Sixteen Years in the Making

Every paper has a story but Sunny Daniel's Arxiv paper from yesterday deserves a blog post.

We begin in 1982 when Ravi Kannan proved that Σ₂ (the problems computable in NP with an NP oracle) cannot have n² size circuits. Kannan's result hold for n^k-size circuits but for this story we'll keep it simple.

Kannan had an ingeniously simple proof. By diagonalization you can create a language L in Σ₄ that does not have n²-size circuits. Now there are two cases:

1. SAT doesn't have n²-size circuits. Since SAT is in Σ₂ we are done.
2. SAT has n²-size circuits. Then by Karp-Lipton Σ₄ = Σ₂ so L is in Σ₂ and we are done.

Kannan's proof is non-constructive and doesn't give an explicit Σ₂ language that we can show doesn't have n²-size circuits. Either SAT or L but one can't be sure which one.

In 1998, Sunny Daniels, a PhD student at Rutgers, took Eric Allender's complexity course. Eric offered up a constructive example of Kannan as an open problem. Sunny came up with a solution. He wrote up a draft in LaTeX but for personal reasons dropped out of academics and never published the paper.

In 2003, Jin-Yi Cai and Osamu Watanabe, not aware of Daniels' paper, came up with their own independent construction and presented their paper at the COCOON conference in Montana. Word got back to Sunny but he thought he had lost the LaTeX file and didn't want to retypeset the whole proof.

Sunny had Iomega Zip Drive cartridges from his Rutgers days. Recently he found someone who had a Zip Drive reader and managed to recover the files. In there he discovered the original LaTeX, cleaned the paper up, and sixteen years after his proof put the paper on ArXiv. Even if you don't care about the math, read the introduction for the complete version of this story.

Kannan's proof actually shows Σ₂∩Π₂ does not have n²-size circuits and this was later improved to S₂^P. Whether we have any constructive language in Σ₂∩Π₂ or S₂^P without n²-size circuits still remains open. 

A Deadly Side of Complexity

Better algorithms can lead to better medicine and save lives. Just today Tim Gowers discusses Emmanuel Candès' ICM Plenary Lecture, which among other things describes how Candès' work on compressed sensing leads to shorter MRI scans for children, greatly reducing the risk of oxygen deprivation. Prove P = NP with a practical algorithm, and you'll conquer that worst of our diseases. Sounds great until you realize what we can't do.

I was talking to a cancer researcher recently and he points out that many of their challenges are indeed algorithmic. But he also brings up the contrapositive. Since we don't have great algorithms now, we don't know how to make sense of DNA sequences and in particular don't know how to map genetic markers to an appropriate cancer treatment. He works with cancer patients, knowing he can't give them the best possible treatment, not because of lack of data, but due to lack of ways to analyze that data. People die because we don't have the ability to break through the complexity of these algorithmic challenges.

Turing's Oracle

My daughter had a summer project to read and summarize some popular science articles. Having heard me talk about Alan Turing more than a few times, she picked a cover story from a recent New Scientist. The cover copy says "Turing's Oracle: Will the universe let us build the ultimate thinking machine" sounds like an AI story but in fact more of an attack on the Church-Turing thesis. The story is behind a paywall but here is an excerpt:

He called it the "oracle". But in his PhD thesis of 1938, Alan Turing specified no further what shape it might take...Turing has shown with his universal machine that any regular computer would have inescapable limitations. With the oracle, he showed how you might smash through them. 

This is a fundamental misinterpretation of Turing's oracle model. Here is what Turing said in his paper Systems of Logic Based on Ordinals, Section 4.

Let us suppose we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of the oracle apart from saying it cannot be a machine. (emphasis mine)

The rest of the section defines the oracle model and basically argues that for any oracle O, the halting problem relative to O is not computable relative to O. Turing is arguing here that there is no single hardest problem, there is always something harder.

