CCC 2016

Earlier this week I attended the 31st Computational Complexity Conference in Tokyo. I've been to thirty of these meetings, missing only the 2012 conference in Porto. Eric Allender has attended them all.

The conference had a 130 participants with fewer women than you can count on one hand and 26 made it from the States. There were 34 accepted papers out of 91 submitted.

The proceedings are fully open access though the Dagstuhl LIPICS series, including the paper by Rahul Santhanam and myself that I presented at the meeting. The best paper by Marco Carmosino, Russell Impagliazzo, Valentine Kabanets and  Antonina Kolokolova drew a surprising strong connection between natural proofs and learning theory. In one of my favorite other talks, John Kim and Swastik Kopparty show how to decode Reed-Muller codes over an arbitrary product set instead of a structured field.

The German government will in the future no longer support LIPICS due to EU rules to prevent unfair competition with the commercial publishers. (Don't shoot the messenger) LIPICS will continue, the conferences will have to spend a little more to use them.

Next year's conference will be in Riga, Latvia July 6-9 right before ICALP in Warsaw. The 2018 meeting is likely to take place closely located to STOC in southern California.

Osamu Watanabe put together this slide show for the conference reception featuring pictures of attendees of the Complexity Conference through the ages, including the authors of this blog.

Peter van Emde Boas forwarded the call for papers and initial letters for the very first conference, originally called Structure in Complexity Theory.

New Ramsey Result that will be hard to verify but Ronald Graham thinks its right which is good enough for me.

If you finitely color the natural numbers there will be a monochromatic solution to

x+2y+3z - 5w = 0

There is a finite coloring of the natural numbers such that there is NO monochromatic solution to

x+2y+3z - 7w = 0

More generally:

An equation is REGULAR if any finite coloring of the naturals yields a mono solution.

RADO's THEOREM: A linear equation a1 x1 + ... + an xn = 0 is regular IFF some subset of the ai's sums to 0 (the ai's are all integers).

Rado's theorem is well known.

What about other equations? Ronald Graham has offered $100 to prove the following:

For all two colorings of the naturals there is a mono x,y,z such that

x² + y² = z² 

I've seen this conjecture before and I thought (as I am sure did others) that first there would be a prove that gave an ENORMOUS bound on

f(c) = the least n such that for all c-colorings of {1,...,n} there is a mono (x,y,z) such that ...

and then there would be some efforts using SAT solvers and such to get better bounds on f(c).

This is NOT what has happened. Instead there is now a paper by Heule, Kullmann, Marek, where they show f(2)=7825.

(NOTE- ORIGINAL POST HAD f(2)-7285. Typo, was pointed out in one of the comments
below. Now fixed.)


It is a computer proof and is the longest math proof ever. They also have a program that checked that the proof was correct.

And what did they get for their efforts? A check from Ronald Graham for $100.00 and a blog entry about it!

While I am sure there proof is correct I wish there was a human-readable proof that f(2) existsed even if it gave worse bounds. For that matter I wish there was a proof tha,t for all c, f(c) exists. Maybe one day; however, I suspect that we are not ready for such problems.

Theory Jobs 2016

In the fall we point to theory jobs, in the spring we see who got them. How is the CS enrollment explosion affecting the theory job market? We've seen some big name moves but that's only part of the picture.

Like last year and years past I created a fully editable Google Spreadsheet to crowd source who is going where. Ground rules:

• I set up separate sheets for faculty, industry and postdoc/visitors.
• People should be connected to theoretical computer science, broadly defined.
• Only add jobs that you are absolutely sure have been offered and accepted. This is not the place for speculation and rumors.
• You are welcome to add yourself, or people your department has hired.

This document will continue to grow as more jobs settle. So check it often.

Edit

My third post on Gathering for Gardners

(Workshop for women in computational topology in August: see here. For a post about these kinds of workshops see here.)


(I have already posted twice on stuff I saw or heard at the Gathering Conference here and here,)

Meta Point- At the Gathering for Gardner conference I learned lots of math (prob more accuarte to say I learned ABOUT lots of math) that I want to tell you about which is why this is my third post on it, and I post more.

The pamplets of  Lewis Carol: Games, Puzzles, and related pieces: This was mostly puzzles that are by now familiar, but one new (to me)  struck me: an aloof word  is a word where if you change any one letter to anything else then its no longer a word. I think aloof is such a word.

Some talk don't know which one was about the Piet Hein Egg, also called a superegg. The talk (which differs slightly from he page pointed to) said it was a solid whose surface has the equation

(x/a)^2.5 + (y/a)^2.5 + (z/b)^2.5

and its an egg which can stand on its end. (Note the x/a,y/a,z/b- that is correct, not a typo).
(Personal Note: Piet Hein invented Soma Cubes which is a puzzle where you put together 3-d pieces
made of small cubes into a large cube or other shapes. I learned about these in a Martin Gardner column and bought a set. I was very good at this- I put together every figure in the booklet within a week. This was the ONLY sign that I was GOOD at math when I was a kid, though there are many signs that was INTERESTED in math. About 30 years ago my girlfriend at that time and I went to a restaurant and there was a SOMA set on the table, assembled into a cube. I took it apart and she said ``Bill, you'll never be able to put it back together!!!'' I then ``tried'' to and ended up putting together instead a bathtub, a dog, a wall, and a W. But gee, it ``seemed'' like I was fumbling around and couldn't get a cube. ``Gee Bill, I think you've seen this puzzle before''. And who is this insightful girlfriend? My wife of over 20 years!)

