Lessons from the Nobel Prizes

We've had a big week of awards with the Nobel Prizes and the MacArthur "Genius" Fellows. The MacArthur Fellows include two computer scientists, Regina Barzilay and Stefan Savage, and a statistician Emmanuel Candès but no theoretical computer scientists this year.

No computer scientists among the Nobel Laureates either but technology played a large role in the chemistry and physics prize. The chemistry prize went for a fancy microscope that could determine biomolecular structure. The LIGO project that measures extremely weak gravitational waves received the physics prize.

In a sign of the times, Jeffrey Hall, one of the medical prize recipients, left science due to lack of funding.

The economics prize went to Richard Thaler who described how people act irrationally but often in predictable ways such as the endowment effect that states the people give more value to an object they own versus one they don't currently have. The book Thinking Fast and Slow by 2002 Laureate Daniel Kahneman does a great job describing these behaviors.

While at Northwestern I regularly attended the micro-economics seminars many of which tried to give models that described the seemingly irrational behaviors that researchers like Thaler brought to light. My personal theory: Humans evolved to have these behaviors because while they might not be the best individual choices they make society better overall.

Michael Cohen

When I first saw posts about Michael Cohen (see here, here, here) I wondered

is that the same Michael Cohen who I knew as a HS student?

It is.  I share one memory.

Michael Cohen's father is Tom Cohen, a physics professor at UMCP.  They were going to a Blair High School Science fair and I got a ride to it (I had some students presenting at it.) In the car with Tom and Michael, Michael began telling is dad that his dad's proofs were not rigorous enough. I was touched by the notion that father and son could even have such a conversation.

Were Tom's proofs rigorous? I suspect that for Physics they were. But the fact that Michael could, as a high school student, read his dad's paper and have an opinion on it, very impressive. And very nice.

Michael was brilliant. It's a terrible loss.

Is the Textbook Market doomed?

STORY ONE:
I always tell my class that its OKAY if they don't have the latest edition of the textbook, and if they can find it a  cheap, an earlier edition (often on Amazon, sometimes on e-bay), that's fine.  A while back at the beginning of a semester I was curious if the book really did have many cheap editions so I typed in the books name.

I found a free pdf copy as the fourth hit.

This was NOT on some corner of the dark web. This was easy to find and free. There were a few things not quite right about it, but it was clearly fine to use for the class. I wanted to post this information on the class website but my co-teacher was worried we might get in trouble for it, and he pointed out that the students almost surely already know, so we didn't. (I am sure thats correct. When I've discussed this issue with people, they are surprised I didn't already know that textbooks are commonly on the web, easy to find.)


STORY TWO:
I know someone who is thinking of writing a cheap text for a CS course. It will only be $40.00. That is much cheaper than the cost of a current edition of whats out there, and competitive with the used editions, but of course much more expensive than free. I think once students start getting used to free textbooks, even $40.00 is a lot.

STORY THREE (What I do): For discrete math we had slides on line, videos of the lectures on line, and some notes on line. For smaller classes I have my own notes on line. The more I teach a course the better then notes get as I correct them, polish them, etc, every time I teach.  Even so, the notes are very good if you've gone to class but not very good if you haven't (that is not intentional- is more a matter of, say, my notes not having actual pictures of DFA's and NFA's).  I have NO desire to polish the notes more and make a book out of them.  Why do some people have that urge? I can think of two reason though I am sure there are more: (1) To make money. If you get a text out early in a field then this could work (I suspect CLR algorithms text made money). I wonder if Calc I books still make money given how many there are. But in any case, this motivation is now gone--- which is one of the points of this post.  (2) You feel that your way to present (say) discrete math is SO good that others should use it also!  But now you can just post a book or notes on the web, do a presentation at SIGCSE or other comp-ed venues. You don't need to write a textbook. (Personally I think this is a bit odd anyway--- people should have their own vision for a course. Borrowing someone else's seems strange to me.)

DEATH SPIRAL: Books cost a lot, so students buy them cheap or get free downloads, so the companies does not make money so they raise the price of the book, so students buy them cheap...(I"m not going to get into whose fault this is or who started it, I'm just saying that this is where we are now.)


With books either cheap-used or free, how will the textbook market survive? Or will it? Asking around I got the following answers

1) There will always be freshman who don't know that books can be cheap or free. This might help with Calc I and other first-year courses, but not beyond that.

2) There will always be teachers who insist the students buy the latest edition so that they can assign problems easier, e.g., `HW is page 103, problems 1,3,8  and page 105 problems 19 and 20. This will help the textbook publishers in that window between the new edition coming out and the book being scanned in. Is that a long window?

3) Some textbooks now come with added gizmos- codes on the web to get some stuff. For the teachers there may be online quizzes. Unfortunately this makes the books cost even more. I personally never found such things useful, but others might.

4) If a student has a scholarship that pays for books, and the students buys the books used on amazon, can the scholarship still pay for them? I ask non-rhetorically. Even if the answer is no, so the student has to buy books at (say) the campus book store (will they still sell books in 10 years?) this is not enough to save the market.

