship is in the harbor, but that’s not what ships are for."<br />
<br />
Whiplash had a different kind of scene that definitely hit home. Andrew could not impress his family with the fact that he was lead drummer in the top college jazz band in the nation. I’ve been there, trying to get my mother excited by the fact that I had a STOC paper early in my career. "That's nice dear".http://blog.computationalcomplexity.org/2015/09/whiplashed.htmlnoreply@blogger.com (Lance Fortnow)6