Myhill Nerode versus Pumping Lemma

I have seen some recent backlash against the pumping lemma for showing that languages are not regular and as I am now teaching regular languages I had to choose should I teach the pumping lemma or Myhill-Nerode to show languages are not regular. Let's review both definitions (taken from Wikipedia)

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Pumping Lemma: If a language L is regular, then there exists a number p ≥ 1 (the pumping length) such that every string uwv in L with |w| ≥ p can be written in the form uwv = uxyzv with strings x, y and z such that |xy| ≤ p, |y| ≥ 1 and uxyⁱzv is in L for every integer i ≥ 0.

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Myhill-Nerode: Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that exactly one of the two strings xz and yz belongs to L. Define a relation R_L on strings by the rule that x R_L y if there is no distinguishing extension for x and y. It is easy to show that R_L is an equivalence relation on strings, and thus it divides the set of all finite strings into equivalence classes.

The Myhill–Nerode theorem states that L is regular if and only if R_L has a finite number of equivalence classes, and moreover that the number of states in the smallest deterministic finite automaton (DFA) recognizing L is equal to the number of equivalence classes in R_L. In particular, this implies that there is a unique minimal DFA with minimum number of states.

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The two basic complaints about the pumping lemma: Five quantifiers and it is not complete--there are nonregular languages that can be pumped. To the first point if you think of the pumping lemma as a game with the adversary choosing p, x, y and z, the quantification is not as confusing as some would think. Myhill-Nerode also has five quantifiers when you spell it out: For all regular L, there exist x₁,...,x_k such that for all y there is an i such that for all z, x_iz is in L iff yz is in L.

As to the second part, the counterexamples are contrived and usually go away with simple closure properties. Consider the one from wikipedia:

Take L ∩ (01(2∪3))* eliminates the strings in the first part of L and now it is easy to pump.

So I don't buy the arguments for Myhill-Nerode over pumping. Nevertheless I'll teach the pumping lemma and Myhill-Nerode because they are both so cool.

Types of questions for exams

QUESTION: Give as many types of exam questions you can, give examples, and comment on if this is a good type of question.

My answer below.

1. A problem that some students can get right even if they never had the course because they have seen it in some other course. EXAMPLE: In a course on Ramsey Theory have a question that uses the Prob. Method. PRO: The question is still in scope for the courses. CON: A bit awkward that someone may have learned the material elsewhere. UPSHOT: This is FINE.
   
2. A problem that some students can get right even if they never had the course because they are quite clever. EXAMPLE: Easy Combinatorics or Probability in a sophomore Discrete Math Course. PRO: The question is still in scope for the courses. CON: A bit awkward that someone may have missed class but still got it right. UPSHOT: This is FINE.
   
3. A rigged question--- students saw two examples in class, two examples on the HW and now have to do one themselves. EXAMPLE: proving numbers irrational. PRO: Clearly in scope and fair. PRO: They will surely understand what you are asking for. CON: They may get it right via memory rather than understanding (they may not even know the difference.) UPSHOT: This is FINE though it requires some planning ahead of time.
   
4. A rigged question with a twist--- students saw two examples in class, two examples on the HW and now have to do one themselves but its DIFFERENT in an important way. EXAMPLE: In class and HW do many problems like Here is the distribution, here is a random var, what is its expected value but on the exam give Here is a random var, here is what we want for the expected value, give a distribution that gives us that. PRO: Harder to memorize template. CON: May be hard to grade as they say odd things. CON: May be confusing to know what you are asking for, even for good students. UPSHOT: This is FINE though it requires some planning ahead of time.
   
5. A problem that requires utter mastery of the material but no creative thought. EXAMPLE: Give the algorithm (that we did in class) for proving that a CFG's are in P. Write it up so that someone who had never seen it can understand it. PRO: Straightforward yet hard to get via memorization. CON: Might be too time consuming for an exam. CON: (From experience) no matter how much you say in bold letters things like Write it up so that someone who had never seen it can understand it. They will skip steps and write it up badly and its hard to tell if THEY really know it. UPSHOT: I do this but only in certain cases.
   
6. A problem that requires them to be creative (this is ill defined but its the opposite of the one above). PRO: If they truly understand the material they can do this. CON: My PRO may be incorrect. UPSHOT: Absolutely fine for HW which are not worth much for the grade anyway and I can enlighten them. I tend to avoid these on exams. Though the line between creativity and standard is a thin one. (Problem for an exam: How thin in millimeters?)
   
7. A giveaway question. When I teach Formal Lang Theory I have (going back to when I was Harry Lewis's TA in 1981) have on the exam Give an example of a string of length 4 over the alphabet {a,b}. An unintended consequence- if they CAN"T do this its a really bad sign. I have asked this question many times and I have literally NEVER seen someone get it wrong and pass the course. I have gotten the following answers: ab*, ababa, and a DFA recognizing aaaa (that I was tempted to give credit to but did not). Incidentally, the most common right answer has always been abab. Second is abba. PRO: I have this one early in the exam to calm them down.
   

