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Dana Randall is an ADVANCE Professor of Computing and also is an Adjunct Professor of Mathematics at the Georgia Institute of Technology. She is a terrific speaker and teacher and leader. Her class ratings are off the charts. See also AMS for her past special talks.
Today I thought we would discuss her research and its connections to complexity theory and to physics and to math in general.
Dana does research into the boundary between math and physics. At the highest level Dana seeks to understand random processes, especially those connected to physical systems. The difficulty, in my opinion, is that sometimes the random system is not artificial. This means that we have no control over the system, and this makes the analysis of its behavior that much harder.
Another way to say this is: we often fare better when we can control the exact random process. When someone else gets to decide on what the process is, we are often in trouble. The system might behave badly, or even worse, might be hard to understand. Nature is often that way—not always thinking about making the analysis of a system easy.
Let’s make this concrete. Suppose that you or Dana were presented with three methods for shuffling a deck of cards—when we play cards is . Imagine the methods are:
The task of understanding these methods is before us. The first (1) is slow, even for modest size . The chance of getting a permutation on a given trial is , which for is , which equals
But it does generate a fair shuffle—all possible ones are equally likely. And the proof of this is easy. The second (2) is more complicated. The final shuffle depends on the number and manner we use to shuffle the deck. The final analysis is messy.
The third one (3) is due to Ronald Fisher and Frank Yates, who discovered it in 1938. It has an elegant, but nontrivial, analysis. It is both exact in that all orderings are equally likely, and it takes time linear in . The history of it is:
The modern version of the Fisher-Yates shuffle, designed for computer use, was introduced by Richard Durstenfeld in 1964 and popularized by Donald Knuth in The Art of Computer Programming as “Algorithm P (Shuffling)” […apparently unawares; Fisher and Yates were acknowledged in later editions of Knuth’s text…]
My guess is that Dana, if given these methods, would not be interested in (1): too slow on one hand and trivial on the other. Nor interested in (2): not elegant and messy. Perhaps (3) would accord with her work: it has a nice analysis and runs in linear time.
As we said earlier, Dana is part of ADVANCE at Georgia Tech:
Georgia Tech’s ADVANCE Program seeks to develop systemic and institutional approaches that increase the representation, full participation, and advancement of women and minorities in academic STEM careers—thus contributing to a more diverse workforce, locally and nationally.
Dana has been and is a leader in helping advance these goals. It is especially relevant since this Monday, March 8th is special. It is Celebrate International Women’s Day with WiDS. See this for details:
Join us for the 24-hour virtual WiDS Worldwide Conference. We’ll follow the sun, bringing you speakers from around the world on International Women’s Day beginning at 1:00 am GMT March 8 (5:00 pm PST March 7).
In making a collage of their speaker page, we have compressed and rearranged it somewhat. And we have added the logo for the regional events happening around the globe (some already past) and one for their sponsors.
If you could not take time to follow all the talks, you might take a few for a sample. Maybe you would figure that taking the first four speakers in alphabetical order, or the last four, would be as random a sample as any—after all, what’s in a name? Well, if you took the last four, you would actually get three of the four keynote speakers. Sometimes procedures that we hope would give “random” samples in fact give special ones. That takes us back to one more topic in Randall’s work.
One benefit of a random process is that under good conditions it can give us an accurate small sampling of a large and complex system. We have mentioned dimension reduction in this context. A simpler task is just to get an approximate count of entities in the system. Sometimes one can control the system, but sometimes not.
One success is represented by a paper with Sarah Miracle of the University of St. Thomas. It is about counting colorings in multigraphs that do not violate simple constraints on the edges . Each edge has a forbidden pair of colors, and a coloring defined on is legal provided it does not have both and . Multiple edges between and can enforce multiple such constraints. Several natural problems can be represented via this one. The paper is one where their Markov Chain methods do well.
A second recent paper uses Markov chains to count elements of a given rank in finite partially ordered sets. The chains should be biased according to the structure of the Hasse diagram of the poset. The trick in the paper is a way to balance the bias so as to prevent states that need to be counted from have too low frequency. This enables a direct analysis of the mixing time. The notable application was the first provably efficient ways to sample uniformly from certain kinds of partitions of , for quite large.
The flip side—a phase change being a kind of a flip—is represented by other work with myriad collaborators that is summarized in a wonderful talk she gave at a workshop marking the 30th anniversary of DIMACS. As with an earlier version at a Schloss Dagstuhl workshop, the talk was titled, “Phase Transitions and Emergent Phenomena in Algorithms and Applications”:
Markov chain Monte Carlo methods have become ubiquitous across science and engineering as a means of exploring large configuration spaces. The idea is to walk among the configurations so that even though you explore a very small part of the space, samples will be drawn from a desirable distribution. Over the last 30 years there have been tremendous advances in the design and analysis of efficient sampling algorithms for this purpose, largely building on insights from statistical physics. One of the striking discoveries has been the realization that many natural Markov chains undergo a phase transition where they change from being efficient to inefficient as some parameter of the system is varied.
Here are a video and slides. The first main slide is about programmable active matter and it interests me especially to see DNA computing included. These systems can have emergent behavior, and while that can ruin randomized procedures that would bank on the system staying stable, it opens other opportunities.
I, Dick, have run into this type of issue before. I have been on the wrong side, with theorems that were weak because we assumed the process could not be changed. Others who followed us changed the process—got stronger results with easier proofs. Is there a name for this?
Take a look at the talks for the WiDS Worldwide Conference this Monday.
Ken also notes another event happening tomorrow: a 10am ceremony for the International Women of Courage Award. His friend the Iranian chess arbiter Shohreh Bayat is among the honorees, after a story told in her own words here in the Washington Post. Ken was working with her, on statistical assurance against cheating in the championship match, at the time. The ceremony starts at 10am ET hosted by the US Department of State, with opening remarks by Dr. Jill Biden.
We must add that there is something that is hard about the type of random processes that Dana studies. We tried to explain what makes her work deep, but perhaps we did not properly explain it.
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Jessica Fridrich is a Distinguished Professor of Electrical and Computer Engineering at Binghamton University. She is an expert on data hiding, that is, steganography. She has over 34,000 citations—impressive. A lot more than most of us. She also has worked on the famous Rubik’s cube.
