Gödel's Lost Letter and P=NP

The Lonely Runner Conjecture

KWRegan — Sun, 29 Jan 2012 04:58:28 +0000

Another elusive open problem

Jörg Wills is a mathematician who has done many interesting things in geometry and related areas. But he is probably best known for a paper he wrote in 1967, “Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen.” In this he created a conjecture, that was later independently discovered by Thomas Cusick, and finally named by Luis Goddyn as the lonely runner conjecture.

Today I (Dick) want to talk about this elusive conjecture that is another disease along with graph or group isomorphism and all the others and others.

So beware: read on only if you are prepared to spent hours thinking about a problem that is simple to state, has been open for forty-five years, and has resisted the efforts of some of the top researchers in the world. You have been warned.

The Problem

Actually unlike some of the other diseases the lonely runner conjecture (LRC) is really an important question in Diophantine approximation theory. Unlike many diseases it has nontrivial consequences and is related to important open questions in, for example, graph theory.

However, the best way to state the LRC is probably with the beautiful language of runners on a circular track rather than fractional parts of numbers. There are runners going around a circular track of unit length, and they are very steady runners: they each have their own distinct paces. They all start together from the same starting spot and continue running at their own unique pace forever. Unlike real runners they never vary their speeds, they never get tired, and they never interfere with each other—even as faster runners pass slower runners.

The question is, will there be a time when no runner will be near the start? More precisely, when each runner will be distance at least from the start?

The reason this is called the “lonely runner conjecture” is that we can imagine that there really are runners, but one runner has pace zero—that’s me. So I stay at the start and become lonely when no one else is near me, either behind or in front. If you think I should try walking, just subtract my walking speed from everyone else’s and we’re back in the case where I stand still. If you get Patrick Makau to run instead, then subtracting his speed is the same as using me and having the others run in the opposite direction. Thus the originally stated conjecture that each of runners gets “lonely” at some time is equivalent to runners and just me being lonely.

Results

There are two types of partial results that are known. There are results that show that the problem holds for small values of . The best of these is , I believe. Again because I’m in every race, the paper’s title refers to seven “runners.” The other results treat the general case of runners but restrict the speeds of the runners. For example, if the runners initially select random paces, then it is possible to prove very strong separation results. Other restrictions assume that the paces of the runners are sufficiently different, i.e. that the ratios of paces are relatively large.

The proofs of the conjecture even for small values of can be quite intricate. If you like hard case-analysis proofs, then these are for you. Take a look. It is hard to imagine how difficult these proofs were to find, and how difficult they are to check for correctness—especially when computer searches are employed.

In order to give a bit of the flavor of the complex case analysis used, here is a diagram from the paper by Tom Bohman, Dan Kleitman, and Ron Holzman that proved the case :

An Approach?

I must admit to have fallen a bit for this disease, but I cannot imagine trying to solve by even more complex case analysis. That is beyond what I can do. But it did occur to me that there might be a different approach to the problem. My approach does not work yet, and may never work. But it could yield some new insights.

The idea is simple: apply the probabilistic method to the problem. Let’s start with a simple lemma:

Lemma 1 For any collection of runners with non-zero paces that need not be distinct, and any interval that includes the starting point, there is at least one time so that at most

runners are in the interval .

Here is the length of the interval. Before we prove the lemma let me give two interesting statements that follow.

Corollary 2 For any and any runners with distinct non-zero paces, there is at least one time so that at most one runner is within of the starting point.

Proof: The corollary follows by letting be the interval . The lemma shows there is a time so that

runners are in . But this quantity is .

Another corollary is this:

Corollary 3 For any and any runners, fix a runner . Then there is at least one time so that at most one runner is within of the starting point.

Note that the interval is slightly smaller than before: the denominator has changed from to . But now we can “control” slightly which runner is allowed to be near the start.

Proof: The proof of the lemma is a simple probabilistic method argument. I cannot find a reference for it, but it must be well-known. In any case let’s go over the proof for completeness.

Define to be the indicator random variable so that it is when the runner is in the interval for a random time . I claim that

for all . The claim follows by noting that as time varies uniformly so does the runner’s position.

Now consider the summation

This is the number of runners that are in the interval at a random time . But the expectation of is . So there must be some time when , since is an integer valued random variable.

Building the Argument

Of course, having runner near the start is not loneliness for me. But we can ask, what about the times when this runner enters, or leaves?

If it is not the case that two other runners were in the interval while he was, then I really was lonely at one of those times. One of those two must have been in the interval while he entered, exiting while he was still in the middle, and the other must have entered before he left. Since the probabilistic argument not only shows there exists a time but gives a set of times with positive measure, we can isolate a block of time when the runner was alone.

Can we apply probabilistic arguments in a compound manner to find times when there are not three runners in this kind proximity? Alas this may still require breaking into cases according to their speed relative to the single runner. One idea Ken and I have explored is cases where some slower runners can be paired with ones moving at (or about?) twice the speed, and using this form of the union bound:

We think this shows loneliness in an interval of width about the origin for in case the speeds are in ratio . We don’t know which such special cases have been analyzed—we wonder if this is a good subject for a computer repository or PolyMath project.

Open Problems

My suggestion is to move on and forget this problem. Yet it seems like the probabilistic idea might be improved just a bit to get the existence of a time when no runners are near the start. Can we use more clever probabilistic ideas? What about higher moments? or using the Local Lemma, or…?

Wait—I see an idea…

Ode To The Math Monthly

rjlipton — Tue, 24 Jan 2012 12:45:54 +0000

Some fun results on matrices

Olga Taussky-Todd was one of the leading experts on all things related to matrix and linear algebra in the middle and late 1900′s. She was born in the Austro-Hungarian Empire in what is now the Czech Republic, and obtained her doctorate in Vienna in 1930. She attended the Vienna Circle while fellow student Kurt Gödel was proving his greatest results, and recalled that Gödel was very much in demand for help with mathematical problems of all kinds. She left Austria in 1934, worked a year at Bryn Mawr near Philadelphia, then held appointments at the universities of Cambridge and London until after World War II, when she and her husband John Todd emigrated to America.

Today I want to present a couple of simple, but very cool results about matrices.

Taussky-Todd once said in the American Mathematical Monthly—from now on the Monthly:

I did not look for matrix theory. It somehow looked for me.

That is to say, her doctorate was on algebraic number theory, and then she progressed to functional analysis. Heading in the direction of continuous mathematics was the available path. According to these biographical notes, the field of matrix theory did not really exist at the time. The notes hint that matrix theory was too light to be a main subject for graduate education unto itself, so perhaps the Monthly was a needed vehicle to help to launch it.

Matrix and Monthly

One of Taussky-Todd’s great papers is “A recurring theorem in determinants,” which proved a variety of simple, but fundamental, theorems about matrices. It appeared, as did many of her papers, in the Monthly. One of these theorems is a famous non-singularity condition:

Theorem: Let be a complex matrix, and let stand for the sum of the absolute values of the non-diagonal elements in row , namely

If is diagonally dominant, meaning.
A_{i}, \ \ i=1,\dots,n, ' title='\displaystyle |a_{ii}| > A_{i}, \ \ i=1,\dots,n, ' class='latex' />

then

I recall as a student “meeting” Taussky-Todd’s work in a book on algebraic number theory. Somehow many of the results in that area could be reconstructed as theorems on matrices, and the resulting proofs sometimes were much more transparent.

Ivars Peterson has a nice discussion of her work here. The same biographical notes referenced above have this revealing passage:

Olga Taussky always wished to ease the way of younger women in mathematics, and was sorry not to have more to work with. She said so, and she showed it in her life. Marjorie Senechal recalls giving a paper at an AMS meeting for the first time in 1962, and feeling quite alone and far from home. Olga turned the whole experience into a pleasant one by coming up to Marjorie, all smiles introducing herself, and saying, “It’s so nice to have another woman here! Welcome to mathematics!”

Let’s look at two simple but I believe interesting results about matrices. One is from the Monthly, and perhaps Olga would appreciate them.

A Matrix Result

Consider the two matrices and over the integers:

The question is it possible for a sequence of ‘s and ‘s to equal another distinct sequence of ‘s and ‘s? For example, is:

The answer is , and see the next paragraph for the cool proof. One way to think about this is that the two matrices generate the free semigroup. There are pairs of matrices that generate the free group, but the proof that they do that is harder, in my opinion. I once used that the free group is generated by matrices to solve an open problem—see here. What is cool is that the proof is quite unexpected.