5) Rent-a-books. I've seen these services. But they still cost too much.

6) e-books. If e-books catch on  then that might get rid of the used-book market. And if they are cheap enough that might help. But the flip side- once e-books are out there  it might be even easier to find a free copy online someplace. (Side note- Many people tell me that math books just don't work as e-books.... yet.)

7) The basic problem is cost. Is there a way for publishers to keep costs down? Or is even that too late as students get used to free or free-ish books?

So I ask again, non-rhetorically- is the textbook market doomed?

Monty Hall (1921-2017) and His Problem

Monty Hall passed away yesterday, best known for co-creating and hosting the game show Let's Make a Deal, a show I occasionally watched as a kid. To the best of my knowledge he's never proven a theorem so why does he deserve mention in this blog?

For that we turn back the clock to 1990 when I was a young assistant professor at Chicago, more than a decade before this blog started, even before the world-wide web. The Chicago Tribune was a pretty good newspaper in those days before Craigslist. Nevertheless, the Sunday Tribune, as well as many other papers across the country, included Parade, a pretty fluffy magazine. Parade had (and still has) a column "Ask Marilyn" written by Marilyn vos Savant, who does not hide the fact that she had the world's highest IQ according the record books in the 1980's.

In 1990, vos Savant answered the following question in her column. Think about the answer if you haven't seen it before.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

This is the kind of deal Monty Hall might have made on his show and so his name got attached to the problem in a 1976 paper in the American Statistician. Marilyn vos Savant claimed it was an advantage to switch. Many mathematicians at the time wrote into Parade arguing this was wrong--either way you have a 50% chance of winning. Even several of my fellow colleagues initially believed it made no difference to switch. Who was this low-brow magazine columnist to say otherwise? In fact, Marilyn was right.

Here is my simple explanation: If you make the commitment to switch, you will win if you pick a goat in the first round, a 2/3 chance of happening. Thinking it makes no difference is a fallacy in conditional probability, not unlike Mossel's Dice Paradox.

Monty Hall himself ran an experiment in his home in 1991 to verify that Marilyn was correct, though modulo the assumption that the host would always offer to make the switch and that everything was chosen uniformly.

Thanks to Bill Gasarch and Evan Golub for some useful details and links. Bill says "history being history, Monty Hall will be remembered as a great mathematician working in Probability." Maybe not, but it does get him remembered in the computational complexity blog.

Tragic Losses

I'd like to remember two young people who's lives were taken way too early. I didn't know either well but both played large roles in two different communities.

Michael Cohen

Michael Cohen, a young researcher in theoretical computer science, passed away. He's had a number of great algorithmic results most notably his solely authored paper giving a polynomial-time algorithm to construct Ramanujan graphs. Luca Trevisan and MSR give their remembrances. Update (10/5): See also Scott Aaronson, which include comments form Michael's mother and father and a Daily Cal article.

Scout Schultz, in a story that made national news, studied computer engineering at Georgia Tech. On Saturday September 16th Scout was shot by a member of the Georgia Tech campus police. A vigil was held the following Monday, quite peaceful until a splinter group (mostly not Georgia Tech students) broke off, marched to the Georgia Tech police department and set a police car on fire. 

Scout Schultz

The death and its aftermath have shooken us all up at Georgia Tech. What has impressed me during this times is the strength of the Georgia Tech student body. Instead of focusing on blame, they have come together to remember Scout, a leader of the LGBQT community on campus. Being in a liberal city in a conservative state, the politics of the student body is quite mixed, but it doesn't divide the students, rather it brings them together. There's hope yet.

Science fiction viewers used to embrace diversity (or did they) and now they don't (or do they)

(This post is inspired by the choice of a female to be the next Doctor on the TV show Dr. Who. Note that you can't say `the next Dr. Who will be female' since Dr. Who is not the name of the character. The name has not been revealed. Trivia: The first Dr. Who episode was the same day Kennedy was shot.)


I give a contrast and then say why it might not be valid:

Star Trek- The Original Series. 1966. There is a black female communications officer, a Russian officer and an Asian officer. And Science Fiction Viewers EMBRACED and APPROVED of this (for the time) diversity.

Modern Time: A black Storm Trooper in Star Wars VII (see here), a black Jimmy Olsen  in Supergirl (see here), female Ghostbusters (see here), a female Doctor on Dr. Who (see here and here) , and even the diversity of ST-Discovery (see here) have upset science fiction viewers.

So what happened in 50 year?

Now I say why this contrast might not be valid.  All items here are speculative, I welcome comments that disagree intelligently. Or agree intelligently. Or raise points about the issue.

1) Science Fiction fans aren't racists and anti-women, they just don't like change. Star Trek: The Original Series didn't have an original cannon to violate. Having a black Captain (ST:DSN) or a female captain (ST:VOY) was a matter of NEW characters and I don't recall any objections. (Were there objections?) If in  the ST reboot they made Captain Kirk black, I suspect there would be objections which the objectors would claim are not racist. Would they be?