Today we look at her work on Rubik’s cube, the WSJ’s interest in Rubik’s cube, and what both say—and don’t say—about fundamental algorithms.
By the way, WSJ stands for the Wall Street Journal—the American newspaper of business. The WSJ has shown great interest in the Rubik’s cube puzzle and has run many articles over the years on it.
Recall the cube puzzle was invented…of course you know all about Rubik’s cube. You probably have owned one at one time. Right. Just for a refresher:
The Rubik’s Cube is a 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. As of January 2009, 350 million cubes had been sold worldwide, making it the world’s top-selling puzzle game. It is widely considered to be the world’s best-selling toy.
But you may not know all about Fridrich.
Fridrich was one of the progenitors of speed cubing. She took part in the First World Championship in 1982 in Budapest, next-door to her native Czechoslovakia. She finished in the middle of the pack with a time of 29.11 seconds from a randomly well-mixed starting cube position. Her thoughts on how the cubes could be better prepared for speed are recorded on her page about the tournament.
At the Second World Championship, she improved her average time to 20.48 seconds and placed 2nd. She had the two fastest solves in the finals but lost on average-of-median-three-of-five. That championship took place in Toronto—in 2003. She is at a loss to explain why there was such a gap. Usually an athlete—in this case a mathlete?—is on the downswing nearing age 40, but even as a self-described “old-timer,” she fended off all but one of a whole next generation.
Much of the credit goes to her solving method. She originated the “O” and “P” parts of the CFOP method. CFOP stands for: Cross, First 2 Layers, Orient Last Layer, Permute Last Layer. Versions of this are used my most top “cubers” to this day, and her name is often affixed to the method. In a 2008 profile of her, the New York Times quoted the 2003 winner as saying that Fridrich found the route up the mountain while the rest of the cubers optimize traversing ledges along it. And in 2012, the NYT quoted Olympic shot-putter Reese Hoffa as wanting “to learn the Fridrich Method of solving the puzzle, ‘which is what all of the best cubers use.'”
At this point, knowing our interest in chess, you might expect a Queen’s Gambit reference. But what we have here is not a story of Beth Harmon coming back from a life adjournment or Roy Hobbs in The Natural rejoining baseball almost 20 years after being shot. It’s about going overseas, earning a PhD, getting two research positions, writing early papers (under the name Jiri Fridrich), transitioning, then getting a faculty position leading to tenure while developing mathematical formulas and writing tons of code for systems to source photos and catch digital pirates and pornographers and other image fraudsters, then coming back to light up an or universe. Not to mention doing her own stunning photo art of the American Southwest.
Since 2003, the championships have been held every other year, thought the 2021 championships set for the Netherlands are uncertain owing to the pandemic. The youngsters soon broke through en-masse, and it strikes me that the cube technology improved so that the cubes are springier and lighter. The winning time fell almost 5 seconds to 15.10 in 2005 and hit 6.74 in 2019. That was not the world record, however—an incredible 3.47 seconds in 2018 by Yusheng Du, beating the previous record of Feliks Zemdegs by a whopping 3/4 of a second.
Fridrich, however, must claim a distinction no one may ever match. She learned how to solve the cube and traced out the performance of methods of doing so in 1981, months before she saw a cube, let alone owned one. Despite the “Bűvös Kocka” (“Magic Cube,” as Rubik called it) having been on shelves in neighboring Hungary for four years, with worldwide marketing by early 1980, they were hard to come by in her home city, Ostrava.
She found an article on solving the cube in a Russian magazine. It laid out the concept of group theory and the role of group commutators, which she learned to apply creatively in order to streamline actions. The first time she touched a cube was to help a friend put his back the way it was. A family visiting from France let her keep one, and later in 1981 she was finally able to purchase a few more. This invites analogy to working out chess without a board on a bedroom ceiling as depicted in The Queen’s Gambit.
We—Dick and Ken—must admit that neither of us has ever done this with the cubes we own, not fast, not slow. Yet we do understand the theory behind it. We believe we do.
I (Dick) plan on explaining the theory by using a new toy that I have invented: The .
We will write the state as where each of is one of 1, 2, or 3. The operations allowed are the cyclic shift , which does
and the flip of the initial two elements:
Note there are 6 possible states. For the real Rubik’s cube, the number of states is just a little bit larger: 43,252,003,274,489,856,000. But the basic concept is the same. Suppose we are given the state
How fast can you get the initial state ? Apply :
This is a special case of the general result that any symmetric group is generated by two operations: a full cycle and a single flip. The key with the actual Rubik’s cube is since the group is larger and it has more operations that can be applied finding the group operations may be more difficult. But there are algorithms that can find them. See this for another article by Keith Conrad.
There are many more pages like that on the cube. But Fridrich still shows the seminal page she posted in “Winter 1996/97.” It links to other pages, ones that also credit other people, such as this explaining the algorithms in great pictorial detail. This was in the infancy of the Internet. Her pages are often credited with spurring the turn-of-the-millennium boom in Rubik’s cube which led to the revival of the championships in 2003. A 2016 New York Post article whose URL is titled, “how the Internet brought the Rubik’s cube back to life,” says:
The seeds for Rubik’s Cube’s rediscovery were sown on the internet. In the mid-1990s, a Rubik’s Cube champion-turned-computer-science professor at SUNY Binghamton posted her secrets of the Cube on a primitive Web 1.0 site on the university’s servers. Jessica Fridrich’s method spread and is today the most widely used technique to solve the puzzle.
See also this telling by the 2003 winner, Dan Knights. This shows how one person using spare time on the Internet can power up business.
The WSJ has had an interest in Rubik’s cube for years. They had a long feature article last week titled, “How to Teach Professors Humility? Hand Them a Rubik’s Cube,” by Melissa Korn. It describes a faculty development challenge among several small colleges in which professors became students again. Last month they also had an article on symmetry by the mathematician Eugenia Cheng that mentioned the cube.