The key is to look at the action of the matrices on positive vectors: the vector is positive provided

where 0}' title='{x > 0}' class='latex' /> and 0}' title='{y > 0}' class='latex' />. Note that both and map positive vectors to positive vectors.

Also define to be those vectors whose first coordinate is strictly larger than its second and define to be those whose second coordinate is strictly larger than its first. Thus,

Let and be distinct sequences of the matrices and that are equal: we plan to show that . If they both start with the same matrix, since the matrices are invertible, we can find shorter such sequences. Thus, we can assume that and for some and . Let be any positive vector. Define

It follows that both vectors and are positive. But we note that is in and is in , which is impossible since by assumption.

This neat argument is due to Reiner Martin answering a question in the Monthly—the question was raised by Christopher Hillar and Lionel Levine.

Another Matrix Result

There are many normal forms for matrices of all kinds. I recently ran into the following question, which was quickly solved by Mikael de la Salle. Let be an matrix. Prove that there is a 0}' title='{\lambda>0}' class='latex' /> and two unitary matrices and so that

This says that any matrix, up to scaling, is the average of two unitary matrices. It seems this should be a useful fact, but I have not applied it yet.

We can find unitary matrices and and a real diagonal matrix so that

This is the famous Singular Value Decomposition. We can assume that the values on the diagonal are all at most in absolute value, by using to re-scale if needed. The key is that

where each and are unitary. This insight is based on the fact that if for a real , then there is a real so that

where has absolute value . This follows since there is a real so that . We can now use this term-by-term on the diagonal of to construct the diagonal matrices and . Then it follows that

Open Problems

Which matrices are averages—or sums—-of some given fixed number of unitary matrices? With , that is. Is there a good way to characterize them?

Do you have your own favorite matrix results that have a simple proof, but may not be well known to all? If so please share them with all of us.

Finally, I would suggest that you read the Monthly regularly, since it is filled with gems.

Is it Time to Declare Victory in Counting Complexity?

rjlipton — Fri, 20 Jan 2012 15:24:10 +0000

The last dichotomy theorem: graph homomorphism with complex values

Jin-Yi Cai is a great theorist, a world expert on Dichotomy Theorems, and a best friend. We have talked about him before here for his work on these types of theorems, and more recently here.

Today is a new type of post for us—a guest post, by Jin-Yi. We have used his suggested post title, but have inserted a couple sections for added background.

Well we always do things differently here at GLL—as Tina Turner says:

You see we never ever do nothing
Nice and easy
We always do it nice and rough.

We have several guests lined up for the new year to write posts, but as always we will add a bit of extra flavor and a few comments—Ken and I hope that they will understand.

What Is A Dichotomy Theorem?

One of the central questions of complexity theory is: given a computational problem, can we tell whether it is easy or hard? For example, is it in polynomial time or not? In full generality this meta-problem of classifying problems is hopeless—beyond hopeless, if that makes sense. How could we possibly take an arbitrary problem and classify it?

Yet we would like to do this automatically and for as large a class of problems as possible. One method is to search the Web and see if it has been studied before; a related one is to use the famous Garey and Johnson book. But that only goes so far.

The central idea of a dichotomy theorem is two-fold, pun intended. It is to restrict both the family of problems and the notions of complexity. For every problem of a restricted kind, it says that either the problem is easy or it is definitively hard.

Definitively hard can mean many things, from -complete, to -hard, to -hard. The power of such a theorem is determined by the “size” of the family of problems—the larger the family, the better the theorem. Eliminating any in-between possibility of hardness is why it is called a dichotomy theorem, since, of course, the meaning of “dichotomy” is the splitting of a whole into two pieces.

A Failed Dichotomy?

Before the guest post, let’s talkabout the famous question that Emil Post raised in the early days of computability theory, in the 1940′s. Following Alan Turing’s seminal result that the Halting Problem is undecidable, people quickly undertook to classify problems as computable not. A problem—or if you prefer a set—is called computably enumerable (c.e.) if it is possible to list its values by a Turing machine. The old style that I learned was to call these sets r.e., but that name is out of date. So use “c.e.” if you want to seem to be au courant, especially if you are talking to logicians.

As people began to classify problems it quickly appeared that every one was either computable or equivalent to the Halting Problem. Somehow they could find nothing in-between. Post asked whether there were problems, natural or not, that were in-between. Essentially he asked: is there a dichotomy theorem for c.e. sets?

This great question led to a flurry of ideas and results, but no solution. Post and others defined a variety of types of sets: simple, hypersimple, hyperhypersimple, and others that attempted to solve his problem. (These are real names—I do not know if there was a hyperhyperhypersimple notion.) The idea was this: suppose that you could define a property of sets so that:

If a set satisfies , then it cannot be equivalent to the Halting Problem.
There are undecidable c.e. sets that satisfy .

Then there would be sets between computable and complete—the dichotomy theorem for c.e. sets would fail. None of these notions worked, since the language they used for defining the property X was not powerful enough. Eventually, in a brilliant argument, at about the same time in the 1950′s, and independently, Richard Friedberg and Albert Muchnik proved that such intermediate sets do exist. The used a method, now called the priority method, to construct the sets. Therefore the dichotomy theorem failed for c.e. sets.

The issue then becomes whether properties that distinguish the priority-constructed sets are “natural.” There is some analogy to the famous theorem by Richard Ladner which we also referenced alongside Jin-Yi earlier, insofar as it produces sets that are “gappy.” Ladner’s technique can be adapted to many cases to break up possible dichotomies, but always with intuitively “unnatural” sets. In any event it’s not clear how much generality one has to sacrifice to get a dichotomy—and the point is that one always has to sacrifice some.

Let’s now turn to Cai’s post on the current status of dichotomy theorems in complexity theory. We have edited it somewhat, and it refers to Jin-Yi in third person in some places to say we concur with some opinions.

The Guest Post

If you care about classification theorems in complexity theory, especially for the complexity theory of counting problems, or the so-called Sum-of-Product type problems (a.k.a. partition functions), you may want to know about the recent result of Jin-Yi and Xi Chen. Even if you do not care, you may still wish to know about this paper.

The paper has a plain title: Complexity of Counting CSP with Complex Weights. But what the paper has done can be the conclusion of an era for the theory of counting problems.

Or is it?

Let me (Jin-Yi) first give you some historical context, starting from the days of Kurt Gödel and Alan Turing and Emil Post.

Complexity theory—in particular Steve Cook’s NP-completeness—essentially springs from a direct adaptation of Post’s ideas with a polynomial time restriction. Post’s problem was so intriguing because most “natural problems” seem to be either decidable or as hard as the Halting Problem. In fact, it was said that one of the psychological barrier to proving Hilbert’s 10th Problem undecidable was the fear that it might be of intermediate computability, and no one knew then how to prove a natural problem undecidable unless it is as hard as the Halting Problem. Of course ultimately Hilbert’s 10th Problem was proved equivalent to the Halting problem, by the combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson, with Matiyasevich providing the last crucial step.

Back in Complexity Theory, the story is different. The analogue of decidable sets is P, and the analogue of c.e. sets is NP. We still can’t prove NP P, whose analogue in Computability Theory is Turing’s undecidability of the Halting Problem. However, assuming NP P, Richard Ladner did show that there are problems of intermediate complexity in NP. They are neither in P nor NP-complete. However, there is one thing in common with Computability Theory: No “natural problem” in NP has been shown to be of intermediate complexity, even assuming NP P. In fact, most “natural problems” in NP seem to be either in P or NP-complete (some exceptions include Graph Isomorphism and Factoring, but there are no proofs.)

Dichotomy for CSPs

Giving this belief a precise form, Tomás Feder and Moshe Vardi formulated a CSP dichotomy conjecture. Here CSP stands for Constraint Satisfaction Problem. This is an extremely broad class of problems expressible as follows: Let be an arbitrary finite set called the domain set. Let be an arbitrary finite set of relations on . Each has an arity . Then denotes the following class of problems: The input is a set of variables over , and a collection of constraints , each applied to a sequence of variables. The question is, is there an assignment that satisfies all the constraints. Feder and Vardi conjectured that for every , is either in P or NP-complete. This is still open.

The Feder-Vardi Conjecture is about decision problems. There are two other versions of Constraint Satisfaction Problems. One is the optimization version: What is the maximum number of constraints that can be satisfied in a particular instance? The other is the counting version: How many satisfying assignments are there? It is in this counting version we wish to report a recent decisive victory.