Magic Magic Square (Sorry, dont know which talk) Try to construct a 4x4 magic square where (as usual) all rows and columns sum to the same thing. But also try to make all sets of four numbers that form a square (e.g., all four corners) also add to that number. Can you? If you insist on using naturals then I doubt it. Integers I also doubt it. But you CAN do it with rationals. How? If you want to figure it out yourself then DO NOT go to the answer which is at this link: here

Droste Effect: When a picture appears inside itself. For an example and why its called that see here

Black Hole Numbers: If you have a rule that takes numbers to numbers, are there numbers that ALL numbers eventually goto? If so, they are black hole numbers for that rule.

Map a number to the number of letters in its name

20 (twenty) --> 6 (six) --> 3 (three) --> 5 (five) --> four (4) --> four(4) --> ...

It turns out that ANY number eventually goes to 4.

Map a number to the sum of the digits of its divisors

12 has divisors 1,2,3,4,6,12 --> 1+2+3+4+6+1+2=19

19 has divisors 1,19 so --> 1+1+9 = 11

11 has divisors 1,11 so --> 1+1+1 = 3

3 has divisors 1,3 so --> 1+3=4

4 has divisors 1,2,4 so --> 1+2+4=7

7 has divisors 1,7 so --> 1+7=8

8 has divisors 1,2,4,8, --> 15

15 has divisors 1,3,5,15 --> 1+3+5+1+5 = 15

AH. It turns out ALL numbers eventually get to 15.

Boomerang Fractions: Given a fraction f do the following:

x1=1,  x2=1+f, x3- you can either add f to x2 or invert x2. Keep doing this. Your goal is to get back to 1 as soo nas possible.  Here is a paper on it: here. This notion can be generalized: given (s,f) start with s and try to get back so s. Can you always? how long would it take? Upper bounds?

Liar/Truth teller patterns on a square plane b Kotani Yoshiyuki. You have an 4 x 4 grid. Every grid point has a person. They all say ``I have exactly one liar adjacet (left, right, up, or down) to me.''
How many ways can this happen.  This can be massively generalized.

Speed Solving Rubit's cube by Van Grol and Rik. A robot can do it in 0.9 seconds: here.

Upfronts

The US television industry has long fascinated me, an entertainment outlet driven by technology. David Sarnoff introduced television at the World's Fair in 1939 and developed the NBC network to provide content so people would buy RCA televisions, much the way Steve Jobs created the iTunes store to sell iPods. For decades television was broadcast over the air funded mostly by commercials. You could only watch a show when it aired and people adjusted their schedules to the broadcast schedule. People stayed home instead of going to the theater, movies, social clubs and restaurants. They all still exist but not to the extent before television. The nature of jobs changed. One funny comedian on TV would make considerable money but would put hundreds of vaudeville comedians out of a job.

In the 70's came cable television to big cities, initially to provide a better signal. But it also provided more stations including stations that were paid explicitly by consumers like HBO and implicitly through cable subscriptions like ESPN. ESPN is the single largest source of revenue for Disney. Eventually we would have hundreds of cable stations, many very specialized.

In the 80's came the VCR, then the DVR. No longer did we need to plan our time around the TV broadcast schedule. Eventually TV shows could have a continuing story line allowing for richer plot and character development.

Then came the Internet and streaming video. You could watch videos from series and movies on Netflix to user generated short pieces on YouTube or shorter still on Vine. People are watching TV not so much on TVs anymore but on their computers and phones. Like many others we have cut the cable cord in the Fortnow household, a trend that the industry still tries to fathom. Every cord cutter is $6 less a month to ESPN and the Disney bottom line.

 Why bring up TV now? This is what used to be the most exciting week for television, the upfronts, where the broadcast networks reveal their new seasons to advertisers and the public at large. The networks are still having their presentations and parties, but the new shows fail to excite and quite a few retreads and revivals including 24, Prison Break, MacGyver, Tales from the Crypt, Gilmore Girls. Do you remember the Muppets returning last year? Neither do I.

We are in a golden age of television. One could take a rich novel and turn it into an equally rich 10-13 episode TV series. There were over 400 scripted TV series, and more really good series than I have time to watch (basically when I run on the treadmill). Meanwhile the networks continue to promote and party though an undercurrent of a very uncertain future. Watching the television industry is itself a never ending story.

Does this leak information?