2) While the fans that are upset get lots of coverage, they might be the minority. I sometimes see more stuff on the web arguing against the racism then the racism itself.  (A friend of mine in South Carolina told me that whenever a Confederate monument is about to be taken down the SAME 12 people show up to protest but get lots of coverage).

3) Science Fiction has gotten much more mainstream, so the notion that `science fiction viewers now do BLAH' is rather odd since its no longer a small community.

4) In 1966 there was no internet (not even in the Star Trek Universe!!) for fans and/or racists to vent their anger.

5) Some of the objections have valid counterparts: "I don't mind Jimmy Olsen being black, I mind him being so handsome, whereas in the Superman Cannon he is not." (Counter: some of the objections are repulsive:; "I don't mind Jimmy Olsen being black, I mind him being a love interest for Supergirl". Gee why is that?)

5a) Another `valid' one `storm troopers were all cloned from ONE white guy so there cannot be a black stormtrooper'.  Racism hiding behind  nitpicking? Actual nitpicking?

6) I give the fans back in 1966 too much credit- it was the showrunners who embraced diversity. The fans-- did they care?

6a) I give the showrunners to much credit. ALL Klingons are war-like, ALL Romulans are arrogant, ALL Vulcans are logical (except during Pon Farr), In the more recent shows like ST-TNG ALL  Ferengi are greedy. So the show accepts that stereotypes can be true.

6b) Women were not portrayed that well in the star trek universe, even in the more recent shows.  See 15 real terrible moments for women on Star Trek

7) Even the 1966 ST  was not as diverse as I make it out to be. I doubt it would pass the Bechdel test


two other points of interest

1) In the 1960's Science Fiction was sometimes  used as a way to talk about current issues since talking about them directly would not have been allowed. We can't really talk about real racism in a TV show so we'll have an alien race where they are all half-black, half-white, but differs on how which side (see here). And now?  Racism, sexism, homophobia can all be talked about freely. Hence other media has moved ahead of Science Fiction for diversity.

2) Also of interest, though not science fiction: The Edward Albee estate blocks a production of Who's afraid of Virginia Woolf that was going to cast a black man as Nick (a supporting character- George is the main male character).  See here.  Why the block? Because that is what Albee (who is now dead) requested. What would he think now? Who knows?

Acronyms and PHP

Whenever I teach discrete math and use FML to mean Formula the students laugh since its a common acroynm for  Fuck My Life. Now they laugh, and I say I know why you are laughing, I know what it means  and they laugh even harder.

BUT it got me thinking: Pigeonhole Principle! There are more things we want short acroynms for then there are short acroynms. Below are some I thought of. I am sure there are others, in fact I am sure there are websites of such, but I wanted to see which ones I just happen to know.

AMS- American Math Society and much much more:see here

DOA-

Dead on Arrival

Department of Aging. Scary!

ERA-

Earned Run Average in Baseball,

Equal Rights Amendment in politics

PCP-

Phencyclidine, a drug that you should never take.

Prob. Checkable Proofs. Obscure to the public but not to us.

ADDED LATER: A reader noted Post Correspondence Problem, a good example of a natural undecidable problem.

IRA- 

Irish Republic Army

Internal Retirement Account

Several companies have had rumors they fund terrorism because they were giving their employees IRA's. The headline `Company X funds IRA's' could be misunderstood.


SAT-

Standard Aptitute Test

Satisfiability (of Boolean Formulas) Obscure to the public but not to us. Actually it may get less obscure as more ``proofs'' resolving P vs NP come out.

SJW

Single Jewish Female (in classified ads- more on that later). I think SJF is more common.

Social Justice Warrior (sounds like a good thing but might not be)

Classified ads are a source of many acronyms which can be used to teach combinatorics.

{S,M,W,D,G}{B,C,H,J,W}{M,F}


S-single, M-married, W-widowed, D-Divorced, G-Gay (this one I've seen alone making me wonder
about S/M/W/D? I've also seen four-letter acronyms to disambiguate).

B- black, C-Christian, H-Hispanic,  J-Jewish, W-White.

M,F- Male, Female, though I am sure there are ways to say other genders.

Great for combinatorics! especially if you add in other ones (like BD)

WTF-

Wisconsin  Tourism Federation

You know what else it means so I won't say it (this is a G-rated blog). When I first saw it I thought `what the fuck?- how could they have screwed up so badly?'


TEACHING TOOL- when teaching PHP (Pigeon hole Principle, not the language PHP which stands for Hypertex PreProcessing, not quite in order, or Personal Home Page) you can use the the fact that

number of concepts GREATER THAN  number of 3-letter combos

leads to some 3-letter combos will be used more than once.

A problem I thought was interesting- now...

On Nate Silver's page he sometimes (might be once a week) has a column edited by Oliver Roeder of problems. Pretty much math problems though sometimes not quite.

Some are math problems that I have seen before (e.g., hat problems). I don't bother submitting since that would just be goofy. I would be  ringer.

Some are math problems that I have not seen before, I try to do, I fail, but read the answer and am enlightened. I call that a win.

But some are math problems that I have not seen before, I try to do, I fail, but when I see the solution its a computer simulation or something else that isn't quite as interesting as I had hoped.