One can express counting CSP problems as follows: Identify each with a 0-1 valued function ; then the conjunction of a collection of constraints can be replaced by their product. Then a counting CSP problem is to evaluate the following so-called partition function on an input instance ,

where has arity and is applied to variables . Of course if is 0-1 valued, this counts the number of solutions.

General Notions

In this Sum-of-Product form, there is a vast generalization where each function need not be 0-1 valued. This heritage comes not from logic, but from statistical physics and elsewhere.

In many physical systems, global behavior can be determined by local interactions, which are naturally expressed in this Sum-of-Product form. Included in this class are the Ising Model, the Potts Model, and many others. There is a long tradition in statistical physics to classify the so-called Exactly Solved Models, as expounded in this book by Rodney Baxter.

In computer science language this translates to polynomial time computable. Physicists did not have a rigorous notion of what is not Exactly Solvable; the proper notion in computer science is #P-hardness.

For the whole class #P, Ladner’s theorem still holds: If #P P, then there are counting problems of intermediate complexity between P and being #P-hard. However no natural counting problems are known to have this intermediate complexity between P and #P-hardness. Again the general belief is that all natural counting problems are either in P or #P-hard.

A succession of beautiful papers can be seen in this light, culminating in the latest Cai-Chen dichotomy theorem. These papers give ever stronger indications to the scope of this general belief.

Graph Homomorphisms: A Natural Case

Over the years, a great deal of effort has been focused on a special case of CSPs: Graph Homomorphisms. This is the case when there is a single binary constraint function . For undirected graphs, as is the basic case, this is a symmetric binary function on and is usually represented as a matrix. This version of the problem was first proposed by Laszlo Lovász in 1967: Given a symmetric matrix , compute

where is any undirected graph, with vertices representing variables, and edges representing constraints.

Let’s list some highlights of this substantial body of work:

For graph homomorphisms, first Martin Dyer and Catherine Greenhill proved (note a complexity dichotomy: For any symmetric 0-1 matrix , is either in P or #P-complete, depending explicitly on . Then Andrei Bulatov and Martin Grohe extended this dichotomy to all non-negative matrices. The next major step was a beautiful paper by Leslie Goldberg, Grohe, Mark Jerrum and Marc Thurley, extending this dichotomy to all real matrices. When constraint functions can take both positive and negative values, it becomes significantly more difficult to prove complexity dichotomies. This is because there is the possibility of cancellation, which makes life much more difficult to analyze. Compare this situation with the permanent versus the determinant, where cancellation actually can make a problem easier. The recent work on holographic algorithms is also a good reminder of this. The situation is somewhat analogous to monotone versus non-monotone circuit complexity.

The final step for was a long and intricate paper by Cai, Chen and Pinyan Lu, which extended the dichotomy to the full generality of all complex-valued matrices. This blog has discussed this paper of over 100 pages before here. The upshot is that they gave a complete classification of to be either in P or #P-hard, with an explicit criterion on .

For directed graph homomorphisms, the matrix is not necessarily symmetric. In this case, the strongest previous result is another paper by Cai and Chen, giving a dichotomy of with a decidable criterion on all non-negative . However, this criterion is more opaque. Before that, Dyer, Goldberg and Mike Paterson did prove a dichotomy for more restricted directed graphs, with a more transparent tractability criterion, called Lovász-goodness. The drawback is that it only applies to special 0-1 matrices that as adjacency matrices define acyclic graphs.

A major accomplishment was a dichotomy by Bulatov, who showed that for every finite set of relations (i.e., 0-1 valued functions) the counting CSP problem is either in P or #P-complete. Bulatov is a master of applying techiques from Universal Algebra to computer science, and this proof uses deep results such as commutator theory and tame congruence theory from Universal Algebra. A major open problem of Bulatov’s theorem is the decidability of the criterion for tractability. This was solved by Dyer and David Richerby, who gave an alternative proof to Bulatov’s dichotomy theorem, which uses very little Universal Algebra. Using a technique of Lovász, they showed that their dichotomy criterion for tractability is decidable. However, even though it is decidable, compared to the Cai-Chen-Lu criterion for Graph Homomorphism, it is still not as explicit.

The Unified Dichotomy Theorem

Now all these dichotomy theorems have become special cases of the following theorem by Cai and Chen.

Theorem: Given any finite and any finite set of constraint functions , where each maps (for some ) to the complex numbers, the problem of computing the partition function is either computable in P or #P-hard.

The tractability criterion for this theorem is explicitly given. However it is not known whether the criterion is decidable. The proof uses ideas from both the Dyer-Richerby paper and the Cai-Chen-Lu paper.

More than the technical issue of decidability of the criterion, there is a very different taste to this dichotomy theorem, compared to the Cai-Chen-Lu paper on Graph Homomorphism. There is a sense in which the Cai-Chen-Lu paper on Graph Homomorphism really lets you understand what makes a problem tractable, and what makes it #P-hard. The criterion is delicate, but roughly speaking, in order to be computable in polynomial time, the matrix must be a rank-one modification of tensor products of Fourier matrices; otherwise it is #P-hard. By contrast, the criterion for the broader CSP dichotomy of Cai and Chen is infinitary: It defines certain infinitary objects from the finite set , and imposes some conditions on them.

One can caricature the situation as follows: In the Cai-Chen-Lu dichotomy, they conquered the land with meticulous care. They catalogued all the flowers and inspected all the caves. In the latest CSP dichotomy, they claimed victory by having encircled the mountain; the land is logically conquered, but one does not really know what treasures lie within.

Open Problems

So is it time to declare victory? You be the judge.

One final comment, from Dick: One simple way to explain the reason for all the above beautiful work shooing that problems are either easy or hard is that . If these complexity classes did collapse that would of course explain away all the dichotomy theorems. The reason there is nothing “natural” in between is that there is nothing there. Just a final thought.

It’s Still the Slime Mold Story

rjlipton — Wed, 18 Jan 2012 00:49:38 +0000

Short paths for love and glory as time goes by

Kurt Mehlhorn is a theorist who has made many contributions to almost all aspects of theory. A recurrent theme is the balance of beautiful theory with potential practical applications—most of his work has some of each. He has made contributions to: data structures, computational geometry, parallel computing, VLSI design, computational complexity, combinatorial optimization, and graph algorithms. He is famous beyond the theory community for his work on LEDA, the Library of Efficient Data types and Algorithms. And he has also been the director of the Max Planck Institute for Computer Science.

Today Ken and I want to talk about yet another paper on slime mold—when will these papers stop?

Just kidding. The paper in question is one of the first to study the general computational power of a slime mold, namely Physarum polycephalum. The paper is joint with Vincenzo Bonifaci and Girish Varma, and is about to appear at this winter’s SODA conference in Japan. While there are many cool papers at SODA 2012—change that to all papers at SODA are cool— this appears to be the only one with the name of a particular type of slime mold in its title.

As a nod to Wikipedia’s “blackout”, here is a picture of black slime mold, along with one of slime mold finding shortest paths in a maze:

I have known Kurt for many years, and have always been a great supporter of his and an admirer of his beautiful work. One of the surprising issues that arose early in the LEDA project was the issue of accuracy. I have talked about it before here: the short story is that even simple computational geometry algorithms can fail in practice, if one is not careful with the issue of finite precision. Quite surprising.

Let’s turn to the computational power of a class of creatures that are called slime molds—a wonderful name.

Slime Mold

When I worked on DNA computing in the late 1990′s, I talked often to biologists. One of the hot areas of research then, and I believe still, is the study of slime molds. I have to admit as a computer scientist that the name of the area did not exactly get me excited, but slime molds are important. There are entire conferences devoted to this single subject, for example.

You may ask why are they so important? A good question. I believe there are several reasons that biologists continue to be so intrigued with slime molds. One is a pragmatic reason: slime molds are easy to work with in the laboratory. They are not hard to grow, to manipulate, or to pay for—mice, for example, are several orders of magnitude more expensive to experiment with. Another is that slime molds have behavior that is more complex than one might imagine, especially given their simple biological structure. Here is a quote on this:

There are few creatures more remarkable than the lowly slime mold.

But is the slime mold really a creature? Or is it a fungus? If it’s not a fungus, is it a single entity—or millions? Or both?

Animal, Vegetable, or Model?

