Electrika source |
Margaret Farrar was the first crossword puzzle editor for The New York Times. Ken fondly recalls seeing her name while watching his father do the daily and weekly NYT puzzles—they were under Farrar’s byline as editor until 1969 when she retired from the Times. More than a maker of countless puzzles, she also created many of the meta-rules for crossword puzzles, which are still used today in modern puzzle design.
Today Ken and I wish to discuss a light topic: how 2 and 3 are different in many parts of theory and mathematics.
What do 2 and 3 have to do with crossword puzzles? Farrar enshrined the “rule of 2 and 3″ while producing the first crossword puzzle book in 1924 for the fledgling publisher Simon & Schuster. The rule says that 2-letter words are forbidden but 3-letter words are fine in moderation. In the crossword game Scrabble, however, 2-letter words are not only allowed but are vital to strategy. So 2 and 3 are different—yes.
Additional meta-rules include this interesting one:
Nearly all the Times crossword grids have rotational symmetry: they can be rotated 180 degrees and remain identical.
When asked why, Farrar said:
“Because it is prettier.”
In other respects crossword puzzles are more liberal than Scrabble rules. Proper names, abbreviations, multiple-word phrases, prominent foreign words, and clean/trendy slang terms are allowed. Clues ending in ‘?’ may have puns and other wordplay. Here is a small example from us at GLL:
While 2 and 3 are different enough between crossword puzzles and Scrabble, they are even more so in mathematics. For example, 2 is magic:
Try that with 3 or any other number. But we are after deeper examples of how 2 differs from 3.
In Number Theory: The number 2 is the only even prime. I recalled here a story of a colleague who works in systems. He was listening to a talk by a famous number theorist. The latter constantly said things like:
Let p be an odd prime and …
My friend asked, “what is an odd prime?”—thinking it must be special in some way. The answer back was: not 2.
In Group Theory: The famous Feit-Thompson Theorem shows that 2 is very special. Any group of odd order—a group with an odd number of elements—must be a solvable group. This was immensely important in the quest to classify all simple groups. Every non-cyclic simple group must have even order, and so must have an element so that .
In Complexity Theory: The evaluation of the permanent is believed to be hard. The best known algorithm still for an permanent is exponential. Yet modulo 2 the permanent and the determinant are equal and so it is easy to evaluate a permanent modulo 2. This relies on the deep insight that
modulo 2.
In Quadratic Forms: The theory is completely different over fields with odd characteristic compared to those with characteristic 2. A neat book by Manfred Knebusch begins with this telling verse:
In Algebraic Geometry: I have talked about the famous, still open, Jacobian Conjecture (JC) many times. It is open for two variables or more. But it has long been solved for polynomial maps of degree at most 2. Degree three is enough to prove the general case:
Theorem 1 If the JC is true for any number of variables and maps of degree at most three, then the general case of JC is true.
In Complexity Theory: Many problems flip from easy to hard when a parameter goes from 2 to 3. This happens for coloring graphs and for SAT—to name just two examples.
In Physics: It is possible to solve the two-body problem exactly in Newtonian mechanics, but not three.
In Diophantine Equations: is solvable in positive integers, but as Pierre Fermat correctly guessed, and all higher powers are not.
In Voting and Preferences: Kenneth Arrow’s paradox and other headaches of preferences and apportionment set in as soon as there are 3 or more parties.
Computing: Off-on, up-down, NS-EW, open-closed, excited-slack, hot-cold, 0-1 is all we need for computing, indeed counting in binary notation.
In Counting Complexity: Although is in polynomial time, the counting version is just as hard as . Even more amazing, remains -complete even for monotone formulas in 2CNF or in 2DNF (they are dual to each other).
In Polynomial Ideals: Every system of polynomial equations can be converted to equations of three terms, indeed where one term is a single variable occurring in just one other equation. The idea is simply to convert into and , and so on. This cannot be done with two terms.
However, systems with two terms, which generate so-called binomial ideals, share all the (bad) complexity properties of general ideals. In particular, testing whether a system forces two variables to be equal is -complete. The proof takes and to be the start and accept states of a kind of 2-counter machine known to characterize exponential space. For example, an instruction to decrement counter , increment counter , and go from state to state yields the binomial . Thus a configuration such as becomes on substituting for .
In Diophantine Equations: Hilbert’s 10th problem is known to be undecidable for equations in 11 variables. A broad swath of classes of 2-variable Diophantine equations have been shown to be decidable, enough to promote significant belief that a decision procedure for all of them will be found. For three variables, however, the situation is highly unknown according to this 2008 survey by Bjorn Poonen. Only the trivial one-variable decidability is known.
In Diophantine Equations, yet again: Poonen also relates Thoralf Skolem’s observation that every Diophantine equation is equivalent to one of degree 4 in a few more variables. One simply breaks down a power like into the terms you had with replaced by . Degree 2 is decidable, but degree 3 is unknown—the feeling is that it’s likely undecidable. All this recalls an old quote by James Thurber:
“Two is company, four is a party, three is a crowd. One is a wanderer.”
What is your favorite example of the is different from phenomenon? Recall that Albert Meyer once said:
Prove your theorem for and then let go to infinity.
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Creating vast beautiful mansions from the becoming of nothing
L’espace d’un homme film source |
Alexander Grothendieck, who signed his works in French “Alexandre” but otherwise kept the spelling of his German-Jewish heritage, passed away Thursday in southwestern France.
Today we mourn his passing, and try to describe some of his vision.
Part of the story of this amazing mathematician is that in 1970 he renounced his central position at the Institut des Hautes Études Scientifiques (IHES) in Paris, and made himself so remote shortly after formally retiring from the University of Montpellier in 1988 that not even family and friends could track him. He boycotted his 1966 Fields Medal ceremony in Moscow to protest the Red Army’s presence in eastern Europe, and declined the Crafoord Prize in 1988.
As captured by this obituary, he had left to seek a society kinder and more just than the ones that killed his father at Auschwitz and convicted him in 1977 of violating a French law dating to 1945 against feeding and sheltering an unregistered alien. More will be told of this story as his voluminous writings from the hinterland are being read. But between 1945 and 1970 he published mathematics of unparalleled sweep and power, conveying escalations of abstraction to the solution of concrete problems, and this is the part we wish to appreciate.
One of humanity’s greatest intellectual tropes is Plato’s “Allegory of the Cave,” which likens what we apprehend through our senses to shadows of forms projected on a wall by a dimly-lit fire. The forms hail from an outside world whose light is blinding to one prisoner unchained and led out of the cave. Although Plato speaking through Socrates addressed all of reality, let us just imagine this outside world as Euclidean space, in which the Platonic solids shimmer in their ideal forms. Then what Grothendieck perceived when he was led into the light is the following:
The outside of the cave is another cave.
In this outer cave the focal point is , zero. This zero is the only solution to the equation . It is also the only solution in that cave to the equation . Likewise , , and so on. These are different equations, but each has only the same single root in the Euclidean space of the outer cave. We can add the words “with different multiplicities,” but what difference do they make to the objects from which we draw our solution?
Perhaps this is but a projection along a beam of elements in a higher space that can furnish different solution structures to these different equations. What Grothendieck regarded as needed for “truly natural methods in geometry,” as related by Jacob Murre quoting a lecture by Grothendieck in 1959, is the employment of nilpotents, that is, elements such that the sequence eventually gives zero. Such elements can be as simple as modulo or the matrix , but organizing them is what unchains us from the single zero.
