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[Axiomdeveloper] Re: FW: data structure vs. mathematical structure
From: 
Gabriel Dos Reis 
Subject: 
[Axiomdeveloper] Re: FW: data structure vs. mathematical structure 
Date: 
16 Nov 2006 01:20:22 +0100 
Ralf Hemmecke <address@hidden> writes:
 > The definition you gave is it: least fixed point of
 > X > 1 + T × X × X

 Hmmm, good question. In Aldorcombinat (AC) we deal with combinatorial
 species. They encode actual structures. The corresponding generating
 series G(x) for binary trees given by your X above has to fulfil the
 equation

 G(x) = 1 + x * G(x) * G(x) (+)

 As a quadratic formula it has at most 2 solutions. And only one of
 those solution is a power series with only nonnegative
 coefficients. Since I don't know what it should mean to say "there are
 5 trees with 3 nodes", it is clear which solution I choose for the
 generating series.

 Assuming that I understand a bit of the theory of species, then there
 is only *one* solution to

 X = 1 + T * X * X.

 We are not yet dealing with "virtual species" which would allow
 negative coefficients in the generating series.
I realize my sentence could be ambiguous: I meant "least fixed point
in the category of continuous partial orders (CPO)." I'm not familiar
with the theory of species. At any rate I did not intend "least" in
terms of numerical coefficient.
 > The interesting bit about those inductive definition is that they come
 > with "fold" (reduce in Axiomspeak) for free.

 Hmmm, lazyness is important here. Unfortunately the internal
 definition of formal power series in AC does not allow to write
 equation (+) as such. One first has to say (something like)

 g: FormalPowerSeries Integer := new();
 set!(g, 1+x*g*g);

 where x corresponds to the inteterminate of "FormalPowerSeries Integer".
Aha! Good point. Now, I just need you to point me to a good
introductory point on species.
 Gaby
 RE: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), (continued)
 RE: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), Bill Page, 2006/11/14
 Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), Gabriel Dos Reis, 2006/11/14
 RE: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), Page, Bill, 2006/11/14
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/14
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/14
 [Axiomdeveloper] RE: FW: data structure vs. mathematical structure, Page, Bill, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
 Re: [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Martin Rubey, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure,
Gabriel Dos Reis <=
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 [Axiomdeveloper] RE: FW: data structure vs. mathematical structure, Page, Bill, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 Re: [Axiomdeveloper] RE: FW: data structure vs. mathematical structure, Martin Rubey, 2006/11/15
 Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), Gabriel Dos Reis, 2006/11/14
 Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), Vanuxem Grégory, 2006/11/14