Renormalization and Computation 4
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 and we take an infimum over paths; in the quantum it’s into the rig 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 form a rig 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 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 depending only on the programming language such that if you run the th program steps and it hasn’t stopped, then the density of times near at which it could stop is roughly
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 to the length of the program, renormalize, and then let 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 define the computably enumerable by if is defined, and 0 otherwise. Now define
When is undefined, which has a pole at When is defined, converges everywhere except Birkhoff decomposition would separate these two cases, though I’m not sure what value would take or what it would mean.
The other construction involves turning into a permutation 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.