In moving essays written by Grothendieck’s friend and colleague Pierre Cartier for the 40th and 50th anniversaries of the IHES, coinciding with Grothendieck’s 70th and 80th birthdays, Cartier did not shrink from invoking Albert Einstein for intellectual comparison. Nor did Grothendieck, as the latter essay relates regarding the approach to space.
Einstein famously derived the core of his physical theories by working out all the logical consequences of the visions in his thought experiments. One of them is that there is no focal point of space. Nor is space a pre-existing entity, as Isaac Newton had posited, but rather space emerges from relational properties of its contents. A manifold is not made by its points, and it need not be determined by the locally Euclidean structure near any one point, but rather by how open sets around points mesh together. But in math, when we have no matter, what can we take as the content that drives the structure?
In executing mathematics we can approach contents only via definitions and formulas and proofs, which are pieces of syntax. Plato was aware of this. In his “Allegory of the Divided Line,” which immediately precedes the “Cave” passage in his Republic, Plato divided the mathematical world internally in the same ratio by which he divided it from the world of sense experience. Mathematical Platonists distinguish themselves from formalists by affirming reality beyond formulas and proofs, which can seem like chains on the intellect. However, all schools can alike acclaim the way major advances in the 20th Century came from treating syntax as objects. One example comes clearest in Leon Henkin’s proof of Kurt Gödel’s Completeness Theorem by employing logical statements as elements of the constructed model.
Cartier’s 1998 essay picks right up from this. Consider a model assigning a truth value to every proposition in some Boolean algebra . Associate to the set of all models that make . These sets obey the rules:
Thus our “points” correspond to special subsets of the space of models. The sets of can be generalized to sets of valuation functions obeying , , and for all ,
giving . In algebraic geometry there is a similar relation between points and special sets of functions giving equations that they solve. The upshot is that if we can identify the “special” property so that other sets besides our original ‘s have it, then from those sets we can harvest more “points.”
Suppose we have a system of equations , where ranges over some space . Consider all objects of the form
where the multipliers are arbitrary functions, including constants. Then any common solution to the equations also makes . The set of such functions can be regarded as the “algebraic consequences” of . is clearly closed under addition and under multiplication by arbitrary elements, so it forms an ideal in the function space.
Now consider any point in an -dimensional Euclidean space . It is the unique solution to the simple system of equations
of course setting each . The ideal is then maximal in the space of polynomials over , meaning that but for any other ideal ,
Every maximal ideal is prime, meaning that if a product belongs to , then either or . If neither were in , then the ideal would properly contain and hence have to be all of , whereupon we could find a scalar such that . But then would belong to after all.
The concepts of ideal and prime and maximal can be applied even in simpler spaces such as the set of integers. Every integer generates the ideal of multiples of . If factors properly as , then and , so is not prime, while and , so is not maximal. But when is prime, is both prime and maximal, and these are the only prime or maximal ideals of . For other spaces such as our polynomials over , however, the concepts of prime and maximal do not coincide. Grothendieck culminated a long list of people who realized that while maximality is the “pointy” property, primality is the “special” one.
With respect to our original points , for any ideal we can identify the set
called the algebraic set or variety determined by . If , then it is enough that be the common solution set of the in . There is something analogous to the above example of Boolean valuations going on, except that the set operations are flipped around:
Here means the ideal closure of , and equals when and . The product also gives .
We must skip over some wonderful finiteness theorems by David Hilbert and his students—and over distinctions such as the “base ring” being a field that is/is-not algebraically closed and projective versus affine space—to say only that algebraic sets are primitively defined in first-order arithmetic and hence are “neat” in many senses. Specially neat are those that cannot be written as in a nontrivial way, that is without one of or being all of . Then we can’t have or in a nontrivial manner, and exactly what this means is that the ideal of all functions that vanish on is prime. Such a is called irreducible, and sometimes the term “variety” is still reserved for this case. In an abstract but natural way, irreducible varieties of all dimensions up to can be made to behave like points. To quote Cartier on “the meaning of the word scheme” (his emphasis):
“One must, of course, understand that the space Grothendieck associated with an algebraic variety is not the set of its own points, but the set of its irreducible subvarieties.”
To see where the nilpotents come in, and how Grothendieck unchained us not only from zero but from Euclidean points overall, we can begin by perceiving how evaluating a function is like doing long division with remainder. With respect to any ideal of , the relation is an equivalence relation, and allows us to write the quotient . For our Euclidean point , evaluating can be achieved syntactically by reducing modulo (the ideal generated by) . Taking modulo by long division works out the same as substituting , and the same goes iteratively with the other elements of .
The reduction process works for any ideal and gives a unique result , provided a special kind of basis giving named after Wolfgang Gröbner is used for the iterated long division. Thus the “evaluation” is well-defined for any ideal and can be carried out by an algorithm that first expands any initial set of generators for into a Gröbner basis. Alas all known algorithms have doubly exponential worst-case time complexity, perhaps unavoidably since deciding whether is complete for exponential space even when is linear and the initial generators for have constant degree. Nevertheless, these algorithms are run all the time for important equation-solving applications, and impress on us this philosophical fact:
We can do richer kinds of evaluation in the space delineated by our syntax than in the external Euclidean space.
This holds even when we return to our simple equations , with , only one variable. The ideal is prime—indeed maximal—but is not prime. Also , and the ideal generated by equals . Nevertheless, when we reduce a polynomial like modulo these respective ideals, we get different results.
Thus we can dispense with the original points, even the origin in . However, we would still like to preserve our primitive idea of “evaluation” in some kind of external space. How can we do this, and in what kind of space?
We cannot squeeze these answers out of our Euclidean space. We can interpret a quotient as endowing with coordinates as a space in its own right, but that only works up to the prime ideals. Once we connected irreducible varieties to prime ideals, that much was one-and-done. We can’t get multiple function values , , out of our single, irreducible zero. There is no “square root of zero” different from zero. To go further, Grothendieck drew inspiration from how multiple-valued complex functions such as square-root and can still be treated as holomorphic, by “snipping” and then “layering” to fan out their branches.
For square-root, let us snip the non-negative real axis out of the complex plane. This leaves an open subset , on which every has a unique square root with positive real part. The function is analytic on , as is the other branch . To get them to coexist as a single entity with the essence of being holomorphic, however, requires a way of building “layers” on , and on other open subsets as needed to cover the part of that was snipped out.
Here is where the edifices become tall and the abstraction too steep to cover in a single post. Considering the infinitely-branching function on one hand, and infinitely many degrees of equations on the other, we can expect that infinite structures will be employed. Indeed, Grothendieck built them above every open subset in a “glued-together” manner. We cannot even easily resort to our usual sign-off to “see the paper for details.”
Yet we can say that the structures carry the idea of “becoming” points in via the concepts of fibres and sheaves, and that nilpotent elements are employed. Einstein’s relational foundation is actuated by what Grothendieck termed his “relative view” of defining morphisms between representations as regulated by category theory, rather than defining stand-alone objects. The category of sheaves is abstracted to topos theory, by which the Greek word for “place” supplants the original idea of “point.” His French word étale described a flat sea as “spreading” like these layerings. It further reflects his heritage by deriving from a German word estal meaning “place,” whereby it also connotes spreading out goods in layers in a stall that can be one of many spread out over a marketplace.
All this became massive, so much that Grothendieck’s manuscripts before and after leaving IHES spread to hundreds and thousands of pages, as well as his personal memoir in the 1980s. Indeed, as related by Winfried Scharlau in a 2008 article for the AMS Notices, some of Grothendieck’s colleagues believed that he tired at the prospect of climbing his own mountains.
