Lessons from Economics

Rakesh Vohra pointed me to some interesting takes on journals in economics. Economics runs on a different model than computer science; conferences are less selective and economists are judged more on the quality of the journals where their papers appear.

The Berkeley Electronic Press offers an electronic subscription-based system for their journals. Look at the B.E. Journals in Theoretical Economics. Here you submit to all four journals at once and your paper gets accepted to one with the highest quality rating that the editors decide is appropriate for your paper.

NAJ Economics is Not A Journal but offers reviews of economics papers. One cannot submit papers but a strong rotating editorial board just finds papers freely available on the internet and post reviews of those they feel are worthy. From the FAQ:

The purpose of NAJ Economics is to work towards replacing the existing commercial system of scientific publication. Because papers published in printed journals are less available than working papers, which are freely available on the Internet, publication in the traditional sense inhibits scientific communication. It also generates additional costs as most printed journals charge high subscription fees, in particular to libraries. However, it does serve the useful purpose of certifying the scientific quality of published work. It also assures that articles remain available regardless of the idiosyncrasies of individual websites and links. Our immediate goal is to provide some of the useful certification functions of current journals at a negligible cost by reviewing papers that we think have substantial merit.

I have some quibbles about the service. Without submissions a lesser known author might have trouble getting his paper reviewed. The editors will have a nightmare keeping links up to date, especially since they seem to link to papers on people's homepages. They also don't have the ability to force improvements in the papers they review the way a journal can.

But perhaps in this age of the internet one needs to separate the refereeing and distribution aspects of a journal. NAJEcon is an interesting step in that direction.

FOCS Accepted Papers

The list of accepted papers for the upcoming FOCS conference in Rome is out. [Thanks Suresh]

A few complexity papers to note: Ran Raz finds easy languages with no log-depth multilinear circuits. Andris Ambainis and Mario Szegedy have separate papers showing nice applications of quantum "random" walks. Barak, Impagliazzo and Wigderson show how to do extract nearly uniform distributions from multiple independent random sources as opposed to one random source and a few truly random bits. And lots more.

Don't Make it Too Easy or Too Much

Two easy ways to improve your paper but lessen your chances of acceptance at a conference: Add more results and simplify your proofs. Adding a result could only increase the usefulness of a paper but program committees see many results in a paper and conclude that none of them could be very strong. One of our students a few years ago had a paper rejected at STOC, he removed one of his two main theorems and won the best student paper award at FOCS.

Given the same theorem, the community benefits from a simple proof over a complicated proof. Program committees look for hard results so if they see a very simple proof, it can count against you.

You need to play the game. If you have many results depending on the situation, you can either split the paper or highlight one result and bury the others. It's a bit unethical to use a hard proof where you know an easy one but many people make an easy proof look harder by adding an unnecessary level of detail or proving a more general but less interesting theorem.

You do what you need to do, within ethical standards, to get your paper accepted. After you get it accepted, remember you have a rewrite for a proceedings version to get the paper written the way it should.

Note from Vereshchagin

I received the following from Nikolay Vereshchagin.

The combinatorial question I have discussed last summer at the rump session at Computational complexity (about partitioning a planar set into a small number of uniform parts) has been answered almost immediately by Ilan Newman and Gabor Tárdos. They have found a pure combinatorial proof. Recently I have written a note on the subject.

Complexity Conference Recap

The Complexity Conference ended yesterday. You can already find the papers on the IEEE site and if you don't have access you can often find versions of the papers on author's homepages.

We had a strong turnout and a nice variety of papers on many different areas of complexity with particularly strong showings in quantum complexity and structural complexity making a comeback.

Amit Chakrabarti asked about group isomorphism. Arvind and Torán showed that solvable group isomorphism is "almost" in NP∩co-NP.

Although I did not have my own talk in the conference, I presented a paper by Buhrman and Torenvliet since they unfortunately could not be in Amherst. I like giving talks on other people's work since you can be honest about the strengths of a paper without having to brag. My favorite result in their paper showed that if you take a many-one complete set for EXP, remove any easily computable set of subexponential size, what remains is Turing-complete for EXP. The proof is a clever recursive algorithm using the set itself to find safe places to map the reduction.

Next year we have our 20th conference in San Jose followed by Prague in 2006.

Rump Session Redux

This week I am in Amherst at the University of Massachusetts for the 19th IEEE Conference on Computational Complexity. Lots of fun papers and complexity theorists. This is complexity heaven.

Like last year, we had a number of interesting new results described at the rump session. Let me describe a couple of them to you.

Scott Aaronson follows up on his guest post about the complexity of agreement. Aumann has a famous theorem that two players who communicate cannot agree to disagree on the probability of some state of the world; after some discussion they will converge to a common probability. Aaronson looked at the complexity of this process and found that convergence comes relatively fast. He defined a notion of (ε,δ)-agreement where the probabilities are within ε of correct with a confidence of 1-δ and shows that such an agreement happens after polynomial in 1/ε and 1/δ rounds.

Neeraj Kayal looked at the complexity of the problem #RA, the number of automorphisms of a ring given by generators. He showed that factoring and graph isomorphism reduce to #RA and #RA sits in AM∩co-AM. As an open question he wondered about the complexity of determining whether a ring has nontrivial automorphisms where one is given tables for addition and multiplication. It remains open even for commutative rings.

Update 6/23: Kayal tells me I didn't accurately capture his rump session talk and sent me the following summary.

We have an algorithm that determines whether a ring has a nontrivial isomorphism even when the ring is given in the form of generators for its additive group and pairwise product of the generators expressed as a linear combination of the generators. (We get this by getting a characterization of all finite rigid rings and it turns out that we can test whether a ring follows this characterization or not without solving integer factoring.) Unfortunately however we do not know of a reduction from Graph automorphism to ring automorphism although we have found a cute reduction from Graph Isomorphism to Ring Isomorphism!

The open problem that I would love to solve is to decide whether two rings are isomorphic or not when they are given in the form of tables (one table each for addition and multiplication.) I do not know how to do this even for commutative rings.

Shimon Even (1935-2004)

Shimon Even was born in Israel on June 15th, 1935. He died on May 1st, 2004. In addition to his pioneering research contributions (most notably to Graph Algorithms and Cryptography), Shimon is known for having been a highly influential educator. He played a major role in establishing computer science education in Israel (e.g., at the Weizmann Institute and the Technion). He served as a source of professional inspiration and as a role model for generations of young students and researchers. Two notable avenues of influence were his PhD students and his books Algorithmic Combinatorics (Macmillan, 1973) and Graph Algorithms (Computer Science Press, 1979).

From a memorial page by Oded Goldreich.

Visa Problems Continue

Wisconsin Professor Dieter van Melkebeek has a paper at the ICALP conference but cannot go to Finland to present it. Why not? Delayed processing of his green card application has led to problems with his current visa putting him in some temporary state of visa hell. Dieter would actually have no trouble attending ICALP; he would just have problems coming back.

Dieter is one of many stories of people changing travel plans and missing conferences because of America's tougher requirements and slower processing of foreign immigration applications. An Indian graduate student with a paper at next week's Complexity conference could not get a visa in time. I would not be surprised if many graduate students will not start the fall semester on time awaiting my government's blessing to come to study here.

This is a story I have told before and will likely tell again. I understand the need for security but most scientific progress happens through collaboration and preventing or delaying this collaboration holds back the advancement of knowledge. Not since the 80's have we seen such a limitation on traveling though this time in reverse. During the cold war several countries would not let many of their best scientists out; these days we don't allow many of the world's best scientists in.

Riemann Hypothesis and Computational Complexity

A commenter asks a good question for a bad reason: Would a proof of the Riemann Hypothesis have any impact on complexity theory?

Rather surprisingly the answer is yes, particularly in the area of computational number theory. In the most famous example, Gary Miller in 1975 gave a polynomial-time algorithm for primality whose correctness could be proven by assuming the Extended Riemann Hypothesis (ERH). Of course in 2002 we had a polynomial-time primality algorithm with no assumption. However the original analysis of the algorithm gave a constant which depends on how ERH is resolved.

There are still many other problems in computational number theory that require ERH. For example, according to Eric Bach, the only polynomial-time algorithm computing square roots modulo p, when p is large relies on ERH. "The idea is to combine an algorithm that uses a quadratic nonresidue, such as Shanks's algorithm (this in Knuth v. 2 I am pretty sure) with a bound on the least quadratic nonresidue mod p (e.g. in my thesis it is proved to be <= 2 (ln p)^2 if ERH is true)."

Special Issues

Journals dominate the non-research talk at STOC. We had a long discussion at the business meeting about the special issue of STOC. A little background: For the past 24 years the STOC program committee selects 6-10 papers from the conference and one of the PC members serves as editor of a special issue of a journal where all these papers are invited to appear. The Journal of Computer and System Sciences (JCSS) has always hosted the special issue for STOC as well as a few other conferences including FOCS and Complexity.

