Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 
Yes, I notice that. Legal constructions in a constraint system are in general "multiway".
I've been looking at various modified NKS systems that incorporate "underdetermined" transitions for independent reasons. A CA with one random output case in its rule table is the simplest example  multiple evolutions are possible from a given initial. Or you can consider a "machine with input" CA or ICA, that switches its rule at each step according to an external sequence. If the rules is switches among, differ only in a single case in their rule table, effectively you are determining the outcome of that case according to an external sequence. If that sequence is random it behaves much like the previous. But you can also manipulate the application order to maximize this or minimize that e.g.
I take your point about Wolfram on constraint systems, that the difficulty of satisfying complicated constraints is not in itself a reason not to look at constraint systems. It is a reason not to expect most natural cases of complexity to be caused by specific, "only this works", satisfaction of multiple constraints. But he also found it was relatively easy to get iterated schemes to approach constraint satisfaction without reaching it. He focused less on constraints so easy to satisfy, that large sets of possible patterns are compatible with the constraints. Although he considered something similar later as multiway systems.
This suggests a couple of additional things you might look at. One, you might look at the variety of typical patterns produce by your tile sets, without the absolute preference for single choice moves or the closest to the center criterion, and without searching for a tiling. Just, if tiles with these edges are thrown together making possible legal moves, what sort of (potentially tangled rather than tiled) shapes result? When you've seen an instance, keep it in a list that you won't stop when you find, and will backtrack from if you get stuck without finding a new pattern. For a given tile set, then ask, what are the first n (limited size) patterns found this way? In other words, just typical evolutions in tilegrowth space, without "trying" to get an Xee or a Yee anything. Empirically, the idea is such shapes would arise naturally from a soup of such fitted parts.
Two, you might look at the shapes you find internally with the existing algorithm, but from which you have to backtrack as things are now. Find large shapes that eventually "get stuck" or "don't fit" somewhere. These would be analogs of approaching constraint satisfaction without actually doing so. Empirically, one might think of this is trying to minimize some energy and getting enough of the system into a state that does, with bits sticking out that fail to, but that are energetically less "expensive" than the gain from the rest of the shape.
I should also say I see real aesthetic potential here. I particularly liked your Celtic knots, for example.
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