I recall several features the WSJ has run on the cube and its solvers. The 2011 article, “One Cube, Many Knockoffs, Quintillions of Possibilities,” led off with the Polish teenager Michal Pleskowicz winning the 2011 world championship with a time of 8.65 seconds, then discussed the performance of pirated cubes: “One reason Mr. Pleskowicz and a new generation of Rubik’s fanatics can solve the notoriously difficult puzzle in record time: They don’t use Rubik’s Cubes at all, instead substituting souped-up Chinese knockoffs engineered for speed…” Their 2014 article, “Rubik’s Cube Proves It’s Hip to Be Square,” profiled both Rubik and speed-solvers.
The feature I recall best was in 2018. It was titled, “A Thinking Person’s Guide to the Rubik’s Cube,” and subtitled, “What’s the best solution method—theory, algorithms or chance?” It was also by Eugenia Cheng. She begins by confessing, “I have always loved playing with a Rubik’s Cube, which combines logic with a satisfying tactile activity. I can solve it—getting each of the six sides to be one color—but not particularly quickly or cleverly.”
They also like its use for analogies. Scrolling through their advanced search—both Ken and I subscribe to the WSJ—we find:
In all, using the WSJ advanced search, we find 239 hits for “Rubik” going back to 1980. We should mention in-passing that one of them is their 7/17/20 obituary for Ron Graham. We also find 7 hits for “Fridrich” over the same span. But they are all about the housing market, involving the Nashville-based realty Fridrich and Clarke.
I am happy to see that the WSJ has published multiple articles on a particular algorithmic task. I like that algorithms have been the center of articles. I wish they would talk more about important algorithms. Solving a Rubik’s cube is not an algorithm that is used every day: What about:
They do have Eugenia Cheng, who wrote a column comparing sorting algorithms. And they have written on algorithms used in trading and on social–media platforms and for policing and parole and bail decisions. But that tends away from fundamental algorithms where the math is the matter.
A 2018 WSJ article by Hannah Fry titled “Don’t Believe the Algorithm,” which begins with flaws in using facial recognition to find wanted suspects, brings us back toward Fridrich’s research. Might this all also raise discussion of “algorithms” for what and whom to cover?
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Joshua Grochow is an assistant professor in Computer Science and Mathematics at the University of Colorado at Boulder. He was a student of Ketan Mulmuley and Lance Fortnow at Chicago; his dissertation and some subsequent papers did much to widen the horizons of the “Geometric Complexity Theory” (GCT) program. He is also a gifted expositor.
Today we will highlight some of his work and some of his exposition of new and old theorems.
An ancient one is Stephen Mahaney’s famous theorem on the nonexistence of sparse -complete sets (unless ). Grochow discusses a simpler proof of the theorem by Manindra Agrawal and gives some further impacts on GCT.
A recent one with Youming Qiao is on an old topic and is in the 2021 Innovations in Theoretical Computer Science conference. It is titled, “On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness,” and grows out of a 2019 paper by the same authors.
This came to my attention through communications with Grochow’s student, Michael Levet. Indeed, Levet is the reason for my putting this all together. He raised through email some questions about an ancient result of mine on group isomorphism. I reported previously:
Long ago Bob Tarjan and Zeke Zalcstein and I made a simple observation: Group isomorphism could be done in time
This relies on the easy-to-prove fact that every group has at most generators. We have discussed this idea earlier here.
Levet raised an issue about related observations of mine—ones that were misleading at best. I think he has a good point and we are still trying to unravel exactly what I meant back then. I applaud him for reading ancient stuff, for trying to extend it, and for working on such problems. I wish him well.
While Levet and I work that out and think about Grochow’s paper on isomorphism problems with Qiao, Ken and I want to highlight a different expository paper by Grochow on news from 2016 that we covered then. Grochow’s paper appeared in the AMS Bulletin and is titled, “New Applications Of The Polynomial Method: The Cap Set Conjecture And Beyond.”
To lead in to the subject, here is a problem from 1917 by the English puzzlemaster Henry Dudeney titled, “A Puzzle With Pawns”:
Place two pawns in the middle of the chess- board, one at Q 4 and the other at K 5. Now, place the remaining fourteen pawns (sixteen in all) so that no three shall be in a straight line in any possible direction. Note that I purposely do not say queens, because by the words ” any possible direction ” I go beyond attacks on diagonals. The pawns must be regarded as mere points in space — at the centres of the squares.
Sixteen is obviously the maximum possible for a standard chessboard because a seventeenth pawn would make three in some row and some column. For an board, the limit is by similar reasoning—this is an example of the pigeonhole principle which we just mentioned.
It is possible to achieve the maximum for all up to and then the only other cases known are , , and . That’s it. Here are solutions for and . The latter was found by Achim Flammenkamp, whose page has encyclopedic information. On the former, the pieces are positioned on gridpoints like stones in Go, which seems a better context for this problem than chess.
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The conjecture is not only that a -size solution exists for only finitely many , but also that the maximum size for all sufficiently large is bounded by with , indeed, with . It is known that stones can always be placed with no three collinear, where the depends on the closeness of a prime to .
The problem can be taken to dimensions that are beyond the plane. We can also extend what is meant by a “line” via various notions of wrapping-around. Then the question is how close the maximum size can stay to being linear in the size of the space—as and/or the size of an individual dimension increase.
The theme of the no-three-in-a-line problem is fundamental to combinatorics. There are tons of problems of the form:
How many objects can one place, so that no pattern of some certain type exists?
In -dimensional space the smallest of interest is . This means playing on higher-dimensional versions of the grid and cube. Then the only Euclidean lines are the kind we know from tic-tac-toe: straight across or down, or diagonal.
For dimension there are other kinds of diagonals, such as within a face or through the center of the cube, but they all win at -dimensional tic-tac-toe. So the problem becomes: what is the maximum number of moves you can make by yourself without creating a win at tic-tac-toe? The cap-set problem adds a twist by extending the notion of what is a line. It is like playing tic-tac-toe on a floor of tiles where a play in one tile is replicated in all of them. Then you can make a line by playing in a corner and in the middle of the two opposite edges, as shown at left in the following diagram (original drawing).
The four orange O’s at right have no 3-in-a-line even with this extended notion of line. Note that the four blank cells in the top two rows also avoid putting 3 in a line. Four is the maximum, however: it is not possible to have a drawn game in extended tic-tac-toe.