We can “morph” the description of Grothendieck’s “rising sea” approach in an essay by Colin McLarty to say that Grothendieck preferred to harness surveyors, engineers, and dam-builders so he could float to the top on rising waters, rather than do the ascent by “hammer and chisel.” He decried Pierre Deligne’s 1974 completion of their program of proving the famous conjectures by André Weil by methods he and others felt were not “morally right” on account of bypassing Grothendieck’s still-open “standard conjectures.”
A 2004 article by Allyn Jackson reproduces a cartooned abstract by Grothendieck for a colloquium in 1971 by which he warned that doing his lecture “in black-and-white detail for Springer Lecture Notes would likely take 400–500 pages,” ending by writing that from “a life-distancing logical delirium” it was “high time to change course.” Today’s mathematical community has in two years still barely touched a similarly-motivated though procedurally different theory erected by Shinichi Mochizuki on foundations named for Oswald Teichmüller, despite its “mere” 512 pages in first drafts.
So what can all this mean for us who work in what Grothendieck described as a “mansion” in which “the windows and blinds are all closed,” while he was one of those “whose spontaneous and joyful vocation it has been to be ceaseless building new mansions”? At least he did not call our dwelling a cave. However, in complexity theory we have it worse than Plato’s cave-prisoners in not merely missing the blinding world outside, but sensing its impact as a negative image in our present ignorance of lower bounds.
Much of complexity theory translates naturally to questions about polynomials over finite fields. This goes not only to for Boolean functions but also to and for higher primes and , which in turn yield questions about Boolean solutions to equations over these fields. There are possible advances to be had by improving the partial correspondence to problems in zero characteristic. The larger program out of the Weil conjectures seeks to transfer geometry to positive characteristic. Can we see how its further development might allow us to extract combinatorial results needed to put bounds on complexity?
Polynomials modulo composite numbers give us a more immediate frontier, one represented by the complexity class , whose nonuniform version was only recently separated from nondeterministic exponential time. These polynomials behave badly in manners stemming from nilpotent elements in the rings for composite . Can we somehow supply “extra points” and valuations to raise their structure toward that of polynomials over finite fields, and thus at least achieve bounds known for polynomials modulo primes?
A third example, my favorite and most immediate for the theme of this post, concerns the famous lower bound of Volker Strassen and Walter Baur on the size of arithmetic circuits computing some natural families of polynomial functions in zero characteristic. Its proof, as we recounted in 2010, turns on a property of geometric degree that pertains only to affine Euclidean space (or to its projective cousin). It employs the ideal generated by
where the “mapping variables” ensure the ideal is prime since the graphs of all mappings are irreducible varieties. Unfortunately, the highest geometric degree attainable for when has ordinary total degree is , whose logarithm for gives the known lower bound. This bound however holds for simple functions such as , while comes nowhere close to the exponential lower bounds we conjecture for functions such as the permanent.
Higher algebraic geometry has yielded notions of “algebraic degree” that can go one or more exponential orders higher, typically . If the Strassen-Baur technique could be transferred to their higher spaces, then we could hope for strong lower bounds. The étale idea and related facets of algebraic geometry and representation theory also animate Ketan Mulmuley’s “Geometric Complexity Theory” programme. I once tried to find a combinatorial shortcut using a degree-like measure of counting “minimal monomials” in ideals, which we described here. It is striking that the determinant polynomials score zero on this measure, whereas the permanents score astronomically even for , but as with the other degree measures , there are counterexamples to being a circuit size lower bound.
Can the answer to versus be caught up in the “rising sea”? Or will it need something even stronger than “hammer and chisel”? What can we learn from his work? A sign of hope is that for all their heft and abstraction, his schemes can be programmed.
Our condolences to his relations and friends.
["life-enrapturing"->"life-distancing"]
Zohar Manna is an expert on the mathematical concepts behind all types of programming. For example, his 1974 book the Mathematical Theory of Computation was one of the first on the foundations of computer programming. He wrote textbooks with the late Amir Pnueli on temporal logic for software systems. As remarked by Heiko Krumm in some brief notes on temporal logic, there is a contrast between analyzing the internal logic of pre- and post-conditions as each statement in a program is executed, and analyzing sequences of events as a system interacts with its environment.
Today I want to talk about an encounter with Zohar years ago, and how it relates to a puzzle that I love.
Zohar did one of the coolest PhD theses ever. It was on schema theory, which we have talked about before here. He was advised by both Robert Floyd and Alan Perlis in 1968—two Turing Award winners. The thesis Termination of Algorithms was short, brilliant, a gem. I recall when I started as a graduate student reading it and thinking this was beautiful. His own abstract is:
The thesis contains two parts which are self-contained units. In Part 1 we present several results on the relation between the problem of termination and equivalence of programs and abstract programs, and the first order predicate calculus. Part 2 is concerned with the relation between the termination of interpreted graphs, and properties of well-ordered sets and graph theory.
I will explain the puzzle in a moment, but first let me describe the encounter I had with Zohar. At a STOC, long ago, I saw him and started to explain my result: the solution to the puzzle. After hearing it he said that he liked the solution and then added that he once had worked on this problem. I said that I thought I knew all his work and was surprised to hear that. He smiled and seemed a bit reluctant to explain. Eventually he explained all.
It turned out that he and Steven Ness had a paper On the termination of Markov algorithms at the 1970 Hawaiian International Conference on System Science, held in Honolulu. Zohar explained the conference was not the “best,” but was held every year in the Hawaiian Islands. In January. Where it is warm. And sunny. It sounded like a great conference to me.
I soon went to a couple of these conferences. I stopped going after they accepted one of my papers, then rejected it—this is a long story that I will recount another time. The “accepted-rejected” paper finally did appear in an IEEE journal.
Zohar explained that he went to the conference for the same obvious reasons. He also explained to me the “three-person rule.” The Hawaiian conference was highly parallel and covered many areas of research. Zohar said that you were always guaranteed to have at least three people in the room for your talk: There was the speaker, the session chair, and the next speaker. Hence the three-person rule.
The issue is the distributive law:
Consider any expression that has only variables and the two operations plus () and times (. Suppose one applies the distributive law to the expression, in any order. One stops when there are no further possible applications. The question, the puzzle, was: Does the distributive law always stop? Of course it does. Or does it? The puzzle is to prove that it actually does stop.
I raised this recently as my favorite result, with a smile, during my Knuth Prize lecture at FOCS. I said I had a nice solution and would be glad either to give it or let the audience think about it. They seem to want to think about it, so I gave no solution.
My lecture had been right before dinner, and the next day I spoke to some people about whether they’d thought about it. A few said they had some idea of how to proceed, but no one seemed to have a proof. The reason the problem is a bit challenging is that the rule increases the size of the expression: two copies of now appear instead of one. This means that any local measure of structure may fail.
Indeed Zohar’s proof uses a well ordering argument that is not too hard, but is perhaps a bit hard to find. Check out his paper with Ness.
The first thing I noticed immediately when I heard the problem—see here for the context—was this: The distributive law preserves the value of the expression. We apply it because it expands the expression but it does not change the value of the expression. A rule like
does not preserve value. So who cares whether it halts or not?
But the distributive law preserves the value. So here is a proof based on that observation. Notice the following two things:
The trick is to replace all the variables by . The value stays the same, but it is easy to argue that if the rule never stops then eventually the value of the expression would increase without bound. This is a contradiction.
Are there other termination problems that can be attacked in this way?
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A pointed question about the plane
Stanisław Ulam was one of the great mathematicians of the last century. We talked about him in a recent post on his prime spiral and other strange mathematical facts. He is associated with several famous problems, including the 3n+1 problem and the Graph Reconstruction conjecture.