JCSS became an Elsevier journal a few years ago when Elsevier bought Academic Press. Elsevier has come under attack over the past few years in our field for their pricing policies, an issue discussed in this weblog before. Some editorial boards have resigned and many others are considering it. The current PC chair (and fellow U. Chicago Professor) Laszlo Babai has strong negative feelings towards Elsevier and spearheaded the issue at the conference.

The STOC Executive Board has final say on the future of the special issue but based on the business meeting discussion, the special issue for STOC will likely move to SIAM Journal on Computing (SICOMP) perhaps as early as this year.

My concern, which I expressed at the meeting, is that we already have a culture where too many papers never appear in a journal, i.e., never get written with full proofs and go through a rigorous refereeing process. The more negative press we give towards journals the more likely authors will take the easy solution of no journal. When was the last time you downloaded the journal paper never written?

Update 6/18: Hal Gabow, chair of SIGACT, has set up a website containing additional information on the meeting and subsequent procedures.

STOC Business Meeting

STOC got underway Sunday with a full slate of talks and a lengthy business meeting last night. I do not have time for a long post now so I will just bring you up to date on some facts from the business meeting.

The attendance was 261 (242 paid + 19 local helpers). Later today I will update the contest post with the results.

STOC 2005 will be in Baltimore May 22-24 and STOC 2006 will be in Seattle. There were announcements of three new journals, the previously mentioned ACM Transactions on Algorithms and two on-line open-access journals Logical Methods in Computer Science and Theory of Computing.

Most of the business meeting was devoted to the future of the special issue and I left around 11 PM last night before this discussion had ended. This discussion will require a post of its own in the near future.

STOC runs through Tuesday. Much more as the week goes on.

Favorite Theorems: Connections

May Edition

I have always loved results that find connections between previously-thought different areas of complexity. This month we highlight one of the best.

Extractors and Pseudorandom Generators by Luca Trevisan

Informally a pseudorandom generator takes a small random seed and generates strings that can fool every probabilistic algorithm. To describe an extractor we start with some distribution D over strings of length n. Let p be the maximum probability of any string in D and let k = log(1/p). An extractor uses D and a small number of truly random bits to create a new uniform distribution of strings of length close to k.

Both pseudorandom generators and extractors have many uses in complexity and many papers in the field show various constructions to improve the parameters of both. Trevisan showed that one can view any pseudorandom generator as an extractor and then derives better extractors from known pseudorandom generator constructions.

Pseudorandom generators fool resource-bounded algorithms while extractors nearly uniform distributions in an information-theoretic sense. That makes this connection all the more amazing. Trevisan's paper has affected the how researchers think about and prove results in both areas.

Win a Gmail Account

My first weblog contest. Guess the paid attendance (including students and postdocs) at next week's STOC conference. Closest to the correct answer receives an invitation for a Beta Gmail account (donated by weblog friend Meridel).

Rules: Send your guess in the subject of an email to stocguess@fortnow.com. Include your name and email in the body of the message. One guess per person. All guesses must be sent by Saturday noon CDT. Closest guess to the attendance announced at the business meeting Sunday night will receive an invitation to open a Gmail account (still in Beta testing). In case of tie, first closest guess received will win. Anyone involved in STOC organization is ineligible. Not responsible for delayed or undelivered email. My decision of the winner is final. Contest not sponsored or affiliated with Google or ACM SIGACT.

Good luck.

Results Update 6/14: Total paid attendance was 242. The closest at 254 was Nanda Raghunathan, second place at 223 was Kamalika Chaudhuri and third at 265 was Chandra Chekuri. We have some extra invites so we've decided to give gmail accounts to all three. Congratulations and thanks to everyone who participated.

Professional Societies

Professional Societies perform valuable roles in academics. They give awards, sponsor conference and publish reasonably-priced journals as well as bulletins, newsletters and reviews. Societies disseminate information among researchers about future activities and the state of the field. They form an advocacy group representing the scientists in government and universities. Most importantly they give a focal point that lets us identify as a community.

Unfortunately in theoretical computer science no single group plays all these roles and thus one interacts with a large number of professional societies during an academic career. Let's look at some of them.

First most comes the Association for Computing Machinery (ACM) as the largest society devoted to computer issues. ACM tries to cover the entire computing profession so computer science research issues do not get center stage. They do publish several journals and give many of the important awards such as the Turing award.

ACM has a number of special interest groups (SIGs). SIGACT, the Special Interest Group on Algorithms and Computation Theory, is the main organization devoted to theoretical computer science in the US. They sponsor STOC and other conferences and publish SIGACT News. Many theorists join SIGACT without joining ACM.

The IEEE Computer Society also deals with computer issues and has a Technical Committee on Mathematical Foundations of Computer Science that sponsors conferences including FOCS and Computational Complexity. Why do we need both a Computer Society and ACM and a SIGACT and a TC-MFCS? Perhaps for the competition?

None of these societies serve as a strong advocate for computer science research and so we have the Computing Research Association. The CRA has as its members not individuals but academic departments and research labs. They have a newsletter, advocate and keep us informed on government policy on computer science, and collect information such as the Taulbee Surveys giving salary and job information in CS research. The CRA also has a strong focus on women's issues in CS research.

Let's not forget the Society for Industrial and Applied Mathematics (SIAM) that helps sponsor some conferences (SODA) and publishes the well-respected Journal on Computing.

The European Association for Theoretical Computer Science (EATCS) covers not just Europe but captures theory from an international perspective. They sponsor conferences like ICALP and publish a hefty bulletin three times a year. Also many countries have their own computer science and/or theoretical computer science societies.

Then based on my research interests I have now or at some time been a member of AMS, MAA, ASL, SIGecom and the Game Theory Society. Where does it all end?

BEATCS Complexity Column

With the June issue, Jacobo Torán takes over the editorial duties of the BEATCS Complexity Column. Following with tradition, he wrote his first column, Space and Width in Propositional Resolution. A strong start to what should be a great run of columns.

Survey Papers

Let's end this week how we started it, with a survey paper. Luca Trevisan has recently posted on ECCC a new survey Some Applications of Coding Theory in Computational Complexity. The survey gives a rather in-depth look at several different types of codes with some connections to private information retrieval, average-case complexity and probabilistically checkable proofs. Trevisan gives a broader and more in-depth look at coding theory than an earlier yet also excellent survey by Madhu Sudan focusing on list decoding.

Survey papers play a valuable role in our field. As computational complexity has broadened over the years, one cannot hope to keep on top of all of the many areas. A survey paper written by an expert in the field can perform many valuable tasks including

• Putting the main results of an area in a common framework. Early work often uses different notation and definitions making it hard to compare one paper to another. Fixing the notation and definitions allow us to easily compare different results. A well-liked survey can also influence future notation.
• Proofs get easier over time and a survey can give easier-to-follow proofs of old results. A survey can also develop a common proof technique useful for many result in the area.
• Giving the author's informed opinion to the importance of different results in an area.
• Stating open problems and directing future research in that area.

In case I've managed to put the survey bug in you, here are two topics where we've seen several recent research papers but lack good surveys that I know of.

1. The complexity of Nash Equilibrium
2. ε-biased Sets

Complexity Registration Deadline

Tomorrow is the last day for early registration for this year's Complexity Conference in Amherst. I promise a good time will be had by all.

IEEE Fellowship at the State Department

Are you an American IEEE member? Now you can help guide American foreign policy. IEEE-USA has announced an Engineering and Diplomacy Fellowship where IEEE members can serve as a Fellow in the U.S. State Department and continue to advise them afterwards. These fellowships are being offered for a few professional societies; perhaps the ACM should try to get in on this.

Some more background from FYI.

Impagliazzo's Five Worlds

Boaz Barak in a comment last week mentioned one of my favorite survey papers, Russell Impagliazzo's A Personal View of Average-Case Complexity presented at the 1995 Complexity Conference. In that paper he describes five possible worlds and their implications to computer science.

• Algorithmica: P = NP or something "morally equivalent" like fast probabilistic algorithms for NP. This was the world I described last week but looking back at Impagliazzo's paper, he does a nicer job.
• Heuristica: NP problems are hard in the worst case but easy on average.
• Pessiland: NP problems hard on average but no one-way functions exist. We can easily create hard NP problems, but not hard NP problems where we know the solution. This is the worst of all possible worlds, since not only can we not solve hard problems on average but we apparantly do not get any cryptographic advantage from the hardness of these problems.
• Minicrypt: One-way functions exist but we do not have public-key cryptography.
• Cryptomania: Public-key cryptography is possible, i.e. two parties can exchange secret messages over open channels.

Impagliazzo does not guess which world we live in. Most computer scientists would say Cryptomania or Minicrypt.
The paper goes on to give one of the better justifications for Levin's definition of average-case complexity.