Another way to regard their complexity is the Pluto effect. Recall Pluto was once considered a planet and now is no longer one. When scientists change their minds about what something is, that suggests to me that something complex must be involved. Slime molds were formerly classified as fungi, but are no longer considered part of this kingdom.

Perhaps the most important reason is they exhibit interesting behavior. For example, if you physically divide a mass of slime mold into two pieces on a dish they generally find they way back together. One of the world experts on them, John Bonner, states that they are:

no more than a bag of amoebae encased in a thin slime sheath, yet they manage to have various behaviors that are equal to those of animals who possess muscles and nerves with ganglia — that is, simple brains.

This is pretty neat. As a computer scientist it sounds like he is saying: here are very simple little computational agents and yet as an ensemble they can “compute” quite non-trivial things. This is exactly the types of problems we love: a simple computational device that exhibits complex behavior. Even more relevant to today’s highly distributed and networked world is the paradox: how can simple local rules afford interesting complex computational behavior? This is where slime molds come in to the picture: indeed here is a picture of one.

Luckily we do not have to look at them too much, or even worse grow them, but we can build a simple mathematical model that appears to give good approximations to their behavior, and thereby try to figure out how they work. This is exactly what Kurt and his co-authors have been able to do, in greater generality than previously established.

The Big Picture

One of the driving forces behind the new computational interest in slime molds is a seminal experiment performed by Atsushi Tero of Hokkaido University. He grew the mold Physarum polycephalum on a standard dish, but placed both attractors and obstacles on the dish. The attractors were food—oat flakes if you must know—and the obstacles were bright light. The mold is attracted to food—who is not?—but is camera shy and tries to avoid bright light. Tero for fun arranged the attractors and obstacles to model the major centers in the Greater Tokyo Area. The mold initially filled the whole dish, but over time evolved into a network that connected the centers in a way that closely approximated the actual Tokyo rail system.

Tero introduced a model that claimed to represent the behavior well. Roughly the model was a time varying electrical network. The next major result was by Tomoyuki Miyaji and Isamu Ohnishi. They proved that Tero’s model converges to an optimal solution in any instance of a planar graph.

What this means is that the slime mold units that are initially on non-optimal edges in the graph eventually gain information about this fact and move off them. In the model this is analogized to “taking the path of least resistance” in an electrical network. Here, however, the network itself is adaptive: the resistance an edge changes with time in connection with the organisms’ movements as well as the fixed cost of the edge itself. Eventually edges that are not part of any optimal solution see their resistance grow unboundedly. The nodes in the graph have derived potentials, and these regulate the flow of current which corresponds to how the organisms congregate.

The big picture is that millions of years of evolution can generate organisms responding to local properties (like “potential”) that still collectively solve global computational problems. We may learn from Nature how best to solve non-trivial problems by cheap local means.

Sketch of Model and Result

As in Tero’s original model, every edge in the instance graph has a positive length cost and a time-dependent state called “diameter.” The resistance at time is . Every node has a potential at time . Current is modeled as a flow in the graph from a source node to a sink node . The conservation of flow dictates that for every node other than or ,

The current technically has a sign according to whatever orientation is given to the edge , but this does not matter to the fundamental dynamical equation, which depends only on the volume of the current:

The magnitudes are normalized by stipulating an overall flow of from to , meaning is set to and is set to . Since these relations are invariant under an additive constant on the potentials, we can finally fix for all . Then the remaining potentials can be computed at any timestep by solving linear equations in terms of the and the fixed values of . This yields values and hence computes the derivative of for each edge , which then tells how to update the state for the next timestep.

At equilibrium the edge derivatives are all zero, so the normalized current in an edge is equal to the diameter, which in turn models how the slime mold is congregating in edge . The question is whether bad edges get zero diameter. Let be the subgraph of formed by all edges that belong to some minimum-cost path from to , and let denote the collection of all normalized flows just through . If the min-cost path is unique then is just a path and is its flow, but otherwise may have flows that combine two or more optimal paths.

Thus far this is all Tero. The achievement of Kurt and his co-authors is to remove the planarity restriction on , with the slight caveat that in the non-unique case they don’t quite prove pointwise convergence to an optimal flow, but only attraction to meaning that for any 0}' title='{p > 0}' class='latex' />, the norm between the vector of values and some member of (which could differ for different ) goes to zero. Their main theorem is:

Theorem 1 For any graph as above, the dynamical system is attracted to . If is a unique path then the dynamics converge pointwise to the flow of value along that path.

The main innovation in the proof is finding the best Lyapunov function to impose on the system while attempting to bound it over all iterations. This was actually done experimentally by regarding various instance graphs as parallel networks. This enabled them to handle arbitrary graphs. As usual the considerable details are in the paper.

I like the fact that this work requires the analysis of a dynamical system. Theory is currently great at many things, but the understanding of dynamic or iterative methods is not one of them. Also we see—is this allowed?—the use of continuous differential equations and related methods in a discrete problem.

They further show that the same dynamic electrical system can be used to solve the transportation problem, which is a generalization of the shortest path problem. This leads us to ask, what else can be done this way?

Open Problems

We quote Kurt:

The Physarum computation is fully distributed; node potentials depend only on the potentials of the neighbors, currents are determined by potential differences of edge endpoints, and the update rule for edge diameters is local. Can the Physarum computation be used as the basis for an efficient distributed shortest path algorithm?

Indeed can these or related local rules be used to solve other problems? For instance, can they be upgraded to evaluate networks of Boolean gates, at least heuristically? Note that the circuit value problem is -complete, so this would be a hard step for any parallel model.

[inserted a missing "(t)", added slime mold photos to intro]

What Makes a Knot Knotty?

KWRegan — Fri, 13 Jan 2012 21:24:51 +0000

Can it be just the not-ation?

Colin Rourke is a topologist at the University of Warwick in England. He is one of numerous mathematicians whose work is surveyed in an entertaining article by Sam Nelson of the Claremont Colleges in California, “The Combinatorial Revolution in Knot Theory” which appeared in last month’s AMS Notices. Rourke’s home page has interesting materials and notes besides a selection of his papers. These include a one-page formulation of Gödel’s first incompleteness theorem and a model for the universe that “overturns nearly every tenet of current cosmology.”

Today Dick and I wish to draw attention to the mathematical structures in Nelson’s article, whose properties have been developed by Rourke and others, and to how syntax may take precedence over the ‘real physical meaning’ of a mathematical concept.

Nelson summarizes the “revolution” in his opening paragraph as follows:

Instead of thinking of knots…geometrically as simple closed curves in , a new approach defines knots combinatorially as equivalence classes of knot diagrams under…certain diagrammatic moves.

Much as the concept of number needed to move from ‘real’ to ‘imaginary’ in order to yield the fundamental theorem of algebra, so are the more general not-ation based not-ions of knots needed to fill out invariants of certain mathematical structures.

Rourke also made an important advance on the Poincaré Conjecture in 1994, about a decade before it was completed to a theorem by Grigory Perelman. He proved that there was an algorithmic procedure that would generate a counterexample to the conjecture if it were false. In logical terms, he proved the negation of the conjecture equivalent to a , i.e. existential, statement of arithmetic. Of course Perelman’s proof obviated this theorem. However, the idea matters to us because is only known to be a statement. Proving it equivalent to a or statement, so that at least one of the ‘equal’ or ‘not equal’ cases would be provable, would be the greatest development in decades.

Knots Into Codes

According to a multiple-sourced legend, Alexander the Great was once challenged to untie a complicated knot that had tied the ox-cart of a peasant-king named Gordias for many years. Alexander either used his sword or pulled out the rod of the cart by which it was tied, thus creating or exposing ends of the string by which to undo it. Pulling out an end of a string to unravel a knot may not yield any useful mathematics, however.

Alexander could have first fluffed the knot so that no more than two segments of string would cross each other at any point. Then the knot’s shadow yields a planar graph of degree 4 in which each vertex has two “over” edges and two “under” edges, and this is the knot’s diagram. If the knot had eleven crossings, the diagram might look like this:

If we trace the diagram beginning at the arrow, assign a number for each crossing in the order encountered, and use a minus sign when traversing an “under” edge, we can abstract the knot into the sequence of numbers

Of course this code is named after Gauss. There are several variations on it. We adopt the “signed Gauss code” in one of Nelson’s attractive knot-theory pages, but using to mean “under” and suggest complex conjugation—in line with a footnote in his survey.