Today we want to talk about one of his oldest conjectures.
The conjecture was first stated in 1945. It is simple to state, seems plausible that it is true, but so far has resisted all attempts at resolution. René Descartes could have stated in the 1600s—well almost.
You can skip this and the next section on Ulam and get right to his conjecture. But its been open almost sixty years, so it can wait a minute or two.
Ulam did amazing work that impacted a vast part of mathematics and physics. He also wrote popular articles that were wonderful to read. The areas include: set theory, topology, transformation theory, ergodic theory, group theory, projective algebra, number theory, combinatorics, and graph theory. One particular example is his invention of the Monte Carlo method in the 1940′s, while working at the Los Alamos National Laboratory. The name, Monte Carlo, is due to Nicholas Metropolis, after the casino, where Ulam’s uncle, Michał Ulam, was supposed to have frequently played.
The reconstruction conjecture states roughly that every undirected non-trivial graph is determined up to isomorphism by its subgraphs. Paul Kelly also gets credit for his slightly earlier work. This conjecture is wonderful in that it seems so plausible: Take a graph and give us all the subgraphs. How can that not yield the original graph? Yet it is still open since around 1960. One aspect that is especially teasing is that it has been solved in the affirmative for regular graphs, which are the hardest case for graph isomorphism, but this does not tell us much about the general case.
In 1976 Ulam published his autobiography: Adventures of a Mathematician. Here is the cover of the 1991 version:
When the book was first published, we in the theory community all read it. It was quite successful, on all measures, since Ulam was a wonderful writer. He mixed stories of people, events, and mathematics; all put together in an engaging way.
The book had another hook: Ulam included an easy-to-state puzzle that many immediately begin to work on and try to solve.
He asked for the maximum number of questions required to determine an integer between one and a million, where the questions are answered “Yes” or “No” only, and where one untruthful answer is allowed.
This is binary search with one lie allowed.
There is a strategy that is not too bad. At worst we could just do binary search asking each question twice. If we get the same answers to a question we know all is well; and if we get two different answers we know one is a lie. So this works in in worst case.
The trick is that it is possible to avoid such a brute force method and come much closer to the usual binary search bound of . Of course we are not worrying about rounding off the value of , which is a small matter.
The problem became a race among many of us and the finish line was hit quickly. His original problem can be done in exactly questions. This is pretty close indeed to the binary search value. The general answer for even any constant number of lies was found to be
The constant depends on the number of lies. Thus lies have a low order effect on the search time. I always liked this result and wonder whether if it has been used in practice? The result is due to Ron Rivest, Albert Meyer, Danny Kleitman, Karl Winklmann, and Joel Spencer. They beat us all—perhaps an unfair fight—with such a powerful group.
Let’s now state Ulam’s really old problem. Not the reconstruction conjecture from the 60′s nor his search problem which was solved quickly in the ’70s, but his question about points in the plane.
Say a subset of the points in the Euclidean plane form a rational-distance (RD) set provided the distance between any two points in is a rational number. Thus the four points at the corners of a unit square is not an RD set. Of course we know that the diagonal distances are which is not rational.
It is easy to construct an infinite set that is a RD set. Take all the points with is rational. These clearly have all rational distances: The distance between and is
This can be done for any line and the same idea can be made to work for circles.
Notice that every line and every circle is a sparse subset of the plane. So Ulam made the natural conjecture:
If is an RD set then it is not dense in the plane.
Recall that a set is dense provided every open disk in the plane contains at least one point from the set. Paul Erdős conjectured that if a set is an RD set, then should be very special. Indeed. None has been found after many attempts. See this for some progress.
Do RD sets exist? What about restricting the distances to lie in other subfields of the reals? I believe that it should be easy to prove there is a proper subfield of the reals so that there is an infinite set of points whose distances all lie in this field. However, to classify these subfields is clearly very hard, since it would solve the Ulam conjecture and more.
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Discussing lower bounds for this blog’s 600th post
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Superman, Batman, and Wonder Woman are three of the best known superheroes in DC Comics books, which were a childhood staple for many of our age. They brought us action in color before our families had color TVs. There is a new nonfiction book on Wonder Woman. She and Superman have super powers: flying, super strength, X-ray vision, magical healing, and more. It is notable that Batman does not. All of his powers come from great training, great study, great equipment, and great helpers.
Today this blog makes post number 600, DC in Roman numerals.
Our joint-author handle “Pip” recently passed 100 posts, and I have 67 more as single author. We now have over 15,000 non-spam comments. Some technically-detailed posts take much of the full average 4–5 day interval to write, while others with lighter fare or discussion points are quicker. We are always open to suggestions for kinds of posts and balance of content.
DC can stand for various things: Washington where power changed hands on Tuesday, direct current, da capo in music meaning to restart from the beginning, even Dyson’s Conjecture. It is also the product code for all vacuums made by James Dyson (no close relation to Freeman Dyson). But we will stay with DC Comics and the question:
Will it require “super powers” to break down the lower-bound barriers?
We have of course discussed barriers often before, but they came up again last week at a workshop at MIT for Michael Sipser’s 60th birthday.
Superman has one weakness of a kind not shared by Wonder Woman or Batman: he becomes powerless in the presence of kryptonite. The name and material come from Superman’s home planet of Krypton, though substances with much the same chemical formula were found to exist innocuously on Earth. There are a huge variety of forms of kryptonite in official DC Superman stories, not to mention others. In complexity theory we also have several forms of kryptonite, of which the three best known are:
New radioactive varieties may yet be enriched from sources such as the hardness versus randomness tradeoff, by which certain strong lower bounds require the ability to de-randomize certain algorithms (and in several senses vice-versa), or sameness of average-case complexity by problems of different worst-case complexity. Weaponry that was supposed to penetrate to the core but was melted or blunted include the Sipser programme for circuit lower bounds and the Fusion Method. The ravages in our knowledge are far worse than ignorance of whether :
The last item hints at a kind of kryptonite that no champion of “truth, justice, and the American Way”—at least not the first of these—can overcome. Perhaps many of the lower-bound statements we have been trying to prove are just false.
Superman’s greatest enemy, Lex Luthor, reminds us that “lex” stands for binary words, which are the Gotham City we are fighting to control. What is it about information that refuses to be known? As is clear from a post made by Bill Gasarch about the Sipser-fest, questions like these filled the halls. A discussion of the Fusion Method with its co-originator, Mauricio Karchmer, led to Karchmer saying the following:
(1) We can’t prove lower bounds because they are false, and (2) we can’t prove upper bounds because we are stupid.
A followup comment by him says he was not kidding.
What makes this double trouble is a theme we have talked about on this blog, that proving a good lower bound often involves finding a new upper bound. Bill gave eleven examples in a post last year titled, “Are most lower bounds really upper bounds?”
The link works in the other direction with Natural Proofs and hardness-versus-randomness: from any lower-bound proof (of a broad kind) for certain problems (again of a broad kind) one can derive a new algorithm for some other problem. For discrete log it can be the same problem, and this leads to ironclad nonexistence of natural proofs of its difficulty. Lance Fortnow has just now remarked the following about Ryan Williams and his circuit lower-bound for , which is Bill’s eleventh example, and which we covered two years ago:
Recently Ryan has turned that process around, for example getting a faster algorithm for all-pairs shortest path using the techniques from the Razborov-Smolensky circuit lower bounds.
