Saturday, September 16, 2017

Cross Orbweaver


I don't claim to be an entomologist, but I am pretty sure this is Araneus diadematus, the Cross Orbweaver. It's not native to Canada, but common in Ontario gardens.

Saturday, September 09, 2017

Are Religious Beliefs Really Off Limits for Voters?


Consider this letter from the President of Princeton University, Christopher Eisgruber.

I am not persuaded at all by these arguments. I wrote the following response.

Dear President Eisgruber:

I could not disagree more strongly with the sentiments expressed in your letter to the Judiciary Committee.

1. Religious beliefs are not, as you claim, "irrelevant" to the qualifications for a Federal judgeship. Would you, for example, be willing to confirm an otherwise-qualified judge who subscribed to the tenets of Christian Identity? (If you are not familiar with this religion, please consult https://en.wikipedia.org/wiki/Christian_Identity .)

2. Beliefs do not magically become off limits to questioning, probing, or otherwise investigating simply because one labels them "religious". As you know, there is significant debate about whether some belief systems, such as Scientology, are indeed religions. There is no bright line separating religion from other kinds of opinions one may hold.

3. It is absurd to claim that Professor Barrett's religious beliefs are not part of her judicial philosophy, when you yourself cite an article of hers that addresses precisely this issue.

4. You also should know that the "religious test" clause in the Constitution was in response to legislation, such as the Test Acts, that made it impossible for members of certain religions to hold public office. This Constitutional clause says nothing at all about whether voters may make up their minds based on a candidate's religious beliefs. Nor does it say that Judiciary Committee members may not evaluate the suitability of a candidate based on what he or she believes.

This kind of posturing is unworthy of you and unworthy of Princeton.

Jeffrey Shallit '79

In addition, I note that polls show that a large percentage of the American public would not vote for an atheist candidate. Why is it that this never merits letters of concern by people like President Eisgruber?

Many religious people want a double standard: the freedom to hold beliefs, no matter how pernicious or unsupported, and the right to never be questioned on those beliefs.

Robert Marks Anniversary: Three Years Later, Still Evading


Three years after I first asked him, the illustrious Robert J. Marks II, intelligent design advocate and professor at Baylor University, is still refusing to back up his claim that "we all agree that a picture of Mount Rushmore with the busts of four US Presidents contains more information than a picture of Mount Fuji."

I guess he must be just too busy to answer. Too busy writing extremely important articles like this one, for example, which is the only article published so far this year in the intelligent design vanity journal, Bio-Complexity. A journal, by the way, for which the illustrious Robert J. Marks II is also the Editor in Chief! The intensive peer review that article must have undergone is rather mind-boggling. I wonder how many times he sent it back to himself for revision.

Previous ducks:

Two years

One year

Six months

Three months

First question

Friday, August 25, 2017

We Don't Need a Wall...


to keep moose out.

Apparently it's hard to recruit border agents at remote US borders. It seems to me the moose would be a major attraction.

Friday, August 18, 2017

I Hope it is not an Alt-Right Moose


From readers JPA and RM comes the story of this rare white moose in Sweden.

Moose are sensible creatures. I suspect that white moose do not believe they are superior to brown ones.

Thursday, August 17, 2017

Eclipses


Since there's a big eclipse coming up on Monday, maybe these old comic strips are appropriate:

July 15 1963

July 16 1963

July 17 1963

July 18 1963

July 19 1963

July 20 1963

A Civil War Monument We Should Leave Up


Here, from Hertford, North Carolina, a town where many of my relatives once lived, is a little-known Civil War monument, one of very few of its kind.

Friday, July 14, 2017

Powers With Repetitions


What's so cool about 574595585933 expressed in base 12400?

And why is something like that really rare?

For the answer, you'll have to read my latest paper, co-written with Andrew Bridy, Robert J. Lemke Oliver, and Arlo Shallit.

Wednesday, July 05, 2017

Using a Decision Method to Prove a New Theorem in Number Theory


The mathematician David Hilbert had a dream, a dream to mechanize mathematics. He hoped that every true mathematical result had a formal proof, and furthermore, that this formal proof would be discoverable by algorithmic methods. All you would have to do is state your theorem in some formal mathematical language, perform the algorithm, and voila! You would have a proof or disproof.