Visas and Titles

Thanks to Technorati I can track who links to this weblog. Recently an Indian student Nitish Korula started a new blog Pseudo-Random Thoughts where he describes the trials of getting a visa so he can start grad school at Illinois in the fall. Good luck Nitish, we're rooting for you.

Meanwhile I agree with Will Baude at Crescat Sententia that most U. Chicago undergrads address faculty as "Professor" rather than say "Mr. Fortnow" as is the official Chicago custom. A decade ago I was more likely to get "Mr. Fortnow" which I never loved since it actually feels more formal than Professor or Doctor.

Graduate students as well as my colleagues call me "Lance," at least to my face. First year grad students sometimes take time to grow out of addressing professors as "Professor." I had this problem myself way back when. A fellow student couldn't shake the professor habit until he started playing sports with them. You just can't say "Throw me the ball, Professor Leighton."

In the end I don't really care that much what you call me. However I do enjoy those letters from Germany that covering all the bases address me as "Herr Dr. Prof. Fortnow."

What if P = NP?

A New York Times essay looks at the hardness of understanding math. The essay quotes from the book The Millenium Problems by Keith Devlin which describes the seven million-dollar Clay Mathematical Institute Millenium Problems including the P versus NP question. So I took a peek into Devlin's book.

Devlin doesn't hide his feelings about the P versus NP problem as "the one most likely to be solved by an unknown amateur." He does make a point that if P = NP we can break RSA and "the current dependence of the Western economies on secure communications over the Internet demonstrates just how high are the P = NP stakes."

Let's play make believe and assume P = NP in a strong way, say that we can find satisfying assignments of Boolean formula in nearly linear time with small constants. It will have a dramatic influence on the Western economy but not at all in the way Devlin perceives. We'll lose public-key cryptography but what we will gain from it will make the whole internet look like a footnote in history.

Learning becomes easy by using the principle of Occam's razor--we simply find the smallest program consistent with the data. Near perfect vision recognition, language comprehension and translation and all other learning tasks become trivial. We will also have much better predictions of weather and earthquakes and other natural phenomenon.

Everything will be much more efficient. Transportation of all forms will be scheduled optimally to move people and goods around quicker and cheaper. Manufacturers can improve their production to increase speed and create less waste. And I'm just scratching the surface.

P = NP would also have big implications in mathematics. One could find short fully logical proofs for theorems but these fully logical proofs are usually extremely long. But we can use the Occam razor principle to recognize and verify mathematical proofs as typically written in journals. We can then find proofs of theorems that have reasonably length proofs say in under 100 pages. A person who proves P = NP would walk home from the Clay Institute not with one million-dollar check but with seven.

Informatics in Indiana

Many universities try to integrate information technology into many different disciplines usually through their computer science departments. Our neighbors to the east are creating a bold experiment in this integration, the Indiana University School of Informatics, with fastly growing departments spread over several of their campuses.

What is informatics? According to Indiana, Informatics is

• understanding the impact technology has on people.
• the development of new uses for technology.
• the application of information technology in the context of another field.

Their research groups already encompass quite a few areas including biological, chemical and social issues of information technology.

I visited the Informatics department in Bloomington a few months ago and sensed an excitement of growing a new discipline and bringing in many information technology researchers from different scientific disciplines. Note the real distinction between computer science that studies and improves the nature of computation and and informatics that aims for integration of information technology between various areas of study.

Mixing researchers from vastly different disciplines has had its shares of successes and failures and only time will tell how successful the Indiana experiment will become. But I'm extremely impressed with the commitment from the University and the state to this area of informatics and I expect we'll hear much more from Indiana in this area.

Comments

Some strong comments on Rocco's post on the recent Columbia theory day. In my own highly biased point of view, I find the study of efficient computation critical in a society that becomes continually reliant on computation on both explicit computers and implicitly in various biological, economic and physical systems. And how can one study efficient computation without developing reasonable models of computation and analyzing those models?

I don't mean to sound so altruistic; I get paid to do what I love. But I do truly believe one needs to understand the mechanisms that make up our world if we wish to improve them. I write this weblog, in part, to educate about the beauty and applications of theoretical computer science.

A comment about comments. I understand that many of you choose to post anonymously rather than register at Blogger and I'm fine with that. If you don't mind please add your name at the end of the comment. I like to know who is behind the comments and its useful to match up different comments by the same person. Of course, I'd rather get your comments anonymously than not at all.

Update 5/21: Stanley Fish, the departing Dean of the Arts and Sciences of University of Illinois at Chicago argues more for a separation of academic research and policy.

I exit with a three-part piece of wisdom for those who work in higher education: do your job; don't try to do someone else's job, as you are unlikely to be qualified; and don't let anyone else do your job. In other words, don't confuse your academic obligations with the obligation to save the world; that's not your job as an academic; and don't surrender your academic obligations to the agenda of any non-academic constituency � parents, legislators, trustees or donors. In short, don't cross the boundary between academic work and partisan advocacy, whether the advocacy is yours or someone else's. Marx famously said that our job is not to interpret the world, but to change it. In the academy, however, it is exactly the reverse: our job is not to change the world, but to interpret it.

A Part-Time Ph.D.?

A question from a reader (slightly edited):

There are no part-time (or even full time) Ph.D. programs at top universities in computer science or mathematics that can be completed by those who work full time. For various personal reasons I find myself in a position that requires me to work full time; however, I am passionate about theoretical computer science/mathematics. Unfortunately, most American schools do not accommodate Ph.D. students under these circumstances. Is this a decision based upon the assumption that those who work full time will not produce good/enough work, or is this a decision based, simply, upon the fact that professors want to work standard hours and teaching a course from 5:30 - 6:20 is quite non-standard?

Courses are not a major issue. The course requirements for a Ph.D. usually do not significantly differ than those for a Masters and many universities offer a Masters program in computer science for full-time workers. I do see two other major barriers to a part-time Ph.D.: Funding and Research.

Nearly all Ph.D. student get funded for tuition and some living expenses via a fellowship, teaching assistantship or research assistantship. Government agencies generally don't give fellowships to part-time students and a TA or RA requires about twenty hours a week, leaving someone who already has a full-time job with no time for actually completing the Ph.D.

But suppose you felt that a Ph.D. was worth the expense or were independently wealthy and for some reason still had to work a full-time job. Ph.D. level research in math and theoretical computer science requires intense background study and long stretches of thinking, understanding the problem and working through many different ideas until one actually makes significant progress toward original work. For this one needs time and the relationship is not linear. Someone who can spend forty hours a week focusing on research will be far more than twice as successful as one who can only spend twenty.

The dominant limitation on number of Ph.D. students in CS departments is funding. If you have a record that would have gotten you in to a top computer science department as a full-time Ph.D. student and you bring your own money to the table, I suspect at many schools you can work out a part-time schedule. But you'll find doing original research on a part-time basis a daunting if not impossible task.

Randomized Blogspace

A report from Theory Day co-organizer Rocco Servedio

On Friday May 14 a special Columbia/IBM Research/NYU Theory Day was held at Columbia University in New York City. The New York area theory days started at Columbia in 1982; this one was a special event to mark both the 25th anniversary of the CS department at Columbia and the 250th anniversary of Columbia University.

More than 280 attendees came out to hear four talks by outstanding theorists:

• Richard Karp (UC Berkeley): Current Challenges in Computational Genomics: Haplotyping
• Shafi Goldwasser (MIT/Weizmann): Proving Hard-Core Predicates using List Decoding
• Prabhakar Raghavan (Verity/Stanford): Finding Information in Networks
• Peter Shor (MIT): Quantum error correction and fault tolerant quantum computation

The day ended with a panel discussion on "The Future of CS Theory." Avi Wigderson (IAS) joined the four speakers for the panel, which was moderated by Mihalis Yannakakis (Columbia). Here is a brief summary of what was said.

Mihalis started things off by observing that over the past 50 years CS theory has enjoyed outstanding successes and has had tremendous impact on computing. Indeed, some of the successes were so profound that they gave rise to whole new fields of computer science (databases, security) that are no longer thought of as "CS theory". Mihalis asked each of the panelists to briefly give their views on the future of CS theory. Some highlights of what they said:

Avi observed that CS theory can (and should) have more impact on early education, starting in high school or even earlier. We can give important insights into fundamental ideas such as adversaries, randomness, learning, recursion, games, proofs, and "getting things done efficiently" (which Avi referred to as "the oldest profession in the world"). He also highlighted some specific goals for CS theory at this point, which included showing that BPP ≠ NEXP; coming up with non-natural proof techniques for circuit lower bounds; discovering new types of quantum algorithms; developing a general theory of what types of algorithms can give optimal approximation ratios; and proving that SL = L and that MATCHING is in NC.