I describe one of those now; however, I ask if it can be made more interesting.

The problems is from this column: here

I paraphrase:  Let A be the numbers {1,2,3,...,100}.  A sequence is nice if  (1) it begins with any number in A, (2) every number is from A and is either a factor of multiple of the number just before it, and (3) no number can appear more than once.  Find the LONGEST nice sequence

Example of a nice sequence: 

4, 12, 24, 6, 60, 30, 10, 100, 25, 5, 1, 97

I worked on it

1) By hand I came up with a nice sequence of length 42. This was FUN! You can either have fun trying to find a long nice sequence or you can look at mine here.

2) I tried to prove that it was optimal, hoping that either I would find its optimal or be guided to a longer sequence. Neither happened. More important is that this was NOT FUN.

3) I looked forward to the solution that would be in the next column and would be enlightening. 

4) The next column, which did have the solution, is here! The answer was a sequence of length 77 found by a program that also verified there was no longer sequence. The sequence itself was mildly enlightening in that I found some tricks I didn't know about, but the lack of a real lower bound proof was disappointing.

They mentioned that this is a longest path problem (Graph is {1,..,100} edges are between numbers that are either multiples of factors) and that such problems are NP-complete. That gave the impression that THIS problem is hard since its a case of an NP-complete problem. Thats not quite right- its possible that this type of graph has a quick solution.

But I would like YOU the readers to help me turn lemon into lemonade.

1) Is there a short proof that 77 is optimal?  Is there a short proof that (say) there is no sequence of length 83.  I picked 83 at random.  One can easily prove there is no sequence of length 100.

2) Is the following problem in P or NPC or if-NPC-then-bad-thing-happen:

Given (n,k) is there a nice sequence of {1,...,n} of length at least k. (n is in binary, k is in unary, so that the problem is in NP.)

I suspect not NPC.

3) Is the following problem in P or NPC or ...

Given a set of numbers A and a number k, is there a nice sequence of elements of A of length at least k (k in unary).

Might be NPC if one can code any graph into such a set.

Might be in P since the input has a long length.

4) Is the following solvable: given a puzzle in the Riddler, determine ahead of time if its going to be interesting? Clyde Kruskal and I have a way to solve this- every even numbered column I read the problem and the solution and tell him if its interesting, and he does the same for odd number columns.

Random Storm Thoughts

It's Monday as I write this post from home. Atlanta, for the first time ever, is in a tropical storm warning. Georgia Tech is closed today and tomorrow. I'm just waiting for the power to go out. But whatever will happen here won't even count as a minor inconvenience compared to those in Houston, the Caribbean and Florida. Our hearts goes out to all those affected by these terrible storms.

Did global warming help make Harvey and Irma as dangerous as they became? Hard to believe we have an administration that won't even consider the question and keeps busy eliminating "climate change" from research papers. Here's a lengthy list cataloging Trump's war on science. 

Tesla temporarily upgraded to its Florida Owners' cars giving them an extra 30 miles of battery life. Glad they did this but it begs the question why Tesla restricted the battery life in the first place. Reminds of when in the 1970's you wanted a faster IBM computer, you paid more and an IBM technician would come and turn the appropriate screw. Competition prevents software-inhibitors to hardware. Who will be Tesla's competitors?

During all this turmoil the follow question by Elchanan Mossel had me oddly obsessed: Suppose you flip a six-sided die. What is the expected number of dice throws needed until you get a six given that all the throws ended up being even numbers? My intuition was wrong though when Tim Gowers falls into the same trap I don't feel so bad. I wrote a short Python program to convince me, and the program itself suggested a proof.

Updates on Thursday: I never did lose power though many other Georgia Tech faculty did. The New York Times also covered the Tesla update. 

The Scarecrow's math being wrong was intentional

In 2009 I had a post about Movie mistakes (see here). One of them was the Scarecrow in The Wizard of Oz after he got a Diploma (AH- but not a brain) he said

The sum of the square roots of any two sides of an isoscles triangle is equal to the square root of the remaining side. Oh joy! Rapture! I have a brain!

I wrote that either this mistake was (1) a mistake, (2) on purpose and shows the Scarecrow really didn't gain any intelligence (or actually he was always smart, just not in math), or (3) It was Dorothy's dream so it Dorothy was not good at math.

Some of the comments claimed it was (2).  One of the comments said it was on the audio commentary.

We now have further proof AND a longer story: In the book Hollywood Science: The Next Generation, From Spaceships to Microchips (see here) they discuss the issue (page 90). The point to our blog as having discussed it (the first book not written by Lance or Lipton-Regan to mention our blog?) and then give evidence that YES it was intentional.

They got a hold of the original script. The Scarecrow originally had a longer even more incoherent speech that was so over the top that of course it was intentional. Here it is:

The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side: H-2-O Plus H-2-S-O-4 equals H-2-S-O-3 using pi-r-squared as a common denominator Oh rapture! What a brain!

While I am sure the point was that the Scarecrow was no smarter, I'm amused at the thought of Dorothy not knowing math or chemistry and jumbling them up in her dream.