To define it formally, choose as the principal eighth root of unity. The crossing elements are now subscripted copies of a power of . If you are following the “over” edge with the “under” edge directed right-to-left, take ; if left-to-right, take . If you are following the “under” edge with the “over” edge going right-to-left use ; if left-to-right then by . If the second crossing has this last case we can write as short for . The code always has a where the original Gauss code had a sign. The above knot gives:

Flipping the knot over conjugates every element, while any reflection (right to left or top to bottom) negates every one. Reversing every edge reverses the sequence but does not change any or . As with the “extended Gauss code” defined in our source for this diagram, this code preserves enough information to reconstruct the knot from the diagram.

The code yields the trefoil knot. Note that in any code, each element appears once with a and once without. Thus the codes are some kind of generalized commutator.

Diagrammer Move Grammar

Suppose the modified Gauss code has the sub-sequence

for some crossing element . This means we’ve drawn a loop over or under in he direction of travel, but the loop isn’t wound around anything. So it can just be pulled taut. Since the crossing elements are complex units, we already have . Note that since the codes are really cycles, and are the same cases.

Now suppose instead that the modified Gauss code has

where and have the same sign, and the subscripts on and differ by . In words this means we have two consecutive over-crossings of the same other segment, or two under-crossings of it. In either case the two segments are merely sitting one atop the other and are not linked. Again these sub-sequences can be erased.

To describe the third move, suppose we originally had 3 segments cross at one point: a lower strand going northwest–southeast, a middle strand going southwest–northeast, and a top strand going east–west. When Alexander fluffs the knot, the top segment can either bend up or bend down. The two resulting equivalent configurations are:

Suppose the topmost segment is first traversed west-to-east, next the bottom one NW-to-SE, and last the middle one SW-to-NE. The corresponding rule for the modified Gauss code is:

There are similar rules for the other arrow directions and orders of visiting the segments. The diagram comes from this article in John Armstrong’s blog “The Unapologetic Mathematician” which we keep in our sidebar, and shows colorings associated to diagrams.

How Knotty is the Not-Knot?

Nelson’s article shows the moving strand in the middle rather than on top. This doesn’t matter—if we were to move the SW-NE strand in the left diagram northwest, we would still get the right diagram. This neat theorem was proved in 1923 by Kurt Reidemeister:

Theorem 1 A knot diagram represents the unknot if and only if all of its crossings can be eliminated by some sequence of the above moves—equivalently, iff its (modified) Gauss code is reducible to 1.

Here is a “Gordian” example. This echoes Hilbert’s Nullstellensatz saying that a system of polynomial equations over has no solution if and only if 1 can be derived from it by Gröbner basis moves.

As shown by Dick’s posts on work with Zeke Zalcstein, we are interested in complexity properties of these kinds of rewriting systems. In particular, the problem of whether a diagram represents the unknot is in , and is in if the Generalized Riemann Hypothesis is true, but has not been proved to be in . GRH is needed only to get good bounds on primes such that certain integer polynomials associated to the knots have roots in .

To Knot or Not to Knot, or N-Knot?

The point of departure in Nelson’s survey is that the rewriting theorems apply even to codes that are not from shadows of “real” knots in 3-space. The simplest such code is the commutator

Any shadow that includes this pattern must have at least one more crossing. However, a diagram giving this code can be drawn on the torus, looping from under and around to join the line into .

His survey goes into a concept of virtual crossings and virtual knots introduced in this 1999 paper by Louis H. Kauffman. There is no Gauss code for the virtual crossing itself, though we have marked it in this diagram:

The point is that every (modified) Gauss code—subject only to the rule of every element and its conjugate appearing once each—has a planar diagram with virtual crossings. This allows spatial intuition and properties about crossings to be applied to all codes.

In passing we ask whether there is any utility in considering in the diagram to be a “nondeterministic crossing” instead—with either strand possibly being the “over” one. This could be coded as

Note that if the SW-to-NE strand in crossing is “over,” then we get the trefoil knot. If “under,” then it becomes which the second Reidemeister move applied to transforms to . (One could also use the move for since it is a cycle.)

Another way to think of an “N-Knot” is that for any element labeled nondeterministic, it is legal to swap and . Is it harder to decide whether an N-Knot can be reduced to than for a regular (or virtual) knot? This is different from computing the unknotting number insofar as only some crossings are nondeterministic. It strikes us that this problem ought to be -complete, which would make it unlikely that an efficient invariant can be found to prove that a given non-trivial N-knot does not reduce to .

Not Knot-Code Codes

It would be nice to avoid the non-local coupling between the two times a crossing is visited. Therefore let us try to abstract the crossing itself. Or, let us abstract the notion of a segment as it goes through one or more crossings. Define to mean that goes under at a crossing. Optionally we can specify to mean that the crossing is positive in our signed Gauss code, and use when it is negative. The three Reidemeister moves then yield these axioms:

That and the last equation with in place of everywhere follow from this. When the distinction between and is ignorable, these define a structure called a kei, else they define a quandle. These rewriting systems tell some already-familar stories.

Keis and Quandles and Racks, Oh My

Here are two examples that distinguish a kei from a quandle. Consider any group, such as one of nonsingular matrices. Define:

The first operation gives a kei because and

The latter two operations give a quandle and not a kei because both are needed to cancel the ‘s, namely

while

We can get the same examples with unitary matrices and for inverses, and this is just one hint of why knots and keis and quandles are relevant to quantum mechanics.

If a quandle is “excused” from obeying the first axiom, then it becomes a rack. Our last example, from Nelson’s survey and quandle page, shows all three. Consider any algebraic structure that admits left-multiplication by objects and , such as a module. For define:

The middle two operations define the Alexander quandle, which needs no modulus, while the first defines only a rack because makes no assumption on .

David Joyce showed in his dissertation in 1980 that two oriented knots are related by a homeomorphism of the 3-sphere if and only if the quandles they generate are isomorphic. Racks go with a further loosening of the idea of knots, and working with Roger Fenn, Colin Rourke proved a similar theorem. Nelson’s survey closes with links to further applications. We are interested in what computational complexity may say about them.

Open Problems

Can we extend the applications of these knot-derived notions?

Is the set of (codes for) unknots in ? Does membership in need GRH?

Are there coding schemes that are both smarter in helping prove theorems and simpler in complexity?

[changed way signs were described in second move; inserted "negation of the" before Poincaré conjecture.]

2012 Survey Results

KWRegan — Tue, 10 Jan 2012 22:54:07 +0000

What would your favorite theoretician be doing?

Antea is a fictional young aspiring mathematical physicist in the epilogue of Roger Penrose’s book The Road to Reality. She experienced the rare green flash phenomenon while watching the sunrise, and “then an odd [remarkable] thought overtook her…”

Today we wish to thank the 259 people who took our survey posted last week, and present the results. Of course Antea could not be among the great thinkers listed, not at that stage, but our purpose was to see where the greats would seek remarkable thoughts today.

The Antea shown in the painting detail is not Penrose’s, but was painted by Giuseppe Mazzola, called Parmigianino, around 1530. She belongs to the Museo di Capodimonte in Naples, Italy, but has traveled much. During World War II she was taken by German forces to a salt mine near Salzburg, Austria, where Ken’s family spent part of a holiday during Christmas week. She was loaned to the Frick Collection in New York City in 2008.

Parmigianino himself pursued materials science, so much as to neglect his contract creating art for a church. Being the 1500s this meant alchemy, but his motive seems to have been art rather than greed for gold. He was one of the first Italians to do etching as an art form. He sought a metal that would respond more easily to the technique than copper.

The Questions and Results

1. What would Euclid be doing in 2012?

Writing the best blog: 16.1%
Writing a new edition to his famous book: 36.9%
Working on factoring: 49.0%
Other: 9.4%. The most amusing answer was “working on non-Euclidean geometry” from two responders. Two people put him on the front line of the Greek protests. The most ironic response was “Formalizing the foundations of ill-defined fields,” a service David Hilbert performed for Euclid. There were also: engineer, solving math, String Theory, breaking RSA, writing an undergraduate text on Category Theory, writing world-class surveys, synthesizing 20th-century mathematics, encouraging young mathematicians a-la Bourbaki, meditating in Tibet, enjoying the beach, re-founding geometry in topos theory, other math logic, running a hedge fund, tax attorney, blogging, using the Coq theorem prover, and the one denotational truth: requiescatting in pace. There was also a neat riff on debating with Cicero on Greek versus Roman contributions, which is welcome to be put in as a comment.