Dick Karp warned against taking anyone's advice or predictions too seriously. That said, he advocated for a healthy balance between foundational questions at the core of CS theory and new questions that arise from the role of computation in the world and the sciences. He highlighted three areas of interest for the future: (1) the study of large scale distributed systems such as the Web, incorporating ideas from economics and game theory; (2) connections with areas of natural science, ranging from statistical physics to quantum mechanics to biology; and (3) the "new face" of AI in which stochastic and graphical models and statistical inference are playing a big role.

Peter also commented on the perils of predicting the future; we sometimes tend to think that there will be no more revolutionary ideas simply because we don't know what those ideas will be. But such ideas will come along from "out of the blue" as they always have. He noted that while past predictions for the future of CS theory have tended to be on the doom and gloom side, things have actually turned out pretty well -- there are interesting jobs and demand for theorists in industry; theory is more and more noticed and used by practitioners; and rather than becoming increasingly recondite and inward-looking, theory is building stronger connections with mathematics, physics, and other disciplines.

Prabhakar observed that what we think of as CS theory is really two main thrusts of work with some overlap: there is the theory of computation as an inherent phenomenon (i.e. when we study MOD 17 gates and what they can do even though nobody will ever build one), and the theory of computation as it is practiced (i.e. most of the world's cycles are spent making a billion people happy rather than crunching data for a few thousand scientists). The Web is a paradigmatic aspect of the second thrust; he noted that in this area economic factors may play a role at least as important as traditional resource bounds like time and space. Prabhakar also stressed the importance of backing up claims of practical relevance for our work (and, on an unrelated note, mentioned this weblog in a slide entitled "Randomized Blogspace".)

Shafi observed that CS theory is having an increasing impact on classical mathematics such as coding theory, number theory, and signal processing. On the other hand, we are also dedicating more energy (and having more success) in solving problems in the real world -- both of these trends are good signs for the field. She advised researchers to follow their own tastes and interests rather than anyone else's recommendations when it comes to "the next big challenge for the field."

After these statements the floor opened up to questions and discussion with the audience. A brief summary:

One questioner noted that the theoretical models of parallelism from 20 years ago don't correspond to how large distributed systems work now, and asked whether a similar phenomenon could be taking place with quantum computation -- are we studying the right model? Some panelists responded that while we aren't likely to end up with quantum computers that correspond exactly to quantum circuits, it seems likely that algorithms developed for the quantum circuit model will prove useful if/when we do get quantum computers in one form or another.

There was quite a bit of discussion about the role of CS theory in the undergraduate curriculum and what undergraduate CS majors should know about theory. Some panelists opined that NP-completeness, undecidability, models of computation, and algorithms are core topics that even high school students perhaps should know. A view emerged that there is real (potential) widespread interest out there in the "gems" of CS theory, and that we should do a better job of explaining what is fascinating and beautiful about our field to students.

In response to a question about the future status of the P=NP question, some panelists observed that other great research communities (mathematics, physics) have tussled with unsolved questions for centuries. We seem to be stuck right now, but on the bright side we have some understanding (natural proofs, for instance) of why we are stuck -- perhaps mathematicians should step back and think about why the Riemann hypothesis is still unresolved.

To close, here are three quotes lifted more or less verbatim from the panel discussion (but left anonymous here):

• "The future for DNA computation is dim" (in response to the question "What is the future for DNA computation?")
• "Polynomial time computation is a complex object to understand."
• "We are so much closer to understanding each other's talks than the mathematicians are."

Cornell's New President

On Friday I went to an alumni reception for Jeffrey Lehman, new president of Cornell University. Besides learning that the cinderblock dorms where I spent my freshman year are finally being demolished, a number of interesting aspects of university life came out of the question and answer session.

One question asked about lack of student activism on campus. Lehman acknowledged the problem outside of environmental issues and told of his plan for a mock presidential election at Cornell before the real election. This seemed like a weak answer--mock elections we had in high school. I doubt college students could get excited about a mock election when most of them can vote in the real thing.

On the other political end was a question about the liberal bias in faculty. Lehman acknowledged this as well but didn't consider it a problem as long as the conservative voice was not silenced. This was a good answer.

On affirmative action he said that Cornell needed more minority applicants and was working on a suggestion to start attracting students even in middle school. And someone asked a question about whether Cornell should have common core courses for the students, an interesting issue for me since even small changes in the University of Chicago's traditionally strong core have caused major controversy. Lehman said that Cornell will continue its tradition of not having any fixed course requirements for all students (besides the swimming test).

Favorite Theorems: Probabilistically Checkable Proofs

April Edition

No single topic has dominated computational complexity over the past dozen years than probabilistically checkable proofs (PCPs). Arora, Lund, Motwani, Sudan and Szegedy, in a paper on my 1994 list, showed that every language in NP has a polynomial-sized PCP that can be verified by probabilistic polynomial-time verifier using O(log n) random coins and some constant number of queries. Well beyond the complexity interest in this result, PCPs give hardness of approximation results for a variety of NP-complete problems.

Researchers in many exciting papers have improved the parameters of the PCP results in order to get improved limits on approximation. But one paper really puts it all together for some tight results.

Some optimal inapproximability results by Johan Håstad, JACM, Volume 48, 2001.

Håstad's paper shows that every language L in NP has a PCP with with O(log n) random coins and 3 queries, where

1. If x is in L then the verifier is convinced with probability arbitrarily close to one.
2. If x is not in L then no proof can convince the verifier with probability more than one-half.

There parameters are the best possible.

The paper gives some optimal approximation results. Consider Max-3-SAT, where one wants to find an assignment that maximizes the number of satisfied clauses of a 3-CNF formula. We can satisfy 7/8 of the clauses by choosing a random assignment, a process we can also derandomize. Håstad's result implies that no better algorithm exists unless NP is easy. The paper also gives improved lower bounds on approximation on problems like vertex cover and max cut.

Håstad's paper pulls in tools from a large collection of research papers. Madhu Sudan's lecture notes describes Håstad's results and the techniques and papers leading up to it. There's also been exciting PCP research since Håstad's paper but I'll have to leave that for another day.

An Auction of Google

For those with an interest in auction theory, the Google IPO auction gives an interesting testbed for auction mechanism design. Instead of having an investment bank set a fixed price for the IPO, instead Google will auction off the shares.

A New York Times article today describes many of the decisions and possible pitfalls of the various kinds of auctions Google might use. Also check out the Google SEC filing. One can learn quite a bit about auctions as well as the business of search engines from this rather informally written document. I have never had so much fun reading a prospectus.

My prediction: Great interest in Google will highly overvalue the stock whatever auction mechanism they will use. If you are interested in investing in Google, hold off until the price settles or you will suffer the dreaded "winner's curse."

Page Charges

The Journal of the ACM has started asking for page charges.

Author's institutions or corporations are requested to honor a page charge of $60.00 per printed page or part thereof, to help defray the cost of publication. Page charges apply to all contributions. Payment of page charges is not a condition of publication; editorial acceptance of a paper is unaffected by payment or nonpayment.

SIAM also recently asked us for $72/page for a Journal on Computing paper.

I despise page charges. Authors do the research, write the papers, give the journals the copyright and now the journals want us to pay for the privilege. I know the charges are optional and come from research funds but we have other needs for the money. The page charges on a moderate-sized paper could, for example, send a grad student or two to a major conference.

We have problems in our field with expensive for-profit journals and papers that never appear in refereed journals at all. We need to encourage authors to send their articles to journals run by the non-profit societies. We should not then send them a bill for doing the right thing.

Games

A readers asked about the complexity of games like Go and Chess. David Eppstein has a nice site giving a short description and references to a number of specific games.

Let us thought put such games in a general framework. We have a board and each player in turn can make one of a list of legal moves that depend on the current placement of pieces on the board. We focus on deterministic games of complete information, as opposed to games like backgammon or poker.

Games like Go and Chess are played on a fixed board, one could just enumerate all of the possible board combinations and perform perfect play in a constant amount of time. So we need to look at generalized versions of Go and Chess where the size of the board and the set of rules can vary.

Let's place this in a general setting. We have a polynomial-time algorithm that given a board and the current player can tell whether the game has ended with its outcome or can give a list of legal moves for the player. Chandra, Kozen and Stockmeyer have a seminal paper on these alternating games: If we restrict the length of the game to polynomial-time, such games characterize PSPACE (problems solvable with polynomial memory and unlimited time). Games with arbitrary long play on polynomial-size boards characterize EXP (exponential time).

So we have results like given an opening position on a generalized Go games, it is EXP-complete to determine if a player have a forced win. But even if the official Go rules allow it, I find it hard to believe that players can play the game for an exponential number of moves. So it makes sense to add some artificial stopping rules that cause the game to end after a reasonable amount of time and such games are usually PSPACE-complete.

