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 provableresults, 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.

Bloom’s Bizarro World

It’s always entertaining to visit Bizarro World, a place where a mediocrity like C. S. Lewis is revered as a deep thinker, where “natural law” is viewed as a serious philosophical and legal theory, and where a 3rd rate college like Hillsdale is spoken of with reverence.

In today’s visit, let us read this piece by one Nathan Schlueter, an academic so forgettable that I have already forgotten how to spell his last name. Prof. Schlueter is under the delusion that Allan Bloom’s The Closing of the American Mind is not only an important book, it’s so important that 30 years after its publication, it merits an entire symposium.

I read The Closing of the American Mind back in 2000, after someone recommended it to me. It was, to put it simply, a disaster. Bloom was a professor at the University of Chicago; we overlapped teaching there for a few years. At the time, focused on my own research, my only knowledge of him was the unflattering stories about him that I had heard from students.

It was apparent from nearly the first page of Closing that Bloom was a deeply troubled individual. His book, ostensibly a critique of higher education, was so clearly a rant based on Bloom’s own intellectual and sexual insecurities that I found it almost painful to read. I wrote the following review for amazon back then, and here it is again, cleaned up a little:

This is not just a bad book. It is a sick one.

Bloom’s obsessions are clear on almost every page: sex and rock music. Although clothed in a pretentious philosophical language, his objections betray what’s really on his mind: he feels left out. The sexual revolution of the Sixties passed him by, and like the child whose playmates decide he’s not good enough to get into the game, he retaliates by labelling everything about his opponents evil.

The poet Philip Larkin had Bloom’s number when he wrote in his poem, Annus Mirabilis:

Sexual intercourse began
In nineteen sixty-three
(which was rather late for me)-
Between the end of the “Chatterley” ban
And the Beatles’ first LP.

Like many neo-conservatives, Bloom doesn’t really understand the principle of free speech. He says it “has given way to freedom of expression, in which the obscene gesture enjoys the same protected status as demonstrative discourse.” In other words, freedom of speech should only apply to the stuff that Bloom approves of. (He also doesn’t apparently know that the Canadian constitution guarantees “freedom of expression”, precisely to avoid arbitrary Bloom-style distinctions.)

Like many professors in the humanities, he is deeply distrustful of science. And so he continues to push thinkers like Plato as essential to understanding the world, displaying no comprehension of the intellectual revolution brought by, for example, Charles Darwin. Is it still fruitful to read the Greeks? Certainly. But to pretend that we have learned nothing in 2000 years, that the insights of science play no part in an informed understanding of the world, is to play the part of the small child who insists, contrary to all evidence, that there is really is a Santa Claus.

Ultimately, this books tells us not about the mind of Americans, but rather the small, sanctimonious, and quite closed mind of one university professor named Allan Bloom.

It is no surprise at all that this Bloom Symposium is being published by Robert George’s Witherspoon Institute, the very same place that funded and guided the Regnerus study on gay parents, a laughable piece of scholarship that just so happened to confirm Robert George’s own negative view of homosexuality.

These folks have very closed minds, and their natural habitat is a setting where their prejudices can be confirmed and clothed in academic respectability.