Information Markets for Fighting Terrorism

A few months ago I had a post describing information markets, a system of buying and selling securities that pay off if a given future event happens. Based on the price of a security, one can get an estimate of the probability that that event will occur. Studies have shown that information markets are better predictors than polls or experts.

Information markets have taken a blow in the past few days. The US Department of Defense has cancelled a program that would have set up limited futures markets on securities based on terroristic activities. They bowed to pressure from senators who consider it morally wrong to bet on events on future terrorist attacks. I understand their concerns but computer scientists and economists have produced what could have been a powerful tool in controlling terrorism and it is quite a shame to see it discarded so easily.

David Pennock sent me some links on a more positive point of view from CNN, Fortune and Wired and a fun CNN piece on the Tradesports Poindexter future.

Update (8/1): A well-written New York Times column A Good Idea with Bad Press and a nicely argued opinion piece by David Pennock.

The Tour

I know this is not a sports weblog and I don't even like bicycling but anytime an American named Lance wins a major championship I can't let it go unnoticed.

The Move

Way back when I was a graduate student, I moved from Berkeley to MIT. I put what few belongings I had into boxes and shipped them via UPS. My brother flew out and we drove across the country together. Those were the days.

Now making the move back to Chicago is not nearly so simple. We have houses to sell and buy. Getting our kids ready for a new school. Real estate agents, lawyers, mortgage and insurance people to deal with. Meanwhile there is academic work that needs to get done before the real move. Conference and grant deadlines don't move to accommodate my move.

So this weblog might get a little spotty until I get settled into Chicago, sometime in mid-September. I'll try to find some time for some posts during that time but don't expect too much. If you are having complexity weblog withdrawal check out the archives. Nice thing about complexity--old stuff doesn't (usually) get stale.

Juntas

Eldar Fischer gave a talk today on his paper on testing juntas. So what is a junta? According to the American Heritage Dictionary, a junta is a "A group of military officers ruling a country after seizing power", named after such groups in Central and South America in the 80's. In this paper a junta refers to a function that depends on a constant number of variables.

Back when I was young (those 80's) we had a different name for a function that depends on a constant number of variables: NC⁰, a specification of NC^k, functions computable in constant fan-in circuits of depth O(log^k n). If k = 0 we have constant fan-in and constant depth, which means it depends on a constant number of variables.

Digression: NC stands for "Nick's Class" named by Steve Cook in honor of Nick Pippenger. Pippenger repaid the favor with the class SC but enough of that for now.

Juntas are just one example I'm seeing of a trend of new names and definitions of concepts defined years ago. Makes me feel old. Of course back then we likely studied concepts were thought were new but might not have been. Just part of the great circle of math.

Citeseer

Perhaps the NEC project most valued by computer scientists is Citeseer developed by Steve Lawrence, with help from Lee Giles and Kurt Bollacker and others. Citeseer is a free web-based service that scans the internet for computer science technical reports, parses the citations and cross-links the papers. You can use Citeseer to see, based on citations, which papers are most relevant to a particular topic. You can sort papers or even computer scientists (not necessarily a good thing). Since the papers are cached you also quickly get hold of old versions of many papers.

With the changes at NEC, I often get asked what will happen to Citeseer. Don't worry--Citeseer will soon have a new safe home and will continue to provide computer scientists with an easy way to track down papers.

Quantum Advice

Another rump session talk by Scott Aaronson showed that BQP/qpoly is contained in EXP/poly. In other words, everything efficiently quantumly computable with a polynomial amount of arbitrarily entangled quantum advice can be simulated in exponential time with a polynomial amount of classical advice.

Let me try to put this in context while avoiding quantum mechanics. Advice is method for encoding a different program for each input length. We define the class P/poly as those languages computable in polynomial time with access to a polynomially-long advice string a_n where the string a_n depends only on the length of the input. P/poly is equivalent to those problems having nonuniform polynomial-size circuits.

Quantum advice is a bit more tricky, since it can be in a superposition of regular advice strings. Formally, quantum advice is an exponentially long vector of numbers β_a where β_a is the amplitude of advice string a. For simplicity let us assume those numbers are real and we'll also have the restriction that the sum of the squares of the amplitudes is one.

You can see there are far more ways to give quantum advice than classical advice. But the quantum machines are limited in how they can use the advice. Harry Buhrman asked whether one can give any limit at all to what one can do with quantum advice. Scott Aaronson gives an answer: No better than classical advice as long as you are allowed (classical) exponential time.

Ideally one would like that efficient quantum algorithms with quantum advice can be simulated with efficient quantum algorithms with classical advice. Still Aaronson's result shows that even with fully entangled advice one cannot get all the information out of it.