The PSPACE-completeness results hold for many very simple games. This mirrors the fact that complexity does not arise from complicated actions, rather from the interactions of many simple actions.

New Web Host

I'm moving my web hosting service--if you can read this you are accessing the new host. I will wait a day or two to post again until the changeover is complete.

Meanwhile enjoy this Guardian column by John Sutherland describing how the British higher education system has evolved over the past four decades (via Crooked Timber). Many of the same issues apply in America and I suspect many other countries as well. Sutherland sums it up nicely.

The big question. Is the whole system in better or worse shape than it was in 1964? I don't know. All I do know is that I'd like to do it all again, and get it right this time.

America Losing Its Edge

Some required reading if you haven't seen it yet, a New York Times article on how America has lost some of its scientific leadership role over the rest of the world.

The article does not go much into the reasons behind the change so let me make some conjectures. For a long while now, the majority of Ph.D. students in the US came from other countries. As the academic job market in the US got tighter, many of these researchers went back to their home countries and established strong research groups there. Also recent technological changes have taken away some comparative advantage of doing research in the states as communication and access to research papers has become a much easier task.

I welcome the added competition, the more globalization of science that we have, the more we will all push one another with scientific research becoming the big winner. In my own field, I like seeing countries like Israel becoming theory powerhouses and definite growth of theory in places as diverse as India and Australia. A few years ago it would have been unthinkable to have STOC or FOCS overseas but recently STOC 2001 was held in Greece and the upcoming FOCS will be in Rome.

Most of all I hope the article serves as a wake-up call to American legislators. Time to give NSF that large budget increase that they've been talking about for several years now.

Karp Symposium

[A report from weblog correspondent Bill Gasarch. Link to Allender's talk added 5/7]

On Wednesday April 28 there was a SYMPOSIUM HONORING DR. RICHARD M. KARP at Drexel University in Philadelphia.

They were honoring him for winning the BEN FRANKLIN MEDAL IN COMPUTER AND COGNITIVE SCIENCE (There are Ben Franklin Medals for Physics, Chemistry, Life Sciences, Earth Science, Computer and Cognitive Science, and Engineering.)

There were three talks:

ERIC ALLENDER: The Audacity of Computational Complexity.

This talk described the basics of complexity theory and mostly focused on reductions. A nice contrast that it made:

1. in the year 2004 we have good reason to think that many problems (e.g., SAT, 3-COL) are hard, except factoring which is still hard to classify.
2. in the year 1970 most problems (including SAT, 3-COL) were hard to classify.

The talk also pointed out some of the problems with Computational Complexity (e.g., "How can you call a n¹⁰⁰⁰⁰⁰ algorithm feasible?") and answered them nicely (e.g., "we want to show problems are hard, so showing its not in P does that.") The talk both began and ended on the topic of CHECKERS and GO being computationally hard problems.

AVI WIGDERSON: The Power and Weakness of Randomness (When you are short on time).

This talk showed several examples of problems where randomness helps (hence randomized algorithms are powerful) but also indicated why there may be reason to think that you can always replace a randomized algorithms with a polynomial time algorithm (hence randomization adds no power). The problems it helped on involved sampling, routing in networks, and mazes.

RICHARD KARP: Even Approximation Solutions can be Hard to Compute.

This talk was about certain problems that can be approximated and certain ones that (it seems) cannot be. A nice contrast was variants of TSP, which ranged from what can be approximated very well, to what can be approximated some, to what can't be approximated. He also brought in randomized rounding as a technique for approximation. The talk ended on PCP (done informally) and how it can be used to show lower bounds for approximation.

OVERALL:
The talks were all well presented and quite understandable. The point of the talks was to expose our area to people outside of theory and perhaps even outside of computer science. As such the theorists in the audience did not learn much new; however, it is still interesting to see someone else's perspective on material that you are familiar with.

Conferences versus Journals

A reader asks why Gafni and Borowski did not publish their paper in a journal and become eligible for the Gödel Prize. I wish this was an isolated incident but it reflects on a sad state of affairs in computer science and theoretical computer science in particular. Too many papers in our field, including many great ones, do not get submitted to refereed journals. In an extreme case, Steve Cook received the Turing Award mostly for a STOC paper.

In most cases, conferences in computer science are more selective than journals. Your reputation in theoretical computer science is measured more by the conferences your papers appear than the journals. In other fields like mathematics, physics and biology, journals have a much greater reputation and most of their papers do appear in refereed form. I believe the reason is historical: computer science started as a quickly changing field and journals could not keep up with the rapidly emerging ideas.

Conference program committee cannot and do not produce full referee reports on conference submissions. Proofs are not verified. Papers are not proofread carefully for mistakes and suggested improvement of presentation. Computer science suffers by not having the permanency and stamp of approval of a journal publication on many of its best papers. The founders of the Gödel Prize put in the journal requirement to encourage potential award winning papers to go through the full refereeing process.

Many papers in our field do appear in journals and some researchers are extremely diligent in making sure all of their work appears in refereed form. Also I know of no computer scientist who purposely avoids sending their papers to a journal. But when we have a system that does not value journal publications, a computer scientist pressed for time often will not make the effort to take their papers past the conference version.

Is Disney World NP-complete?

The Unofficial Guide to Walt Disney World 2004 gives a lesson on complexity by describing the optimal tour of the Magic Kingdom as a traveling salesman problem. Some excerpts:

As we add more attractions to our list, the number of possible touring plans grows rapidly...The 21 attractions in the Magic Kingdom One-Day Touring Plan for Adults as a staggering 51,090,942,171,709,440,000 possible touring plans...roughly six times more than the estimated number of grains of sand in the whole world...Fortunately, scientists have been hard at work on similar problems for many years..finding good ways to visit many places with minimum effort is such a common problem that it has its own nickname: the traveling salesman problem.

The book goes on to describe the computer program they use to approximate the optimal tour. Read more here (which I found by searching within the book for "traveling salesman" on the Amazon site). You'll need to be a registered user of Amazon.com to read it.

Gödel Prize

From the PODC (distributed computing) mailing list via Harry Buhrman. Usually the winners are kept secret until the ICALP or STOC conference but the PODC mailing list has already broken the news.

It has been recently announced that this year's winners of the Gödel Prize are

• Maurice Herlihy and Nir Shavit for their paper "The topological structure of asynchronous computability" (J. ACM, 1999), and
• Mike Saks and Fotios Zaharoglou for their paper "Wait-Free k-Set Agreement is Impossible: The Topology of Public Knowledge" (SIAM J. Computing, 2000).

As we all know, the result was initially published simultaneously in STOC 1993 also by Eli Gafni and Liz Borowski, but the Gödel Prize is awarded only to journal articles.

Congratulations to the winners!

Note that for the second time, the Gödel's Prize honors a core PODC topic (in 1997, Joe Halpern and Yoram Moses won the prize). This is a sign both of the scientific quality of the PODC community, as well as the respect it wins in the theoretical CS world at large.

In case you are counting, that's Complexity 5, PODC 2.

Theory Girl

From Bill Gasarch: There are some more novelty songs about theory (aside from THE LONGEST PATH) from the Washington CSE Band. The best one is THEORY GIRL.

A Few Short Announcements

Alan Kay will receive the 2004 Turing award. It can't always be a theorist.

Registration is open for the 2004 Conference on Computational Complexity. The final schedule will be posted soon. Also keep in mind STOC 2004 right here in Chicago.

The list of accepted papers for ICALP is up.

Finally, next Wednesday the 28^th in Philadelphia, Drexel is hosting a symposium on computational complexity honoring Richard Karp.

Are There #P Functions Equivalent to SAT?

Help me solve this problem, write the paper with me, get an Erdös number of 3 and it won't cost you a cent.

We can have #P functions hard for the polynomial-time hierarchy (Toda) or very easy but can they capture exactly the power of NP?

Conjecture: There exists an f in #P such that P^f=P^SAT.

There is some precedence: counting the number of graph isomorphisms is equivalent to solving graph isomorphism.

The conjecture is true if NP=UP, NP=PP or if GI is NP-complete. I don't believe any of these. Does the conjecture follow from some believable assumption or perhaps no assumption at all? We don't know if there exists a relativized world where the conjecture does not hold.

Even the following weaker conjecture is open: There exists an f in #P such that NP⊆P^f⊆PH.

A good solution to these conjectures might help us settle the checkability of SAT

 

Asian Food for Thought?

Many years ago, an Israeli graduate student made the rounds and gave talks at several US universities. When he arrived in Chicago, he asked me if Americans only eat Chinese food. I told him he hadn't seen a random sample of Americans and took him out for some good Chicago ribs. Afterwords he told me he preferred the Chinese food.

At a logic conference at Notre Dame, I ate dinner with a small group at one of the few Chinese restaurants in South Bend. Surprisingly no other mathematicians were eating in the restaurant. Just as we noticed this, the waiters started putting tables together and about five minutes later in walk about 20 logicians for dinner.