I try to ask some of each type on an exam. However, sometimes a question can be easier or harder than you intended, or be harder to grade then you thought, or not be in category you thought it would be in. The hardest line to draw is which questions are a matter of mastery and which are a matter of creativity? Another issue- some students can abstract better than others.

When teaching a large course such as Sophomore discrete math (150-200 students) I tend to get a uniform distribution skewed a bit on the high side. More precise: I tend to get at roughly 10 students in EVERY 10-point interval: 0-10, 10-20, 20-30,..., 90-100, with less on the low side and more on the high side. The benefit of this is that the students who get (say) less than 40 CANNOT say Well--- everyone did badly. They really are send a signal to either work harder or drop (I tell them this directly as well). I don't understand profs who give exams where nobody cracks 50/100 (I have heard this is common in Physics). They are wasting half of the grade spectrum.

The Dream

I have this theory that everybody's notion of "recent history" starts not from their memories but from their birth date. Case in point: Billy Joel's We Didn't Start the Fire. The first major event of my then very young life came from an oppressed people making their voices heard. The newspapers in the early days of my life were full of fear of violence that might come from the upcoming march on Washington. But 200,000 souls came out fifty years ago today in a peaceful demonstration asking for the basic freedoms the rest of America had.

Having moved to the birthplace of Martin Luther King, Jr from the hometown of the first black president, I know much has improved in the last fifty years. But we know King's dream is far from fulfilled, obvious to us from the paucity of African-Americans in our conferences and classes.

Take a moment of your day, watch the greatest speech of the 20th century, and remember how far America has come, and how far America has yet to go.

What are Galois Games?

How are math concepts named?

1. After the people who was involved with it. Examples: The Cook-Levin Theorem, Goldbach Conjecture, Ehrenfeucht-Fraisse games,
   Banach-Tarski Paradox.
   
2. A descriptive name:
   Examples: Chromatic Number; Girth of a graph (length of shortest cycle). This resembles the definition of Girth in English though I have only heard the word used in mathematics;
   Duplicator-Spoiler games.
   
3. A name that conjures up a nice image. Examples: Dining Philosophers problem;
   The Monty Hall Paradox (though future historians will think he was a great Probabilist).
   
4. Name may have very little connection to the concept. Example: The Pell equation.
   

I saw an article whose title was Greedy Galois Games. I wondered what this game could be.

1. Do the players alternate picking polynomials and if the composition is solvable by radicals then (say) Player I wins.
   
2. Did Galois invent some game?
   

The first game I thought of might be interesting; however, the paper was not about that. Nor was it about some game Galois invented. So---what is a Galois game? Aside from being a mathematician what else is known about Galois:

He died in a duel!

In the article Greedy Galois Games they study a DUEL between two BAD DUELISTS. The idea is that if both have prob of hitting p (and p is small) and they want to make it fair, first Alice shoots, then Bob shoots the min number of times so that the prob of Bob winning exceeds Alice's, then Alice shoots a number of times so that her prob of winning exceeds Bob's, etc. The paper ends up involving the Thue-Morse sequence. They are NOT using the name Galois the way we use Banach in Banach-Tarski Paradox, nor the way we use Monty Hall in The Monty-Hall Paradox. The fact that Galois was a mathematician has nothing to do with the naming,  The authors are using  Galois because he is a  famous duel-loser. They could have used Alexander Hamilton (who lost a Duel to Aaron Burr) and then called them Greedy Hamiltonian Games, in which case I would assume that the game involved
Hamiltonian cycles or Quaternions.

P = NP and the Weather

In the Beautiful World, my science fiction chapter of The Golden Ticket where P = NP in a strong way, I predicted that we could predict weather accurately enough to know whether it will rain about a year into the future. Besides putting Novosibirsk on the wrong side of Moscow, my weather prediction prediction has drawn the most ire from my readers.

Here was my thinking: Weather forecasting comes down to modeling. Find a good model, use the current initial conditions and simulate the model. P = NP can help dramatically here by making what should be the hardest part, finding the right model, easy. P = NP would help create much better models and should lead to far more accurate and deep forecasts than before. A year ahead prediction of weather didn't seem out of the realm of possibility.

As my readers point out, one cannot put in all of the initial conditions which would involve too much data even if we could get it, and small random events, the so-called butterfly effect, could dramatically change the weather in even a short period of time. Dean Foster, a Penn statistician, wrote me a short piece giving an analogy to a game of pool over time changed by the gravity generated by a single proton.