To our mind, this sends up a natural challenge, at least as far as making survey charts of objectives to guide people entering our field:
Which upper bounds seem hard to prove because we might be stupid, and which seem hard to prove because they imply lower bounds that might be false? And vice-versa…
As we said in our intro, Batman is distinguished by not needing to rely on any super powers, just diligence and training and study and a sonar sense of where to attack. A new movie, tentatively titled “Batman v Superman: Dawn of Justice,” is in production for a 2016 release. So what chance does a Batman have in our arena?
Dick and I have tried to offer a “progressive” outlook that at least frames the issue of what an algorithm might be said to “learn,” and trial notions of constructiveness and succinctness. These may be mere gadgets, but Batman after all keeps a large utility belt. There are of course many other ideas out there, and Batman has no illusion that just using any one of them will bring the hard cases down.
What we suspect is needed, however, is a deeper strategy—a new algorithmic idea. We are struck that many breakthroughs in the past—including several mentioned above—have come from perception and application of a new procedural idea. Even diagonalization is algorithmic—given a function from a set to its power set, it shows how to find a subset of that is not in the range of . Indeed, for any function on , we can define , and there is still scope for interest in how the diagonal sets one obtains depend on the function as well as on .
What we wonder is whether there is greater scope for “introspection” of the process of executing a proof, specifically the effect of complexity limitations on the process. Suppose a deduction hinges on whether certain functions on a family of sets are 1-to-1. Suppose this may in fact be false, but is a cryptographic hash function whereby finding colliding pairs such that is hard. Already we have tools for relating this kind of hardness to independence in (weak) formal systems, and many papers on the proof-theoretic power of various degrees of the pigeonhole principle have faced this issue. We still feel this is a cave worth delving deeper. Independence results are also mentioned in Bill’s post, quoting Steven Rudich as telling him at the Sipser workshop that they too are alas not close.
In DC Comics, Batman was eventually able to seize Lex Luthor’s kryptonite ring, though he has used it only in case Superman goes wrong. Perhaps with more-clever self-reference we can turn it around and find a sense in which a little kryptonite can be exploited for good?
Can Batman make progress, or will we need a new super power idea? What do you think?
We also salute Mike on his 60th.
[minor word and symbol tweaks]
By permission of Renee Bolinger, artist : source |
Kurt Gödel left a large amount of unpublished writings and notebooks and preserved correspondence. Called his Nachlass, German for “after-leavings” or bequest, these writings were catalogued and organized by several—including his first biographer, John Dawson, for a heroic two years. Those of highest scientific and general interest were published in volumes III, IV, and V of Kurt Gödel: Collected Works. Among them was a list of 14 numbered assertions titled “My philosophical viewpoint” but without elaboration. They are believed associated to a lecture Gödel started preparing in the early 1960s but never gave, whose draft is in the Nachlass.
Today we are delighted to have new communications from Gödel, as we have previously received around Halloween and All Saints’ Day, so we can continue our series of interviews with him.
What the Nachlass shows clearly is a perfectionist at work. Dawson’s biography relates that a two-year process of trying to publish a lecture that Gödel did give bogged down so much that in the manuscript found in his Nachlass:
…several whole pages are crossed out and two series of interpolations have been added, the first consisting of insertions to the body of the text, the second of footnotes. The sequence in which the interpolations occur in the text is chaotic, and the interrelations among them are byzantine: There are insertions within insertions, insertions to the footnotes, footnotes of insertions, and even footnotes to other footnotes. The situation finally became so confusing that Gödel himself found it necessary to draw up a concordance between the interpolation numbers and their locations.
The Harvard philosopher Warren Goldfarb, who corresponded with Gödel and later became one of few people to find and fix a mathematical mistake by Gödel, was a co-founder of the Collected Works project. He prefaced a lecture he gave in 2011 on Gödel’s 14 points by saying:
When Gödel does express his own outlook, he tends to do so in suggestive assertions rather than developed exposition. Detail, both about the content of his viewpoints and his reasons for holding them, is lacking. So amplification would be needed if there’s to be a real account of his general philosophical position.
Immediately next, however, Goldfarb related that only a trace of at least 25 notebooks in the Nachlass devoted to philosophy have been transcribed out of the old and difficult shorthand used by Gödel. He admitted that the body of his lecture would be conjectural, and might need revision as further pages are brought to light.
Dick and I realized we might not need to wait. Since Gödel had written and left notebooks, perhaps he would share their contents with us directly if we asked him. However, owing to physical issues like those already affecting our earlier communications with Gödel, we wound up doing much the same kind of “amplification” as Goldfarb posited.
First, here are Gödel’s 14 assertions as newly transcribed by Eva-Maria Engelen for her 2014 paper on Gödel’s philosophy. I have deviated slightly from her translation in points 7, 9, and 13.
To these one could add a 15 point, preceding them: Gödel went even beyond the controversial principle of sufficient reason to maintain that existence is undergirded by purpose. This underlay his belief in both an afterlife and a before-life in point 5.
Gödel had broken off our previous session with the warning that we would “need new science” to hear from him again. Happily, new developments regarding black-hole firewalls emerged in time, and by wormhole mechanisms described in this New York Times article, we were able to re-establish contact. Or rather, he did.
We had fortunately already told Gödel that we would like to ask him about the 14 points. For all his reputation of strangeness, he was most obliging personally and saw response as duty—if he could be sure of his response.
The trouble is that the wormholes “split into a zillion spaghetti-like strands” as the article says. However, because they are “geometric manifestations of quantum entanglements,” we could hope to collect them with a quantum computer. We bought time on two D-Wave machines, one to filter large amounts of SETI data and the other—the one owned by Google—to filter Web data as a control. The SETI data showed a systematic anomaly for which there could only be two explanations: either it was proof of higher beings on other worlds (per Gödel’s point 4) or it was Gödel himself (which could be the same thing).
Transcribing what he sent was still a problem. We needed to deduce from the firewall theory how to invert his channel, generate and encode lots of candidate plaintexts for what he could say, and pick the one with the highest correlation. Random trials and trigram cryptanalysis showed that Gödel was using a code based on the shorthand system. We used a correlation program Dick had obtained from the engineering anomalies and ESP detection project run by his former dean at Princeton. We still got only a fraction of what he sent.
We have arranged the few fragments from Gödel in the order of his 14 points. Some points are left out. We had wanted to hear more on point 3 since it seemed to contradict the essence of his incompleteness theorems, that David Hilbert’s programme of a systematic method to solve mathematical problems must fail. And what are “art problems”? But at least he began with point 1. It is only Gödel speaking—we hope by next year that physicists will have worked out the firewalls problem enough for us to transmit and make a proper interview again.
Gödel: 1. Die Welt ist vernünftig. That should be obvious of course, but people have high appetite and forget the ‘negative’ logical consequences of this. One is that rationality requires privation. If you don’t deprive possibilities then you have the chaos of the quantum microworld where everything that can happen does happen. We can make sense of quantum probability only from higher structure, and using it to process information needs clever interference, to make more things impossible. On the human scale, the action by which rationality is grown is learning, but this must start from privation. Also, as pre-Renaissance scholars maintained, privation is the gateway to evil, which is thereby unavoidable. In this I follow the older [Gottfried] Leibniz, as with much else.
Learning is the essence of the human condition, and I believe—at least hope—it is unlimited. Otherwise there is a shoreline of elementary truths about numbers that we can never learn, but why should it discriminate only us? For example, every consistent formal system has a sharp finite limit on the length of sequences it can prove to be even minimally random, although maximally random sequences abound at all lengths. Well, even if we have a limit, there is no reason for the Universe to be so artificially constrained, hence higher beings must exist. If we have no limit then we must become the higher beings. Either way my point 4 is true, but because the universal computation which we apprehend is absolute, and because I agree with Emil Post about the implication of mine and [Alonzo] Church’s and [Alan] Turing’s work for creativity, I believe also 5.