Why do mathematicians and computer scientists eat so much Asian food? Not just Chinese but Japanese, Thai, Korean, Vietnamese, Indonesian, Ethiopian (not Asian but close enough) and of course Indian (northern and southern). Not that I don't enjoy Asian food but what's wrong with a good hamburger?

Favorite Theorems: Primality

March Edition

Primality is a problem hanging onto a cliff above P with its grip continuing to loosen each day. - Paraphrased from a talk given by Juris Hartmanis in 1986.

It took sixteen more years but the primality problem did fall.

PRIMES is in P by Manindra Agrawal, Neeraj Kayal and Nitin Saxena.

This paper gave the first provably deterministic polynomial-time algorithm that could determine whether n is a prime given n in binary. The theoretical importance cannot be overstated. But why do I consider the paper a complexity result instead of just an algorithmic result?

Manindra Agrawal had already a strong reputation as a complexity theorist. The proof involves a derandomization technique for a probabilistic algorithm for primality. But more importantly primality had a long history in complexity.

Primality is in co-NP almost by definition. In 1975, Vaughn Pratt showed that PRIMES is in NP. In 1977, Solovay and Strassen showed that PRIMES in co-RP and testing primality became the standard example of a probabilistic algorithm. In 1987, Adleman and Huang building on work of Goldwasser and Kilian showed that PRIMES is in RP and thus in ZPP. In 1992, Fellows and Koblitz showed that PRIMES is in UP∩co-UP. Finally in 2002 came AKS putting PRIMES in P.

A runner-up in this area is the division problem recently shown to be in logarithmic space and below.

The Cost of Textbooks

The University of Chicago Bookstore has asked for textbook requests for the fall quarter by the middle of next month instead of during the summer as in past years. The reasoning: A burgeoning used textbook market. If the bookstore knows what books faculty will use in the fall, they can offer higher prices to pay for used books at the end of the spring quarter.

This is just an indication of the problems of higher textbook costs. CALPIRG has a recent extensive report on this topic. Textbook costs add to already spiraling increases in tuition and other college expenses.

In addition, I have more griping than usual about buying the textbook from students in my class though the book, Homer and Selman's Computability and Complexity Theory lists new for $50, under even the average used price mentioned in the CALPIRG report.

What should I do as a faculty member? Should professors strive to reuse the same textbook each year so student's can buy and sell used versions to keep their costs down? That can lead to courses getting stale very fast.

Or should I even forgo textbooks completely and rely on less organized material freely available on the internet? I already do this for graduate courses where strong up-to-date textbooks simply do not exist.

The View of a Science Writer

A friend of mine from college became a science writer for various newspapers and magazines. Once he told me about his two biggest complaints about scientists.

1. Scientists want everyone who works on a project to be named in an article.
2. Scientists want every detail in an article to be complete and correct.

You might initially take the side of the scientists. But the science writer does not write for the scientists but for the general public.

Put yourself in the position of the reader. The reader doesn't want to read through a long list of names that they won't remember anyway. The average reader also just wants an overview of the research and its importance. If removing some technical caveats and slightly oversimplifying the research achieves a better level of understanding to the reader, so be it.

Remember next time you read a science article in the popular press or get interviewed for such an article, the goal of the article is not to pass a serious referee review but to give the general public some glimpse into an important research area.

Blum Complexity Measures

The Blum speed-up theorem states that there exists a computable language L such that if L is in time t(n) then L is in time log(t(n)). The log function can be replaced by any arbitrarily slowly growing computable function. Instead of time one can use space or any other measure Φ that fulfills these properties:

1. Φ(M,x) is finite if and only if M(x) halts, and
2. There is a computable procedure that given (M,x,r) can decide if Φ(M,x)=r.

These are known as Blum axioms and measures that fulfill them are known as Blum complexity measures. They were developed by Manuel Blum in the late 1960's.

The Borodin-Trakhtenbrot Gap Theorem states that given any computable function g(n) (e.g. g(n)=2^n), there exists a function t(n) such that every language computable in time g(t(n)) is also computable in time t(n), i.e., there exists a gap between these time classes. Once again the theorem holds for any Blum complexity measure.

We don't see much of the Blum complexity measures these days for a few reasons.

1. The only truly interesting Blum measures are time and space.
2. The functions and languages that one gets out of the speed-up, gap and related theorems are usually quite large and artificial.
3. Many measures that we are interested in today, like the number of random coins used by a probabilistic Turing machine, do not fulfill the Blum axioms.

In 1991 I saw Manuel Blum give a talk discussing a new complexity measure, something about mind changes, that did not fulfill his axioms. So we had a Blum complexity measure that was not a Blum complexity measure and as Douglas Adams would say Manuel Blum "promptly vanishes in a puff of logic." [Just kidding-we like Manuel]

More News from Dagstuhl

Another Guest Post from Dieter van Melkebeek

Thursday morning, Shuki Bruck gave the first talk at the workshop that dealt with actual Boolean circuits. He pointed out that cyclic circuits can be combinational and may allow us to realize Boolean functions with fewer gates and/or less delay. Consider the following circuit with inputs x1, x2, x3, and outputs f1, f2, f3, f4:

 |-----------------------------------|
 |                                   |
 |    x1       x2     -x1       x3   |
 |    |        |       |        |    |
 |    |        |       |        |    |
 |    v        v       v        v    |
 |                                   |
 |-> \/ ----> /\ ----> \/ ----> /\ --|

      |        |        |        |
      v        v        v        v

      f1       f2       f3       f4

Although the circuit is topologically cyclic, the outputs are well-defined and only depend on the inputs. (Look at the cases x1=0 and x1=1 separately.) A careful analysis shows that every acyclic circuit that outputs f1, f2, f3, and f4 needs at least 5 nonunary gates. Thus, circuits with feedback allow us to gain a factor of 4/5 in terms of number of gates needed to compute these functions. (As usual, we do not count negations.) Shuki presented a sequence of Boolean functions for which the reduction in the number of nonunary gates asymptotically reaches 1/2 if we only allow gates of fanin at most 2. He raised the question how significant the reduction can be if we allow larger fanin.

Thomas Thierauf presented an NC² algorithm for unique perfect matching. A perfect matching in a graph is a collection of disjoint edges that cover all vertices. It is known for some time how to decide the existence of a perfect matching and how to construct one in randomized NC²:

1. Assign random weights from a small range of integers to the edges of the graph such that with high probability there is at most one minimum weight perfect matching. If we are in the situation with a unique minimum weight matching M, we can decide whether a given edge belongs to M by evaluating two determinants of matrices with integer entries that are exponential in the weights. Since the weights are small, we can do the latter in NC².
2. Run the NC² algorithm on all edges in parallel and verify that the result is a perfect matching M.

It is open whether perfect matchings can be constructed deterministically in NC.

To decide whether a graph G has a unique perfect matching, Thomas first runs step 2 above (with unit weights). If that test fails, the algorithm rejects since G either has no perfect matching or has more than one. If the test is passed, the algorithm additionally verifies that G has no perfect matching M' other than M. Such an M' exists iff G contains a cycle that alternates between edges from M and edges in G-M. The latter can be cast as a reachability problem in a graph that is roughly a concatenation of directed copies of M and G-M. Since directed graph reachability can be computed in NC² and the input to the reachability problem can be computed in NC² by step 2 above, the additional test runs in NC², as does the entire algorithm.

On Friday, Oded Lachish discussed the current records on unrestricted circuit lower bounds for explicit functions in n Boolean variables. For circuits that can use any binary gate, the record dates back to 1984 and stands at 3n. For circuits that can use any binary gate except parity and its negation, the record has recently been improved from 4n - O(1) to 5n - o(n). Both records use the technique of gate elimination, and Oded conjectured that the 3n result can be improved along the lines of the recent 5n - o(n) result.

The workshop ended at noon on Friday. One statistic: among the 33 talks, 3 were blackboard only, 5 used handwritten slides, 1 printed slides, and 24 were computer presentations.

Finally, I have one suggestion for those readers who have attended a Dagstuhl seminar in the past. In a response to changes in financial support, the Dagstuhl office is requesting information about research publications that grew out of or have otherwise been significantly influenced by a Dagstuhl seminar. If you are an author of such a publication, please send the information to office@dagstuhl.de. Let's try to keep the wonderful tradition of Dagstuhl alive!

A Free Lunch Theorem For Circuit Complexity

A guest post from Dieter van Melkebeek

This week, about 50 computer scientists gather at Schloss Dagstuhl for a seminar on "Complexity of Boolean Functions." The setup follows a long tradition that started back in 1944 at Dagstuhl's mathematical sister institution in Oberwolfach: a flexible program of talks, ample time for discussion, and Deutsche Gruendlichkeit in a wonderful setting.