So how far can you predict the weather if P = NP? A month? Of course we'll probably never find out since I doubt P and NP are the same. In retrospect I shouldn't have put in such an aggressive weather forecasting because it detracts from other great things that happen if P = NP such as curing cancer.

When Lance was 10 years old..

In honor of Lance's 50th birthday I ask the following: When Lance was 10 years old which of the following were true?
(Disclosure- some of the below are from a birthday card.)

1. A REMOTE meant a secluded spot off the beaten path.
   
2. CABLE was something that supported a bridge.
   
3. A VIDEO GAME was trying to make out what fuzzy images were on a snowy black and white 10 inch TV screen.
   
4. A CELL PHONE was what you used to make one phone call from jail.
   
5. A CALCULATOR was the accountant who did your parents taxes.
   
6. AN AIRBAG was someone who talked too much.
   
7. DIGITAL COMPUTING was counting on your fingers.
   
8. HIGH SPEED ACCESS was an on-ramp to the freeway.
   
9. SURFING was something done on a board in the ocean.
   
10. A BIRTHDAY was something Lance looked forward to.
    
11. A MOUSE was something you didn't want in your house.
    
12. A SPAM ASSASSIN was someone who killed people by giving them poisoned spam.
    
13. A WEB was what spiders wove.
    
14. A BUG was what spiders ate.
    
15. AMAZON meant where some big rain forest is (smaller now).
    
16. GOOGLE was an obscure term used by some math folks for the number 10¹⁰⁰.
    
17. BING had no meaning.
    
18. APPLE was either a fruit or the record company founded by the Beatles. (There really WAS a legal name-issue when Apple-the-computer-company got into music see here .)
    
19. It was impossible to have 10,000 friends.
    
20. There were only three Network channels and a few local ones.
    
21. Music was on Vinyl records.
    
22. You went to the bathroom during commercials.
    
23. Johnny Carson joked that couples had sex during commercials on his show. (Ask your grandparents who Johnny Carson was, what commercials were, and what sex was.)
    
24. People read books written on paper.
    
25. Computer Science was not available as a major at most schools.
    
26. When people said you sound like a broken record they actually knew what a broken record sounded like.
    
27. People really would DIAL a phone number.
    
28. People would have to actually stop at toll booths instead of using easy-pass.
    
29. Long running TV shows would have one (or at most two) Christmas episodes since there were no arcs, hence an episode could be inserted into any season at any time. Contrast: M*A*S*H in its 11 seasons and 256 episodes had TWO Christmas episodes, where as 30 ROCK its 7 seasons and 131 episodes had FOUR Christmas episodes. (This may be THE least important consequence of the new technology.)
    
30. There were bar room fights over trivia since you couldn't just look it up on Google. The Guinness Book of World Records was supposed to cut down on bar fights, but it didn't quite work.
    
31. People knew how to read maps and get a sense of where things were instead of relying on technology. That's why today the number of hikers who get lost has skyrocketed.
    
32. If MTV existed they would still be playing music videos. The question Why doesn't MTV show Music Video's anymore has been asked so often it is now Cliche. But the above video provides an answer.
    
33. Lance did not recognize the importance of NP-completeness. Then again, neither had the math community, the non-theory computer science community, and Probably parts of the theory community.
    
34. To find out what time it was you couldn't look at your cell phone, TV set, or Microwave. You had to go outside and look at your sundial.
    
35. TV shows may have pilot episodes, or may not, but they didn't bother with explaining everything. Thought experiment: If Mr. Ed was on today
    they would explain how he could talk (A government experiment gone wrong? gone right?) rather then the ONE line by Mr. Ed in the first episode: Don't try (to understand why I can talk)--- its bigger than both of us.
    
36. We all watched a TV show the same night. Contrast- last month I watched Firefly.
    (If you are a fan of firely check this out.)
    Bizarre result of this--- since people can't find people to talk about shows as much as the used do, there is now a show called TALKING BAD where people on the show TALK ABOUT Breaking Bad
    
37. The final Jeapordy theme music didn't have lyrics. Now it does: here.
    
38. When you heard a mnemoic device like Kids Prefer Cheese Over Fried Green Spinach it was hard to find out what it meant- now its easy (just use Google!)
    

True story: On March 9, 1967 John Smith (not his real name) wanted to watch Star Trek (Episode: Devil in the Dark) but his parents wanted to take the family out for dinner. So he pretended to be sick so he could watch it- because, as he puts it, if I don't see it now I will NEVER GET TO SEE THE EPISODE, EVER!!!!!. Imagine a world without DVR, DVD, TIVO, On-Demand, Hulu. He doesn't have to imagine it. Our younger readers do.


I think SURFING, MOUSE, and SPAM really have changed primary meanings. FRIENDS may have also.