The theorem proved by Jordan Ellenberg and Dion Gijswijt in 2016 is that the upper bound is not only a vanishing fraction of the size of the space as grows, it is bounded by where . Namely:
Theorem 1 Every cap set in the -cube has size at most .
There is a simple way to express the extended notion of “line” that works for all dimensions : Number the coordinates of each dimension . Make the space with addition modulo , that is, make it . Then the condition for three points to be in a line is simply
It is easy to write polynomial equations over the field to express the property of a set having such a line. What was unexpected, until Ernie Croot, Vsevolod Lev, and Péter Pál Pach solved a related problem with , was that there would be
“an ingeniously simple way to split the polynomial[s] into pieces with smaller exponents, which led to a bound on the size of collections with no [lines].”
The quotation comes from an article by Erica Klarreich for Quanta right then in 2016. A 2016 AMS Feature column by David Austin covers how to make this say a set of points is a cap set modulo 3:
where we (not Austin) write to mean the set of sums where and . If there is an element in the intersection then , and since , we get with in and all distinct, a contradiction. (If then , so .) Let stand for the number of monomials of degree at most in the variables. The key first insight is:
Lemma 2 If a polynomial of degree vanishes on , then for all but at most points of .
One could first try to interpret this as saying that “looks like” to polynomials of “low” degree . However, if stays low relative to then the “if” part would hold vacuously, opposing the goal of bounding and making the whole idea self-defeating. In fact, the important tension comes when is intermediate: , which for makes and neatly occupy the middle of the range .
The proof also uses the trick that if a product of two monomials has degree then one of them must have degree at most . As I (Ken writing these sections) wrote about it back in 2016, this reminds of Roman Smolensky’s degree-halving trick in his celebrated 1987 theorem on lower bounds for mod- versus mod-. This trick, however, runs from to for all moduli.
In any event, the 2016 papers were a new form of the polynomial method that led to striking new results. What Grochow’s survey does for us now is bring out wider implications of this ingenuity.
Grochow’s four application areas in section 4 of his survey are:
We say a little more about the last of these. For any matrix and define the rigidity to be the minimum number of entries in which differs from some matrix of rank (at most) . The highest possible rigidity for rank is , since zeroing out an block leaves a matrix of rank at most . Sufficiently random matrices meet this upper bound with high probability, but the best lower bounds for explicit families of matrices are , which is only quasi-linear when is close to . The question is whether we can inch this up to for some .
Definition 3 A family of matrices is significantly rigid if there is an such that taking makes .
The interest in this definition comes from a lack of lower bounds on linear algebraic circuits computing natural families of linear transformations that seems even more extreme than our lack of super-linear lower bounds on Boolean circuits, nor better than for general algebraic circuits computing polynomials in variables of degree . It is still consistent with our knowledge that every natural family can be computed by linear algebraic circuits of size and depth. Leslie Valiant in 1977 proved the following sufficient condition to improve this state of affairs.
Theorem 4 Every significantly rigid family cannot be computed by linear algebraic circuits of linear size and logarithmic depth.
So for coming on half a century the question has been:
Can we construct a natural explicit family of significantly rigid matrices?
Beliefs that the Hadamard matrices provided such a family were refuted by Josh Alman and Ryan Williams at STOC 2017, and known results for Vandermonde matrices do not have close enough to .
One hope had been to derive such matrices from explicit functions over for prime by taking and defining
Unfortunately, the polynomial method for cap sets shows that no such attempt can work. Zeev Dvir and Benjamin Edelman proved that no matter how and are chosen, there is such that for all large enough ,
This means we cannot get for , indeed, far from it. What is most curious to us is that for matrix multiplication, the cap-set related technique frustrates a better complexity upper bound, whereas here it frustrates a better lower bound.
What further applications can we find for the polynomial method?
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Everything that can be invented has been invented—Charles Duell, Commissioner, U.S. Office of Patents, 1899
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George Cantor has been featured here and here and here before on GLL. Of course, he invented modern set theory and changed math forever. His birthday is soon, so we thought we would talk about him now—he was born on March 3rd in 1845.
Today we thought it might be fun to have a quiz on math quotes.
Wait. Cantor did not invent quotation marks, nor is he known for many quotes. He does of course have many famous results, and they will live forever. But his results were subject to immediate horrible criticism and therefore memorable quotes.
Leopold Kronecker was a particular source of barbs. For example: “What good is your beautiful proof on the transcendence of ? Why investigate such problems, given that irrational numbers do not even exist?”
As a complexity theorist I must say that Kronecker has a point when he also said:
“Definitions must contain the means of reaching a decision in a finite number of steps, and existence proofs must be conducted so that the quantity in question can be calculated with any degree of accuracy.”
David Hilbert defended Cantor and said: “No one shall expel us from the paradise that Cantor has created.”
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On to the quiz. Each quote is followed by two possible authors in alphabetical order. You should pick the one you think is correct. The players are:
1. Douglas Adams 2. Bernard Baruch 3. Eric Temple Bell 4. Raoul Bott
5. Paul Erdős 6. Richard Hamming 7. Godfrey Hardy 8. David Hilbert
9. Admiral Grace Hooper 10. Alan Kay 11. Donald Knuth 12. John von Neumann
13. Alan Perlis 14. Henri Poincaré 15. Srinivasa Ramanujan 16. Marcus du Sautoy
17. Raymond Smullyan 18. Alan Turing 19. Moshe Vardi 20. Andrew Wiles
“I always have a quotation for everything—it saves original thinking.”
—Dorothy Sayers
Here are the answers:
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Sergei Voronin was an expert in number theory, who studied the Riemann zeta function, but who sadly died young over twenty years ago. We discussed his amazing 1975 result about the Riemann zeta function here. Others call the result the amazing theorem. I (Dick) am getting old—I almost forgot that we did a post on his theorem again over four years ago.
Today I thought we would recall his theorem, sketch why the theorem is true, and then discuss some extensions.
Starting with Alan Turing we have been interested in universal objects. Turing famously proved that there are universal machines: these can simulate any other machine on any input. Martin Davis has an entire book on this subject.