Statistics on my dead cat policy- is there a correlation?

When I teach a small (at most 40) students I often have the dead-cat policy for late HW:

HW is due on Tuesday. But there may be things that come up that don't quite merit a doctors note, for example your cat dying, but are legit for an extension. Rather than have me judge every case you ALL have an extension until Thursday, no questions asked. Realize of course that the hw is MORALLY due Tuesday. So if on Thursday you ask, for an extension I will deny it on the grounds that I already gave you one. So you are advised to not abuse the policy. For example, if you forget to bring your HW in on thursday I will not only NOT give the extension, but I will laugh at you.

(I thought I had blogged on this policy before but couldn't find the post.)

Policy PRO: Much less hassling with late HW and doctors notes and stuff

Policy CON: The students tend to THINK of Thursday as the due date.

Policy PRO: Every student did every HW.

Caveat: The students themselves tell me that they DO start the HW on Monday night, but if they can't quite finish it they have a few more days. This is OKAY by me.

I have always thought that there is NO correlation between the students who tend to hand in the HW on Thursday and those that do well in the class.  In the spring I had my TA keep track of this and do statistics on it.

The class was Formal Lang Theory (Reg Langs, P and NP, Computability. I also put in some communication complexity. I didn't do Context free grammars.)  There were 43 students in the class. We define a students morality (M) as the the number of HW they hand in on Tuesday. There were 9 HW.

3 students had M=0

12 students had M=1

9 students had M=2

5 students had M=3

4 students had M=4

4 students had M=5

1 student had M=6

1 student had M=7

2 students had M=8

2 students had M=9

We graphed grade vs morality (see  here)

The Pearson Correlation Coefficient is 0.51. So some linear

The p-value is 0.0003 which means the prob that there is NO correlation is very low.

My opinion:

1) The 5 students with M at least 7 all did very well in the course.This seems significant.

2) Aside from that there is not much correlation.

3) If I tell my next semesters class  ``people who handed the HW in on tuesday did well in the class so you should do the same'' that would not be quite right- do the good students hand thing in on time, or does handing things in on time make you a good student? I suspect the former.

4) Am I surprised that so many students had such low M scores? Not even a little.

Rules and Exceptions

As a mathematician nothing grates me more than the expression "The exception that proves the rule". Either we bake the exception into the rule (all primes are odd except two) or the exception in fact disproves the rule.

According to Wikipedia, "the exception that proves the rule" has a legitimate meaning, a sign that says "No parking 3-6 PM" suggests that parking is allowed at other times. Usually though I'm seeing the expression used when one tries to make a point and wants to dismiss evidence to the contrary. The argument says that if exceptions are rare that gives even more evidence that the rule is true. As in yesterday's New York Times

The illegal annexation of Crimea by Russian in 2014 might seem to prove us wrong. But the seizure of Crimea is the exception that proves the rule, precisely because of how rare conquests are today.

Another example might be the cold wave of 2014 which some say support the hypothesis of global warming because such cold waves are so rare these days.

How about the death of Joshua Brown, when his Tesla on autopilot crashed into a truck. Does this give evidence that self-driving cars are unsafe, or in fact they are quite safe because such deaths are quite rare? That's the main issue I have with "the exception that proves the rule", it allows two people to take the same fact to draw distinctly opposite conclusions.

NOT So Powerful

Note: Thanks to Sasho and Badih Ghazi for pointing out that I had misread the Tardos paper. Approximating the Shannon graph capacity is an open problem. Grötschel, Lovász and Schrijver approximate a related function, the Lovász Theta function, which also has the properties we need to get an exponential separation of monotone and non-monotone circuits.

Also since I wrote this post, Norbert Blum has retracted his proof.

Below is the original post.

---

A monotone circuit has only AND and OR gates, no NOT gates. Monotone circuits can only produce monotone functions like clique or perfect matching, where adding an edge only makes a clique or matching more likely. Razborov in a famous 1985 paper showed that the clique problem does not have polynomial-size monotone circuits.

I choose Razborov's monotone bound for clique as one of my Favorite Ten Complexity Theorems (1985-1994 edition). In that section I wrote

Initially, many thought that perhaps we could extend these [monotone] techniques into the general case. Now it seems that Razborov's theorem says much more about the weakness of monotone models than about the hardness of NP problems. Razborov showed that matching also does not have polynomial-size monotone circuits. However, we know that matching does have a polynomial-time algorithm and thus polynomial-size nonmonotone circuits. Tardos exhibited a monotone problem that has an exponential gap between its monotone and nonmonotone circuit complexity. 

I have to confess I never actually read Éva Tardos' short paper at the time but since it serves as Exhibit A against Norbert Blum's recent P ≠ NP paper, I thought I would take a look. The paper relies on the notion of the Shannon graph capacity. If you have a k-letter alphabet you can express kⁿ many words of length n. Suppose some pairs of letters were indistinguishable due to transmission issues. Consider an undirected graph G with edges between pairs of indistinguishable letters. The Shannon graph capacity is the value of c such that you can produce cⁿ distinguishable words of length n for large n. The Shannon capacity of a 5-cycle turns out to be the square root of 5. Grötschel, Lovász, Schrijver use the ellipsoid method to approximate the Shannon capacity in polynomial time.