2. What would Gauss be doing in 2012?

Working on quantum computation: 36.1%
Working on complexity theory: 29.0%
Working on P=NP: 31.0%
Other: 21.0%, making 53 responses. A general drift was that this universal mind would not be confined to computer-science topics; one had him working on what we take to be mag-lev trains, recalling that “gauss” is a unit of magentism. Riemann, Navier-Stokes, and ‘all’ Clay problems came up. The most wicked answer was that he would name all arXiv published papers after himself. The most “real” answer IMHO (Ken) was “Doing the Langlands program,” while someone else had him zoned in on the Collatz problem, and another had him making good on a proof by constructing a regular 65,537-gon. One had him fixing the world’s economy, while another anointed him chair of the Federal Reserve, not too far from a job in the British Government actually held by our next man.

3. What would Newton be doing in 2012?

Working on string theory: 39.8%
Creating a new branch of mathematics: 56.1%
Working on P=NP: 4.1%
Other: 15.4% Lots of economic vocation for this former Master of the Mint—one connected his concern for counterfeit and shaved-down coins into work on unforgeable quantum money. Some had him into polemics against quantum theory and Leibniz. Others noted his paranormal and alchemy interests. Teaching theology was a given—Newton wrote over a million words on theology. The most topical response was that Newton would have worked on the 3-body problem—note that our next man identified a meaty special case that is solvable in closed form. However, what we note most is that the founder of so much continuous mathematics drew no truck with a ‘discrete’ problem like P=NP.

4. What would Euler be doing in 2012?

Working on everything: 73.6%
Solving the Navier-Stokes Clay problem: 20.1%
Working on P=NP: 6.3%
Other: 5.0% One of the twelve “Other” responses said “everything and two or three other things too.” Another said he would co-author a paper with Erdős “(pretend Erdős is still alive)” but the qualifier was hardly necessary—Erdős has maintained a stellar publication record since his death in 1996. This all creates a higher polymath impression than Gauss. One had him helping with global-warming satellites, another with elementary school math teaching, and another living like Grigory Perelman in St. Petersburg. I once wrote a short story in Italian about a man who dies trying to move a rock with his mind, and someone (else) put Euler up to that, while another said worse. The most elevative respondent had him resolve P vs. NP from results on complex power series. But no mention of theology, despite Euler being the only mathematician on the Lutheran Calendar of Saints.

5. What would von Neumann be doing in 2012?

Working on peta-scale computing: 27.1%
Working on quantum computing: 55.0%
Working on P=NP: 17.9%
Other: 9.2% More “everything” including housing computer server rooms on the Moon for better thermal regulation. When not working on “something beyond our imagination,” he campaigns for Barack Obama and becomes his science advisor or runs for his job. Global-scale synthetic biology and regenerative medicine, patterns of resemblance, computational fluid dynamics, alternative energy sources, people got real specific with Johnny. Only one “pure math” topic came up, non-commutative geometry, though two mentioned game theory of course. However, the main categories show the degree to which quantum computing rules!

6. What would Gödel be doing in 2012?

Thinking: 58.3%
Finally writing up his solution to the P=NP question: 46.0%
Working on string theory: 7.9%
Other 9.1% Theology made a comeback with three responses, in-tandem with proving his Third Incompleteness Theorem or the Comtinuum Hypothesis. The cleverest logical answer was “He would prove that deciding whether P = NP would imply that P = NP iff P != NP”—though this is equivalent to others’ responses that he’d prove P vs. NP undecidable. Honorable mention to “Using the fact that Euclid works on Non-Euclidean geometries as a contradiction to prove that P=NP.” Others addressed his paranoia and conspiracy theories, including the pithy reply, “Dieting.” Another had him viewing endless reruns of the movie “Inception”—while another put it more philosophically: “Applied transcendental epoché.” Category theory, new foundations for mathematics, and ordinal analysis of (strength of) theories harked back to what for him was his real world, along with “depression at having killed Hilbert’s program” and “hiding in his office at the Institute.” The most upbeat answer was, “Co-authoring some papers with ME!” Note also that the answers for Gödel add up to more than 100%, as also with Euclid but not the others.

7. What would Turing be doing in 2012?

Working on AI: 65.8%
Working on the Riemann Hypothesis: 6.9%
Working on P=NP : 27.3%
Other: 9.0% Being able to marry legally and greeting former Prime Minister Gordon Brown who delivered a formal British Government apology for the treatment leading to his suicide topped the list topically. One respondent with several “Creating a new branch of…” responses followed here with simply “…science,” while four others noted a branch Turing did help create: the computational study of morphogenesis. He was tabbed for (CEO of) Google or Apple or Facebook, or somewhere he could “invent the Turing machine.” Straight-arrow answers had him working on AI/robotics/brain science, and this gives me opportunity to note that Scott Aaronson has posted the culmination of his Philosophy and TCS course, including a student project showing the Turing Test remains very much alive. Skiing, winning second-tier marathons, and hobnobbing with celebrities seemed to be wishes for the “good life” unknown, though one can follow Turing’s real life in the online scrapbook maintained by his biographer Andrew Hodges. Another had him attracted to parallel computing, which made me think more of how Turing would have weighed in on Jim Backus’ 1977 Turing Award article, “Can Programming be Liberated from the von Neumann Style?” And a healthier percentage than anyone except Gauss thought he would take a crack at the time-bounded NTM version of the Halting Problem, our last subject.

8. Will 2012 see the end of the P=NP question?

Yes: 5.9%
No: 49.4%
Perhaps: 9.9%
No way: 28.5%
I hope so: 6.3%

I’d be curious to know what the 15-of-253 answering “yes” may have up their sleeve—after all, even Donald Rumsfeld famously skipped over the case of “unknown knowns.” But most were confident in the question’s staying power, and nobody mused that extra-terrestrials might give the answer away. Instead, having belatedly realized we could have invited other luminaries to be mentioned in a catchall “Other” field, I became the only one to mention someone of the same gender as Antea.

Other Folks, Other Strokes

These were the 18 general suggestions, including one from the comments.

Ramanujan doing in 2012 : Celebrating National Mathematical Day in India, creating more number theoretic formulas.
Hilbert: would attack Perm vs Det via polynomial ideal theory, with help from Em. Lasker and Em. Noether
Leibniz: working on Lie groups to show the identity of indiscernibles
What would Claude Shannon be doing in 2012?
I would ask what John Horton Conway would be doing, but he is still with us.
Which part of AI would Dijkstra be studying? None of it..
Whether P = NP is only the beginning of the P vs. NP question.
Maybe a proof will be found, but we would have to wait more than a year for validation.
None of the seven Euclid–Turing will be working on any standard problem—they are too creative for that.
Hilbert would design his own automated proof assistant.
Depends on how long it takes me to flesh out all the details in my paper.
P=NP question will see the end of the world! Remember it’s 2012! :-p
Only dishonest people would say it is still unsolved.
I think it’s more likely that society crumbles, rendering this question irrelevant [plus some encomium for this blog]
I will prove that parity games can be solved in polynomial time. Mark my words.
Perhaps you can add Hilbert and Einstein to the list.
[Terence] Tao will be working on things I don’t understand
[On P=NP ending] I hope not.

Although our survey was meant to be lighthearted, there were interesting correspondences between a researcher’s proclivities and future areas. We are all for diversity in science: different strokes from different folks.

Open Problems

What areas might be primed for a stroke of creativity? What should priorities be?

What would your favorite research mentor be doing?

We thank all those who responded to the survey.

Predictions For 2012

KWRegan — Tue, 03 Jan 2012 05:59:07 +0000

Possibly excepting the last 10 days

Joseph T. Goodman became the owner and editor of a newspaper in Virginia City, Nevada called the Territorial Enterprise, a few years after its founding in 1858. He furthered the circulation of the newspaper throughout the Western territories, and used it as a mouthpiece for positions such as supporting the Union in the Civil War. In his day one could say that this amounted to a prediction. One of his later efforts was to grow grapes in Northern California, and though his own effort seems to have had no memorable yield, one can say he predicted the area would be good for wine.

Today we wish to go over last year’s predictions, and offer a mostly new slate for 2012. The supposed Mayan prediction of the end of the world on December 21st, 2012 makes us hedge by not venturing anything beyond that date.