Here are four fictional stories though inspired by real world events or TV shows (I forget which is which). My question is, was a confidence broken or was some information leaked that should not have been? I do not have answers.

Tenure: The candidate DOES find out the vote (e.g., 18 yes, 2 no) but DOES NOT find out who voted what. But what if the vote is 20 Yes 0 No. Then the candidate DOES know the vote. (Worse if it was 20 NO, 0 yes). I am sure this has been studied in crypto. Here is one solution: randomly flip one bit.

Lawyers:

CLIENT: I have roughly X dollars counting all of my assets. Are you the right firm to handle my estate?

LAWYER: Yes

CLIENT: Do you always say that?

LAWYER: No. If you had log(X) money then we would recommend a cheaper firm since your estate would not need our complex services. And if you had X^{10} money then there are other firms that are more familiar with investments at that level.

CLIENT: So, for example, Mitt Romney is not a client.

LAWYER: That is correct.

Did the lawyer break a confidence by saying that Mitt Romney was NOT a client? Could CLIENT goto lots of law firms and play this game and eventually find out Mitt Romney's  lawyer?

Nobel Prize: If he committee leaks that the winner has been notified THAT he or she won, but not WHO it was, is that a breach?

Someone has confessed to a priest that he murdered someone (a staple of TV shows and movies). The wrong man is in jail, whose  name is Bob.

PRIEST TO COP: You have the wrong man.

COP: How do you know.

PRIEST: I can't say how I know, but I know.

COP: Oh, It must be that the guilty man confessed to you but you can't break the seal of the confession. I won't ask you to. But here is a question: Has Bob been to confession lately?

PRIEST: No! (and he seems relieved to have gotten the message through)

Did the Priest betray the killers confidence?

People in Crypto (and elsewhere) define information, Knowledge, Security, similar terms formally so they can have protocols and try to prove things. Are their defintinitions realistic? In the above scenario's, are the above cases breaches or not? Is that even a rigorous question?

The Challenges of Smart Cities

Earlier this week I attended the CCC workshop Computing Research: Addressing National Priorities and Societal Needs (video). The workshop covered a large collection of topics, highlighting challenges of big data, privacy, security, sustainability, education, the future of work, CS funding and partnerships and more.

I'd like to highlight the challenges of Smart Cities, addressed in a panel Monday Morning and a talk by Keith Marzullo on Tuesday afternoon. Roughly a smart city is using technology to improve services, for example, sensors everywhere or preparing cities for autonomous vehicles. The speakers highlighted a number of major challenges.

• There are 382 Metropolitan Statistical Areas in the US from New York to Carson City that totals 84% of the US population and 91% of GDP. Many cities share similar problems but how easy can one port hardware and algorithms from one area to another? How do you scale smart cities without reinventing the wheel each time?
• Who pays for the infrastructure? Sometimes one can get research grants or federal help to start new projects, but these projects need continual maintenance afterwards. Are researchers just in it to start a project, write a paper and get out? How do we keep the advantages going in the long run?
• How do you keep the public's trust that the information collected will help the city and not just keeping track of everyone a la Orwell's 1984?
• How do we make sure we tackle the problems of the general public and not just the researchers and those who help fund? A great quote: We need to make sure we are focused more on mass transit than on how to make parking the Tesla easier.
• If we use big data to predict crime and position police in response, could that cause discrimination and harassment?
• How do we keep our research relevant?

Rural areas got their due as well. Interesting presentations on how farmers can use sensors and machine learning to optimize crops, fertilizer and water to use just the right amount needed for each segment of the farm. 

To paraphrase Tip O'Neill, all computing is local, but we face many challenges taking our broad tools of cloud, big data, machine learning, automation and internet of things and apply them in our own neighborhoods.

Math lessons from the Donald Trump Nomination (non-political)

There may be articles titled Donald Trump and the Failure of Democracy. This is NOT one of them. This is about some math questions. I drew upon many sources but mostly Nate Silver's columns:Donald Trump's Six Stages of doom, How the Republican Field Dwindled from 17 to Trump (a collection of article),Four things I learned from the Donald Trump Primary. For the best news piece of the year on Donald Trump see John Oliver's

1) Trends. Since 1972 (the beginning of the modern era of prez primaries) the republicans, have ALWAYS (with one exception I"ll get to) nominated someone who was either PRZEZ or a sitting or former Gov, Senator, or VP who had ALSO been a serious candidate in a prior primary-prez race. The only exception is W who was a sitting Governor but had never run before, though he of course had name recognition. In short, someone FAMILIAR. This also fits our image of the Republicans as an old boys network (Dole got the nomination in 1996 because it was his turn). Hence most pundits expected the same this year.

a) The old ML maxim: Trends hold TILL THEY DON"T.

