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.

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.

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.