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# Boolean network as a gauge field

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.)