Dick and I guess that points 2 and 3 (and later 8) had to do with more reasons for “no limit” and creativity, though we still wish we’d found what Gödel meant about “art problems.” As my dog stared at me asking to do tricks for treats, I wondered whether the same argument would prove “higher dogs,” but I guess dogs do not apprehend universal computation. Since math knowledge is a-priori, his point 6 seems obvious, though we wished to know why “incomparably.” Point 7 is covered copiously early in Goldfarb’s lecture even though he had the older shorthand transcription (einsichtige, “intelligible,” instead of einseitige, “one-sided”—away from a-priori knowledge toward positivism which Gödel deplored as self-contradictory and attempting to “prove everything from nothing”).
Our next fragment hit points 9–11, evidently in mid-stride:
Gödel: Again we must be logical about the consequences of what we subscribe to. If we really hold “it from bit” then information not material is primary. Only information and mathematical consequence can destroy sameness—when helium transmutes into carbon it does so because of equations. If we derive unlimited other worlds, as [Hugh] Everett did, then we have worlds with identical configurations of matter to us and our environment. If these Doppelgängers are not us then something besides material is primary to identity; but if they are us then we must mysteriously think of the configurations as spread out beyond causal reach yet automatically united in identity. In either way, materialism is untenable, so for something to be a whole, it must have a separate object nature. You can make a good analogy to “object” in programming, as in Simula or Smalltalk: its nature is apart from any instance. Formal correctness and the moral idea of formal right behavior can be put on the same scientific ground this way, and it is the formal not the material component that actuates it, that creates what we know in our reason as reality.
Also we should not disparage analogy to insist always on proof. As I wrote, we do not analyze intuition to see a proof but by intuition we see something without a proof. Intuition comes first by senses and by analogy—by patterns—and analogy is the only ratchet for becoming a higher being. Association is more powerful than composing rules—I believe you will find this also with databases.
Dick and I were dizzied by the swing from quantum many-worlds to concrete topics in software systems, but we had to give Gödel credit for concreteness, and he made points 9-10-11 hang together. Point 12 was obvious since we, like he, are Platonists, and we didn’t miss it. We wish we could hear more about Gödel’s opinions on Everett than those at the very end of these notes by Rudy Rucker of his conversations with Gödel. Everett not only was a student of John Wheeler in Princeton, but also advanced the following, which intersects with Gödel’s version (via Leibniz) of St. Anselm’s “Ontological Proof” of the existence of God:
source |
This scrap was found among Everett’s papers only a few years before it appeared in his biography by Peter Byrne. Byrne’s speculates in his main text that it was answering Gödel, but instead I agree with two others acknowledged in a footnote that it came earlier and was a short form of Everett’s “Universal Existence Theorem” from the 1950s, when he might have been provoked by Bertrand Russell’s respect for the argument. Whether Gödel heard of the issue raised by Everett through the Princeton grapevine or not, I suspect he grasped this clash between existence as a predicate and under quantification, and sought to mitigate it by limiting his deduction to “possibility implies necessity” for any essence with abstract properties ascribed to God. Our last fragment from Gödel seems to pick up from all this:
Gödel: What we commonly regard as objects have concrete boundaries and hence do not have the highest abstraction. Sets have boundaries in how they are defined, and this explains the failure shown by Russell in trying to regard “the set of all sets” as both an object and the highest abstraction. This also supports my belief that there is no “shoreline” for mind in mathematics, and that the human mind suffices to partake of this unboundedness—whereas if it is captured by a bodily organ then it is bounded. It is rather the possibility to make the “all” into an object that drives our expandability, much as the ordinals in set theory transcend all boundaries. To echo Thomas Aquinas, this conveys the conception of God as—if only as—possibility, and frees us from worry that this conception is logically inconsistent.
For example, I was guided concretely by this analogy to my formulation of set theory with Johnny [von Neumann] and [Paul] Bernays, in which “all sets” has the highest and last place and is not bounded. It is not “the all” but plays that role with regard to sets. Mathematics benefits from well-guided purpose the same way Albert [Einstein] established correct physics by taking deeply abstract consequences in his Gedankenexperimente, and as I wrote, this can be fruitful for all the sciences.
Dick and I had wondered at some religious allusions spoken by Gödel in our last interview about his set theory, and while they are absent from his known writings, we cannot read his points 13 and 14 other than that Gödel was influenced by such thinking.
To quote Goldfarb’s final words in his lecture,
Can any sort of view of this type be developed in a plausible way, whether or not Kurt Gödel himself got there?
We hope we can learn more, from further transcribed writings or from “amplification” and coherence testing, how far he got or could go.
[received permission for artwork at top, minor word changes]
Elie Track and Tom Conte were co-chairs of the recently-held Third IEEE workshop on Rebooting Computing. Tom is a computer architect at Georgia Tech. A year ago he became the 2014 President-Elect of the IEEE Computer Society, which according to both the press release and Conte’s Wikipedia page entails that he “currently serves as the 2015 President.” I guess that is one way to stay focused on the future. Track is president of the IEEE Council on Superconductivity. A year ago he founded nVizix, a startup to develop photovoltaic power, and serves as CEO. He is also a senior partner at Hypres, a superconducting electronics company.
Today I wish to relate some of what happened last week at the meeting.
The meeting was held in Santa Cruz and had about forty participants. I was honored to be one of them. The problem we were given to consider is simple to state:
Computers are growing in importance every day, in all aspects of our lives. Yet the never-ending increase in their performance and the corresponding decrease in their price seems to be near the end.
Essentially this is saying that Moore’s Law, named for Gordon Moore, is about to end.
Well “about” is a bit tricky. Some would argue it has already ended: computers continue to have more transistors, but the clock rate of a processor has stayed much the same. Some argue that it will end in a few decades. Yet others claim that it could go on for a while longer. Lawrence Krauss and Glenn Starkman see the limit still six hundred—yes 600—years away. This is based on the physics of the Universe and the famous Bekenstein bound. I wonder if this limit has any reality when we consider smartphones, tablets, and laptops?
In 2005, Gordon Moore stated: “It can’t continue forever.” Herbert Stein’s Law says:
“If something cannot go on forever, it will stop.”
I love this law.
The workshop was divided into four groups—see here for the program.
Let me say something about each of these areas. Security is one of the most difficult areas in all of computing. The reason is that it is essentially a game between the builders of systems and the attackers. Under standard adversarial modeling, the attackers win provided there is one hole in the system. Those at meeting discussed at length how this makes security difficult—impossible?—to achieve. The relationship to the end of Moore’s Law is weaker than in the other groups. Whether computers continue to gain in price-performance, security will continue to be a huge problem.
Parallelism is a long-studied problem. Making sequential programs into parallel ones is a famously difficult problem. It is gaining importance precisely because of the end of Moore’s Law. The failure to make faster single processors is giving us chips that have several and soon hundreds of processors. The central question is, how can we possibly use these many processors to speed up our algorithms? What I like about this problem is it is really a theory type question: Can we build parallel algorithms for tasks that seem inherently sequential?
Human Computer Interaction (HCI) is about helping humans, us, to use computers to do our work. Gregory Abowd of Georgia Tech gave a great talk on this issue. His idea is that future computing is all about three things:
Of course “clouds” refers to the rise of cloud computing, which is already an important area. And “crowds” refers not only to organized crowdsourcing but also to rising use of social media to gather information, try experiments, make decisions, and more.
