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Re: FW: data structure vs. mathematical structure (was: [Axiom-developer
From: |
Gabriel Dos Reis |
Subject: |
Re: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory) |
Date: |
14 Nov 2006 04:16:55 +0100 |
[ Is it normal that I don't see Ralf's messages? ]
"Bill Page" <address@hidden> writes:
[...]
| So is computational complexity a relative thing, i.e. we take
| the operations from the representation as units and then express
| the complexity of the operations of the new domain in terms of
| these? Or is it an absolute thing that must be expressed in
| terms of the most primitive domain (Lisp) or even deeper in
| terms of a given hardware architecture?
relative.
| > After all that is a distincion between classic mathematics
| > and constructive mathematics.
| >
|
| I am not sure that I would agree that this is the right distinction.
| Ultimately in computer algebra systems I think all mathematics must
| be constructive. In a sense, that is what we mean by "algebra". An
| existence proof or a proof by contradiction is not of much use in
| such a system.
|
| > Example. You want to implement quicksort. Would someone choose
| > lists to do that instead of arrays?
|
| In Axiom this is not really an appropriate question. quicksort
| would be implemented in a generic manner so that it can be applied
| to any domain that provides the necessary methods. In fact in the
| Axiom book contains exactly this example (bubble sort actually).
However, to call it quicksort, you have to meet some algorithmic
complexity guarantee.
| > Both basically represent the mathematical idea of tuples.
|
| No, I don't think so. As mathematical objects lists, arrays and
| tuples are all quite distinct however I agree that they all provide
| the necessary methods on which sort may operate.
The notion of sorting is defined terms of order and permutation.
[...]
| > > For example, is an Axiom set, i.e. a member of the domain Set,
| > > a data structure or a mathematical structure?
| >
| > How expensive is it to add a new element to an existing set of
| > type Set(T) where T has no ordering function? How expensive is
| > it to add a new element to Set(T) if T provides an ordering
| > function?
| >
|
| All domains that have SetCategory are required to have a hash into
| SmallInteger.
That is not a mathematical requirement.
| I suppose that this means that the cost of adding
| new elements is independent of Type T.
Why?
- Re: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Gabriel Dos Reis, 2006/11/13
- FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Bill Page, 2006/11/13
- Re: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory),
Gabriel Dos Reis <=
- RE: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory), Bill Page, 2006/11/13
- 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), 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