But Hilbert's dream was killed by Kurt Gödel and Alan Turing in the early 20th century. Gödel showed that, given any sufficiently powerful axiom system (roughly speaking, it's enough to be able to define addition and multiplication of integers), there would be true results lacking any formal proof. And Turing showed that, even if we restrict ourselves to provable results, there is no algorithm that, given a mathematical statement, will always halt and correctly report "provable" or "unprovable".

Nevertheless, there are some logical theories in which (a) every well-formed statement is provable or disprovable and (b) there is, in fact, an algorithm to find a proof or disproof. Such a theory is sometimes called decidable and the corresponding algorithm is called a prover. The first-order theory of the natural numbers with addition, often called Presburger arithmetic, is one such theory; it has a prover. Unfortunately, Presburger arithmetic is not very powerful, and although it has some practical applications, I am not aware of a single significant mathematical theorem proved using a Presburger prover.

However, when Presburger arithmetic is augmented with a certain additional function on natural numbers called Vk, for a fixed integer k ≥ 2, it remains decidable. And now it is powerful enough to prove things people really want to prove! My former master's student, Hamoon Mousavi, wrote a prover called Walnut for this bigger theory, and with it we have proven many new theorems of genuine mathematical interest. And we also reproved many results in the literature for which the only proofs previously known were long, tedious, and case-based.

Recently, my co-authors Aayush Rajasekaran and Tim Smith used a different method to algorithmically prove a new result in number theory. It concerns palindromes, which are numbers that, when considered as a string of digits in some integer base b ≥ 2, read the same forward and backwards. For example, the number 717 is a palindrome in base 10, but it is also a palindrome in base 2, since in binary it is 1011001101.

Last year, the number theorist William D. Banks started studying the additive number theory of palindromes. He proved that every natural number is the sum of at most 49 decimal palindromes. More recently, this result was improved by Florian Luca, Lewis Baxter, and the late Javier Cilleruelo, who proved that every natural number is the sum of at most 3 palindromes in base b, for all b ≥ 5. However, it seems that so far, nobody proved any bound at all for bases b = 2, 3, 4.

Here is our result: every natural number is the sum of at most 9 binary palindromes. The bound "9" is probably not the best possible result, as empirical results suggest the best possible bound is probably 4. Probably somebody will improve our bound soon! What makes our result interesting, though, is how we did it. Instead of the heavily case-based approach of Banks and Cilleruelo-Luca-Baxter, we used a decision method: we recoded the problem as a formal language theory problem, and then used the fact that this formalism has a decidable logical theory associated with it. Then we used publicly-available software to prove our result.

Here are the details: we created a certain kind of automaton, called a nondeterministic nested-word automaton, that takes integers n, represented in base 2, as input. Given an input representing an integer n, our automaton "guesses" a possible representation as a sum of palindromes, and then "verifies" that its guess is correct. Here the "verifies" means checking that the summands are indeed palindromes (read the same forwards and backwards) and that they sum to n. If the guess succeeds, the automaton accepts the input. Then the "sum of palindromes" theorem we want to prove amounts to claiming that the automaton accepts every possible natural number as input.

Luckily for us, the so-called "universality problem" (does a given automaton accept every input?) is actually decidable for nested word automata, a result proved by Alur and Madhusudan. We used the ULTIMATE automata library to then check the universality of the automaton we created. For more details, see our preprint.

Could other theorems in number theory be proved using this method? Yes, we proved a few more in our preprint. The holy grail would be a decidable logical theory strong enough to express traditional theorems about primes. If such a theory existed, we could, at least in theory, prove statements like Goldbach's conjecture (every even number > 2 is the sum of two primes) purely mechanically, by expressing them in the proper formalism and then running a prover. But currently we do not even know whether Presburger arithmetic, together with a predicate for primality, is decidable.

What's next? Well, a lot! But that will be the subject of Aayush's master's thesis, so you'll have to wait to find out.