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Let’s Mention Foundations

September 18, 2014


Congratulating Dick on the 2014 Knuth Prize

LiptonSecondary
Cropped from source

Dick Lipton is of course the founder and driving writer of this weblog. He is also a computer scientist with a great record of pathbreaking research. The latter has just been recognized, I am delighted and proud to say, with the award of the 2014 Knuth Prize. The prize is awarded jointly by the ACM Special Interest Group on Algorithms and Computation Theory and the IEEE Technical Committee on the Mathematical Foundations of Computing, and was instituted in 1996, shortly after the formal retirement of the great—and very much active—Donald Knuth.

Today I am glad to give my congratulations in public, and also my thanks for a wonderful and long association.

Only in reading around while writing this post did I learn that Dick won another major honor earlier this year: election to the American Academy of Arts and Sciences. The end of April was crazy with personal and unusual professional matters for both of us, and it didn’t come up. In the 2014 list for CS he is alongside Jennifer Chayes, Ron Fagin, David Harel, Daphne Koller, Leslie Lamport, Leonid Levin, and Mendel Rosenblum.

The Knuth Prize is “for outstanding contributions to the foundations of computer science.” The description also says it may be partly based on educational contributions such as textbooks and students, but “is not based on service work for the community, although service might be included in the citation for a winner if it is appropriate.” The ACM release does not mention this blog. So just for now I will imitate Basil Fawlty’s advice in a famous episode of the immortal 1970’s BBC TV sitcom “Fawlty Towers”:

Don’t mention the blog.

But I will mention foundations, since that is an aspect of the recognition that comes with a price but is a longstanding purpose of this—wait, don’t mention the…

Ideas and Foundations

The release includes some of Dick’s major contributions, and some others have been noted on StackExchange. I can add the first work by Dick that I was aware of: two papers in 1979–80 with Rich DeMillo—also now at Georgia Tech—which were part of various efforts to answer questions about {\mathsf{P =?\;NP}} via logic and model theory. There is also his involvement with time-space tradeoffs for {\mathsf{SAT}}, which represent the best lower bounds—well, in a broad sense, kind-of lower bounds—known for any major {\mathsf{NP}}-complete problem. Most recent and potentially important in my purview—but you might have to comb through his DLBP page to realize they’re there—are papers on multivariable polynomials modulo composite numbers, which are at the frontier of circuit complexity lower bounds.

What generates all this is a powerful beacon of ideas that have had a photoelectric effect on many parts of computer science, more than a typical theoretician encounters. I should know—it has been my privilege to do radiographic processing for many more, even some that haven’t yet appeared on this—oops—anyway, more ideas than I can often keep up with. Not to mention that I’m also pursuing (and opening) some ideas of my own. The point is that these ideas are almost all aimed at foundations. The examples above are aimed at foundations of complexity theory. That goes double for some ideas that haven’t progressed well, at least not yet.

The Problem

The problem—with the problem and much more—is that the foundations of complexity are set in sturdy concrete. They basically haven’t budged. Beams of ideas bounce off or just get absorbed. Major related parts of algorithms and crypto and other fields are likewise shielded. This is polar the opposite situation of what Bill Thurston recounted about his field of manifold foliations in his famous essay, “On Proof and Progress in Mathematics.”

The difficulty of direct progress has been so discouraging that through the theory applications grapevine I’ve heard whispers that one can phrase as, “Don’t mention foundations…” We tried to present aspects of this more positively in a recent post featuring Chayes.

However, if you aim a beam hot enough and keep it focused for long enough, events do happen. The question should not be whether it’s worth aiming the beam, but rather what’s needed to establish and maintain the intensity and focus. What we suspect is that just as in particle physics, a century on from when the likes of Joseph Thomson and Ernest Rutherford and Robert Millikan could work alone or with one associate or two, progress will require greater co-ordination of multiple researchers and shared focus on ideas than our field is yet accustomed to. Hence this…well, we can mention that various people we know and admire are trying to foster this in their own ways, not just proposed but some activated to a wonderful degree.

