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[Axiom-developer] Re: FW: data structure vs. mathematical structure
From: |
Gabriel Dos Reis |
Subject: |
[Axiom-developer] 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 Aldor-combinat (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 non-negative
| 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 Axiom-speak) 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: [Axiom-developer] Graph theory), (continued)
- RE: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Bill Page, 2006/11/14
- Re: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Gabriel Dos Reis, 2006/11/14
- RE: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Page, Bill, 2006/11/14
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/14
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/14
- [Axiom-developer] RE: FW: data structure vs. mathematical structure, Page, Bill, 2006/11/15
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
- Re: [Axiom-developer] Re: FW: data structure vs. mathematical structure, Martin Rubey, 2006/11/15
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
- [Axiom-developer] Re: FW: data structure vs. mathematical structure,
Gabriel Dos Reis <=
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
- [Axiom-developer] RE: FW: data structure vs. mathematical structure, Page, Bill, 2006/11/15
- [Axiom-developer] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
- Re: [Axiom-developer] RE: FW: data structure vs. mathematical structure, Martin Rubey, 2006/11/15
- Re: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Gabriel Dos Reis, 2006/11/14
Re: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Vanuxem Grégory, 2006/11/14