If you take O to be the usual halting problem then a Turing machine equipped with O can solve the halting problem, just by querying the oracle. But that doesn't mean that you have some machine that solves the halting problem for, as Turing has so eloquently argued in Section 9 of his On Computable Numbers, no machine can compute such an O. Turing created the oracle model, not because he thought it would lead to a process that would solve the halting problem, but because it allowed him to show there are problems even more difficult.

Turing's oracle model, like so much of his work, has played a major role in both computability and computational complexity theory. But one shouldn't twist this model to think the oracle could lead to machines that solve non-computable problems and it is sacrilege to suggest that Turing himself would think that.

Complexity versus Algorithms: The FOCS Challenge

In recent years, I've heard complaints from my complexity colleagues that FOCS and STOC are mostly algorithms and from the algorithm buddies that STOC and FOCS are mostly complexity. What exactly counts as a complexity or algorithms paper has become quite blurred in recent years. So let's try an experiment. Below is a poll I've created using titles from the upcoming FOCS conference. Which of these papers do you consider complexity? Does complexity in the title make them a complexity paper? 

If you are interested, you can find the manuscripts for most of these papers on the FOCS accepted papers list.

survey tools

Disclaimer: This is a completely non-scientific poll solely for the interest of the readers of this blog. The results will have no effect on future conference papers.

Favorite Theorems: Limited Independence

When can limited randomness act as well as true random bits?

Polylogarithmic independence fools AC⁰ circuits by Mark Braverman (JACM 2010)

To explain this result consider choosing uniformly from among the following four strings:

000 110 101 011

If we look at any two of the bits, say the first and third, all four possibilities 00 10 11 01 occur. The sequence is thus 2-wise independent. We can get 2-wise independence using only two random bits to choose one of the four strings. In general one can get k-wise independent in n-bit strings using O(k² log n) random bits.

Braverman shows that any polylogarithmic independent distribution fools polynomial-size constant-depth circuits (AC⁰), or more precisely for a size s depth d circuit C, for k=(log (m/ε))^O(d2), the probability that C will output 1 with uniformly-random inputs will differ at most ε from the probability C outputs 1 from inputs chosen from a k-wise random distribution. Braverman's result followed after Bazzi and Razborov proved a similar result for depth-2 circuits (CNF formulas).

Another nice result along these lines: Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio and Emanuele Viola show that Bounded Independence Fools Halfspaces. A half-space is just a weighted threshold function (is the sum of w_ix_i at most some given θ). Diakonikolas et al. show that one can fool halfspaces with k=O(ε^-2log²(1/ε)), in particular for constant ε, k is a constant independent of the number of variables.

Subhash Khot wins Nevanlinna

At the opening ceremonies of the International Congress of Mathematicians in 2014, Subhash Khot was awarded the Rolf Nevanlinna Prize, given every four years to an under-40 researcher for outstanding contributions in Mathematical Aspects of Information Sciences. Subhash's citation reads

Subhash Khot is awarded the Nevanlinna Prize for his prescient definition of the “Unique Games” problem, and leading the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems; his work has led to breakthroughs in algorithmic design and approximation hardness, and to new exciting interactions between computational complexity, analysis and geometry.

Khot's work has indeed generated a large research agenda over the last decade. I highlighted his work in March's favorite theorems post.

In other big news, we have our first female Fields Medalist Maryam Mirzakhani for contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces. Still no female winners among the nine Nevanlinna winners. Artur Avila, Manjul Bhargava and Martin Hairer also received Fields medalists. Stanley Osher won the Gauss Prize, Phillip Griffiths the Chern Medal and Adrián Paenza the Leelavati Prize.

Pictures and press releases and citations of all the prize winners.

Questions that arose teaching High School students crypto

I taught a 3-week, 3-hours-a-day course to High School student titled

Computer Science: A Hands Off Approach.

Given that time constraints and the fact that some already know (say) Java and some don't know any language, this seemed like a good choice.