Flash Gordon

We watched the movie Ted last week but this post isn't about that movie. The movie has several references to the 1980 movie Flash Gordon including an extended cameo by Sam Jones who played Flash.

Flash Gordon and its soundtrack from Queen saved me senior year of high school--whenever I felt down I would listen to the album and run the movie through my head escaping reality for a little bit. These were the days before videos and CDs, now I've rewatched the movie several times on DVD.

Flash Gordon was not a great movie by any means but it resonated with me with its action sequences, great music and corny lines like "Flash, I love you, but we only have fourteen hours to save the Earth!". The stars of the movie Sam Jones and Melody Anderson were and still are relatively unknown but it had a great supporting cast.

Topol, best known as Tevye in Fiddler on the Roof, played a scientist who many mocked for his crazy (but true) ideas of what was happening in outer space. Basically the same character as when he played Galileo.

Timothy Dalton played Prince Barin and would go on to be James Bond and the Max von Sydow, who played chess against Death in The Seventh Seal, was the Ming the Merciless.

What does this all have to do with computational complexity? Absolutely nothing. But today I turn 50, it's my party and I'll post what I want to.

How much Trig does your governor know?

How much math should our public officials know? Basic probability and statistics so they can follow the arguments that their science advisers give them. And they should hire good objective science advisers and listen to them.

How much Trigonometry should a Governor know? Should a Governor know the angles of a 3-4-5 triangle? The following true story is paraphrased from Somewhat more than Governors need to know about Trigonometry by Skip Garibaldi.

In June 2004 Governor Jeb Bush of Florida was giving a talk to promote state-wide annual testing of students in public schools. A high school student asked him What are the angles in a 3-4-5 triangle? He responded I don't know. 125, 90, and whatever is left to add up to 180. Note that (1) he knew that 3-4-5 triangle has a 90 degree angle, (2) he knew that the angles of a triangle add up to 180, but (3) he didn't realize that 125+90 > 180. Still, I suspect most governors would do worse. The real answer is 90, 53.1 (approx), 36.9 (approx). A retired math professor was later quoted as saying I would not expect many mathematicians to know that.

The paper then proves the following:

The Governors Theorem: If a right triangle has integer
side lengths then the acute angles are irrational when measured
in degrees.

When politicians say things that contradict current science (e.g., on evolution or global warming) I wonder if they know the truth and are lying to please their voters, or if they honestly don't know the truth.I also wonder which one is worse. In the case above I think Jeb honestly didn't know, and that's fine.

Don't Have an End Game

As a young professor, I wrote a grant proposal and took it to a senior theory professor for comments. He told me to take out the line "The ultimate goal of computational complexity is to settle the P versus NP problem." He agreed with the line, he just said that if we make these claims to the NSF then what happens after someone proves P different from NP? Nothing left to fund in complexity.

There was precedence here. In the 70s and 80s algebraists had the great goal of classifying all the finite simple groups. Once they were done, then what? Other examples are sending a man to the moon in the 60's or having a computer that beats the best human chess player.

Having an ultimate goal can be very motivating but quite limiting if that goal is actually reached. Luckily for us the P versus NP problem is a goal which will not likely be reached for a very long time.

Longest time between posing a math problem and it being answered?

(We were asked to remind you: ITCS 2014 Call for papers: call for papers.)


What problem in math had the longest time between POSING IT and SOLVING it? This might not be a well defined question since the notion of when was it posed? might be murky. For some problems even when it was solved? might be murky. Nevertheless I have a candidate:

Is there a straight-edge and compass construction that will, given a square, produce a circle with the same area. (This problem is often called Squaring the circle..)

Wikipedia says that Oenopides was the first person to pose construction problems and that he posed this one. He was born in roughly 500 BC. Even back then there were people who thought it could not be done. However, it was proven impossible when pi was shown to be transcendental in 1882 by Lindemann. (This was one of the motivations for Lindemann.)

This problem was open for roughly 2300 years.

1. Is there any solved problem that was open for longer?
   
2. Is there any open problem that has been opened for that longer?
   
3. If you polled people in 400 BC what they would have guessed for which way it would go and when it would be solved?
   

Will P vs NP take that long?

Why is Multiplication Hard?

Quick. What is 879544 * 528045? Unless you used a calculator or was some sort of savant you it would take you a couple of minutes to figure out a solution. Of course a computer can calculate this very quickly.

But what a computer can't do easily is learn how to multiply. If we feed in triples of numbers, (879544,582045,464438811480),  (541535,711245,385164061075), (230589,481621,111056504796), ..., into any machine learning algorithm it's doubtful the algorithm could take a new pair (666750,313009) and produce its product 208698750750. For if it could, then we should be able to use a similar algorithm to figure out how to factor numbers, which we believe a computationally difficult talk.