The Shannon capacity is anti-monotone, it can only decrease or stay the same if we add edges to G. If G has an independent set of size k you can get kⁿ distinguishable words just by using the letters of the independent set. If G is a union of k cliques, then the Shannon capacity is k by choosing one representation from each clique, since all letters in a clique are indistinguishable from each other.

So we have the largest independent set is at most the Shannon capacity is at most the smallest clique cover.

Let G' be the complement of a graph G, i.e. {u,v} is an edge of G' iff {u,v} is not an edge of G. Tardos' insight is to look at the function f(G) = the Shannon capacity of G'. Now f is monotone in G. f(G) is at least the largest independent set of G' which is the same as the largest clique in G. Likewise f(G) is bounded above by the smallest partition into independent sets which is the same as the chromatic number of G since all the nodes with the same color form an independent set. We can only approximate f(G) but by careful rounding we can get a monotone polynomial-time computable function (and thus polynomial-size AND-OR-NOT circuits) that sits between the clique size and the chromatic number.

Finally Tardos notes that the techniques of Razborov and Alon-Boppana show that any monotone function that sits between clique and chromatic number must have exponential-size monotone (AND-OR) circuits. The NOT gate is truly powerful, bringing the complexity down exponentially.

either pi is algebraic or some journals let in an incorrect paper!/the 15 most famous transcendental numbers

Someone has published three papers claiming that

π is 17 -8*sqrt(3) which is really =3.1435935394...

Someone else has published eight papers claiming

π is (14 - sqrt(2))/4 which is really 3.1464466094...

The first result is closer, though I don't think this is a contest that either author can win.

Either π is algebraic, which contradicts a well known theorem, or some journals accepted some papers with false proofs. I also wonder how someone could publish the same result 3 or 8 times.

I could write more on this, but another blogger has done a great job, so I'll point to it: here

DIFFERENT TOPIC (related?) What are the 15 most famous transcendental numbers? While its a matter of opinion, there is an awesome website that claims to have the answer here. I"ll briefly comment on them. Note that some of them are conjectured to be trans but have not been proven to be. So should be called 12 most famous trans numbers and 3 famous numbers conjectured to be trans. That is a bit long (and as short as it is only because I use `trans') so the original author is right to use the title used.

1) pi YEAH (This is probably the only number on the list such that a government tried to legally declare its value, see here for the full story.)

2) e YEAH

3) Eulers contant γ which is the limit of (sum_{i=1}^n  1/i) -  ln(n). I read a book on γ  (see here) which had interesting history and math in it, but not that much about γ . I'm not convinced the number is that interesting. Also, not known to be trans (the website does point this out)

4) Catalan's number  1- 1/9 + 1/25 - 1/49 + 1/81  ...  Not known to be trans but thought to be. I had never heard of it until reading the website so either (a) its not that famous, or (b) I am undereducated.

5) Liouville's number 0.110001... (1 at the 1st, 2nd, 6th, 24th, 120th, etc - all n!- place, 0's elsewhere)
This is a nice one since the proof that its trans is elementary. First number ever proven Trans. Proved by the man whose name is on the number.

6) Chaitian's constant which is the prob that a random TM will halt. See here for more rigor. Easy to show its not computable, which implies trans.  It IS famous.

7) Chapernowne's number which is 0.123456789 10 11 12 13 ... . Cute!

8) recall that ζ(2) = 1 + 1/4 + 1/9 + 1/6 + ... = π²/6

ζ(3) = 1 + 1/8 + 1/27 + 1/64 + ... known as Apery's constant, thought to be trans but not known.

It comes up in Physics and in the analysis of random minimal spanning trees, see here which may be why this sum is here rather than some other sums.

9) ln(2). Not sure why this is any more famous than ln(3) or other such numbers

10) 2^sqrt(2) - In the year 1900 Hilbert proposed 23 problems for mathematicians to work on (see here for the problems, and see here  for a joint book review of two books about the problem, see  here for a 24th problem found in his notes much later). The 7th problem  was to show that a^b is trans when a is rational and b is irrational (except in trivial cases). It was proven by Gelfond and refined by Schneider (see here). The number 2^sqrt(2) is sometimes called Hilbert's Number. Not sure why its not called the Gelfond-Schneider number. Too many syllables?

11) e^π  Didn't know this one. Now I do!

12) π^e (I had a post about comparing e^π to π^e  here.)

13) Prouhet-Thue-Morse constant - see here

14) i^i. Delightful! Its real and trans! Is it easy to show that its real? I doubt its easy to show that its trans. Very few numbers are easy to show are trans, though its easy to show that most numbers are.

15) Feigenbaum's constant- see here

 Are there any Trans numbers of which you are quite fond that aren't on the list?

If you proof any of the above algebraic then you can probably publish it 3 or 8 or iⁱ times!

 I can imagine a crank trying to show π or maybe even e is algebraic. ζ(3) or the Feigenbaum's constant, not so much.