b) Nobody quite fit the pattern. The only ones who had run before were Rick Santorum, Rick Perry, and Mike Huckabee. Rich S and Mike H were niche candidates, Rick P had wider appeal in 2012 but entered late and stumbled (the WHOOPS moment, though more on that later).  Jeb was like W, Former Gov with family name. So based just on Trends Perry or Jeb should have been the nominee, but its not that strong a match. (Added Later- A commenter says that John K ran briefly in 2000. My criteria was had been a serious candidate- not quite well defined, but John K would not have qualified. Even so, Governor and ran a bit, so he also was close to the criteria.) IF YOU HAVE A TREND AND USE IT TO PREDICT MAKE SURE THE DATA YOU HAVE FITS THE TREND.

c) I WILL NOT claim that I predicted any of this but there is an inkling of what happened in my post The law of the excluded middle of the road republicans where I pointed out for each candidate (including Trump) why they couldn't win. IF YOU HAVE A LARGE NUMBER OF LOW PROB EVENTS WHERE ONE WILL HAPPEN ITS HARD TO PREDICT WHICH ONE.

d) The pattern itself is only based on 11 data points and you might not want to count  the four where there was a republican prez running for re-election. And 11 data points is not the full story--- the political situation from 1972 to 2016 changed dramatically. So these are data points on a moving target. Perhaps they should use papers like New analysis and algorithms for learning with drifting distributions. ITS HARD TO DO ANY REAL DATA ANALYSIS WHEN YOU DON"T HAVE ENOUGH DATA AND IT CHANGES OVER TIME.

2) Domination: Republican primary voters (in the past) wanted a candidate who was both conservative and electable. But what combination? I read that Chris Christie had no chance since it was thought that Jeb, S Walker, and Rubio were all MORE electable and MORE conservative- so they dominated him. Hence Donald Trump couldn't win since he was (though to be) less electable and his prob less conservative though that's hard to tell since he never held office. Hence he can't win. But some voters were tired of voting for electable as  McCain and Romney were allegedly electable. And some were just plain angry. If you think your problems are because of immigrants vote Trump, if you think your problems are because of Wall Street then Feel the Bern. VOTERS DO NOT CARE ABOUT CONVEXITY AND DOMINATION.

3) Nate Silver. He's the Pollster who is NOT a pundit, does NOT let who he wants to see win affect what he predicts, wrote a great book about predictions: The Signal and the Noise: Why so Many Predictions Fail But Some Don't and got many predictions right in recent years. He wrote an excellent article Donald Trumps six stages of Doom in Aug 2015 which said what the obstacles are to the nomination and giving the nomination a 2% chance. To his credit he has owned this prediction in that later columns have told us where he went wrong. (Most pundits never say `Gee I was wrong')  So why did his prediction not pan out?

a) They did in a sense. All of the problem he pointed out that Trump would have, Trump DID have- for example, Trump did not have a good organization to control delegates, and the party did try to stop him. So in a strange sense Nate was right. Except that he was wrong.

b) Back to Nate's 2%. Bill James (Baseball Stats guru) wrote (I am paraphrasing) If you are given odds of 500-1 that some awful team will win the world series  than TAKE THAT BET. People have a hard time telling unlikely from REALLY unlikely. And the NY Mets did win the 1969 world series. (A quote from 1962: There will be a man on the moon before the Mets win the world series- true by two months). Also note that the the Leicester Soccer Team won this year despite being (literally) 5000-1 underdogs (see here). WAS NATE WRONG? If you give an event 2% chance and it happens I can't say you are wrong. In fact, if most everyone else gave it less than 2% or even 0  (which is the case here) then you are... less wrong.

4) Bill Gasarch. Based on TRENDS above I predicted Paul Ryan (and I owe Lance a dinner). My mistake was betting Ryan-I win, ANYONE ELSE-Lance wins (oddly enough, with a contested convention I might have still won that bet) . I should have made Lance name 5 candidates, and if any of those five win, he wins, but if its Ryan I win. I  doubt he would have named Trump.

5) Game Theory: Lance has posted about Primary Game Theory. The main issue for a Trump voter might be `Gee, if I vote Trump he is not electable event though I like him, so I'll vote for X instead who is more electable' But voters are not game theorists. Plus they voted for John McCain and Mitt Romney based on that and they lost. So when Rubio said A Vote for Trump is a Vote for Hillary he may be right but the voters are not listening.  Plus since Little Marco only won Minnesota and Puerto Rico (they have a primary! who knew!) he was not positioned to talk about electability. Plus one could argue that VERY few of the candidates could beat Hillary. In an early Column Nate thought only Jeb, Little Marco, and Scott Walker (remember him?). So once Rubio dropped out the electability argument was useless.

6) More Game Theory: Many of the candidates wanted someone ELSE to go after Trump so they went after each other.

7) The Pledge: For fear that Trump would run third party they all signed a pledge promising to support whoever got the nomination. When they signed it they never imagined that Trump would be the nominee.

8) Prediction Markets: They did pretty well, in about March they came around. Last week David Brooks maintained that Trump would not be the nominee, but he was kidding. I think.