A Combinatorial Theorem Proven by Kolmogorov Complexity

During the rump session of complexity, Nikolai Vereshchagin presented a combinatorial theorem that he proved using Kolmogorov complexity. Let A be a finite subset of N×N where N is the set of natural numbers. Let m be the size of A, r be the number of nonempty rows of A and c the number of nonempty columns.

We say A is good is every nonempty row has m/r elements and every nomempty column has m/c elements of A. A rectangle has this property, as does a diagonal. We say A is k-good if every nonempty row has at most km/r elements and every nonempty column has at most km/c elements. A is good if it is 1-good.

Vereshchagin's Theorem: There is a constant c such that for all finite subsets B of N×N with n = log |B| there is a partition of B into at most n^c sets each of which is n^c-good.

Vereshchagin asks whether there is a purely combinatorial proof of this theorem. If you know of one let me know.

For those who know some Kolmogorov complexity, let me sketch the proof: We label each point (x,y) of B with the following five values: K^B(x,y), K^B(x), K^B(y), K^B(x|y) and K^B(y|x). We partition the points into sets with the same labels. Standard counting arguments from Kolmogorov complexity show that each partition is n^c-good for some c.

Update

The Rump Session

One of the nice aspects of the Complexity Conference, the rump session, I don't see at many other conferences. Here anyone who wants to, mostly students, get ten minutes to lay out their latest research.

This year we had one of the best rump sessions ever with five really neat results, listed below. Don't worry if you don't immediately understand the results, I will talk about some of them in more detail in future posts. All but Vereshchagin are students.

1. Nikolai Vereshchagin: A combinatorial theorem proven by Kolmogorov complexity with no known direct combinatorial proof.
2. Troy Lee: For every finite set A and x in A, CND^A(x) ≤ log |A| + O(log³|x|). CND is the nondeterministic version of Sipser's distinguishing complexity.
3. Scott Aaronson: Everything efficiently quantumly computable with a polynomial amount of arbitrarily entangled quantum advice can be simulated in exponential time with a polynomial amount of classical advice.
4. Kristoffer Hansen: Constant-width planar circuits compute exactly ACC⁰, constant depth circuits with a mod gate for some fixed composite.
5. Samik Sengupta: If co-NP has an Arthur-Merlin game with a polylogarithmic number of rounds then co-NP is in NP with quasipolynomial advice and the exponential hierarchy collapses at the third level.

Coming Home

Howdy from Aarhus, Denmark where I am attending the 18th Annual IEEE Conference on Computational Complexity. As readers of this weblog probably have figured out, this is, by far, my favorite conference. Complexity is smaller and more relaxed than the broader theory conferences like STOC and FOCS. I know most of the participants and have a genuine interest in the most of the papers here. While elsewhere theorists try more and more to find connections to applied areas, here we are happy to focus on the fundamental power of efficient computation.

During the rest of the year I attend some conferences in broader areas or in areas that are not in my major focus. But coming to Complexity is coming home, and it always feels great to be back where I belong.

Complexity Abstracts

In the early years of the complexity conference, some researchers complained that if they had new results or their papers were not accepted in the conference, their results would not be known. So we started up Complexity Abstracts, an electronically available document where anyone can publish a one-page abstract of their work. The abstracts were made available right before the conference.

Last year, the first abstracts editor, Bill Gasarch, resigned his job after 11 years of collecting the abstracts. Personally I thought that in this new internet age, with sites like ECCC, the Complexity Abstracts were no longer as valuable a resource. But at the business meeting of last year's conference the Abstracts were greatly supported and Steve Fenner volunteered to take on the job of editor.

Fenner's first collection has just been posted ahead of the Complexity Conference next week in Denmark.

For the Love of Math

A doctor, lawyer and mathematician were discussing whether it was better to have a wife or a girlfriend. The doctor said it was better to have a wife because it is medically safer to have a single partner. The lawyer said it was better to have a girlfriend to avoid the legal hassles of marriage. The mathematician said it was better to have both.

"Both?" said the doctor and the lawyer. "Yes," said the mathematician, "That way the wife thinks I'm with the girlfriend, the girlfriend thinks I'm with the wife and I can do some math."

I was reminded of that joke by the recent New York Times article Pure Math, Pure Joy and the accompanying slideshow. Those pictures look all too familiar.

The greatest lovers of math though are not the famous mathematicians at places like Berkeley and Harvard. Rather the mathematicians who take low-paying jobs with high teaching loads at less-strong colleges or move from visiting position to visiting position just to have some occasional time to do math. They have a dedication (or perhaps an addiction) I can never fully appreciate.