When you look at what machine learning seems to do moderately well: spam detection, face recognition, language translation, voice-to-text and self-driving cars, these are things that humans with a reasonable amount of training, can do very well.

Is this some philosophical argument that our brain works like machine learning algorithms? Think of it more as an observation.

Certifying primality in a CONSTANT number of operations

For this post I will only count the operations PLUS, MINUS, MULT. They may be done on rather large numbers.

Recall that from the work coming out of Hilberts 10th problem we know the following: For every c.e. set (used to be called r.e., some people still do) there is a polynomial f in 13 or less variables (we'll assume 13) with coefficients in the integers such that

x in A iff (∃ a1,...,a13))[f(x,a1,...,a13)=0]

In an article about Hilbert's 10th problem written in 1974 by Davis-Matiyasevich-Robinson they note that by this result there is a FINITE number M such that, for ALL primes p, there is a certification that p is prime that uses at most M operations: given p a prime let a1,...,a13 be such that f(p,a1,...,a13)=0. The certification that p is prime is just the evaluation of that polynomial and seeing that its 0.

Is this still the only proof that one can certify primality in a CONSTANT number of operations?

Primes is irrelevant to all of this--- any c.e. set would work. (The result for c.e. sets may qualify as a theorem that is less interesting because its more interesting.) But for primes I am wondering if there is another way to do this- perhaps using number theory, perhaps with a smaller value of M. For the explicit poly for primes, due to Jones, see here.

Ph.D. Attrition

Leonard Cassuto writes in the Chronicle an article Ph.D. Attrition: How Much Is Too Much? He presupposes the answer with the subtitle "A disturbing 50 percent of doctoral students leave graduate school without finishing".

The 50% goes over all fields but the numbers in computer science are somewhat in that range. Computer Science has different issues than humanities and theoretical CS has not quite the same issues as the rest of CS. Certainly we lose several students to start-ups and high-paying jobs. But what about the ones that just have trouble in grad school.

Cassuto writes

Perhaps they lack the temperament to work on their own (which undergraduate work does not test as severely as graduate school does), or perhaps they lack, say, the mathematical chops necessary to succeed at advanced physics. But there will be a number—and if admissions committees do a good job, it will be very small—who won't be able to finish because they're not up to the demands of the task.

Having read through many graduate applications through the year there are very few, perhaps on average one or two a year, that will clearly succeed through graduate school. Almost without exception those students go to MIT or Berkeley.

For the rest of us, you have a choice. You can either take someone who will probably work their way to a Ph.D. but with uninspired research, or those you can take a risk with a student who might have strong potential. Some of those students become great scientists, some of them flame out. You get a higher attrition rate by taking risks but that's not a bad thing.

If you do take a risk in admissions you need to encourage students to "pursue other opportunities" once you realize they won't make it. That's a process that too many of us try to avoid, so we don't take those risks as much as we should.

I gave a poster session at Erdos 100- so how did it go?

In a a prior post I suggested that STOC perhaps have people give posters instead of talks. While I doubt this will ever happen I think its worth thinking about, especially for future conferences that may be founded. I also noted that the NIPS conference they do this.

But enough theory- at the Erdos 100th I GAVE a poster. Here are my thoughts.

1. The paper I did a poster I posted on here and I posted to arxiv here. A bit awkward in that it was submitted to ERDOS 100 as USING THE ERDOS-RADO CAN RAMSEY THEOREM ON A PROBLEM ERDOS ASKED AND A PROBLEM ERDOS SHOULD HAVE ASKED, but by the time the conference came I had much better results (due mostly to co-authors of which I went from 1 to 4) and no longer used ERDOS RADO CAN RAMSEY. Do I do the Poster on what was submitted or what I have now? I picked a very nice proof to concentrate on for the poster that was new but still in the spirit of what was submitted. (The final version is being written- I'll post on this blog about it later.)
   
2. The posters were for TWO days, for TWO hours after lunch. Since it was after lunch they didn't serve food. This seemed to work. They were in two shifts-- some did Tu-Wed and some did Th-Fri (I did Th-Fri).
   
3. My actual Poster was terrible. But me talking about it and pointing to things was good. This was true in general- other peoples posters were hard to understand if the person wasn't there to clarify and explain, but was pretty good if they were. And it was nice to be able to ask questions directly and interrupt, unlike talks.
   
4. As someone LISTENING to a poster talk it was better than a real talk. In one case I listened, went home that night,
   wrote some things down, realized I missed a point, and asked him again the next day.
   
5. As someone GIVING a poster talk... it was very odd. I explained my results and a simple proof of one of them about 40 times in a 2 day period. I happen to like this (note that I've taught VDWs theorem at least W(6,2) times). But even though I like it, it was tiring. You know how it is ---- the first 35 times you explain a theorem you're excited about it, but then it got to be old hat (which would have been fine if it was a talk on a hat problem).
   