I decided to teach mostly pre-RSA crypto with the following theme: Alice and Bob want to pass secret messages. How do they do it? I cover Shift, affine, general sub, Vigenere, Matrix, 1-time pad, Diffie-Helman (a highpoint of the course since Alice and Bob don't have to meet in a dark alley). In also did secret sharing with polynomials, error correcting codes (elementary), Huffman codes, and some applications of mod arithmetic.

While teaching this course some points of interest came up. I suspect most are know and I appreciate polite comments telling me so.

1. A student suggested this cipher:   code a,b,c,...,z into a 100-letter alapahbet and map each letter to a set of symbols that is the size of the freq. For example, if e occurs 9% of the time then map e to 9 letters. Then use those letters at random. This would seem to foil freq analysis? Does it?  Has it been used? What are the downsides.
2. Many students suggested using Vigenere but instead of having every xth letter be done by a different shift, have it be affine or general. Of course this can be cracked the same way Vig is cracked. But it does raise an interesting point: Which ciphers are used and not used can be the based on when things were discovered. Martians may very well have used some kind of Vig where every xth letter is a different gen sub cipher.
3. Wikipedia and other sources say that the Vig cipher was unbroken for 300 years. A student pointed out that it might have been broken but the one who broke it didn't say. Jon Katz (crypto prof at UMCP) can't believe it wasn't broken immediately, but of course hindsight is 20-20.
4. (I have commented on this before) A matrix cipher with a 10x10 matrix seems uncrackable using plaintext only.  I have made this question rigorous here.
5. I made the comment that 1-time pads are not used much (is this even true? Might depend on the definition of ``must'') because getting a perfect source of randomness is hard. During WW II they also would be hard to use because its hard to carry around a billion bits. But now that would not be a problem. Again--if history had been different we may use 1-time pads, or quasi-random ones today!
6. I told the students about arithmetic mod n. One of the students really really didn't like that (say) in mod 7 we have 1/3 = 5. He REALLY wants 1/3 to be between 0 and 1. I suspect he didn't care much for discrete logs either. This was a GOOD student- so his objection was not that it was hard.
7. For some of the students their first exposure to matrices was matrix codes over mod 26. I hope they can recover from that.
8. Most of the students know what logs were, but weren't that comfortable with them. And here I go and teach them discrete logs! I hope they can recover from that.
9. I showed the students that there were quadratic equations over mod 12 with more than 2 roots and challenged them to see how many roots they could come for other mods. One of my students ran massive computer tests and found stuff and in the end had a result that didn't need all of his computations:  x^2 \equiv 0 mod n^2 has n roots. And I later had on a HW x^a \equiv 0 mod n^a. I am sure none of this is new, but its new to him when he discovered it and of course new to the class when I taught it.
10. I taught the class the Baby-Step/Giant-Step Discrete Log algorithm which has sqrt(p) prepocessing an sqrt(p) running time. Its not used because it also takes sqrt(p) space; however, it was good to show them that Discrete Log can be done in sqrt(p) time, much better than p time--- hence Alice and Bob need to pick their parameters larger than they may have thought when doing Diffie-Helman. That night I easily worked out that it can be modified to do p^{2/3} preprocessing (and space) but just p^{1/3} time. HW was p^a prep, p^{1-a} time. One of the students inquired if this algorithm has a name. I then looked over the web but couldn't find it anywhere so I told them to call it  The Gasarch Variant of Baby-Step, Giant-Step. I also quickly told them to NOT be impressed--- and this helped me make a point I had made often, that CS is a NEW field, so NEW that one can present new or newish results to HS students. I also made the point that I am sure this variant is KNOWN to anyone who would care, but (1) they may not care since it takes even more space if x is larger than 0.5  and more time if x is less than 1/2  and (2) not everything is no the web. That last point freaked them out more than the quadratic equation mod 12 that had more than two roots.

UPSHOT- teaching HS students CS can bring up some issues that you had not thought of. Or at least that I had not thought of.

The n^{1/3} barrier for 2-server PIR's broken! A lesson for us all!

Recall: PIR stands for Private Information Retrieval. Here is the model: A database is an n-bit string (my wife tells me this is not true). The user wants to find the ith bit without the database knowing what bit the user  wants. The user COULD just request ALL n bits. Can the user use less communication? See here for a website of many papers on PIR.