Universal objects are basic to complexity theory. Besides Turing’s notion, a universal property is key to the definition of NP-complete. A set in NP is NP-complete provided all other sets in NP can be reduced to in polynomial time. Michael Nielsen once began a discussion of universality in this amusing fashion:
Imagine you’re shopping for a new car, and the salesperson says, “Did you know, this car doesn’t just drive on the road.” “Oh?” you reply. “Yeah, you can also use it to do other things. For instance, it folds up to make a pretty good bicycle. And it folds out to make a first-rate airplane. Oh, and when submerged it works as a submarine. And it’s a spaceship too!”
In 1975 Voronin had the brilliant insight that the Riemann zeta function has an interesting universality property. Roughly speaking, it says that a wide class of analytic functions can be approximated by shifts with real . Recall
for , and it has an analytic extension for all other values but .
The intense interest in the function started in 1859 with Bernhard Riemann’s breakthrough article. This was the first statement of what we call the Riemann Hypothesis (RH).
In over a century of research on RH before Voronin’s theorem, many identities, many results, many theorems were proved about the zeta function. But none saw that the function was universal before Voronin. Given the zeta function’s importance in understanding the structure of prime numbers this seems to be surprising.
Before we define the universal property I thought it might be useful to state a related property that the function has:
Theorem 1 Suppose that is a polynomial so that for all ,
Then is identically zero.
Since is a single variable, this says that and its derivatives and do not satisfy any polynomial relationship. This means intuitively that must be hypertranscendental. Let’s now make this formal.
Here is his theorem:
Theorem 2 Let . Let be an analytic function that never is zero for . Then for any there is a real so that
See the paper “Zeroes of the Riemann zeta-function and its universality,” by Ramunas Garunkstis, Antanas Laurincikas, and Renata Macaitiene, for a detailed modern discussion of his theorem.
Note that the theorem is not constructive. However, the values of that work have a positive density—there are lots of them. Also note the restriction that is never zero is critical. Otherwise one would be able to show that the Riemann Hypothesis is false. In 2003, Garunkstis et al. did prove a constructive version, in a paper titled, “Effective Uniform Approximation By The Riemann Zeta-Function.”
The key insight is to combine two properties of the zeta function: The usual definition with the Euler product. Recall the Riemann zeta-function has an Euler product expression
where runs over prime numbers. This is valid only in the region , but it makes sense in a approximate sense in the critical strip:
Then take logarithms and since are linearly independent over , we can apply the Kronecker approximation theorem to obtain that any target function can be approximated by the above finite truncation. This is the basic structure of the proof.
Voronin’s insight was immediately interesting to number theorists. Many found new methods for proving universality and for extending it to other functions. Some methods work for all zeta-functions defined by Euler products. See this survey by Kohji Matsumoto and a recent paper
by Hafedh Herichi and Michel Lapidus, the latter titled “Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality.”
Perhaps the most interesting question is:
Can universality be used to finally unravel the RH?
See Paul Gauthier’s 2014 IAS talk, “Universality and the Riemann Hypothesis,” for some ideas.
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]]>Joe Biden is the 46th president of the USA. Note is called a centered triangular number. These numbers obey the formula:
and start with The previous one, the , was Herbert Hoover, hmmm. Biden has promised to make controlling the Covid-19 pandemic one of his top priorities.
Today I thought we would discuss how he might use computer technology to help get the virus under control.
First, we thank the drug companies since we now have vaccines that work against the virus. Without these we would have little chance to bring the pandemic under control at all.
Second, we must state that we are worried that the virus is mutating and this may render the current vaccines less useful, if not useless. We hope this is not happening, or that the drug companies will be able to respond with vaccine boosters. Today there seems to be good news and bad news.
Results will fluctuate, but in any case, vaccines will definitely play a key role in defeating the pandemic. We want to ask the same about computing technology.
There are many web sites that discuss how computing technology can play a role in defeating the pandemic. Here are some of the main points:
Tracking People: Many places are interested in tracking who are sick. Tracking can by itself help stop the spreading of the virus, and thus help save lives. For example, IEEE says:
“We believe software can help combat this global pandemic, and that’s why we’re launching the Code Against COVID-19 initiative…,” said Weiting Liu, founder and CEO of Arc. “From tracking outbreaks and reducing the spread to scaling testing and supporting healthcare, teams around the world are using software to flatten the curve. The eMask app (real-time mask inventory in Taiwan) and TraceTogether (contact tracing in Singapore) are just two of the many examples.”
Changing Behavior: A powerful idea is to avoid human to human contact and thus stop the spread of the virus. For example, here are examples from a longer list of ideas:
Changing Health Delivery: An important idea is how can we reduce the risk of health delivery. A paradox is that health care may need to be avoided, since traditional delivery requires human contact. There are many examples of ways to make health care online, and therefore safer. Shwetak Patel won the 2018 ACM Prize in Computing for contributions to creative and practical sensing systems for sustainability and health. He outlined here CCC blog how health care could be made more online.
The above ideas are fine but I believe the real role for computing is simple:
Make signing up and obtaining an appointment for a vaccine easier, fairer, and sooner.
In the US each state is in charge of running web sites that allow people to try and get an appointment for a vaccine shot. Try is the key word. Almost all sites require an appointment to get a shot—walk-ins are mostly not allowed.
I cannot speak for all states and all web sites, but my direct experience is that the sites are terrible. Signing up for a vaccination shot is a disaster. The web sites that I have seen are poorly written, clumsy, and difficult to use. They are some of the worst sites I have ever needed to use, for anything. Some of the top issues:
Repeating (1,2,3) is a poor joke, but one that reflects reality.
If Amazon, Google, Apple had sites that worked this way, they would be out of business quickly. Perhaps this is the key: Can our top companies help build the state sites? Is it too late to help? See here for a New York Times article on this issue:
When you start to pull your hair out because you can’t register for a vaccine on a local website, remember that it’s not (only) the fault of a bad tech company or misguided choices by government leaders today. It’s a systematic failure years in the making.
Also is the issue of algorithmic fairness relevant here? We know that it is unfortunately easy to have web sites that are unfair—that assign vaccine sign up dates unfairly, that favor one class of people over another.
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A special journal issue in his honor
Elvira Mayordomo, Mitsu Ogihara, and Atri Rudra are going to be the editors of a special issue of the journal Theory of Computing Systems dedicated to Alan Selman. Alan passed away this January 2021.