6. There were 60 posters.
   

This was overall a positive experience but, again, tiring.

So would this work for STOC/FOCS or other existing conferences? We would have to adjust our mentality to thinking that posters were not less prestigious. I don't think this will happen. But what about a new conference? If some new conference in theory gets started perhaps they should look into this model. A new conference does not have to follow the STOC/FOCS model.

A(nother) nice use of Gen Functions

In a prior post I tried to give a simple example of a proof that uses Gen Functions where there was no other way to do it. For better or worse, before I posted it, my HS student Sam found a better way and I posted both proofs.

I have another example. Noga Alon showed this to be over dinner at the Erdos 100th Bday conference. (He claims that the proof he showed me is NOT his but he doesn't know whose it is. I will still call it Noga's Proof for shorthand.)

Let

A+A = { x+y : x,y ∈ A}

A+*A = { x+y : x,y ∈ A and x ≠ y }

We take both to be multisets.

Assume A is a set of natural numbers. When does A+*A determine A?

If A is of size 2 then NO, A+*A does not determine A as we could have x+y=5 but not know if A is {1,4} or {2,3}.

What if A is of size 3? Then YES:

First determine S=((x+y)+(x+z)+(y+z))/2=x+y+z.

Then determine

x = S - (y+z)

y = S - (x+z)

z = S - (x+y)

What if A has four elements? Does there exists A,B of size 4, different, such that A+*A=B+*B?

YES:

A = {1,4,12,13}

B = {2,3,11,14}

For which n does does A+*A, where A is of size n, determine A?

Selfridge and Strauss showed that this happens iff n is NOT a power of two. I have a write up Noga's proof. The original proof, in this paper, does not use gen functions and also applies to sets of complex numbers. I think Noga's proof can be modified to apply here. Which proof is better? A matter of taste; however, Noga's proof can be sketched on a greasy paper placemat in an outdoor restaurant in Budapest while the original proof cannot.

DUMP YOUR TABLES! (the moral of my story that started with a hat problem)

Recall from my last post:

PROBLEM 1: There are n people sitting on chairs in a row. Call them p1,...,pn. They will soon have HATS put on their heads, RED or BLUE. Nobody can see their own hat color. pn can see p(n-1),...,p1. More generally, pi can see all pj j < i.

Here is the game and the goal: Mr. Bad will put hats on people any way he likes (could be RBRBRB..., could be RRRBBB, could be ALL R's - like when a teacher has a T/F test where they are all FALSE.)
Then pn says R or B, p(n-1) says R or B, etc. When people say the color everyone else can hear it.
They want to MAXIMIZE how many of them say THEIR hat color. The people can meet ahead of time to discuss and agree on a strategy.
Mr. Bad knows the strategy the people will use.

What is the best they can do? Answer: n-1:

pn says RED if the number of REDS he sees is EVEN, BLUE if the number of REDS he sees is ODD. p(n-1) sees all ahead of him, knows the parity of all of them, can deduce his own hat. So can everyone ahead of him- KEY is that they use BOTH what they heard from the people who already spoke and what they see ahead of them. So can do n-1. (NOTE- a nice but not-optimal solution that some people have told me is to use the first log n people to code how many of the remaining hats are RED- this yields n- log(n) correct.)

PROBLEM 2: Same as Problem 2 but now there are c colors of hats.

That hats are colors 0,1,...,c-1. p(n) SUMS up all of the hats ahead of him MOD c. He says that number. p(n-1) heard that answer, See's whats ahead of him and sums that, and can deduce his own color. Again n-1 get it right.

OKAY, that's the problem and the answer. NOW my story and point:

I once had a group of College Students in a summer program working on PROBLEM 1. The plan was that they would first do the people-in-a-row-2-colors version, then people-in-a-row-c-colors version, then other versions. One can learn much math from looking at many variants. They began with the 2-color case and begun working out some examples. They had these tables (Note the word TABLES for later) for the n=3 case - really large decision trees- that (I think) did yield 2 people correct. They then had a table for n=4 where (I think) 2 people correct. They worked out a few more as well, perhaps getting up to n=8. The tables got larger and larger and more complicated. I never did quite understand their tables; however, they may have been doing an ad hoc version of the strategy where
n-log(n) people get the correct hats.)

I let them go on (perhaps too long) since they kept telling me NO BILL, DON"T TELL US HOW TO DO IT, IF YOU KNOW. And I was hoping they would have a breakthrough. But by the end of the second week they still hadn't gotten it (NOTE- this is not an indication that they were bad students--- its hard to tell how hard it is to see the trick once you know it) and asked me if I knew how to do it. I told them the solution above using Parity. I THOUGHT they would say OH, that's very nice, now lets see if we can do something similar for c-colors. But no. They insisted that their solution using tables was more intuitive or more informative or more ... something. None of that is remotely true. What is true is that by that point they were emotionally invested in their tables.

