rule sets and patterns, was ...Re: [ Homological Algebra in Maxima]

Some of your stuff could be done with pattern matching,
via tellsimp and matchdeclare
some of it is not sufficiently defined, e.g.

tor(a+b,g) --->  tor(a,g) +  something.

Does b have to be nonzero?  if not, this is an infinite
loop.

  is tor(x+y+45) possible?

you can write functions like  tor(a,g):= if inpart(a,0)="+" then
   map(lambda([r],tor(r,g)), a) else ....

which is not quite pattern replacement syntax, but may actually
do the job a lot better.



Carl McTague wrote:
> Hi again,
> 
> I posted the following message yesterday, but I fear it was buried by
> the tremendous volume of GUI discussion.  As an additional question:
> if this isn't easy to do in maxima, can anyone suggest another
> approach with free software?  Could prolog, for instance, be put to
> this sort of use?  (I don't know prolog, but am considering learning
> it and would be very curious to know if it might be helpful for this
> sort of thing).
> 
> Thanks,
>   Carl
> 
> ----- Forwarded message from Carl McTague <mctague@santafe.edu> -----
> 
> Date: Wed, 11 Jun 2003 00:19:44 -0400
> From: Carl McTague <mctague at santafe>
> To: maxima@www.math.utexas.edu
> Subject: Homological Algebra in Maxima
> User-Agent: Mutt/1.4i
> X-Operating-System: FreeBSD 4.7-RELEASE i386
> 
> Hi Maxima Gurus,
> 
> in order to finish my topology final, I wanted to check my
> computations with the K\"unneth theorem etc. with the computer.  I
> wanted to do it with maxima, but couldn't think of how.  I was able to
> do it very simply in math'a.  Can any of you help me figure out how to
> do the following sort of thing with maxima in the future?
> 
> First, I set up some rewrite rules for the tensor product, Tor, Hom
> and Ext (by no means exhaustive; these simply sufficed for the
> computations I needed to carry out---I was doing everything with Z
> coefficients and all the multiplicand spaces had homologies built from
> Z and Z_n).  Here, I'm using Z[n] to denote Z_n.  I didn't even
> implement bilinearity fully, just the transformations that arose in my
> problems:
> 
> Tens[0, G_] := 0;
> Tens[G_, 0] := 0;
> Tens[Z, G_] := G;
> Tens[G_, Z] := G;
> Tens[Z[n_], Z[m_]] := Z[GCD[n, m]];
> Tens[A_ + B_, G_] := Tens[A, G] + Tor[B, G];
> Tens[A_, G_ + H_] := Tens[A, G] + Tor[A, H];
> 
> Tor[Z, _] := 0;
> Tor[_, Z] := 0;
> Tor[0, _] := 0;
> Tor[_, 0] := 0;
> Tor[Z[n_], Z[m_]] := Z[GCD[n, m]];
> Tor[A_ + B_, G_] := Tor[A, G] + Tor[B, G];
> Tor[A_, G_ + H_] := Tor[A, G] + Tor[A, H];
> 
> Hom[Z, Z] := Z;
> Hom[Z[n_], Z] := 0;
> Hom[Z, Z[n_]] := 0;
> Hom[0, _] := 0;
> Hom[Z[n_], Z[m_]] := Z[GCD[n, m]];
> Hom[A_ + B_, G_] := Hom[A, G] + Hom[B, G];
> Hom[A_, G_ + H_] := Hom[A, G] + Hom[A, H];
> Hom[n_Integer A_, G_] := n*Hom[A, G];
> 
> Ext[Z, _] := 0;
> Ext[0, _] := 0;
> Ext[Z[n_], Z] := Z[n];
> Ext[Z[n], Z[m]] := Z[GCD[n, m]];
> Ext[A_ + B_, G_] := Ext[A, G] + Ext[B, G];
> Ext[A_, G_ + H_] := Ext[A, G] + Ext[A, H];
> Ext[n_Integer A_, G_] := n*Ext[A, G];
> 
> The idea is to represent a chain complex just as a list, and to
> implement Kunneth as a function that maps two lists to a single one.
> I assume in the following that the product space has a single
> component and hence homology Z in dimension 1 (again, I thew this
> together very hastily):
> 
> kunneth[C_List, D_List] :=
>   Block[{l = Length[C] + Length[D] - 2},
>     Block[{CC = Join[C, Table[0, {2*l}]], DD = Join[D, Table[0, {2*l}]]},
>       Prepend[
>         Table[
>           Sum[Tens[CC[[i + 1]], DD[[n - i + 1]]], {i, 0, n}]
>             + Sum[Tor[CC[[i + 1]], DD[[n - 1 - i + 1]]], {i, 0, n - 1}],
>           {n, 1, l}], Z]]]
> 
> So, for example, to compute H_*(S^1 \times S^1), I just do
> 
> kunneth[{Z,Z},{Z,Z}] => {Z, 2 Z, Z}
> 
> (I'm using the built-in addition to represent direct products of
> groups so 2 Z means Z \oplus Z.)
> 
> Similarly, I can implement the universal coefficient theorem to
> compute cohomology from homology with
> 
> uctCohomology[C_List] :=
>   Block[{CC = Append[C, 0]},
>     Prepend[
>       Table[Hom[CC[[n + 1]], Z] + Ext[CC[[n - 1 + 1]], Z], {n, 1, Length[C]}],Z]]
> 
> Now, by Poincare duality, this usually will just give you homology in
> reverse, but in principle it should be useful in more general
> settings...
> 
> Anyway, it's not a super-sturdy, general system.  The point was that I
> just needed to check some tedious algebra quickly.  For instance, I
> had to compute the homology of the product of lens spaces:
> 
> kunneth[{Z, Z[nn], 0, Z}, {Z, Z[mm], 0, Z}]
>   => {Z, Z[mm] + Z[nn], Z[GCD[mm, nn]], 2 Z + Z[GCD[mm, nn]], Z[mm] + Z[nn], 0, Z}
> 
> So, I just would like to be able to throw together naive little
> specialized rewrite systems quickly with maxima, if possible.
> 
> Any ideas?
> 
> Thanks,
>   Carl
> 
> ----- End forwarded message -----
> 
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