Open Questions

Through the years I've mentioned a few of my favorite open problems in computational complexity on this blog that have perplexed me through the years. Let me mention a few of them again in case they inspire some of the new generation of complexity theorists.

1. Does the polynomial-time hierarchy look like the arithmetic hierarchy? I mentioned this in the centenary post for Mostowski. Probably not because it would imply factoring in P (since NP∩co-NP would equal P) but we have no proof of separation and no oracle that makes them look the same.
2. Does UP = NP imply the polynomial-time hierarchy collapses? What are the consequences if SAT had an NP algorithm with a unique accepting path? Remains open, again even in relativized worlds. 
3. Do rational functions that agree with Boolean functions on the hypercube have low decision tree complexity? I really expected someone to have come up with a proof or counterexample by now. 
4. What happens if two queries to NP can be simulated by a single query? Does S²=ZPP^NP? Both questions asked in a post on S₂^P.
5. Separate NP from Logarithmic space. I gave four approaches in a pre-blog 2001 survey on diagonalization (Section 3) though none have panned out. Should be much easier than separating P from NP.

Some more bits from the Gathering for Gardner

I posted about the Gathering for Gardner conference and about some of the talks I saw here. Today I continue with a few more talks.

Playing Penney's game with Roulette by Robert Vallin. Penney;'s game is the following:  let k be fixed. Alice and Bob pick different elements of {H,T}^k.  They flip a coin until one of their sequences shows up, and that person wins. Which sequences have the best probability of winning?

New Polyhedral dice by Robert Fathauer, Henry Segerman, Robert Bosch. This is a good example of how my mentality (and possibly yours) differs from others. When I hear ``60-sided dice'' I think ``p1,...,p60 where are all between 0 and 1 and add up to 1'' I also thought that only the platonic solids could be usedvto form fair dice (so only 4-sided, 6-sided, 8-sided, 12-sided, and 20-sided dice can be made). NOT so. These authors actually MAKE real dice and they do not have to be platonic solids. Here is their website.

Numerically balance dice by Robert Bosch (paper is here). Why do dice have the opposite sides sum to the same thing?  Read the paper to find out!

Secret messages in juggling and card shuffling by Erik Demaine. Erik Demaine was one of about 4 theoretical computer scientists I met at the conference, though Erik is so well rounded that calling him a theoretical computer scientist doesn't seem quite right. I had never met him before which surprised me. In this talk he showed us some new fonts- one using juggling. See here for an example of juggling fonts, co-authored with his father Martin.

Fibonacci Lemonade by Andrea Johanna Hawksley. Put in the leomon and sugar in fib number increments. Here is their website. In my first post I said the talks were on a variety of topics and then presented mostly math talks. This talk is an example of that variety. There were other talks involving the Fib numbers. I was surprised by this since they aren't that special (see here).

Penis Covers and Puzzles: Brain Injuries and Brain Health by Gini Wingard-Phillips. She recounted having various brain injuries and how working on mathematical puzzles, of the type Martin Gardner popularized as HELPING HER RECOVER! As for the title- people with brain injuries sometimes have a hard time finding the words for things so they use other words. In this case she wanted her husband to buy some condoms but couldn't think of the word so she said Penis Covers instead.

Loop- Pool on an Ellipse by Alex Bellos. Similar in my mind to the Polyhedral dice talk (you'll see why). We all know that if you built an elliptical pool table with a hole at one of the foci then if the ball is placed at the other foci and hit hard enough it WILL go into the other hole. But Alex Bellos actually MAKES these pool table (see here if you want buy one for $20,000). He told us the history- someone else tried to make one in 1962 but nobody bought them (I wonder if anyone are going to buy his), and Alex had problems with friction as you may recall that it only works on a frictionless surface. So his game does require some skill. The similarity to dice is that I (and you?) are used to thinking about dice and ellipses abstractly, not as objects people actually build.

This post is getting long so I'll stop here and report more in a later post. Why so mny posts? Six minute talks that I an actually understand and are delighted to tell you about!

Claude Shannon (1916-2001)

Claude Shannon was born hundred years ago Saturday. Shannon had an incredible career but we know him best for his 1948 paper A Mathematical Theory of Communication that introduced entropy and information theory to the world. Something I didn't know until looking him up: Shannon was the first to define information-theoretic security and show that one-time pads are the one and basically only code that secure.

Entropy has a formal definition, the minimum expected number of bits to represent the output of a distribution. But I view information as a more abstract concept of which entropy is just one substantiation. When you think of concepts like conditional information, mutual information, symmetry of information, the idea of an underlying distribution tends to fade away and you begin to think of information itself as an entity worth mentioning. And when you look at Kolmogorov Complexity, often called algorithmic information theory, the measure is over strings, not distributions, yet has many of the same concepts and relationships in the entropy setting.