I kept saying DUMP YOUR TABLES now that you have a better way of doing it. They never did. But the phrase DUMP YOUR TABLES I now
use to mean DUMP SOME OLD WAY OF DOING THINGS THAT YOU ARE EMOTIONALLY ATTACHED TO BUT REALLY DOES NOT WORK.
Once you are aware of this phenomena you can see it often.

1. You have a proof that uses a certain technique that you like (in my case perhaps Ramsey Theory) but then a better proof comes along. You have to admit that the new proof is better. DUMP YOUR TABLES.
   
2. Your proof idea is beautiful but it just doesn't work. SHOULD YOU DUMP YOUR TABLES? Hard to tell- might work later.
   
3. You get emotionally attached to a certain way to teach a course. Times change, technology changes, and perhaps you should DUMP YOUR TABLES.
   
4. I have an idea for a blog entry that I think is really good and I begin writing it, and it just isn't working. I SHOULD DUMP MY TABLES.
   
5. Sometimes in a story there is ONE really good idea and the rest is crap. This might be that the author had ONE really good idea
   but could not build a good story around it. He should have DUMPED HIS TABLES.
   
6. You have a phrase that you are fond of but it distracts from the point you are trying to make. You should
   DUMP YOUR TABLES
   (See Here For a case).
   

A problem and later a story and a point.

I have (1) a math problem to tell you about (though I suspect many readers already know it), (2) a story about it, and (3) a point to make. TODAY I'll just do the math problem. Feel free to leave comment with solutions--- so if you haven't seen it before and want to try it, then don't look at the comments. Tommorow or later I will tell you the story and make my points.


PROBLEM 1: There are n people sitting on chairs in a row. Call them p1,...,pn. They will soon have HATS put on their heads, RED or BLUE. Nobody can see their own hat color. pn can see p(n-1),...,p1. More generally, pi can see all pj j < i. They CAN meet ahead of time to discuss strategy.

Here is the game and the goal: Mr. Bad will put hats on people any way he likes (could be RBRBRB..., could be RRRBBB, could be ALL R's - like when a teacher has a T/F test where they are all FALSE.) Then pn says R or B, p(n-1) says R or B, etc. They want to MAXIMIZE how many of them say THEIR hat color. Assume that Mr. Bad knows the strategy the people will use.

What is the best they can do?

Here is a strategy: pn says R if the MAJORITY are R, and B if the MAJORITY are B, and then everyone says what pn says. They are guaranteed around n/2 correct.

Here is a strategy: Assume n is even. pn says the color of p(n-1). p(n-1) then says what pn said and gets it right. then p(n-2) says what p(n-3) has. Then p(n-3) gets it right. You are guaranteed to get around n/2 right.

GEE- can we do better than n/2? Or can one prove (perhaps using Ramsey Theory, perhaps something I learned at Erdos 100 over dinner) that you can't beat n/2 (or perhaps something like n/2 + log(log(n))).

PROBLEM 2: Same as Problem 2 but now there are c colors of hats.

NOTE- there are MANY hat problems and MANY variants of this scenario--- some where you want to maximize prob of getting them all right, some where everyone sees everyones hat but their own. These are all fine problems, but I am just talking about (1) people are in a row, (2) Want to maximize how many they get right in the worst case.

ADDED LATER- WARNING- THE ANSWER TO PROBLEM 1 IS IN THE COMMENTS NOW.
SO IF YOU WANT TO SOLVE IT YOURSELF DO NOT LOOK AT THE COMMENTS.

Combinatorics use to not get any respect. But because of Erdos...

(This blog is based on things I heard at the Erdos 100th Bday Conference)

I have spend the last week at the Erdos 100th bday conference. One point that was made many times: the acceptance of Combinatorics by the mathematics community and Erdos's effect on that.

In the 1950's combinatorics was seen as recreational but not as serious math. In the 1970's you could get a PhD in it but it was still seen as suspect. Even at the time of Erdos's death (September 1996) it was still not that well regarded. Now it is, as evidenced by Szemeredi getting the Abel Prize (Gowers and Tao getting the Fields Medal is also evidence, though not as strong since one could argue that they are not really combinatorists). What changed?

1. I would have thought Szemeredi's theorem (1975) would have turned people around on combinatorics. It didn't. Roth proved the k=3 case in the 1950's, using Fourier Analysis (``Real Math'') but Szemeredi's proof of the general case was ``purely combinatorial'' and hence of less interest. Furstenberg's proof that used Ergodic theory helped put it on the mathematical map (is the Mathematical map a bijection?) but combinatorics still was not well regarded.
   