As I said, aiming at foundations has a price—and the payment is alas more evenly distributed among those who try than the rewards. That’s why it is good to remember that ever so often, in small as well as big, a prize comes with the price. The above, too, has an added price, but perhaps it will be ushered in with a new model for appreciating the credits.

Open Problems

The Knuth Prize has the unusual period of being awarded every 1-1/2 years officially. The last five awards are dated 2010, 2011, 2012, 2013, and now 2014. Can this seeming contradiction be resolved mathematically? Perhaps the “1-1/2″ is governed by left-and-right approximands in the manner of John Conway’s surreal numbers, which Knuth described in an engaging book. A left-bound of 1.25 would just barely allow it. However, I suspect that any such involvement by Knuth would have instead fixed the period at {2^9 = 512} days, and then I don’t see a solution…unless maybe arithmetic is inconsistent after all.

Of course we congratulate Dick without any inconsistency.

Some Strange Math Facts

September 14, 2014


Things we did not know

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Ulam holding a strange device

Stanislaw Ulam was a Polish-American mathematician whose work spanned many areas of both continuous and discrete mathematics. He did pioneering research in chaos theory and Monte Carlo algorithms, and also invented the concept of a measurable cardinal in set theory. His essential modification of Edward Teller’s original H-bomb design is used by nearly all the world’s thermonuclear weapons, while he co-originated the Graph Reconstruction conjecture. His name is also associated with the equally notorious 3n+1 conjecture. Thus he was involved in some strange corners of math.

Today Ken and I want to talk about some strange facts observed by Ulam and others that we did not know or fully appreciate.
Read more…

A Challenge From Dyson

September 9, 2014


A reversal question

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Freeman Dyson celebrated his {\text{90}^{th}} birthday last December. He is world famous for his work in both physics and mathematics. Dyson has proved, in work that was joint with Andrew Lenard and independent of two others, that the main reason a stack of children’s blocks doesn’t coalesce into pulp is the exclusion principle of quantum mechanics opposing gravity. He shaved a factor of {\sqrt{2}} off the exponent for bounds on rational approximation of algebraic irrationals, before the result was given its best-known form by Klaus Roth. He has received many honors—recently, in 2012, he was awarded the Henri Poincaré Prize at the meeting of the International Mathematical Physics Congress.

Today Ken and I want to talk about one of his relatively recent ideas, which is more mathematics than physics. Perhaps even more theory than mathematics.
Read more…

Enriching the Frankl Conjecture

September 2, 2014


See a number, make a set

BruhnSchaudt

Henning Bruhn and Oliver Schaudt are mathematicians or computer scientists, or both. They are currently working in Germany, but wrote their survey on the Frankl Conjecture (FC) while working together in Paris. Schaudt is also known as an inventor of successful mathematical games.

Today Ken and I wish to talk about the Frankl conjecture and a principle of mathematics. Read more…

Who Invented Pointers, Amortized Complexity, And More?

August 26, 2014

Some algorithmic tricks were first invented in complexity theory

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Andrey Kolmogorov, Fred Hennie, Richard Stearns, and Walter Savitch are all famous separately; but they have something in common. Read on, and see.

Read more…

Why Is 290 Special?

August 23, 2014
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How exceptions in theorems may affect their complexity

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Cropped from India Today source

Manjul Bhargava is a mathematician who just won one of the 2014 Fields Medals. We offer our congratulations on this achievement. He is an expert in number theory, which makes him special to us among Fields medalists. His Fields citation includes his doctoral work on a powerful reformulation and extension of Carl Gauss’s composition law for quadratic forms. He also proved a sense in which 290 is special to us among numbers, since we have been thinking recently about quadratic forms as tools in complexity theory.

Today we talk about his “290 Theorem” with Jonathan Hanke, which is quite accessible, and also raise complexity-related questions about this result. Read more…

The Derivative Of A Number

August 19, 2014

Are you kidding?

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Edward Barbeau is now a professor emeritus of mathematics at the University of Toronto. Over the years he has been working to increase the interest in mathematics in general, and enhancing education in particular. He has published several books that are targeted to help both students and teachers see the joys of mathematics: one is called Power Play; another Fallacies, Flaws and Flimflam; and another After Math.

Today I want to discuss his definition of the derivative of a number, yes a number.
Read more…

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