The last, “shrouds,” is a strange use of the word, I think. Gregory means by “shrouds” the computerized smart devices that we will soon be surrounded with every day. Now it may just be a smartphone, but soon it will include watches, glasses, implants, and who knows what else. Ken’s recent TEDxBuffalo talk covered the same topic, including evidence that human-computer partnerships made better chess moves than computers acting alone, at least until very recently.
The last area is randomness and approximation. This sounds near and dear to theory, and it is. Well, the workshop folks mean it in a slightly different manner. The rough idea is that if it is impossible to run a processor faster, then perhaps we can get around that by changing the way processors work. Today’s processors are made to compute exactly. Yes they can make errors, but they are very rare. The new idea is to run the processor’s much faster to the point where they will make random errors and also be unable to compute to high precision. The belief is that perhaps this could then be used to make them more useful.
The fundamental question is: If we have fast processors that are making lots of errors, can we use them to get better overall performance? The answer is unknown. But the mere possibility is intriguing. It makes a nice contrast:
Today we operate like this:
Design approximate algorithms that run on exact processors. Can we switch this around: Design algorithms that operate on approximate processors?
I note that in the narrow area of just communication the answer is yes: It is better to have a very fast channel that has a reasonable error rate, than a slower exact channel. This is the whole point of error correction. Is there some theory like this for computation?
There are several theory-type issues here. What really are the ultimate limits of computation? Can we exploit more parallelism? This seems related to the class , for example. And finally, can we exploit approximate computing to solve traditional problems?
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For Martin Gardner’s 100th birthday
Global Science source |
Martin Gardner introduced many including myself to the joys of Discrete Mathematics. His glorious monthly column “Mathematical Games” for Scientific American included some continuous mathematics too, of course; one could say it was on “Concrete Mathematics.” However, I conjecture—based on a quick flip of the several books I own of his columns—that the symbols in a calculus context never appeared in them.
Yesterday was the 100 anniversary of Gardner’s birth. Dick and I wish to join the many others marking this centennial and thanking him for all he did to make math fun for so many.
His feature kicked off in 1956 with the famous column on hexaflexagons, which I will talk about in a moment. Gardner related in his autobiography, which was assembled three years after his death in 2010, how important this column was as his “break.” However, the column that made the most lasting impression on me began with the words:
The calculus of finite differences, a branch of mathematics that is not too well known but is at times highly useful, occupies a halfway house on the road from algebra to calculus.
This discrete “calculus” enables one to calculate a formula for any polynomial sequence given enough values for It also led to my favorite “visual proof” that : For any integer , if you write out the powers going across and take differences of adjacent values repeatedly to make an infinite equilateral triangle pointing down, the left side has the powers of . Iterating this gives you the powers of , but the entry for as counts down to steadfastly remains .
Tributes have been gathered all during this centennial year. Scientific American observed yesterday by posting a review of ten of Gardner’s most appreciated columns. Bill Gasarch’s post yesterday links to some of his and Lance Fortnow’s previous items on Gardner, and further to a site where anyone can contribute a testimonial.
Frederic Friedel, who co-founded the chess-database company ChessBase three decades ago, knew Gardner personally from 1979 as a fellow original member of the Committee for Scientific Investigation of Claims of the Paranormal (CSICOP, now CSI). The committee remains housed in my town of Amherst near Buffalo, now at the Center for Inquiry (CFI Western NY) which is across Sweet Home Road from my university campus. Friedel has described to me cold days in Buffalo and round tables with Carl Sagan and other luminaries. All this was before my own arrival in 1989.
Friedel was also among the column readers with whom Gardner interacted from the beginning in the 1950s. His awesome tribute yesterday includes appreciation of Gardner’s book Fads and Fallacies in the Name of Science, which also made a strong impression on me, and other links. Dick recalls the great chapter of that book that starts with Gardner saying that this next crazy claim cannot be disproved. It was that the universe was created recently with a full fossil record that makes it look much older. Indeed, it could be a so-called “Boltzmann Brain”—and a point made in this NY Times article is that it’s crazy that this is not crazy.
I never had any contact with Gardner, despite making a few visits to CFI; it ranks among numerous lost opportunities. I could mention many other influences from his columns, and looking through his book Mathematical Circus just now reminded me that his chapter on “Mascheroni Constructions” was my first knowledge of what I called the “class ” in my “STOC 1500″ post with Dick. I had a similar experience to what Douglas Hofstadter told in his own tribute in May 2010: I started off interested in Math+Physics, intending to take the latter as far as quantum mechanics and particles at Princeton. But I found advanced mechanics and electrodynamics tough going, and am ever grateful for being allowed to parachute out of the latter at midterm into Steve Maurer’s Discrete Mathematics course, in which I knew I’d found my métier. As I could have realized from my love of Gardner all along.
I’ve wanted to make a post on my hexaflexagon twist when I had time to create nice pictures, but now I will have to make do by referring to the fine illustrations in Gardner’s original column, which is freely available from the M.A.A. here. It requires making the standard “hexa-hexa” as shown in Gardner’s Figure 2. For best effect, in addition to numbering the faces 1–6 as shown there (and using a solid color for each face), label the six components of each face A–F in the left-to-right order given there.
The “Twist” is always applicable from one of the three inner faces (1, 2, or 3); finding when it applies from one of the outer faces and from the configurations that follow is more of a challenge. Instead of flexing as shown in Figure 3, follow these directions:
What you will get is a flexagon with the colors on its faces jumbled up—if you’ve used the lettering, you will have ’1C’, ’5B’, and ’2F’-’2E’-’2D’-’2C’ clockwise from upper right. You will still be able to flex it the standard way, but only exposing one other face—that is, you will have something isomorphic to a tri-hexaflexagon.
Now the real fun is that you can iterate this process. For one thing, you can invert it to restore your original hexa-hexaflexagon (teasing ’2E’ and ’2F’ forward and folding in ’1C’). But you can also find other places from which to initiate another “Twist,” and these will lead to more tri-hexa configurations. One is to flip it over, rotate once counterclockwise so you fold backwards with ’6B’ and ’3C’ at right, tease forward ’3E’-’3D’, tuck ’3C’ into the bowl atop ’1D’, collapse and grab at the other end of ’2A’-’6A’, lift flap ’2D’ out of the bowl, and unfold to see ’2D’-’4D’-’6A’-’2A’-’3E’-’3D’. Then you can flip over, rotate once more counterclockwise, and iterate—but there are other twists too.
Some will lump up thick wads of paper on three triangles of each face, so be ginger about it. Finally, after much exploration, you may come upon the “Dual Hexa.” This has six faces, in which the inner three alternate colors. It is, in fact, the configuration you would build if you first rotated the top part A of Gardner’s Figure 3 by 180 degrees. Then you may find a way to go from the primal to the dual and back by a long regular pattern of repeated twists.
As a high-school student in 1976, I attempted to map out the entire space of reachable configurations by hand, but made some bookkeeping errors and gave up. I wanted to write a computer program to simulate my twists, but did not make the time.
Can you do the “Twist”? The space of configurations you can explore is much larger than the “Tuckerman Traverse” of the standard hexa-hexa shown in Gardner’s Figure 4. Can you map it all out? Has anyone previously known about this?
[some format and word changes, updated to include letters of facets.]
Waterloo Mathematics source |
Michael Rubinstein is an expert on number theory, who is on the faculty of the University of Waterloo. He is one of the organizers of a 61-birthday symposium being held December 15–19 in Princeton for my friend and former colleague, Peter Sarnak. I guess it is a matter of taste for a number theorist whether to observe a birthday with a lot of factors (60) or a prime (59 or 61). Rubinstein also does extensive experimental mathematics and lists several code libraries below his publications on his website, which also has interesting articles on the math, history, and practice of musical tuning.