2. Erdos got many people interested in combinatorics and the connections of it to other areas such as number theory. He had incredibly good taste in problems in that the problems he suggested often lead to deep mathematics of interest, and to more problems of interest. His emphasis on asymptotics, which now seems so natural, was revolutionary at the time and later had applications to computer science. His constant pushing for better and better results, his concept of Proof from THE BOOK his encouraging epsilons and deltas to pursue mathematics, all had a profound affect on mathematics and mathematicians.
   
3. One of the reasons for the disdain was that it was seen as recreational math. This was damming for two reasons (1) the problems were not important, and (2) the proofs were easy. Both are unfair. This may have been true at one time but they became less true over time.
   
   1. Problems not important: P vs NP is certainly important. Ramsey Theory reveals hidden
      regular structure and is important. Much of the work that has gone into better bounds
      on the VDW numbers is very important and involves deep mathematics.
      
   2. Proofs are easy: People are using Fourier analysis and ergodic theory and others tools that are rather difficult. Here we have the No true Scotsman Fallacy where people claim that if it uses these tools then its not combinatorics. This raises the question of if a field is defined by its methods or by its problems. In any case, people are solving problems in combinatorics using hard methods. But even among so-called easy proofs, they often exhibit the NP-phenomena where they are easy to verify and hence LOOK easy, but are hard to come up with.
      
4. One of the reasons for the respect is computer science. Just as Continuous math was just the right tool for physics, discrete math is just the right tool for computer science. This lead to a rich source of problems for combinatorists that in turn lead to interesting techniques.
   
5. Erdos stressed asymptotics which was just the right approach for computer science.
   

How much was Erdos responsible for the respect combinatorics has now? For those who believe in The Great Person theory of history, one person CAN make a difference and perhaps Erdos is one of those people. Would combinatorics have moved into the mainstream without Erdos? I think combinatorics would have gotten respect in year n where n might be large. With Erdos, n is smaller. Perhaps much smaller. I leave it to the reader to work out the proper asymptotics.

AltaVista versus Google

Today Yahoo is closing AltaVista, the best search engine before Google. The news caught me by surprise, AltaVista still existed? A number of commentators attribute bad management for AltaVista losing its dominance to Google. But it was an algorithm that killed the search engine.

AltaVista made its claim to fame in the mid-90's by indexing a large number of web pages. AltaVista did very well for obscure search terms like "fortnow" but didn't do so well for more common searches. I used to run a test on search engines by looking for "Holiday Inn", a popular hotel chain in the US. When you search AltaVista for Holiday Inn, the first thing listed was a Holiday Inn in Buffalo, New York. The Holiday Inn home page was nowhere to be found on the search results.

For searches like Holiday Inn, one had to use Yahoo, which back then was not a search engine but a directory tree of web sites. We needed our own directories as well. Ian Parberry maintained the TCS Virtual Rolodex, a list of home pages of theoretical computer scientists, most of which had names common enough that AltaVista wouldn't find them.

A Stanford professor (I can't remember which one) came to give a talk at the University of Chicago around 1997 and he mentioned a research project at Stanford developing a new search engine known as Google. I tested Google with my Holiday Inn test and was in shock when the Holiday Inn home page showed up as the first time. Google passed every other test I could throw at it and I've rarely used any other search engine since. Google made AltaVista, the Yahoo directory and the TCS rolodex irrelevant. Google's PageRank algorithm simply took search to a new level, like the way that Steve Jobs didn't create the first smart phone but completely changed the game with the iPhone. AltaVista managed to survive for another 15+ years but never recovered market share.

The AltaVista story leads to a lesson we still tackle today. Collecting and storing big data is a huge technical challenge but data by itself is of limited value without the algorithms to find the important parts among the muck.

Computability in Europe

Bill and I are both in Europe this week. I'm in Milan at Computability in Europe and Bill is 500 miles away in Budapest for the Paul Erdős Centenary. The US 4th of July holiday doesn't seem to sway the the Europeans from holding workshops. Bill will report on the star-studded Erdős celebration when he gets back.

So what is "Computability in Europe"? Don't the Europeans use the same Turing machines that we do? Wasn't Turing European?

Or course computation is the same, whether we do it in the US or Europe, Japan or Jupiter, but the emphasis is different. In the US we typically deal with traditional models of computers and see how much time and memory we need to solve various problems. The theme of this year's CiE is "The Nature of Computing" with "nature" being the key word. The conference is co-located with the Unconventional Computation and Natural Computation conference that focuses on different models of computing, especially those that rise from nature like biological computing. The two tutorials this week come from Grzegorz Rozenberg, talking on computing modes based on living cells and Gilles Brassard (whom I didn't recognize without his trademark beard) on quantum models.

Me, I like my computation served straight up on Turing machines, thank you very much.