Today we circulate their request for contributions.
The details of the call say: This special issue celebrates Alan’s life and commemorate his extraordinary contributions to the field. The topics of interest include but are not limited to:
Please look at this for details—the deadline for submission is 31st July 2021. You have 164 days to write your paper. Which is 3936 hours or 236160 minutes.
Please send a contribution.
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Mathematics is based on the application of simple ideas over and over: From tiny nuts do big trees grow.
Jorgen Veisdal is an assistant professor at the Norwegian University of Science and Technology. He is also the editor in chief at Cantor’s Paradise, which is a publication of math and science essays.
Today I thought we would discuss a post of his on the famous Pigenhole Princeiple (PP).
Recall the PP states that if items are put into boxes, with , then at least one box must contain more than one item.
The paradox in my opinion is that this idea has any power at all. I wonder if I could explain why it was stated as an explicit principle by the famous Peter Dirichlet under the name Schubfachprinzip (“drawer principle” or “shelf principle”) in 1834.
Parts of mathematics not only use PP, but could not live without it. Other parts of mathematics—I believe—are almost untouched by it. Am I right about this? Number theory and combinatorics especially Ramsey theory could not survive without it. What happens in your favorite area? Is there some area of math that is almost untouched by PP?
The main issue is why is PP so indispensable to some areas of math. But I though it might be fun to give a sample type of proof that uses PP.
Prove that however one selects 55 integers
there will be some two that differ by 9, some two that differ by 10, a pair that differ by 12, and a pair that differ by 13. Surprisingly, there need not be a pair of numbers that differ by 11.
Let be the number of collisions when placing into . Claim:
for and and
for .
Note the first is really simple. Consider the first pigeons. They are placed into places and the inequality follows. The second is about the same. Consider the first pigeons. There are two cases. They are all placed in places. Then we are done. So there must have been some placed into the last place. But if two are there then we are also done. So are placed into . But where does the last one go? In either case we are done.
Are there areas that almost never use the PP? I would like to hear about areas that just do not use PP.
[some word fixes]
Leopold Kronecker was one of the great mathematicians of the 19th century. He thought about foundational issues deeply. We highlighted him before—well not deeply.
Today I thought we would talk about some core math ideas arising out of Kronecker’s work.
Kronecker’s angle as an early leader in the modern foundations of mathematics was on which aspects are helpful in concrete analysis. He no doubt would be comfortable with complexity theory, with our interest in not just existence proofs, but in concrete algorithms for the construction of objects. He famously said:
God made the integers, all else is the work of man.
His was a philosophy that would agree with our view of complexity theory. Maybe he would say now:
God made the binary strings, all else is the work of people.
In any part of mathematics we are often interested in operations that take objects and make new objects. These operations are important as they allow us to build new interesting objects.
There are a wealth of such operations on graphs. One is called Cartesian product, one the strong product, one the direct product. A trouble, in my opinion, is that the names and the notation for these operations is not uniform. There are alternative names for almost all the major operations: For example,
The Cartesian product of graphs is sometimes called the box product of graphs—see here. The notation that has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs.
Confusing, no?
Ken notes that it gets even more confusing when one teaches the Cartesian product construction of finite automata. Each automaton has a state graph. In general the state graph is directed and the edges are labeled by characters, but one can make cases where the graph is undirected and there is only one character. The state graph of the product machine is in general not the Cartesian product of the graphs of the individual machines. What product is it? Let’s look at graph operations.
Let us give a uniform definition of three basic graph products. Let be undirected graphs. Then
is the graph with vertices so that each is a vertex in , and the vertices and are connected provided for exactly indices , and are an edge in and for the rest .
It is usually written as
It is usually written as
It is usually written as
Here is a comparison of four graph products.
The answer to Ken’s question is that you get the tensor product, which is Kronecker’s product on the adjacency matrices of the graphs. This is because each step of the product requires an action by each constituent machine.
Of course then we get other types of products for other values of . Are any of these interesting? We get intermediate notions up to Kronecker’s product. Can the be put to any natural use?
An application of graph products is that they yield some quite compelling conjectures. The conjecture due to Stephen Hedetniemi in 1966 is one example. This states that
Here is the number of colors needed to color . The conjecture is false–see Counterexamples to Hedetniemi’s conjecture by Yaroslav Shitov. Also see Gil Kalai’s post about this news.
A potential application of graph products is to the famous Four Color Theorem (4CT)—see here. In a paper of mine with Atish Das Sarma, Amita Gajewar, and Danupon Nanongkai we show:
Theorem 1 (An Approximate Restatement of the Four-Color Theorem) Suppose every two-edge connected, cubic, planar graph can be edge 3-colored with errors. Then the Four Color Theorem is true.
The proof is simple. We assume that some graph is a counterexample to the 4CT. Then form a kind of product of . To be honest we did not see the proof exactly as this, but is essentially what we did. The is a huge product of many of the copies of , with some small modifications. Then we show if we could almost four color then we would be able to exactly color .
Meanwhile, the paper showing that Hedetniemi’s conjecture is false contains an important lesson for mathematicians, Noga Alon says,
Sometimes, the reason that a conjecture is very hard to prove is simply that it is false.
[restored missing line in second sentence, other fixes]
2017 Who’s Who award |
Alan Selman, my longtime friend and colleague, passed away last Friday, from the one illness that is most cruel for someone who has excelled by brainpower and effervescent joie de vivre alike.
Today, Dick and I join others in mourning and also in appreciation.
I last saw Alan with his wife Sharon in May 2019 at the dinner for the NP50 celebration of Steve Cook in Toronto. There were inklings of his affliction but nothing that kept him from engaging in dinner talk about how our field has progressed, about reunited friends, about the convivial atmosphere and culture on display during the celebration, and even my own goings-on with chess and quantum complexity and all our grown children. Alan was still going out to concerts last March until the pandemic stopped all that kind of activity.