I'll highlight an aspect of roughly one talk per day. Given the wide variety of topics, the selection is idiosyncratic rather than representative. For a full list of the talks, check out the seminar web page.

On Monday, Philipp Woelfel discussed time-space tradeoffs for integer multiplication. Every program computing the product of two n-bit integers in time T and space S has to satisfy TS = Ω(n²), and this lower bound is tight. One may expect that the same time-space tradeoff holds if we're only interested in the i-th bit of the product, where i is part of the input. However, Philipp showed a randomized program (with polynomially small error) that does the job using only O(n log n) time and O(log n) space, for a product TS of O(n log² n). It remains open whether the TS = Ω(n²) lower bound for the simpler problem holds in the deterministic setting.

I stole the title for this weblog entry from Peter Bro Miltersen. Right before lunch on Monday, he presented his "free lunch theorem": Lower bounds for circuits that consist of a gate C applied to symmetric functions of ANDs (type I) imply lower bounds for circuits that consist of a gate C applied to symmetric functions of AC⁰ functions (type II). The proof outline goes as follows. Consider a circuit of type II. By the switching lemma, hitting it with a random restriction transforms each of the AC⁰ functions into small decision trees, each of which can be written as a small OR of small ANDs. For any given decision tree, at most one of its ANDs can be true. It follows that a symmetric function of decision trees is a symmetric function of all the ANDs involved in these decision trees. This transformation gives us a circuit of type I that isn't much larger than the original circuit.

Part of Tuesday was devoted to quantum computing. Andy Yao presented an approach to unify and generalize the known quantum lower bounds for (i) locating an item in a sorted list of n elements and (ii) sorting a list of n elements. Both in the classical and in the quantum setting, we know that (i) takes Ω(log n) comparisons and (ii) takes Ω(n log n) comparisons. Problems (i) and (ii) are instantiations of the following more general problem, which is parameterized by a partial order P on n elements: Using comparisions only, determine an unknown linear order of n elements that is guaranteed to be consistent with P. We obtain problem (i) by setting P to be a linear order on all but one element, and (ii) by making P empty. If we denote by e(P) the number of linear extensions of P, we have that e(P) = n in case (i) and e(P) = n! in case (ii). Since there are e(P) different outcomes and each classical comparison gives us at most one bit of information, we need at least log e(P) such comparisons. Thus, one obtains the classical lower bounds for (i) and (ii) in a uniform way. The simple information theoretic argument breaks down in the quantum setting. Nevertheless, using the notion of graph entropy, Andy proved a lower bound of Ω(e(P)) - O(n) for the number of comparisons in the quantum setting. He conjectured that the O(n) term can be dropped, which would yield a uniform proof of the quantum lower bounds for (i) and (ii).

On Wednesday morning, Omer Reingold talked about recent progress towards a simpler or more combinatorial proof of the PCP Theorem. In particular, he presented a more modular way of composing proof systems, a crucial step in the known proofs of the PCP Theorem. Wednesday afternoon was kept free for a hike in the woods - one of the nice traditions at Dagstuhl.

Changes in Introductory Theory

Comments to my last post basically ask how has the introductory courses in theory has changed over the years. My first reaction: remarkably little. Theoretical models of computation do not depend on the current hot technology, particularly at the undergraduate level. Many of the basic results we teach today were also taught say 25 years ago. But without doubt theory courses have changed their emphasis on various topics over the years.

Every professor teaches a theory course differently so there is no fixed answer to what has changed. But here are some trends that I have seen (from a distinctly American point of view):

• Less emphasis on automata theory, particularly for context-free languages. Many schools do away with automata theory all together.
• Less depth in computability theory. Most courses will cover undecidability but you'll less often see the recursion theory or even Rice's theorem taught.
• Does anybody still teach the Gap, Union and Speed-Up Theorems and Blum complexity measures anymore?
• Only one new theorem since the mid-70's has become a fundamental part of an undergraduate complexity course: The 1988 Immerman-Szelepcsényi Theorem stating that nondeterministic space is closed under complement.
• There has been a trend in adding some recent research in complexity as the end of a course based on the interests of the instructor: Randomized computation (though recent algorithms for primality might change how it gets taught), cryptography, interactive proofs, PCPs and approximation, quantum computing for example. Parallel computation has come and gone.

But remember these are exceptions. Basic topics like Turing machines, undecidability, NP-completeness, Savitch's theorem and time and space hierarchies still get taught much the way they were taught in the 70's.

Opening Day

Today starts the spring quarter at the University of Chicago and I start teaching undergraduate complexity. Many of the most beautiful concepts in theory get taught in the course: The Church-Turing thesis, universal Turing machines and undecidability, the P versus NP problem and much more.

Today's students have an understanding of computers that come from exposure at an early age that I cannot imagine. Still you cannot truly view computer science as a science until you learn its mathematical foundations. This course gives that foundation and uses it to pose (and sometimes answer) many basic questions: What is a computer? What can we compute? What can we compute quickly?

As computers become more and more part of our daily lives, these basic questions take on greater importance and I'm excited, as always, to tackle them with a new group of students.

Teaching High School Physics

In a comment on my last post, Suresh Venkat said "On the other hand, we teach school-age children Newtonian physics without laying out a careful argument why the thesis must hold."

This caught me as strange so I asked one of our Indian graduate students how he learned physics in school. He said they were given the appropriate theory and formulas. I asked if they did experiments. He said they were given descriptions of experiments on exams and had to predict the outcome but they never actually performed any experiments.

This is in sharp contrast to my high school physics class in New Jersey. We did many experiments in small groups as well as some class demonstrations to show that the predictions of the theory roughly corresponded to reality. My favorite demonstration simulated the following thought experiment: If a person aims a gun directly at a monkey in a tree and the monkey, scared of the sight of the gun, falls out of the tree at exactly the time the gun was shot, the bullet will hit the monkey since gravity affects the horizontally moving bullet and the vertically moving monkey exactly the same.

My physics teacher attached a stuffed monkey to the ceiling via an electromagnet. He had a device that fired a metal ball at the monkey that was rigged so the magnet would cut out and the monkey would fall at the same time as the ball was fired. True to the theory, the ball hit the monkey in mid-air. Of course there was that hole in the blackboard from the one year the monkey didn't fall.

Which teaching method is superior? In India they can go into more depth in the theory since they don't spend time on experiments. However I don't think you truly get an understanding for a scientific principle without getting your hands dirty.

Update: Venkat responds on his weblog. Perhaps I shouldn't have generalized Indian education from one data point.

Evolution

Should public schools in the US teach creationism in addition to or in place of evolution? As a scientist I have to say "no," though I'm preaching to the choir in this weblog.

Often in the news we hear of states and school districts that try to pass laws to teach creationism in schools? We should fight these attempts but we need to do so in a careful manner. Scientists should not impose the truth on school-age children, that will make us no better than the creationists who wish to impose their version of the truth. Instead we need to explain the reasoning behind evolution, the same holds for any scientific principle we teach. For example, I can't expect students to trust me when it comes to the Church-Turing thesis but instead I need to lay out a careful argument why the thesis must hold.

One should not force students to accept evolution, rather lay out the arguments and let the students learn to believe evolution on their own. Only then will they become true believers.

Is Satisfiability Checkable?

Time for another of my favorite open questions: Is Boolean Formula Satisfiability (SAT) checkable?

The best notion of program checking comes from a paper by Manuel Blum and Sampath Kannan. Let P be a program claiming to compute a language L. A program checker M for L is a probabilistic polynomial-time Turing machine with access to P as an oracle that outputs either "P(x)=L(X)" or "P incorrectly computes x on some input."

We say L is checkable if for all oracle P and inputs x,

1. If P(x)≠L(x) then with high probability M^P(x) outputs "P incorrectly computes x on some input", and
2. If P=L then with high probability M^P(x) outputs "P(x)=L(x)".

If P is correct on x and incorrect somewhere else, M^P(x) can output either answer.

Blum and Kannan show a nice connection to interactive proofs. We say a language L has a function-restricted interactive proof (FRIP) if there is a PCP for L where the proof for x in L is computable with an oracle for L. We have the following equivalence for all languages L

1. L is checkable.
2. Both L and L have FRIPs.

Checkable languages include Graph Isomorphism, the Permanent and all of the PSPACE-complete and EXP-complete sets.

Back to whether SAT is checkable. SAT has a FRIP by using self-reduction. So whether SAT is checkable is equivalent to whether SAT has a FRIP.

All of the known PCPs for SAT seem to require counting, a prover hard for #P or at least Mod_kP for some k. Whether one can find a PCP for SAT that is even in the polynomial-time hierarchy remains open.