1. If there is just one copy of the DB and there are no comp. constraints on the computational power of the DB then ANY protocol requires n bits comm.
2. If there is one copy of the DB then, with some comp constraints, you can do better. I state one result: if quadratic residue is hard then there is a protocol using n^epsilon bits of comm. (Kushilevitz and Ostrovsky). The rest of this post is about the info-theoretic case, so the DB has no comp. constraints.
3. If there are two copies of the DB then there is a protocol that uses n^{1/3} bits. (Chor, Kushilevitz, Goldreich, Sudan)
4. If there are three copies of the DB then there is a protocl that uses n^{1/32582658} bits. Really! (Yekhanin).
5. If a 2-server protocol is a bilinear-group protocol (which all prior constructions were) then it must  take n^{1/3}.(Razborov and Yekhanin)
6. Most known constructions put into a geometric framework (Woodruff and Yekhanin). 

Recently Dvir ahd Gopi posted a paper  here which gives a 2-server PIR protocol that uses n^P bits where P= sqrt( (log log n)/log n)). That breaks the barrier proven by RY! How is that possible?

The barrier result was only for a certain type of PIR. It DID cover all the known PIR schemes at the time. But it does not cover this one.

This is how Barrier SHOULD work--- they should not discourage, but they should point to difficulties so that someone can overcome them.

Also note, the new result used the some of the Framework of Woodruff and Yekhanin. So good to unify and obtain  new proofs of old results.

How I know Moneyball is a true story. Do we clean up theorems and life too much?

A few years ago I saw the movie Moneyball about how the Oakland A's used intelligent statistics to... win?
No, but to do better-than-expected.  Even if I didn't know it was a true story I would have assumed it was because the following are very rare or odd in a fictional story:

1. At the end of the team doesn't win- it just does better than expected. In the typical sports movie the underdog pulls it all together and wins. In some the underdogs loses but they are now better people or something. In an episode of Cheers where they were the underdog to Gary's Tavern in a bloody mary making contest  the Cheers gang cheats and wins. But in Moneyball, and in NO other sports (or contest) movie that I know of,  does the underdog do better-than-expected in  an undramatic matter. This is NOT a complaint- just note that its real life.
2. In Moneyball the General Manager wants to try out mathematical methods and the Manager resists. In most movies its the suits that are wrong and the people on the ground that are right. This is even a theme of many articles about business that I read in magazines on airplanes. So this inversion is odd - but again, you can't argue that a true story is unrealistic or undramatic.
3. Bill Beane, the one who wants to use math techniques, thinks that what Baseball scounts look for is the wrong thing. In fact, they misjudged him when he was a young player. But in what direction?--- they thought he was BETTER than he was. If this was a fictional story surely the scouts would think he was WORSE than he was.

I am glad that (as far as I know) neither the book nor the movie tried to make the story more satisfying or dramatic.

In academia we do clean things up for a better story line. If the true motivation for working on a problem doesn't really make sense when you see the final paper,  we change our motivation. Our original proof is intuitive but ugly, so we change it to be polished but less clear where it came from. Historians often simplfiy to make sense of things. I am NOT complaining- merely asking, do we do it too much?

When I was in ninth grade and was told that you could solve a quadratic equation (I rederived the quadratic formula once a month to make sure I could), a cubic, a a quartic, but not quintic, I immediately said "I want to goto College to learn why you can't solve a quintic" That sparked my interest in math.

Is the above story true? I am sure that in ninth grade I did learn that the quintic was unsolvable and that was of great interest to me, and I really did rederive the quadratic equation once a month. And I was interested to learn that the quintic was not solvable. But  I really doubt the story is as clean as presented above.  Even so, the story is true in spirit. However, I would not want to push the point.

How about you? Do you tell stories about yourself or about others that are just a little too polished? Not so much false, and not even to put yourself in a better light, but just a little to clean to have really happened.