I’ve known Alan in person for almost 40 years, starting a decade before he became my Chair for 6 years and colleague for 18 years more of them before his retirement and move to New Jersey. This was all through my graduate study at Oxford. It began at ICALP 1982 in Aarhus after my first year. During a December 1983 workshop at Brooklyn College he shared a problem with me whose story I told here. He visited me in Oxford in early 1986 accompanied by his son Jeffrey, whose eulogy in this past Sunday’s burial service was especially moving. Alan gave me other helps in 1985–86 that aided my “repatriation,” so to speak. My point is that he took great interest in builders of complexity theory even as students—Uwe Schöning noted that he visited him in 1982 while he was a student as well. My 2014 post on Alan’s retirement party expressed how this continued throughout.
As others have remarked, Alan’s cars had a license plate that declared P NE NP. Over a quarter century I regularly saw it in the parking lot serving Bell Hall, and after 2012, the same lot for our department’s new digs in Davis Hall. My colleague Sargur Srihari relates seeing someone approach Alan in the lot and say, “Yes, but can you prove it?”—adding that it was “probably the only thing in complexity that Alan couldn’t do.” An irony I will say here now is that the Dean of Engineering and Applied Sciences whom Alan worked with during much of his time as Chair, Mark Karwan, claims the opposite. It would have been fun to see a dueling license plate saying P EQ NP.
Synthesized via ACME license-plate composing app |
Long followers of this blog know that Dick and I have embodied this duel between ourselves. Only recently has Dick swung toward P NP, while some have moved the other way.
But what we think everyone would agree on is that we cannot point to any definite progress in the past dozen-plus years, at least not directly on the question. This struck me particularly from the Cook workshop, and this would have been the lead theme of a post on it, had I not been sprinting toward an effective June deadline to complete a major upgrade to my statistical chess model which I achieved two months later. No one attested real progress in comments to our P=NP status post a year ago. I wrote to Sharon two weeks ago that I wished I could tell Alan of progress but no.
Where Alan, Dick, I, and many of us are completely aligned is on these two points:
We put in the mouth of a fictional STOC 1500 keynote the position that relations between problems should be the field’s fundamental objects. There is incredible beauty in these objects, as was the theme of Lane Hemaspaandra’s tribute to Alan on his retirement surveying gems of Alan’s work, and it was my elation to be captivated by them beginning in my graduate years of the 1980s.
Alan’s Hebrew name was Eliyahu, Elijah. In the Bible, Elijah is a traveling prophet: he not only goes from Zarephath to Beersheba but all the way south to Mount Sinai, where he is commanded to return all the way north to Damascus. The mission from Sinai is conveyed not in cloud and fire as with Moses, nor in wind or earthquake, but famously in a still small voice.
That was the kind of voice in which Alan during his many travels spread the love of complexity theory. I was glad to experience his kindness, good crisp advice, and also the giving of freedom to pursue things with zeal—a trait that Elijah had and recognized. I was not so fast to recognize good food and fine culture. I remember once when Alan was especially happy about securing tickets to see the 86-year-old legendary jazz violinist Stéphane Grappelli perform in Buffalo and I did not know who that was. I was, however, able to reply that I’d once seen Chet Baker live while standing in a small room in London. He and Sharon and my wife Debbie and I shared other musical interests. All of what I implied about whole-person education in my memorial for Peter Neumann applies to Alan. In Bell Hall we had only a mid-size seminar room for gatherings, yet Alan right away recognized the importance of making it like Oxford’s Mathematical Institute tea room—but with wine included—in weekly Friday afternoon “TGIF” gatherings open to faculty and graduate students, for which I did the shopping those mornings at Wegmans.
Elijah is also known as a keeper of promises. I won’t go into the gruesome biblical details here, but rather note that Alan shepherded a promise of a different kind: the notion of a promise problem, following on from his famous 1984 paper with Shimon Even and Yacov Yacobi. This is a structural complexity concept whose usage has grown, besides the original technical challenges which I recounted here. I’ve sometimes felt that wider uses have run risk of getting burned by inexactitudes, such as in the notion of promise-completeness. Alan’s exactness was one trait that made me feel at home.
Elijah also founds a school called the “sons of the prophets,” passing his mantle to Elisha before being taken up in a chariot of fire. It is not for me to say who might pick up Alan’s mantle in structural complexity. But many in his train have said wonderful things in tribute over the weekend, and I wish to pass along some of them, with quotes from those who have given permission or have already written in public, in particular as comments to Lance Fortnow’s tribute to Alan.
The news was passed around our department in Buffalo. Our Chair, Chunming Qiao, mourned the loss of a great colleague, a former Chair, and a respected scholar. Sargur Srihari told the parking lot story above and and noted how Alan recruited a star to the department in the person of Jin-Yi Cai. Stuart Shapiro, who preceded and succeeded Alan as Chair, noted Alan’s higher personal standards for research and publishing. Russ Miller, writing earlier this month when we had news of Alan’s being in hospice care, hailed him as “a terrific person,” meeting the standard of a “true gentleman and scholar,” and hailed his communication skills with colleagues and the university administration alike. Jinhui Xu expressed thanks for Alan’s thoughtfulness and role as a mentor. I can say all this from my own experience. Many of us also took part in the earlier outpouring of affection, including photos and memories of Alan’s time in Buffalo, that was communicated to Sharon.
Mark Karwan adds, “Alan continued to win awards and accolades throughout my 12 years as Dean which ended in 2006. All of his professional accomplishments provided a great sense of pride and international recognition for UB. Alan was truly a senior leader and mentor for so many. He spoke to me as an advocate for his department and for the young rising faculty and the need to nourish and retain them. His wise counsel was always given in the most gentlemanly manner. When I picture Alan today, I see the twinkle in his eyes and his contagious joy of life. I only started working relatively recently with a colleague in our field of Operations Research on matters in complexity with implications to support P = NP. It would have been such fun to bring Alan into our conversations! We will all miss Alan and will always appreciate the great legacy he created here at UB.”
Jin-Yi, of course, is well known to all who knew Alan in the theory community and readers of this blog. He called Alan a great structural complexity theorist, and wrote in one of many e-mails I’ve been copied on: “Alan [was] a wonderful colleague of mine, and had a lifetime contribution to complexity theory that will certainly live on. He had a dry wit, enigmatic smile, and most of all, a truly kind heart. We will always remember him in our fond memory.”