Computational Complexity owes much to Shannon's information. We can use information theory to get lower bounds on communication protocols, circuits, even upper bounds on algorithms. Last spring the Simons Institute for the Theory of Computing had a semester program on Information Theory including including a workshop on Information Theory in Complexity Theory and Combinatorics. Beyond theory, relative entropy, or Kullback–Leibler divergence, plays an important role in measuring the effectiveness of machine learning algorithms.

We live in an age of information, growing dramatically every year. How do we store information, how do we transmit, how do we learn from it, how do we keep it secure and private? Let's celebrate the centenary of the man who gave us the framework to study these questions and so much more.

Some short bits from the Gathering for Gardner Conference

I attended G4G12 (Gathering for Gardner) a conference that meets every 2 years (though the gap between the first and second was three years) to celebrate the work of Martin Gardner. Most of the talks were on Recreational mathematics, but there were also some on Magic and some are hard to classify.

Martin Gardner had a column in Scientific American called Mathematical Games from 1956 to 1981. His column inspired man people to go into mathematics. Or perhaps people who liked math read his column. The first theorem I ever read outside of a classroom was in his column. It was, in our terminology, a graph is Eulerian iff every vertex has even degree.

For a joint review of six G4G proceedings see here. For a joint review of six books on recreational math including three of Gardner's, see here. For a review of a book that has serious math based on the math he presented in his column see here.

The talks at G4G are usually 6 minutes long so you can learn about a nice problem and then work on it yourself. Their were a large variety of talks and topics. Many of the talks do not have an accompanying paper. Many of them are not on original material. But none of this matters--- the talks were largely interesting and told me stuff I didn't know.

64=64 and Fibonacci, as Studied by Lewis Caroll, by Stuart Moshowitz. This was about a Lewis Caroll puzzle where he put together shapes in one way to get a rectangle of area 65, and another way to get a square of area 64, The following link is NOT to his talk or a paper of Moshowitz, but it is about the problem: here

How Math can Save your life by Susan Marie Frontczak. This was part talk about bricks and weights and then she stood on the desk and sang this song (thats not her signing it).

Twelve ways to trisect and angle by David Richeson. This was NOT a talk about cranks who thought they had trisected and angle with straightedge and compass. It was about people who used ruler, compass, and JUST ONE MORE THING. I asked David later if the people who trisected the angle before it was shown impossible had a research plan to remove the ONE MORE THING and get the real trisection. He said no- people pretty much knew it was impossible even before the proof.

The Sleeping Beauty Paradox Resolved by Pradeep Mutalik. This paradox would take an entire blog post to explains so here is a pointer to the wikipedia entry on it: here. AH, this one DOES have a paper associated to it, so you can read his resolution here

Larger Golomb Rulers by Tomas Rokicki. A Golomb Ruler is a ruler with marks on it so that the all of the distances between marks are distinct. The number of marks is called the order of the ruler. Construction a Golumb ruler is easy (e.g., marks at the 1,2,4,8,... positions I think works). The real question is to get one of shortest length. They had some new results but, alas, I can't find them on the web.

Chemical Pi by John Conway.  There are people who memorize the first x digits of pi. John Conway does something else. He has memorized the digits of pi and the chemical elements in the following way:

HYDROGEN  3.141592653 HELIUM next 10 digits of pi LITHIUM etc

that is, he memorized the digits of pi by groups of 10 and separated by the chemical elements in the order they are on the Periodic table. He claims this makes it easier to answer questions like: What is the 87th digits of pi. He also claims it gives a natural stopping point for how many digits of pi you need to memorize (need? maybe want). (ADDED LATER WHEN I CORRECTED HELIUM TO HYDROGEN: here are some mnemonic devices:  here.

This post is getting long so I may report on more of the talks in a later post.

The Master Algorithm

We see so few popular science books on computer science, particularly outside of crypto and theory. Pedro Domingos' The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake the World, despite the hyped title and prologue, does a nice job giving the landscape of machine learning algorithms and putting them in a common text from their philosophical underpinnings to the models that they build on, all in a mostly non-technical way. I love the diagram he creates:

Working out from the inner ring are the representations of the models, how we measure goodness, the main tool to optimize the model and the philosophies that drove that model. The book hits on other major ML topics including unsupervised and reinforcement learning.

In the bullseye you can see the "Master Equation" or the Master Algorithm, one learning algorithm to rule them all. The quest for such an algorithm drives the book, and Domingos describes his own, admittedly limited attempts, towards reaching that goal.

I diverge from Domingos in whether we can truly have a single Master Algorithm. What model captures all the inner-ring models above: circuits. A Master Algorithm would find a minimum-sized circuit relative to some measure of goodness. You can do that if P = NP and while we don't think circuit-minimization is NP-hard, it would break cryptography and factor numbers. One of Domingos' arguments states "If we invent an algorithm that can learn to solve satisfiability, it would have a good claim to being the Master Algorithm". Good luck with that.

Its hard to tell if a problem is hard. Is this one hard?

Here is a problem I heard about at the Gathering for Gardner. Is it hard? easy? boring? interesting? I don't know.

