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Background:

The question I'm about to ask is a follow-on question resulting from a discussion with Paul Sinclair and Henning Makholm in this thread here:

Permutation of 0,,,n-1 as two vectors with n components or n vectors with 2 components

Question:

Suppose it can be shown that any right(left)-regular finite-state grammar G implicitly generates a "hypar" surface ("hypar" = "hyperbolic paraboloid"), where the surface results from a simple algorithm applied to the derivation tree of a derivation in G. (For the sake of clarity, it is important here to note that because of the restricted nature of the production rules in finite-state grammars, there is really only one such derivation tree, up to: i) labelling of vertices; ii) number of non-maximal vertices intervening between the root of the tree and the deepest maximal vertex in the tree; iii) the obvious difference between derivation trees in right-regular and left-regular finite-state grammars.)

Would you suspect an odd and completely meaningless coincidence, i.e. an example of a "pseudo-result" of the sort which JH Conway once wonderfully characterized by saying to me that "the Devil bears gifts also".)

Or, would you suspect that something deeper might actually be going on?

What would your own mathematical instincts tell you?

11/25/2017 11;38 EDT US - edited to add:

I should add that the algorithm also generates a hypar surface when applied to the the derivation tree of a context-free derivation which differs from a finite-state derivation only in that the last rule applied in the derivation was of the form A -> a b rather than A -> a. (Technically, such a derivation is not finite-state because the last rule produces two terminal symbols in the grammar's alphabet.)

David Halitsky
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  • What do you mean by 'generates' here? There are many equivalent ways of presenting a set of coefficients, for instance, and a priori there's no reason to expect any one of them to yield more insight than another. – Steven Stadnicki Nov 26 '17 at 04:32
  • @StevenStadnicki - Thanks for taking the time to ask for clarification. I mean that the application of the algorithm to the derivation tree generates a set of points in 3-space which always lie on a hyperbolic paraboloid. – David Halitsky Nov 26 '17 at 04:35
  • I should add that the algorithm also generates a hypar surface when applied to the the derivation tree of a context-free derivation which differs from a finite-state derivation only in that the last rule applied in the derivation was of the form A -> a b rather than A -> a. – David Halitsky Nov 26 '17 at 04:37
  • See also this follow-on question: https://math.stackexchange.com/questions/2543857/hyperboloids-of-one-sheet-hyperbolic-paraboloids-and-hilberts-famous-three-s – David Halitsky Nov 30 '17 at 02:22

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