Perhaps one can show some consequence of the checkability of SAT perhaps that the polynomial-time hierarchy collapses. Bogdanov and Trevisan have the best result in this direction; they show that if SAT has a non-adaptive self-corrector then PH collapses to third level. Though many checking results use self-correction there still could be some completely different way to show SAT is checkable.

AT&T Research

The Newark Star-Ledger has an article about the downfall of AT&T research. Quantum Algorithms has some follow-up quotes by Bjarne Stoustrup.

No doubt that these industrial research labs can produce great ideas and results especially at the scale of AT&T or Bell Labs before the split. But the business model doesn't work; AT&T failed to capitalize on most of the innovations of its labs nor has any corporate labs with an open and unfettered research staff produced valuable intellectual property for that company. If a corporation tries to limit an open and unfettered environment, the best scientists will often leave for other labs or academia.

Bell Labs/AT&T had a lengthy history helped along by a telephone monopoly. But until someone finds the right business model, we will never see a truly self-sustaining industrial basic research lab.

Computer Science Unplugged

Can you teach basic computer science concepts to children? Without a computer?

Computer Science Unplugged by Tim Bell, Ian Witten and Mike Fellows has a wonderful collection of games and activities designed to teach young people about basic CS ideas like binary numbers, searching algorithms, text compression, information theory and much more. Some of the activities are available online. My favorite: Sorting Networks that kids can run through and find themselves ordered.

An Unnatural Post

When we see "natural" in a computer science papers it usually reflects an informal idea of realism, i.e., Clique is a natural NP-complete problem while 1-in-3 SAT is not. Sometimes though researchers use "natural" in a defined term. Generally this should be avoided--no definition can prevent artificial examples but more importantly perfectly reasonable notions that do not fit the rule are, by definition, not natural.

I work with bits and usually take my logarithms base 2, an unnatural logarithm. I use diagonalization to prove lower bounds on Turing machines, an unnatural proof technique applied to an unnatural computing model. I have even been known to use unnatural numbers, like 1/2.

What do you expect since I study an unnatural science?

Favorite Theorems: Derandomization

As I had mentioned earlier, this year I plan to write My Favorite Ten Complexity Theorems of the Past Decade II. I decided to reveal the choices one per month through the end of 2004.

For March I will go with derandomization, an area where we have seen amazing progress in recent years. My favorite derandomization result in the past decade is

P=BPP unless E has subexponential circuits: Derandomizing the XOR Lemma
by Russell Impagliazzo and Avi Wigderson, STOC 1997

The title both describes the main result and the technique use to prove it. Informally this result says that under a believable hardness assumption one can get full derandomization. Formally, if there exists a language L computable in DTIME(2^O(n)) such that there exists an ε>0 such that for all n, there are no circuits of size 2^εn that compute L on inputs on length n then pseudorandom generators exist and P=BPP.

This paper marks the culmination of a series of papers to derandomize BPP. We have also seen many papers since giving connections between derandomization and other recent areas in complexity like extractors and error-correcting codes as well as other applications for derandomization. I can't mention all of these results in this post but I recommend the survey of Valentine Kabanets and the book chapter of Peter Bro Miltersen for a broader background on derandomization.

March Madness

America's Favorite Binary Tree, the 2004 College Basketball Brackets have been released. This week last year had several posts related to the brackets intermingled with ICALP and a war.

Publishing Papers from Iran

A Chicago Tribune editorial describes an incredibly bad restriction on publishing from Iran. The U.S. Treasury Department's Office of Foreign Assets Control (OFAC) is warning publishers that they may face serious legal repercussions for editing books, papers or manuscripts from Iran or any other country that is under economic sanctions, on the grounds that such editing amounts to trading with the enemy.

Academics have always led the way in establishing relationships between politically antagonistic countries. Scientists often have the same research goals even if their politics or the politics of their countries differ. Preventing publication of their work (or in this case editing of their work) will unnecessarily restrict the communication between scientists and make opening these doors between countries harder.

More from the IEEE Spectrum.

Outsourcing and the Future of Computer Science

How will the trend in outsourcing programming work affect computer science departments in America? In the short term not good. A lesser need for programmers and continued slow growth in the technology sector will keep undergraduate enrollments down and CS departments will have less expansion. We are still a decade or two away from large retirements of the first wave of computer scientists so for the most part new faculty get hired mostly on CS department expansion.

In the long term outsourcing will lead to much stronger computer science departments. Programming skills alone will not necessarily lead to success and technology professionals will need a deeper and broader view of the tools and ideas in computer science. CS departments will have to provide courses that cover these concepts requiring a faculty that covers many areas and knows them well. Departments will have to expand to meet these growing needs with active researchers in a broad range of expertise. As a result we will see many more universities with a strong and vibrant research-oriented CS department.

Seeing the Same People in Different Places

This week I'm visiting the University of Calgary and although I have never been here before it seems like a homecoming. They have a strong quantum computing group with several people out of my past.

• Richard Cleve - I first got interested in quantum computing when Cleve had a short visit to CWI in Amsterdam during my sabbatical there in 1997. But our true bond comes from being stranded together in Tokyo after 9/11.
• John Watrous - The reason I drink my coffee black.
• Peter Høyer - Cleve and I were the foreign committee members at Høyer's Ph.D. defense in Denmark.
• Hartmut Klauck - Klauck had a postdoc at IAS while I was at NEC nearby.
• Hein Röhrig - A new postdoc in Calgary fresh from his defense in Amsterdam. Röhrig also was a summer intern at NEC.

Seeing the same faces in different places. Yet another oddity of the academic life.

SIGACT News

The first issue of 2004 of SIGACT News is out. The complexity column has part 2 of last issues' article on constraint satisfaction problems. More exciting is the list of upcoming columns: Ambainis on quantum, Guruswami on codes and Hitchcock, Lutz and Mayordomo on dimension.

Some interesting pieces in the back of the issue including information on the recent move of the editorial board of Elsevier's Journal of Algorithms to the new ACM Transactions on Algorithms and David Johnson announcing the revival of his NP-completeness column.

Right before that is a cute paper on the complexity of the peg hopping game found at Cracker Barrel (restaurant chain that serves fine American comfort food).

Remember that you can join SIGACT, support theory and get SIGACT news even if you don't belong to ACM for only $18, $9 for students.

On a different topic, the number of comments on my posts have gone up dramatically. I'm not sure why but I appreciate your feedback. I read every comment though usually restrain from responding unless specifically asked a question. I've had my say and you have your say and let's leave it at that. Often I learn something new from your comments like the Stern-Brocot tree. Keep those comments coming.

Persi Diaconis

Persi Diaconis once again shatters our belief in generating randomness; this time showing, with Susan Holmes and Richard Montgomery, that flipping coins does not usually produce uniformly random bits.

Diaconis has an impressive resume of magic, mathematics and psychic debunking. Around 1990 he visited Chicago and taught a course on Markov chain analysis spending the first half of the course on the following problem: Given n cards, pick two at random and swap them. Repeat. How many swaps do you need to get a nearly randomly shuffled deck? Answer: About n log n. The upper bound used representation theory and took several weeks to prove.

During that year, a Chicago Tribune editorial mentioned another Diaconis result showing that one needs seven standard shuffles to get a deck of cards close to random. I found the beginning of the editorial online:

And you always thought mathematicians were serious people. Especially those at Ivy League universities like Harvard and Columbia. Well ...

Dr. Persi Diaconis and Dr. Dave Bayer have just come out with a study that may give you pause. They have found, after no end of riffling and counting, that it takes exactly seven ordinary, careless shuffles to give a totally random mix to ...

Getting back to coin flipping, you can always use the von Neumann coin-flipping trick to correct for the unknown bias.

Counting the Rationals Quickly

In the November 1989 issue of American Mathematical Monthly Yoram Sagher presented a note "Counting the Rationals" giving a simple 1-1 mapping from the positive rationals onto the positive integers. Let m/n be a rational with gcd(m,n)=1. Let q₁, ..., q_k be the prime factors of n. Sagher defined his 1-1 mapping as f(m/n) = m²n²/ (q₁q₂··· q_k) With this mapping, Sagher notes you can easily determine the 10¹⁵th positive rational as 10^-8.

Unfortunately inverting Sagher's function appears to require factoring. Can one find a 1-1 mapping from the positive integers onto the positive rationals that is easy to compute in both directions? Think about it or keep reading for my solution.

Let p(i,j) = i + j(j-1)/2. The function p is an easily computable and invertible bijection from pairs (i,j) with 1≤i≤j to the positive integers. We define our 1-1 mapping from the positive integers to the positive rationals by the following algorithm.

1. Input: n
2. Find i and j such that n = p(i,j).
3. Let g = gcd(i,j) (easily computable via Euclid's algorithm)
4. Let u = i/g and v=j/g.
5. Output: g-1+u/v

Since 1≤i≤j we have 1≤u≤v making the output unique and the function easily invertible.