Let N={1,2,3,...}

PROBLEM: parameters are s (start point) and f (not sure why to call it f). both are in N

Keep in mind the sequence, in order, of operations:

DIVIDE BY f, SUBTRACT f, ADD f, MULTIPLY by f.

form the following sequence of numbers in N

a(0)= s

Assume a(0),...,a(n) are known. Let A = {a(0),...,a(n)}. N-A are the elements in N that are NOT in A.

If a(n)/f is in N-A then a(n+1)=a(n)/f

Else

If a(n)-f is in N-A then a(n+1)=a(n)-f

Else

If a(n)+f is in N-A then a(n+1)=a(n)+f

Else

If a(n)*f is in N-A then a(n+1) = a(n)*f

Else

If none of the above holds then the sequence terminates.

Lets do an example! Let a=14 and f=2

14, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 24, 22, 11, 9, 18, 16, 32, 30, 15, 13, 26, 28, 56, 54, 27, 25, 23, 21,

19, 17, 34, 36, 38, 40, 20,  STOP since 10, 18, 22, 40 are all on the list.

Lets do another example! Let a=7, f=2

7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 9, 11, 13, 15, 17, ... (keeps going)

If f=2 and you get to an odd number x so that ALL of the odds less than x have already appeared but NONE of the odd numbers larger than x have appeared, then the sequence will go forever
with x, x+2, x+4, ...

QUESTIONS and META QUESTIONS

1) Can one characterize for which (s,f) the sequence stops.

2) Is it decidable to determine for which (s,f) the sequence stops.

3) Both (1) and (2) for either fixed s or fixed f.

4) Are the above questions easy?

5) Are the above questions interesting?

There are four categories:

Easy and Interesting- Hmmm, if its TOO easy (which I doubt) then I supposed can't be interesting.

Easy and boring.

Hard and interesting. This means that some progress can be made and perhaps connections to other mathematics.

Hard and Boring. Can't solve and are not enlightened for the effort.

Who Controls Machine Learning?

After AlphaGo's victory, the New York Times ran an article The Race Is On to Control Artificial Intelligence, and Tech’s Future.

A platform, in technology, is essentially a piece of software that other companies build on and that consumers cannot do without. Become the platform and huge profits will follow. Microsoft dominated personal computers because its Windows software became the center of the consumer software world. Google has come to dominate the Internet through its ubiquitous search bar. If true believers in A.I. are correct that this long-promised technology is ready for the mainstream, the company that controls A.I. could steer the tech industry for years to come.

I then tweeted "Can a company control AI? More likely to become a commodity." The major machine learning algorithms are public knowledge and one can find a number of open-source implementations including Google's own TensorFlow that powered AlphaGo. What's to stop a start-up from implementing their own machine learning tools on the cloud?

Some of my readers' comments forced me to rethink my hasty tweet. First, Google, Microsoft and Amazon can create ML infrastructure, cloud hardware that optimizes computational power and storage for machine learning algorithms to get a level of data analysis that one couldn't replicate in software alone.

More importantly, Google etc. have access to huge amounts of data. Cloud companies can provide pretrained machine learning algorithms. Google provides image classification, voice transcription and translation. Microsoft offers face and emotion detection and speech and text analysis. One could imagine, in the absence of privacy issues, Google taking your customer data, matching with data that Google already has on the same customers to draw new inferences about how to market to those customers better.

With almost all our computing heading to the cloud, cloud computing providers will continue to compete, and provide continuing better tools in machine learning and beyond. Eventually will one company "control AI"? That would surprise me but we may still end up with an AI oligarchy.

What Rock Band Name would you choose?

I looked up my colleague Dave Mount on Wikipedia and found  that he was a drummer for the glam rock band Mud. He informed me that (a) on Wikipedia he is David Mount and (b) if he had a rock band it would be named Fried Purple Ellipsoids.

This  set off an email discussion where people said what their rock band name would be. I noticed that many ideas for names had variants. For example, my favorite for Ramsey Theorists: The Red Cliques could be

The Red Cliques

Red Clique

Bill Gasarch and the Red Cliques!

Clique!

So below I list one variant of each name but keep in mind that there are others.

 The Hidden Subgroups

 Amplitudes with Attitude 

 Schrodinger's cat (I wonder if this IS a rock band already)

 The Red Cliques

 Fried purple ellipsoids

 Fried green ellipsoids

 BIG A-G-T

 The Biconnected Sets

 PRAM!

BPP!  (I wonder if any complexity class would work.)

SAT (I wonder if other one-word problems would work. TSP!)

Karp and the reductions

Avi and the derandomizers 

 Aravind and the  Expanders

(Could replace Karp, Avi, and Aravind with others, but these are the first that
came to mind. Plus THE EXPANDERS was Aravind Srinivasan's idea.)

The MIT Logarhythms (This is a real acapella group see here.)

The Discrete Logarhythms

RSA!

The Oracles

The Interactive Proofs

The Natural Proofs

Fried Green Proofs

If we expand to include math we get lots more, so I'll just mention one real one: The Klein Four, an acapella group.

