Question

Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boolean functions of two arguments. The nodes undergo an evolution (ie, follow a trajectory in state space) driven by the transition functions.

Further assume, initially, that the set of transition functions can change arbitrarily from one time-step to the next; the evolution is unconstrained and non-deterministic as possible, given the grounding assumptions.

Now, consider that the node states have names or labels (eg, "0" and "1"). The labeling is arbitrary, so that a description of such a system has a dual, obtained by globally exchanging state label "0" for "1" and "1" for "0", and each transition function by its dual, given that the functions are specified in terms of the labels. For example, AND(x,y) means a mapping {0,0:0; 0,1:0; 1,0:0; 1,1:1}, and OR(x,y) means a mapping {1,1:1; 1,0:1; 0,1:1; 0,0:0}. Thus, an exchange of label in the given mappings amounts to an exchange of AND for OR and vice versa.

We have a global symmetry of the description, reflecting the purely conventional choice of labeling for the node states. This symmetry does not imply any constraint on the dynamics. Suppose we would like to replace this global symmetry by a local symmetry, that allows local or "selective" relabeling of node states, both across the node collection, and through "time". How does this constrain the dynamics, and what is or should be invariant in this dynamics? For that matter, what is required to make the problem well-posed?

My question: Does the above sketch correspond to a developed area of mathematical investigation? In asking this question, I have in mind the observation that the set of N boolean transition functions can be regarded as an abstract simplicial complex; the binary-state nodes are vertices, and the transition functions are edges. I am therefore implicitly asking a question about the dynamics of this simplicial complex when bound to the dynamics of the node states in the way I have described. (This is the real motivation for my interest in this question.)

Answer 1

Thanks, jc. My timing turned out to be unfortunate; sorry for not following up sooner.

What I am considering is an analogy between the system described above and gauge fields well known in theoretical physics. Take the most familiar and oldest example, the electromagnetic field. A matter field is coupled to a potential field—the electromagnetic scalar and vector potentials. The matter field is a complex quantity, and is characterized in part by a phase. A global change of the phase leaves all physical observables associated with the matter field unchanged. Similarly, a global shift in the components of the potential—a global change of "reference" potential—results in no observable changes to the electromagnetic field components derived from the potential.

However, in order to allow invariance of observables with respect to local shifts of the phase and the potential, the evolution of the matter field and the electromagnetic potential must be coupled in a particular way that tightly constrains the dynamics of the combined system, in order to allow local shifts of the phase to be compensated by local shifts of the potential so as to leave physical observables unchanged.

So the correspondence I'm proposing is this:

• matter field: the binary-state nodes
• field phase redefinition: local re-labeling of the states of the nodes

• potential field: the transition functions (one per node) which act on the binary-state nodes (generating state transitions at successive "time" steps)
• potential field redefinition: local replacement of transition functions by their boolean algebraic duals

For the boolean network the question now becomes: What invariant or "absolute" structure is associated with the local gauge symmetry being outlined here? To put it another way, what is the nature of the field that we take to be "observable", and how is its dynamics constrained?

I would like to suggest, roughly, that the right answer to this question is a causal structure implicit in the boolean network's evolution that is "compatible" with a smooth 4-dimensional pseudo-Riemannian manifold. More specifically, I have in mind what has come to be known as a causal set—a discrete skeleton for a spacetime. In this context, recall the observation in my initial post, that "the set of N boolean transition functions can be regarded as an abstract simplicial complex; the binary-state nodes are vertices, and the transition functions are edges."

I might mention that I've derived considerable conceptual inspiration from a recent paper by Hendryk Pfeiffer (arxiv:gr-qc/0404088), which contains the following remark:

If quantum general relativity in d = 3+1 is indeed a PL-QFT, the following two statements which sound philosophically completely contrary,

• Nature is fundamentally smooth.
• Nature is fundamentally discrete.

are just two different points of view on the same underlying mathematical structure: equivalence classes of smooth manifolds up to diffeomorphism.

Answer 2

It sounds like what you're after is some type of a lattice gauge theory. I wasn't able to figure out what you wanted from your question or answer, but I think the area of study you want to learn about is basically this one.

There's a very readable review by Kogut here, which also discusses spin systems (2D Ising model in some detail) and the "QC" correspondence between statistical mechanics and quantum mechanics. I found it fascinating and highly recommend it to anybody interested in these types of models, some of which have recently come into vogue in the physics of topological quantum computation.

Answer 3

Q. Does the above sketch correspond to a developed area of mathematical investigation?

More of a "developing area" than a "developed" one, maybe. Here are some papers and web articles on Differential Logic that may have a bearing on your question:

• Differential Logic : Introduction
• Differential Logic and Dynamic Systems
• Differential Propositional Calculus @ PlanetMath (read under "page image" mode)
• Differential Propositional Calculus @ PlanetPhysics (try this if the other fails to load)