Atri Rudra worked with Alan on many projects including organizing and securing funding for the Eastern Great Lakes (EaGL) Theory Workshops along with Bobby Kleinberg of Cornell. Here are two photos from the first one in 2008:
Composite from 2008 EaGL photo page; Kleinberg to Alan’s right and Atri (obscured) behind. |
Atri writes, “I still remember being taken directly from the airport to Alan’s favorite restaurant, Trattoria Aroma, during my interview at Buffalo. I was a bit intimidated by Alan during the dinner but we bonded over the fact that we were both married to epidemiologists. After I joined Buffalo, Alan’s sage advice helped me throughout my tenure process. Alan was a giant in the department and having him in my corner did not hurt. [H]e had a wicked sense of humor as well.”
Mitsunori Ogihara succeeded Alan as Editor-in-Chief of the journal Theory of Computer Systems and blossomed from being a postdoc recruited by Alan at UB to being Chair of Computer Science at the University of Rochester in a few short years. The story of my taking Mitsu from the airport on a snowy night will be told another time. He is writing his own tribute to appear in the journal, from which I quote just a few words: “While being a scholar of uncanny vision and incredible ingenuity, Alan was a lovely human being. He was generous with his time for his students and colleagues, always willing to offer consultation and advice. The field of theoretical computer science has lost its giant. Alan is no longer here to lead us, but his legacy will live.”
Lane Hemaspaandra, Mitsu’s colleague at UR, writes: “Very briefly put, Alan’s research handiwork and vision is ingrained in the shape of the field. I was lucky enough to through his generosity know Alan across decades: we co-edited a book, wrote two research papers together, even wrote—Alan had a real love of writing and language—a note on writing in theoretical computer science, and had a long tradition of research seminars and theory days that brought together the University at Buffalo, RIT, and University of Rochester theory groups. Alan was also my wife’s wonderful postdoctoral advisor. And, throughout all that, and coexisting with Alan’s technical artistry and amazing taste in theory research, Alan’s love of and expertise in the ‘real’ world was quiet yet luminous. Alan loved the theater, and spoke warmly of his beloved Shaw Festival at Niagara-on-the-Lake. He loved and knew food; any restaurant commended by Alan was going to be an experience.”
Ashish Naik noted that he was Alan’s first student in Buffalo: “[I] still remember going to his office to ask if he would take me as a student, somewhat intimidated because of his stature in the field. Alan he put me at ease immediately with his sense of humor (I still remember the cartoon clippings on his office door) and his generosity. Despite a busy schedule as the Chair, Alan always had time for me — introducing me to the field, sharing interesting problems and his ideas, and teaching me how to write well and asking the right questions, skills that have stayed with me forever.”
D. Sivakumar was my student at the same time as Ashish, and began his own comment in Lance’s tribute by noting how Alan’s early work with Neil Jones on a logical characterization of nondeterministic exponential time was a precursor to Ron Fagin’s 1974 characterization of NP. He continued how at the end of his first year in Buffalo, “Alan encouraged us students to attend the Structures conference in Boston even though only one of us a theory-committed student. Alan drummed up some money to pay our registration, and we rented a car and stayed with friends in Boston. Structures’92 was magical for me—my first academic conference—everything about the atmosphere, the ideas, the people was fascinating and I decided to work in that field. All the work in my thesis (sparse sets, measure theory,…) were on topics I heard about for the first time at Structures’92—without Alan’s nudge, my life would likely have taken an entirely different trajectory. … Alan’s wit, dry humor, and his generosity are the things I’ll remember forever.”
Pavan Aduri was also Alan’s student in the 1990s and, speaking during a shiv’ah service with the family on Monday (via Zoom), noted Alan’s quest for simplicity in complexity theory. “Whenever I started to present a proof, Alan would ask, ‘Can you simplify the proof?’ ” Now at Iowa State, he recalled that he would frequently reach out to Alan for advice on various matters, even on what would be a more appropriate title for a research proposal. “Alan was always very generous with his time and advice.” Frederic Green related at the shiv’ah how Alan helped him migrate from physics to complexity and once reviewed every single step of four pages of notes on a complicated proof to optimize it for presentation. Steve Fenner told of Alan’s hosting him in Buffalo right after Steve’s PhD and putting him at ease with small talk about the permanently temporary quarters in Bell Hall. Steve Homer and his wife told of being travel partners exploring castles and flower markets, and the long gestation of the Homer-Selman textbook, Computability and Complexity Theory.
Harry Buhrman writes: “Alan has been an inspiration for me from the time I started working as a PhD student in complexity theory. My first encounter was his beautiful work on reductions and P-selective sets. I jokingly used to call them P-Selman-sets. I consider myself very lucky that he came over to my PhD defense in 1993 in Amsterdam. It was a joyous gathering.” I remember the photos Alan showed of that defence as well as Edith Hemaspaandra’s: grand pageantry on a scale I did not see even at Oxford, and led by Harry’s co-advisor Peter van Emde Boas, whose note in Lance’s tribute recalled the genesis of the “Structures” conferences in 1985–86 and the 1994 edition in Amsterdam and both hosting and visiting Alan and Sharon for dinners.
My retired colleague Mike Buckley wrote, “I taught with Alan at UB, and he taught me how to teach. Students sought him out until the day he left. He put as much into the undergrads as the PhDs. His presence at faculty meetings meant that no decision was made until he had his last comment. He was a giant in the field, but also the world’s nicest man. I kept in touch and we talked stereos and jazz. I miss him already.”
Arnold Rosenberg and Paul Spirakis hailed Alan as “a true pioneer and leader in the area of logic-based CS” and noted their long cooperation on the editorial board of the Theory of Computing Systems journal. Arnold continued: “He played vital roles in both research and education in CS. He approached every professional challenge with vision and unswerving devotion to the highest standards of our field. Our journal and our field have lost a great person.” Rod Downey included special thanks for Alan’s early support of parameterized complexity and the journal’s support, for instance in special issues.
Readers are welcome to place words of grace in comments; more may be added above as well.
Our embraces go to Sharon, to Jeffrey, to their daughter Heather—who also works in medicine—and to all the family.
[Cantor for the service was not Heather Wargo but Sandra Messinger Aguilar; some word fixes; linked license-plate app]
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