Blog from February, 2008

The Collatz conjecture

This afternoon, lecture on termination in my course Logic and Computation (we're using ACL2 as a logic and theorem proving engine, and ACL2 requires a termination proof to admit a definition as a new axiom). And this allowed me to use the Collatz conjecture as an example of the non-triviality of establishing termination even for simple recursive functions.

The Collatz conjecture, also known as the 3n + 1 conjecture, the Ulam conjecture, the Syracuse problem, the hailstone sequence, is named after the German mathematician Lothar Collatz, who proposed it in 1937. The conjecture is that for every positive natural number a[0], the sequence a[0], a[1], a[2], a[3],... given by:

  a[n+1] = 1            if a[n] is 1
  a[n+1] = a[n] / 2     if a[n] is even
  a[n+1] = 3a[n] + 1    if a[n] is odd

always eventually hits 1.

Casting this as a termination problem, of course, is just asking whether the function f given by

  f(1) = 1
  f(n) = f(n / 2)       if n is even
  f(n) = f(3n + 1)      if n is odd

terminates on all positive natural number inputs.

No one knows whether this is true or not. Experimental evidence indicates that yes, but that's almost meaningless. In particular, we know that conjecture holds for all inputs up to more than 10^16. A good overview can be found in Weisstein's MathWorld:

Because it is so simple to state, and is extremely easy to understand, the problem has been the focus of attention for many amateur mathematicians and number-dabblers. In fact, I believe it to be almost a rite of passage to spend some time playing with the conjecture when first going through a math degree. (Thankfully, my own flirtation with the problem amounted to wasting no more than two weeks trying to formalize the "pattern" I saw in the sequences produced. Ah, the folly of youth...)

Naturally, professional mathematicians have also looked at the problem, and have come up with a surprising number of connections with seemingly unrelated topics. Best known is Jeffrey Lagarias, now at U. of Michigan, but before that researcher at AT&T Bell Labs.

Quite readable is his paper giving a survey of the problem from a mathematical standpoint:

Also of interest are his annotated bibliographies on the problem:

Both can be found on the arXiv.

Now, I seem to remember someone establishing a connection between this conjecture and the distribution of prime numbers, but I have not been able to retrace this. If anyone knows what I am thinking of, please drop me a line.

Notices of the AMS for March

Well, between holiday lethargy (thanks Shawn Standefer), the beginning of the winter semester here at Northeastern, the CSF deadline a couple of weeks back and the upcoming UAI deadline, I have found precious little time to post since the beginning of the year. But I'm back, and ready to tackle the backlog.

I just received the March issue of the Notices of the AMS in the mail yesterday, and two articles caught my eye.

B. Poonen talks about "Undecidability in Number Theory", surveying the solution to Hilbert's Tenth problem by Davis, Putnam, Robinson, and Matiyasevich - coming up with the main theorem that a set is recursively enumerable if and only if it is diophantine (very roughly, it can be obtained as the solution to a diophantine equation). This amount, interestingly enough, to programming using diophantine equations. Applications of the result beyond Hilbert's Tenth's are discussed.

Also in the issue, J. Ewing talks about "Where Are Journals Headed? Why We Should Worry About Author-Pay", yet another article discussing the journal subscription prices crisis. The author makes the interesting point, which I had never thought of before, that mathematics (and CS) are hardly in any position to dictate policy, given that we have such a tiny proportion of the published journals. Biomedical sciences, with half the journals, drive the show. And the reason why this is important is that the publication model for biomedical sciences is quite different than that for the core sciences.

I will let you explore those articles in your copious free time. The issue is freely available: