Over Christmas i took the time to code up a π-calculus representation of the Fano plane. [1] From a layman's point of view, you can think of this as a very specific way to write a computer program that represents the information of the Fano plane. What is novel about this is that the program can evolve. As it evolves it can turn into other projective geometries. This blog records some of my motivations for engaging in this kind of activity.

Imagine that the issues around the reconciliation of quantum mechanics and general relativity have to do with a sort of impedance mismatch of the devices used to capture the intuitions -- not the intuitions themselves. i've long thought that Einstein's intuitions would be better expressed in a different sort of formalism. In particular, i think certain computational formalisms in which there is a different account of the structure-behavior relationship have the promise of being better containers for Einstein's intuitions. Remember -- after all -- Mr E had to essentially invent much of the apparatus he eventually used to express his ideas. Often the first container is not the sleekest or best suited to the job. Compare the model T to the Prius or the Kittyhawk to the 787.

Certain models of computation have hit on a novel combination. To understand the technical and historical novelty you need to understand that in classical approaches to modeling the relationship between structural (or even kinematic) aspects and behavioral aspects of physical phenomena there is a clear dividing line. Algebraic structure encodes structure, and maps or functions between algebraic structure encodes behavior. The classic example of this is linear algebra. Vector spaces are where you encode state. Maps between vectors spaces are where you encode behavior.

Now, certain families of computational calculi -- notably lambda and π-calculi -- do not endure this division of labor. Rather, behavior is "folded into" the specification of the structure. You have a 3-fold spec: an algebra (the grammar of the terms in the model), a structural equivalence (saying which "syntactic differences" can be treated as noise), and a reduction relation -- which says how terms (aka programs) behave or evolve. (See, for example, this specification of the variant of the π-calculus i used to construct my encoding.)

This is a novel way to package structure and behavior. The novelty has had some interesting technical and cultural consequences, but that's beside the point. The main point is that this packaging makes these calculi intriguing candidates for encoding Einstein's intuitions about the relationship of geometry, gravity, mass and energy.

Of course, one of the tricky things about these calculi is finding notions of geometry inside them. Direct (but not entirely simple-minded) interpretations of the topological structure of these formalisms get you things that are not even T-2 -- that is, do not even enjoy certain basic separation properties that we think must hold in models of physical space. So, how do you get geometry to show up in these types of formalisms?

One approach is to encode the geometry. On one level we know this must be possible, because both the lambda calculus and the π-calculus are Turing complete. So, since we know we can encode convincing simulations of (the geometry of) physical space as ordinary programs, we know we can encode them in the lambda or π-calculus. However, the question remains whether that encoding bears any relationship to the fundamental machinery of the computational model. In other words will it be of any use to encode these notions in the model, or will it just be another formal representation -- potentially with more baggage to push around.

That's where employing certain principles -- principles we have picked up from working with computation -- comes into play. Notably, when the structure of the encoding is in alignment with something called the Curry-Howard isomorphism, it enjoys certain properties. It's another long discussion to try to explain what those properties are and what they offer to this endeavor. Suffice it to say that encodings enjoying this discipline are more easily aligned with a whole host of other structures and so are more easily probed and more easily beaten into layers of abstraction that simplify pushing baggage around.

So, i've now exhibited a very simple-minded encoding of the simplest of projective geometries, the Fano plane, into one of the computational calculi, the π-calculus. The encoding aligns with the Curry-Howard isomorphism in a specific way. From a layman's point of view, you can think of this as a very specific way to write a computer program that represents the information of the Fano plane.

What is novel about this is that program can evolve. As it evolves it can turn into other projective geometries. That's starting to feel more closely aligned with Einstein's program -- while being far from a full expression of his ideas. i can't begin to calculate the precession of the perihelion of Mercury. On the other hand, i have a good handle on how to import the recent work by Abramsky and Coecke on axiomatic quantum mechanics into this framework -- a pay-off of respecting Curry-Howard.

The 'Doh!' moment i had over Christmas was that this sort of task was in my grasp years ago. i've already demonstrated a somewhat more challenging encoding -- encoding knots as processes in the π-calculus. But, it just never occurred to me to start with the Fano plane and work out the simplest encoding that exhibited all the desiderata. Again, this simple-minded effort raises many, many more questions than it answers, but it's still pretty cool, nonetheless.

There is an intriguing connection with biology that has occurred to me ever since i began this research program. i confess the idea is not mine, but Christopher Alexander's. In his seminal work, The Nature of Order, this world class architect suggests that 'life' is an objective property of space. He suggests that there are objective, consensually validatable properties of space that allow us to gauge which configurations of space have more life.

Since i began investigating geometry itself as an outcome of information-processing behavior i reached the startling conclusion that there might be something to his idea. Specifically, a number of folks from Cardelli to Priami, have shown that biological processes -- from signal-processing to immunological behavior -- have relatively straightforward representations in the formalism i have been using to represent geometry.

To reiterate, Priami, Regev, Silverman, et al, have represented biological processes as π-calculus processes. i now have in hand representations of both geometry and topological properties such as knottedness in terms of properties of π-calculus processes. This suggests the very real possibility that there might be overlap in the kinds of processes inhabiting the target of these encodings. In other words, could we find encodings of geometries that are also encodings of biological processes? Could we find encodings of biological processes that are also encodings of geometries?

This leads to the startling thought: space itself might be alive!

Again, this is fully in line with both credible lines of investigation and with credible methods of investigation. Wheeler, for example, coined the phrase "it from bit" to describe the research program of physics arising from information-processing. What is more natural, and more in alignment with a reductionist view point than that biology ought to lie wholly inside physics? So, if physics arises from information processing and life arises inside physics, then life arises from information processing.

Ironically, however, this provides an intriguing escape hatch for life: abstraction. The abstract properties we use to characterize those processes that are alive allow it to climb up out of any specific physical container and inhabit any formal dynamical system of sufficient complexity. Life once again shows itself to be uncontainably resilient and robust!

[1] If you want me to publish the encoding, drop a comment. If i get enough requests (say 3), i will put up the encoding. Actually, here is a more interesting artifact -- the synchronization skeleton of the encoding. It has all the connectivity of the Fano plane, but what happens after interaction is yet to be specified. There are solutions for Pij, Qij that regenerate the plane, so that the artifact is static. However, there are other definitions that generate new structures.

## 2 comments:

It rarely happens that I read blogs containing a sound mixture of things like topology, geometry, computer science, physics, biology, ...

If only more research money could be spend on investigating relationships between those fascinating disciplines

All to often money is spend on 'research' that does not have such a sound basis

cool stuff!

Thanks Luc!

Post a Comment