Goodman’s most lasting work came late in life: he deciphered the notation system used in the Mayan calendar. Juan Martinez-Hernandez and Eric J. Thompson later expanded and confirmed this work, so that the system is now known by their initials GMT—which is neatly suggestive of our own time standard.

As editor as well as owner, Goodman paid attention even to the “Letters” page of his paper. He noticed that one unsuccessful, worldly-wise prospector in Aurora, Nevada, wrote in often with a lively pen. He offered the young man, Samuel Clemens, a job on his paper instead. The stories by Clemens gained enough circulation to earn him a reputation, and today we know him as Mark Twain.

The Mayan Y2K Problem

What Goodman and his followers solved can be abstracted as a constraint-satisfaction puzzle. Most Mayan “long count” inscriptions have five entries in a recognizable tally system, so we’ll call them the coefficients . The questions are what units they multiplied and whether this was radix notation with a single base or multiple bases which we’ll call . The data could be given the form

d ' title='\displaystyle a_5 v + a_4 w + a_3 x + a_2 y + a_1 z + C > d ' class='latex' />

from cases where the inscription could be sequenced with a historical event whose Gregorian date was known. This could come along with other information conferring similar equations with or approximate equality in place of }' title='{>}' class='latex' />.

It was clear that the ones place had units of days, and that the Mayans liked to count by or . Thus and , but it was found that , not as in base 20. The switch to base-18 seems surprising but makes closer to a solar year of days. Then (called a katun) and (called a baktun) resumed the base-20 pattern.

The hardest variable to solve was , called the “correlation.” Indeed Goodman, Martinez-Hernandez, and Thompson came up with values apart. Goodman’s value gave a “Day Zero” of August 11 in 3114 BCE, and this was adopted. Thus today’s date (as I, Ken, write, not post date) in Mayan is

Note that the and places are maxed out at , so after more days, the coefficient will tick over.

The issue is whether the base for the place was meant to be base 20 or base 13. If the latter, this is the end my friend. Evidence for the latter is that some Mayan inscriptions indicate more than five places but pad them with 13′s that play the role of zeroes. One noted in Wikipedia’s article here is

whose last five places denote October 19 in the year 744. The inclusion of the leading places, however, grants about 4 billion more years. Another inscription has nineteen leading 13′s. Thereby the Mayans over-predicted the life of the world, as all leading physical theories suggest that at least our branch of the cosmos will freeze or be annihilated long before those places are filled out.

Thus December 21 will end a Mayan cycle of days giving about of our years, but we see nothing more than a limit of 5-place syntax. This is analogous to the two decimal place shortcut that led to the Y2K Problem, in which predicted techno-apocalypse was averted. Well in case we are wrong, at least the 354 days left constitute a lunar year, so we are still offering a year’s worth of predictions.

Reviewing Our 2011 Predictions

Here are the predictions from last year, together with the results.

No circuit lower bound of or better will be proved for SAT.
Fish-in-a-barrel.

A Computer Scientist will win a Nobel Prize.
Not yet.

The above are two holdovers from last year. We will let them “ride” and make them again. Note there is no CS Nobel, so what the latter means is that among those cited for science or economics will be someone trained or currently employed in computer science.

The complexity class , bounded-error quantum polynomial time, will be shown to lie in the polynomial hierarchy. This is rather nervy, since contrary to the clear drift of [we cited several recent papers].
Wrong, but some progress has been made—at least we think we have.

A conceptually new algorithm will be discovered for an important practical problem; it will be a breakthrough in asymptotic time but in concrete terms will be even more galactic than previous ones.
Unclear. We would like to claim credit for prognosticating the new bounds on matrix product, but the most we can say is that the jury is still out on whether the algorithms are “conceptually new.”

At least five claims that and five that will be made.
Correct. According to the 13 items numbered 70–82 by this source, there were 7 claims of equal and 7 claims of not equal.

That seems not to add up, but one claimant ‘proved’ it both ways, depending on the presence or absence of certain axioms for reasoning about certain representations of natural numbers.

Graph Isomorphism will be proved to be reducible to the Graph Reconstruction Problem. For background, see this paper.
Not yet.

Apple will invent a new type of alarm clock: the iClock. Pressing the “snooze” button will take you back one hour via your own personal time travel device. It will require a monthly subscription fee and will be wildly successful—however, see this for a caveat.
Does the iPad count?

The world’s population at exactly 12:00am GMT on April will be even.
We think so—but who can prove us wrong?

A “provably” secure crypto-system will be broken.
Of course. Should be disallowed as a prediction, like saying there will be gravity next year.

A Clay problem will be solved by two independent researchers at about the same time.
Wrong.

2012 Predictions

Some of our new predictions may turn out to exemplify another episode from Joseph Goodman’s life. In 1872 his newpaper opposed William Sharon’s run for the U.S. Senate, Nevada having become a state in 1864. Sharon lost but resolved to try again in 1874. To make sure the paper’s next “prediction” would be to his liking, he bought the paper—and won. That is to say, we hope to promote work on some of our predictions, if we cannot do them ourselves.

Our predictions for the coming year are the following—note that the first two are repeats:

No circuit lower bound of or better will be proved for SAT.
A Computer Scientist will win a Nobel Prize.
Someone other than a Williams will release a headline-making result in November.
A new set-theoretical relation will be found between one or more of these classes which are all hard for Discrete Log or Factoring (perhaps in a promise-problem form, per this paper) but not known to be hard for :
, , , , ,

and one or more of these classes which are -hard (at least under randomized reductions) but not known to be hard for the second level of the polynomial hierarchy:
, , , , ,

(All but the last class name is clickable to get its definition at the Complexity Zoo.) We could throw in some other classes for each of these two “regions,” both of which are heavily populated with little knowledge even conditionally. A new binary relation (most likely a new containment) would score a big hit, a new ternary relation a little hit, a new oracle relation we’ll punt.
Tangible new evidence against the Unique Games Conjecture will be posted (i.e., published at least on the arXiv) by Dec. 21, 2012.
Tangible new evidence supporting the Unique Games Conjecture will be posted by Dec. 21, 2012.
These last two predictions do not contradict each other. To check whether any published evidence is new or substantial, the Mayan cutoff date allows some pre-holiday time—we hope.
Tangible new evidence against the Exponential Time Hypothesis will be posted by Dec. 21, 2012.
Tangible new evidence supporting the Exponential Time Hypothesis will be posted by Dec. 21, 2012.
For a general science bent, we offer:
The Higgs boson will close the year with a stated confidence above from a single team, but strictly less than from any one team.
An Earth-sized planet will be detected orbiting within the habitable zone of its single star. (Currently “Earth-sized” and “habitable zone” have been achieved by different exo-planets.)

We’ve thought to add predictions about new conditional lower bounds, improved quasi-randomness properties of simple hash functions, and scalability of quantum computing, but these can come out later and we’ll keep it at 10.

Open Problems

What are your best predictions? We further predict that next year many of our predictions will be wrong.

Happy New Year.

[fixed bases to u,v,w,x,y not v,w,x,y,z; fixed a couple words including "one or more" in prediction 4, which is a less-"nervy" analogue of last year's prediction about BQP]

A 2012 Survey

rjlipton — Sat, 31 Dec 2011 17:45:40 +0000

Questions we must ask

The Harrisburg Pennsylvanian, a newspaper, conducted one of the earliest political polls. Their poll in 1824 correctly showed Andrew Jackson leading John Quincy Adams by 335 votes to 169 in the contest for the United States Presidency.

Today I want to discuss the future of GLL, since 2012 is potentially a special year.

Note and is a prime number. The actually. I have no idea what this has to do with the 2012 predictions, but I thought noting it’s a sizable prime multiplied by a power of 2 would add to the mix.

The Survey

The questions are all about what famous mathematicians would be doing in 2012, with one exception.

Click here to take survey

Open Problems

Have a great New Year celebration. Be safe and ready for a great new year.

[Expanded poll's "Other" option to allow saying what your own favorite historical mathematician would be doing in 2012---e.g. as prompted by Gaurav in comments, Ramanujan:Creating numerical formulas for instances of #P functions; tweaked wording]

The Name Game

rjlipton — Tue, 27 Dec 2011 14:42:17 +0000

A little trick with Nick

John Doe is not a theorist, but was once the name used as a place holder for everyone.

Today Ken and I want to thank everyone who reads GLL, including John Doe.

We love writing GLL, but with your support the writing is easier, it is more fun, and seems more worth the interruption that it sometimes causes with other duties. Thanks again to all.

