Renormalization and Computation 4

Posted in Category theory, Math, Programming, Quantum by Mike Stay on 2009 October 15

This is the fourth in a series of posts on Yuri Manin’s pair of papers. In the previous posts, I laid out the background; this time I’ll actually get around to his result.

A homomorphism from the Hopf algebra into a target algebra is called a character. The functor that assigns an action to a path, whether classical or quantum, is a character. In the classical case, it’s into the rig (\mathbb{R} \cup \{\infty\}, \min, \infty, +, 0), and we take an infimum over paths; in the quantum it’s into the rig (\mathbb{C}, +, 0, \cdot, 1), and we take an integral over paths. Moving from the quantum to the classical case is called Maslov dequantization.

Manin mentions that the runtime of a parallel program is a character akin to the classical action, with the runtime of the composition of two programs being the sum of the respective runtimes, while the runtime of two parallel programs is the maximum of the two. A similar result holds for nearly any cost function. He also points out that computably enumerable reals \mathbb{R}^{c.e.} form a rig (\mathbb{R}^{c.e.}\cup \{\infty\}, \max, \infty, +, 0). He examines Rota-Baxter operators as a way to generalize what “polar part” means and extend the theorems on Hopf algebra renormalization to such rigs.

In the second paper, he looks at my work with Calude as an example of a character. He uses our same argument to show that lots of measures of program behavior have the property that if the measure hasn’t stopped growing after reaching a certain large amount with respect to the program size, then the density of finite values the measure could take decreases like 1/\log(t). Surprisingly, though he referred to these results as cutoffs, he didn’t actually use them anywhere for doing regularization.

Reading between the lines, he might be suggesting something like approximating the Kolmogorov complexity that he uses later by using a time cutoff, motivated by results from our paper: there’s a constant c depending only on the programming language such that if you run the nth program cn^2 steps and it hasn’t stopped, then the density of times near t > cn^2 at which it could stop is roughly 1/\log(t).

Levin suggested using a computable complexity that’s the sum of the program length and the log of the number of time steps; I suppose you could “regularize” the Kolmogorov complexity by adding \Lambda \log(t) to the length of the program, renormalize, and then let \Lambda go to zero, but that’s not something Manin does.

Instead, he proposed two other constructions suitable for renormalization; here’s the simplest. Given a partial computable function f:\mathbb{Z}^+\to \mathbb{Z}^+, define the computably enumerable \bar{f}:\mathbb{N}\to\mathbb{N} by \bar{f}(k) = f(k) if f(k) is defined, and 0 otherwise. Now define

\displaystyle \Psi(k,f;z) = \sum_{n=0}^{\infty} \frac{z^n}{\left(1+n\bar{f}(k)\right)^2}.

When f(k) is undefined, \Psi(k,f;z) = 1/(1-z), which has a pole at z=1. When f(k) is defined, \Psi(k,f;z) converges everywhere except z=\infty. Birkhoff decomposition would separate these two cases, though I’m not sure what value \Psi_+(f,k;1) would take or what it would mean.

The other construction involves turning \bar{f} into a permutation (x,y) \mapsto (x+\bar{f}(y),y), and inventing a function that has poles when the permutation has fixpoints.

So Manin’s idea of renormalizing the halting problem is to do some uncomputable stuff to get an easy-to-renormalize function and then throw the Birkhoff decomposition at it; since we know the halting problem is undecidable, perhaps the fact that he didn’t come up with a new technique for extracting information about the problem is unsurprising, but after putting in so much effort to understand it, I was left rather disappointed: if you’re going to allow yourself to do uncomputable things, why not just solve the halting problem directly?

I must suppose that his intent was not to tackle this hard problem, but simply to play with the analogy he’d noticed; it’s what I’ve done in other papers. And being forced to learn renormalization was exhilarating! I have a bunch of ideas to follow up; I’ll write them up as I get a chance.


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