Kurtz-Fest

Stuart Kurtz turned 60 last October and his former students John Rogers and Stephen Fenner organized a celebration in his honor earlier this week at Fenner's institution, the University of South Carolina in Columbia.

Stuart has been part of the CS department at the University of Chicago since before they had a CS department and I knew Stuart well as a co-author, mentor, boss and friend during my 14+ years at Chicago. I would have attended this weekend no matter the location but a total eclipse a short drive from Atlanta (which merely had 97% coverage) certainly was a nice bonus.

Stuart Kurtz brought a logic background to computational complexity. He's played important roles in randomness, the structural properties of reductions, especially the Berman-Hartmanis isomorphism conjecture, relativization, counting complexity and logics of programs. I gave a talk about Stuart's work focusing on his ability to come up with the right definitions that help drive results. Stuart defined classes like Gap-P and SPP that have really changed the way people think about counting complexity. He changed the way I did oracle proofs, first trying to create the oracle first and then prove what happens as a consequence instead of the other way around. It was this approach, focusing on an oracle called sp-generic, that allowed us to give the first relativized world where the Berman-Hartmanis conjecture held.

The Crystal Blogaversity

A joint post from Lance and Bill

This blog started fifteen years ago today as  "My Computational Complexity Web Log". Bill came on permanently in 2007 after Lance retired from the blog, a retirement that didn't even last a year. We've had over 2500 posts and 6 million page views. We've highlighted great results, honored 100th birth anniversaries, mourned the passing of far too many colleagues and talked about the joys and challenges of being an academic and a theoretical computer scientist.

During the time of this blog, Lance held jobs at four different institutions, several positions in the theoretical computer science community and watched his daughters grow up. Besides his wife, perhaps the only constant in his life is this blog, and no matter how busy things get he still aims to post once a week. Writing keeps him sane.

Bill, who is somewhat of  Luddite, has seen technology change so much around him that he needs something to stay the same. This blog has kept him sane. Or at least more sane.

Computing has seen dramatic changes in the past fifteen years driven by cloud computing, big data and machine learning. Computing now drives society and we've only seen the tip of the iceberg so far. Precious few of these developments are grounded in theory and our community will have a large role to play in understanding what is and isn't possible in this brave new computational world.

Education has changes as well. The number of people majoring in Computer Science has skyrocketed, crashed, and is now skyrocketing again. We teach large lectures with PowerPoint and other technologies for both good and ill. Some people get their degrees online for both good and ill. We comment on all of these developments for both good and ill.

We're not done yet with the blog. We'll keep writing and hope you keep reading. To the next fifteen.

What is unusual about this MIT grad student in Applied Math?

(Thanks to Rachel Folowoshele for bringing this to my attention)

John Urschel is a grad student in applied math at MIT. His webpage is here.

Some students go straight from ugrad to grad (I did that.)

Others take a job of some sort and then after a few years go to grad school.

That's what John did;  however, his prior job was unusual among applied math grad students







He was in the NFL as a professional football player! See here for more about the career change, though I'll say that the brain-problems that NFL players have (being hit on the head is not a good for your brain) was a major factor for doing this NOW rather than LATER.

How unusual is this? Looking around the web I found lists of smart football players, and lists of football players with advanced degrees (these lists were NOT identical but there was some overlap) but the only other NFL player with a PhD in Math/CS/Applied math was

Frank Ryan- see his wikipedia page here. He got his Phd WHILE playing football. He was a PhD student at Rice.

I suspect that a professional athlete getting a PhD in Math or CS or Applied Math or Physics or... probably most things, is rare.  Why is this? NOT because these people are dumber or anything of the sort, but because its HARD to do two things well, especially now that both math and football have gotten more complex. Its the same reason we don't have many Presidents with PhD's (Wilson was the only one) or Presidents who know math (see my post on presidents who knew math: here) or Pope's who are scientists (there was one around 1000 AD, see here).

If you know of any professional athlete who has a PhD in some science or math, please leave a comment on such.

(ADDED LATER- a commenter pointed out Angela Merkel who has a PhD in Physical Chemistry,
is chancellor of Germany, and there is a musical about her, see here.)

(ADDED LATER- some of the comments were for Olympic Athletes and hence not professional and another comment pointed this out. So I clarify: Olympic is fine too, I really meant serious athlete.)

Wearable Tech and Attention

Remember the Bluetooth craze where it seemed half of all people walked around with a headset in their ear. Now you rarely do.

Remember Google Glass. That didn't last long.

I remember having a conversation with someone and all of sudden they would say something nonsensical and you'd realize they are on the phone talking to someone else. Just by wearing a Bluetooth headset you felt that they cared more about a potential caller than the conversation they were currently having with you.

Google glass gave an even worse impression. Were they listening to you or checking their Twitter feed? [Aside: I now use "they" as a singular genderless pronoun without even thinking about it. I guess an old dog can learn new tricks.]

When you get bored and pull out your phone to check emails or put on headphones to listen to music or a podcast, you give a signal that you don't want to be disturbed even if that isn't your intent. Wearing a Bluetooth headset or Google glasses gave that impression all the time, which is why the technology didn't stick.

What about smart watches? You can certainly tell if someone has an Apple watch. But if they don't look at it you don't feel ignored. Some people think they can check their watch without the other person noticing. They do. I've been guilty of this myself.

What happens when are brains are directly connected to the Internet? You'll never know if anyone is actually listening to you in person. Of course, at that point will there even be a good reason to get out of bed in the morning?

Should we care if a job candidate does not know the social and ethical implications of their work (Second Blog Post inspired by Rogaway's Moral Character Paper)

Phillip Rogaway's article on the

The Moral character of Cryptographic Work (see here)

brings up so many issues that it could be the topics for at least 5 blog posts. I've already done one here, and today I'll do another. As I said in the first post I urge you to read it even if you disagree with it, in fact, especially if you disagree with it. (Possible Paradox- you have to read it to determine if you disagree with it.)

Today's issue:

Should a department take into account if someone understand the social and ethical issues with their work?

1) I'll start with something less controversial. I've sometimes asked a job candidate `why do you work on X?' Bad answers:

        Because my adviser told me to.

        Because I could make progress on it.

        Because it was fun to work on.

People should always know WHY they are working on what they are working on. What was the motivation of the original researchers is one thing they should know, even if the current motivation is different. If its a new problem then why is it worth studying?

2) In private email to Dr. Rogaway he states that he just wants this to be ONE of the many issues having to do with job hiring (alas, it usually is not even ONE). As such, the thoughts below may not be quite right since they assume a bigger role. But if you want to make something a criteria, even a small one, we should think of the implications.

3) In private email to Dr. Rogaway I speculated that we need to care more about this issue when interviewing someone in security then in (say) Ramsey theory. He reminded me of work done in pure graph theory funded by the DOD, that is about how to best disable a network (perhaps a social network talking too much about why the Iraq war is a terrible idea). Point taken- this is not just an issue in Security.

4) What if someone is working on security, funded by the DOD, and is fully aware that the government wants to use her work to illegally wiretap people and is quite okay with that. To hold that against her seems like holding someone's politics against them which I assume all readers of this blog would find very unfair.. OR is it okay to hire her since she HAS thought through the issues. The fact that you disagree with her conclusion should be irrelevant.

5) What if she says that the DOD, once they have the tech, will only wiretap bad people? (see here)

6) Lets say that someone is working on cute crypto with pictures of Alice and Bob (perhaps Alice is Wonderland and Bob the Builder). Its good technical work and is well funded. It has NO social or ethical  implications because it has NO practical value, and she knows it. Should this be held against her? More so than other branches of theory?

7) People can be aware of the social and ethical issues and not care.

8) The real dilemma: A really great job candidate in security who is brilliant. The work is top notch but has serious negative implications. The job candidate is clueless about that. But they can bring in
grant money! Prestige! Grad Students! I don't have an answer here but its hard to know how much to weigh social and ethical awareness versus getting a bump in the US News and World Report Rankings!

        What does your dept do? What are your thoughts on this issue?

What Makes a Great Definition

Too often we see bad definitions, a convoluted mess carefully crafted to make a theorem true. A student asked me though what makes for a great definition in theoretical computer science. The right definition can start a research area, where a bad definition can take research down the wrong path.

Some goals of a definition:

• A great definition should capture some phenomenon, like computation (Turing machines), efficient computation (P), efficient quantum computation (BQP). Cryptography has produced some of the best (and worst) definitions to capture security concerns.
• A great definition should be simple. Defining computability by a Turing machine--good. Definition computability by by the 1334 page ISO/IEC 14882:2011 C++ standard--not so good.
• A great definition should be robust. Small changes in the definition should have little, or better, no change in what fulfills the definition. That is what makes the P v NP problem so nice since both P and NP are robust to various different models of computing. Talking about the problems solvable by a 27-state Turing machine--not robust at all.
• A great definition should be logically consistent. Defining a set as any definable collection doesn't work.
• A great definition should be easy to apply. It should be easy to check that something fulfills the definition, ideally in a simply constructive way.

A great definition drives theorems not the other way around.

Sometimes you discover that a definition does not properly capture a phenomenon--then you should either change or discard your definition, or change your understanding of the phenomenon.

Let's go through an interesting example. In 1984, Goldwasser, Micali and Rackoff defined $k$-bits of knowledge interactive proof systems. Did they have good definitions?

• The definition of interactive proof systems hits all the right points above and created a new area of complexity that we still study today. 
• Their notion of zero-(bits of) knowledge interactive proofs hits nearly the right points. Running one zero-knowledge protocol followed by another using the GMR definition might not keep zero-knowledge but there is an easy fix for that. Zero-knowledge proof systems would end up transforming cryptography. 
• Their notion of k-bit knowledge didn't work at all. Not really robust and a protocol that output the factors of a number half the time leaked only 1-bit of knowledge by the GMR definition. They smartly dropped the k-bit definition in the journal version.

Two great definitions trump one bad one and GMR rightly received, along with Babai-Moran who gave an alternative equivalent definition of interactive proofs, the first Godel Prize.