Special thanks to all we have written about—we appreciate you forgiving our sometimes inaccuracies. Special thanks to all who have commented in public–we appreciate your lively comments, even the ones that we do not necessarily agree. Special thanks to all those who directly have helped and have commented in private.

Names

John Doe, Joe Bloggs, John Q. Public, Joe Public, Luther Blissett, Mary Major, and Nomen nescio are just some of the examples of names that are used when a real name is unknown. Curiously, one of the driving forces behind such names is the legal world. They sometimes need to refer to a person that is unknown or unnameable: Nomen nescio, it is used to name an anonymous person in legal terminology, abbreviated to N.N., it comes from Latin “nomen” for name and “nescire” for not known. One of the names listed above is also a New York City restaurant, of some fame.

Our friends at Wikipedia say: The name “John Doe”, often spelled “Doo,” along with “Richard Roe” or “Roo” were regularly invoked in English legal instruments to satisfy technical requirements governing standing and jurisdiction, beginning perhaps as early as the reign of England’s King Edward III (1312—1377).

The Name Game

Shirley Ellis is famous for the song “The Name Game.”

Katie,
Katie,
bo-batie,
Banana-fana fo-fatie Fee-fi-mo-matie Katie!

Go here to listen to it for a bit of fun:

Open Problems

Please keep reading and commenting. We greatly appreciate all your kind support. We will be back soon with lots of technical stuff and strange symbols of all kinds. Thanks again.

[changed Joe Doe to John Doe]

Proofs, Proofs, and Proofs

rjlipton — Wed, 21 Dec 2011 14:43:04 +0000

Your proof is not my proof

Jennifer Chayes is theorist who has made fundamental contributions in two different ways, some directly as a researcher and some indirectly as a manager. She is the director and co-founder of the Microsoft research lab in Cambridge—the one in New England—not England. Her main research contributions are in the area that spans theory and physics, including phase transitions in discrete systems such as networks. Her main administrative contributions have been in the creation of world class teams. Each of these types of contributions is impressive and each is hard—I think it takes a unique person to be able to do both so well.

Today I want to talk about the notion of proof, with all its various meanings.

Jennifer and Dana Randall helped create the area that now joins theorists and physicists who work on various Markov type systems. They started with a lone workshop, and now the area is a thriving part of both disciplines. When they started what was a proof in one area was not necessarily one in the other. This, Dana tells me, often led to comments like this being made during one of her presentations:

But Dana that is not an open problem, X “proved” it years ago in

One of the great achievements of Jennifer and Dana, with help from many others, is to reduce this dissonance about what proof means. Those from theory and those from physics, who work on Markov systems, may still not completely agree on what is a proof, but the differences are at least now acknowledged.

This made me think about what kinds of proofs there are and how they differ.

Types of Proofs—Silly?

Proof Reading: Ken and I try hard to proof-read these pieces for typos, math errors, and general examples of poor communication. We try very hard, but we know that we often fail short. One type of activity that has “proof” in it is the art of proof reading. Or is it proofreading? I have noticed that the New York Times almost never has spelling errors these days—thanks to spellcheckers—but grammar errors are another story. Oh well, I guess that’s progress.

Proof of Vodka

In the spirit of the holiday season we thought we would mention the notion of “proof” as used in the US to measure the content of alcohol in a beverage. I no longer drink—long story—but still appreciate that many do enjoy wine, beer, or even more adult type beverages. According to our friends at Wikipedia: Alcohol proof in the United States is defined as twice the percentage of alcohol by volume. Consequently, 100-proof whiskey contains 50% alcohol by volume (abv); 86-proof whiskey contains 43% alcohol. The terminology used in the United States is “n” proof, where “n” is a number—not “n” degrees proof.

Why “proof”? The proof that alcohol wasn’t overly watered down was that gunpowder would burn in it rather than be doused by the water. For rum this threshold was measured at 57.15% abv, so that became “100% proof.” The accurate British multiplied abv by 7/4 to get proof for any abv, but the rough Americans multiplied by 2. Now both countries require abv to be stated on bottles, proof being optional.

Legal Proof

Perhaps it is best to skip this one, since it is so unclear. What is “proof” in the legal sense seems to be dependent on obvious stuff like country—legal rules differ from place to place. But even within the same country with the same rules, the notion of legal proof seems to depend on time, and perhaps on some hidden random variables. The definition linked to by Wikipedia is a 190-page PDF file with an awkward space in its URL. It defines legal proof as what the laws that define the status quo require in order to change the status quo. So let’s skip it.

Proof of Pudding

This is also the season for various treats that are called puddings. I especially like bread puddings of various kinds. The phrase “the proof is in the pudding” is a misquote of a misquixote—something mis-attributed to Miguel de Cervantes, Don Quixote (1615). The form credited to Cervantes is correct but is said by this site to date to the 1300′s and first appear in print in a 1605 work by William Camden: “The proof of the pudding is in the eating.” Here “proof” has its root meaning from Latin proba as “test.”

There also is the Yale University “Proof of the Pudding,” an all-female a cappella group specializing in jazz and swing. As an ex-professor of that great institution I agree that this is another fun use of the notion of proof of the pudding.

Math and Physics—Serious?

Math Proof: We could go on for days about what is a math proof. In many of our posts we discuss specific proofs, or we discuss general issues about proof. So I will not spend much time on it now. I do note that there still is a long trail of comments on whether or not the reals are countable or uncountable. So the notion of mathematical proof is not as clear-cut as some would like to believe.

Physics Proof The real point today is to return back to Jennifer and Dana examples of the disconnect between mathematical types of proofs and physics type proofs.

The simple point is this: physics accepts as a “proof” things that we would not and do not. Some examples to make the point are in order.

Physics Proofs?

An Old Old Example: The Dirichlet Principle, named of course for Lejeune Dirchlet, is the “obvious” notion that certain functions exist that obtain a minimum value. Note we are talking about functions that have some minimum value, not points.

There is a book that is out of print that whose title summaries the issue nicely:

Dirichlet’s principle: A mathematical comedy of errors and its influence on the development of analysis, by Antonie Monna.

The great Bernhard Riemann named the principle after Dirchlet and believed it was obvious, from a physics argument, that there had to always be a solution. Karl Weierstrass later showed that there were reasonable problems of this type without any minimizer—that the principle was false in full generality.

The reason it was thought to be “obvious” was the analogy to calculus. We are probably familiar with the idea of finding the real number so that some given function is minimized for in , for example. A typical calculus problem might be: Find the value so that

is smallest for in . This is the one of the prime applications of calculus.

Things get more difficult when we replace by a function and by a functional and ask for the that makes

the smallest possible. An ancient example is to ask for the curve that has length and encloses the most area in the plane: the functional determines the area of the curve . This is the famous isoperimetric problem, whose answer is well known to be the circle with circumference .

The problem is that it is not hard to guess that the circle is the solution, proving that it is the solution is much harder. Jakob Steiner in 1838, showed that if there was a minimum, then it was the circle. He did this by a very clever argument based on taking any non-circle and making it more symmetric, and making the area not decrease. The difficulty with this argument is the proof that there is a minimum. That is the hard part of the proof—see this for a discussion.

Finally see the following for a silly solution in the spirit of the holidays:

Consider the large class of animals capable of changing their ratio of surface area to volume. (And note that these animals live — approximately — in Euclidean 3-space.) What do these animals do when it’s cold? They curl up into a ball! More precisely, they assume the closest approximation to a ball that they can manage. This is because any exposed surface area is a place where heat is lost, and curling up into a ball minimizes that surface area. So a spherical or “ball” shape keeps animals warmer. Now, here’s how these ideas can be turned into a 4-line “biological proof” of the above proposition.

Proof by Physical Impossibility?

Suppose one can prove that the feasible ability to travel back in time and kill one’s grandfather. Is this a proof of ? A similar and more plausible case is when the ability to transmit information at faster than light speed. A controversial example of such an is discussed here.

Are there theorems in quantum information theory for which it is felt that the “nicest” proof is by “reduction to Einstein” rather than from axioms? We note that this paper by Ulvi Yurtsever implies that the ability to maintain a prediction advantage over quantum sources that are adduced to be genuinely random implies the ability to communicate faster than light. We invite readers to comment on other examples and the formal status of this kind of proof.

Open Problems

What is a proof? Must a proof be checkable by a human being unaided by a computer? Note that we could be talking about a checker for the proof, not the proof itself.