Today Ken and I wish to discuss a paper of his on one of my favorite problems: integer factoring.
The paper appears in the 2013 volume of the electronic journal INTEGERS, one of whose sponsors is the University of West Georgia. It is titled, “The distribution of solutions to with an application to factoring integers.” He studies the structure of solutions to (mod a), and uses this to prove the following result:
Theorem 1 For any , there is a deterministic factoring algorithm that runs in time.
The “” hides sub-linear terms including coming from the divisor bound, logarithmic terms from the overhead in nearly-linear time integer multiplication, and related sources.
Factoring algorithms partition into two types: unprovable and provable. The unprovable algorithm usually use randomness and/or rely for their correctness on unproved hypotheses, yet are observed to be the fastest in practice for numbers of substantial size. The provable algorithms are usually deterministic, but their key feature is that their correctness is unconditional.
For those trying to factor numbers, to break codes for example, they use the fastest unprovable algorithms such as the general number field sieve (GNFS). The cool part of factoring is that one can always check the result of any algorithm quickly, so anytime an unprovable algorithm fails, it fails in a visible manner.
Why care then about slower algorithms that are provable? Indeed. The answer is that we would like to know the best provable algorithm for every problem, and that includes factoring. We are also interested in these algorithms because they often use clever tricks that might be useable elsewhere in computational number theory. But for factoring there is a special reason that is sometimes hidden by the notation. The GNFS has postulated runtime
where . This is not a similar order to . To see this more clearly, let be the length of in binary, so . Then the GNFS is roughly and slightly above , but is , which is not only miles bigger, it is a properly exponential function.
Nobody knows a deterministic factoring algorithm that beats properly exponential time.
The unprovable algorithms taunt and tease us, because they hold out the promise of being able to beat exponential time, but nobody knows how to prove it. Indeed, as Rubinstein remarks in his intro, each “teaser” is a conceptual child of one of the exponential algorithms. A reason to care about his new algorithm is its potential to be a new way to attack factoring, even though it currently loses to the best known provable methods. These are all slight variations of the Pollard-Strassen method, all running in -type times; there are also algorithms assuming the generalized Riemann hypothesis. See this for details.
Most factoring algorithms—even Peter Shor’s quantum algorithm—use the idea of choosing a number and working modulo . If and share a factor then we can quickly find it by Euclid’s gcd algorithm, while otherwise the problem of finding such that provides useful information. The starting point of Rubinstein’s algorithm is to do this for values that are right near each other, and compare the values obtained.
In its outermost structure, it uses this fact: Suppose where and are primes of the same length, so . Suppose . Then if we write
with (note they are relatively prime to or else gcd gives us a factor) we get that and are fairly small, about . They are also roughly bounded by and by . Multiples of them will stay small. So now let us work modulo for . We have:
so
Now an issue is that the values giving are far from unique. However, among them as will be the collinear points
The increments are small enough that the points stay close to each other in Euclidean space. If we divide the space into boxes of the right size and offset, they will all stay inside one box. If we can search a box well enough to find them and get all of exactly, then we get and . Moreover—and this is why I like factoring—once we find them we can verify that we have the right ones; if the check fails and we were fooled by -many other points we can move on.
Using , Rubinstein is able to show that the amortized number of “foolers” in a square is . Since there are -many such squares and , we get the target runtime. Note this is amortized, not just expected, so the algorithm is deterministic. The most clever and painstaking aspect is that estimates of the asymptotic convergence of solutions of to uniform distribution on are needed to get the “amortized” part. The details become complicated, but Rubinstein writes in an engaging top-down manner, and he includes a section on how this might—might—be used to break the sub-exponential barrier.
I found this work interesting not just because it is a new approach to factoring. I have tried in the past to prove the following type of result: Is there an asymmetric cryptosystem that is breakable if and only if factoring can be done in polynomial time?
I want to replace systems like AES with ones that uses the hardness of factoring for their security. Systems like AES rely on intuition and experimental testing for their security—there is not even a conditional proof that they are secure.
My goal is really trivial. Any public-key system can be viewed as an asymmetric system. But what I want is that the encryption and decryption should be very fast. This is one of the reasons that modern systems use private-key systems to create symmetric keys: performance. Using public-key systems for all messages is too slow.
My idea is not far in spirit from what Rubinstein does in his factoring algorithm. This is what struck me when I first read his paper. His algorithm is slow because it has no idea which “box” to look at. Can we share some secret that would allow to be factored faster, yet still make it hard for those without the secret?
Can we extend Rubinstein’s algorithm to break the barrier? Can his methods be used as described above to create asymmetric systems that are based on factoring? What is the real cost of factoring?
[fixed sentence about private-key/symmetric]
Adrienne Bermingham is the manager of this year’s TEDx Buffalo event, which will be held this Tuesday at the Montante Center of Canisius College in downtown Buffalo.
Today I wish to proudly announce that our own Ken Regan is one of the presenters at this year’s event.
Bermingham is the head organizer of the TEDxBuffalo events, which started in 2011. This year’s theme is In Motion. When not organizing she works in Anthrozoology, which is the study of how we, humans, interact with animals. As one who daily interacts with our golden retriever, I would love to hear any advice on how to make that better.
The TED organization is dedicated to getting information out to the world: ideas worth spreading. TEDxBuffalo is an example of a local group working with them to put on an TED event. Their event, TEDxBuffalo 2014, is suppose to be relevant to Buffalo, by Buffalonians. It has no keynotes, panels, or any of the usual stuff we see at conferences. No parallel sessions. Just a day of “engaging and refreshing your brain.” Of course as a non-Buffalonain I would expect that the talks, while focused locally, will still be interesting to the rest of us. Its a twist on the famous phrase “think globally, act locally.”
Quoting them:
Applying a theme to our TEDx event allows us to highlight a strength we’ve identified in our community, curate a series of talks that have the ability to build off of one another, and send a clear, powerful message to members of our community and TEDx video viewers across the globe.
This year’s theme builds upon TEDxBuffalo 2013, which celebrated our city’s “Renaissance Citizens”. Now that we’ve acknowledged our city’s renaissance, it’s time to recognize those who are hard at work bringing about positive change in our community—those who are truly “In Motion”.
The day starts at 9am and goes to 4pm. It consists of a dozen talks. Go here for the exact time schedule and also more information on the talks.
A little secret: Ken cropped me out of the photo he used. It was taken at a Barnes and Noble in Ann Arbor when we attended the Michigan “Coding, Complexity, and Sparsity” workshop. Oh well.
The talks are described by bios of the speakers. I must say that Ken’s talk I get completely: it will be on his research into chess. The others are less clear—I guess that is part of the fun of a TEDx event. The talks are special, surprising, and should all be fun. Here are very short descriptions of the talks. Very short.
Maybe I don’t get Ken’s talk. His title is, “Getting to Know Our Digital Assistants.” I thought Ken was involved in making sure people don’t use digital assistants. I guess the only way to know is to watch.
The TEDx program with be broadcast on this Tuesday. General information is found here and go here for the live feed. Ken is talking in the middle of the 10:40am–noon session, perhaps shortly after 11:10am. As usual if you miss the live broadcast the talks will be still available on-line.
If you live nearby you may still be able to get there in person. For the rest of us I look forward to see Ken and the others in motion.
Update 10/16/14: Upon being reminded by John Sidles’ comment that YouTube and other video URLs have fields for jumping to a given time or frame, I (Ken) indexed all the talks and other segments of the day, and the direct links are now posted on the livestream page.