SO- what would YOUR rock band name be?

It's All About the Jobs

In the April CACM Moshe Vardi asks Are We Headed toward Another Global Tech Bust? I agree with some of Vardi’s points, mostly that VC money chasing after unicorns (potential billion-dollar start-ups) will not continue at its heavy pace and we’re already seeing a slow down. But I disagree with Vardi’s assessment that “we should brace ourselves for another global tech and enrollment bust” in computer science. I suspect we’ll see more of a reality check, but that reality looks extremely strong.

Vardi claims that “It is the dream of joining a unicorn that probably attracts many students to study computing”. It’s not just the unicorns bringing students to computer science, but essentially a 100% employment rate for CS graduates looking for a job in the field, many receiving six-figure starting salaries. Few, if any, other disciplines can claim full employment after the bachelor’s degree. Industry is desperate to hire computing professionals in machine learning, cloud computing, cybersecurity, mobile computing, automation, robotics and data science, among others. Not just the computing companies but every industry that deals with data, which is pretty much every industry. Unicorns may become rarer but we won’t see a decline in demand for computer science students until we automate ourselves out of a job.

Take a look at this chart from Ed Lazowska's Where The Jobs Are – 2016 Edition. Those CS jobs won't fill themselves.

Are Perfect Numbers Bigger than Six initial sums of odd cubes (answered)

(NONE of this is my work. In fact some of it is on Wikipedia.)

In my last blog I noticed that

28 = 1³  + 3³

496= 1³ + 3³ + 5³ + 7³

noting that 28 and 496 are the 2nd and 3rd perfect numbers.

I asked if 8128, the next perfect number is also an initial sum of odd cubes. It is!

8128 = 1³ + 3³ + ... + 15³

I also asked if there was something interesting going on .The answer is YES but not that interesting.

All of the math with proofs are  here. I sketch below.

Known Theorem  1: n is an even perfect number iff n is of the form (2^p-1)(2^p- 1) where 2^p-1 is prime.

Known Theorem  2: 1³ + 3³ + 5³ + ... + (2(m-1)+1)³ = m²(2m²-1).

Interesting theorem: if n is an even perfect number larger than 6 and p is the p from Known Theorem 1 then n is the sum of the first  2^(p-1)/2 odd cubes.

Why this is less interesting: The proof does not use that n is perfect. It holds for any number of the form 2^p-1(2^p-1) where p is odd.

So the theorem has nothing to do with perfect numbers. Oh well.

Are perfect numbers bigger than 6 initial sums of odd cubes?

I pose two questions today (Monday April 4).

I will post the answers tomorrow (Tuesday April 5).

Feel free to comment about the answers. If you don't want clues look at the comments.
If I need to clarify something I will do it in the main post So, to reiterate- feel free to leave spoilers but if you want to avoid reading them, don't read the comments.

Note:

The first four perfect numbers are 6, 28, 496, 8128

28 = 1³ + 3³

496 = 1³ + 3³ + 5³ + 7³

Is 8128 the sum of the first six odd cubes? No, and that is not one of my questions.

Questions:

1) Is there a k such that 8128  is the sum of the first k odd cubes?

2) Is there something interesting going on here?

The Machine Learning Turk

Google's AlphaGo took the world by storm when it won its match with Lee Sedol but Demis Hassabis now acknowledges the dark truth. Google wanted to promote its cloud computing division as Amazon AWS and Microsoft Azure have quite the head start. Google needed a killer app that would bring users to Google Cloud and decided they could win if they had the best machine learning tools. They bought Deepmind, run by Hassabis, and needed a showcase event and decided to focus on Go, a game yet to be conquered by computers. Hassabis and his team used clever machine learning techniques on top of Monte Carlo Tree Search but only made mild improvements to the game. Google was growing desperate so a plan was hatched.

Using a modern version of the mechanical turk, an 18th century chess playing automaton that secretly hid a human inside playing the game, Hassabis enlisted Japanese Go player Yuta Iyama to secretly choose the moves for AlphaGo. Iyama, who worked with Google when they agreed to remove Iyama's embarrassing Karaoke videos from YouTube, didn't have to physically be in the machine but relayed the moves by a method Hassabis wouldn't reveal. AlphaGo, secretly getting its moves from Iyama, easily dispatched the European champion in October.

Hannabis and his team wrote up their failed algorithms and found it shockingly easy to fool the Nature editors and reviewers. Yann LeCun of Facebook looked at the Google's team's Nature paper and didn't see that much different from what Facebook had tried. "I just figured Google had chosen better parameters to make their program successful. At the time I should have realized what Google was up to."

Google took a risk challenging Lee Sedol but Sedol, not realizing he was really facing Iyama, played the wrong style of game and lost the match four games to one.

Will this revelation hurt the future of AI? "Machine learning continues to change society, but when it comes to Go," said LeCun, "Alpha fools".