<Section Label="WhyNotMaple">
<Heading>Why was &homalg; discontinued in <Alt Only="Text">&Maple;</Alt><Alt Not="Text"><URL Text="Maple">http://www.maplesoft.com/</URL></Alt>?</Heading>
The original implementation of &homalg; in &Maple; by Daniel Robertz
and myself hit several walls. The speed of the Gröbner basis routines
in &Maple; was the smallest issue. The rising complexity of data
structures for high level algorithms (bicomplexes, functors, spectral
sequences, ...) became the main problem. We very much felt the need
for an object-oriented programming language, a language that allows
defining complicated mathematical objects carrying properties and
attributes and even containing other objects as subobjects. <P/>
As we were pushed to look for an alternative to &Maple;, our wish list
grew even further. Section <Ref Sect="WhyGAP4"/> is a summary of this
wish list.
<Subsection Label="OpenGAP">
<Heading>&GAP; is free and open software</Heading>
In 1993 J. Neubüser <Alt Only="Text">addressed</Alt>
<Alt Not="Text"><URL Text="addressed">http://www.gap-system.org/Doc/Talks/cgt.ps</URL></Alt>
the necessity of free software in mathematics: <P/>
<Quoted>You can read Sylow's Theorem and its proof in Huppert's book
in the library without even buying the book and then you can use
Sylow's Theorem for the rest of your life free of charge, but - and
for understandable reasons of getting funds for the maintenance, the
necessity of which I have pointed out [...] - for many computer
algebra systems license fees have to be paid regularly for the total
time of their use. In order to protect what you pay for, you do not
get the source, but only an executable, i.e. a black box. You can
press buttons and you get answers in the same way as you get the
bright pictures from your television set but you cannot control how
they were made in either case.<P/>
With this situation two of the most basic rules of conduct in
mathematics are violated. In mathematics information is passed on free
of charge and everything is laid open for checking. Not applying these
rules to computer algebra systems that are made for mathematical
research [...] means moving in a most undesirable direction. Most
important: Can we expect somebody to believe a result of a program
that he is not allowed to see? [...] And even: If O'Nan and Scott
would have to pay a license fee for using an implementation of their
ideas about primitive groups, should not they in turn be entitled to
charge a license fee for using their ideas in the
implementation?</Quoted>
<Par></Par>
I had the pleasure of being one of his students. <P/>
<Subsection Label="ExpertGAP">
<Heading>&GAP; has an area of expertise</Heading>
Not only does &GAP; have the potential of natively supporting a wide
range of mathematical structures, but finite groups and their
representation theory are already an area of expertise. So there are
at least some areas where one does not need to start from
scratch. <P/>
But one could argue that rings are more central for homological
algebra than finite groups, and that &GAP4;, as for the time when the
&homalg; project was shaping, does not seriously support important
rings in a manner that enables homological computations. This drawback
would favor, for
example, <Alt Only="Text">&Singular;</Alt><Alt Not="Text">
<URL Text="Singular">http://www.singular.uni-kl.de/</URL></Alt> (with
its subsystem &Plural;) over &GAP4;. Point <Ref Sect="GAP-IO"/>
indicates how this drawback was overcome in a way, that even gave the
lead back to &GAP4;. <P/>
One of my future plans for the &homalg; project is to address moduli
problems in algebraic geometry (favorably via orbifold stacks), where
discrete groups (and especially finite groups) play a central role. As
of the time of writing these lines, discrete groups, finite groups,
and orbifolds are already in the focus of part of the project: The
package &SCO; by Simon Görtzen to compute the cohomology of orbifolds
is part of the currently available &homalg; project. <P/>
For the remaining points the choice of &GAP4; as the programming language for developing &homalg; was unavoidable.
</Subsection>
<Subsection Label="GAP-IO">
<Heading>&GAP4; can communicate</Heading>
With the excellent
&IO; <Alt Not="Text"><URL Text="package">http://www-groups.mcs.st-and.ac.uk/~neunhoef/Computer/Software/Gap/io.html</URL></Alt>
<Alt Only="Text">package</Alt> of Max Neunhöffer &GAP4; is able to
communicate in an extremely efficient way with the outer world via
bidirectional streams. This allows &homalg; to delegate things that
cannot be done in &GAP; to an external system such as &Singular;,
&Sage;, &Macaulay2;, &MAGMA;, or &Maple;.
</Subsection>
<Subsection Label="Objectify">
<Heading>&GAP4; is a <E>mathematical</E> object-oriented programming language</Heading>
The object-oriented programming philosophy of &GAP4; was developed by
mathematicians who wanted to handle complex
mathematical <Alt Only="HTML"><Ref Text="objects"
Sect="Creating New Objects" BookName="Reference"/></Alt>
<Alt Not="HTML">objects</Alt>
carrying <Alt Only="HTML"><Ref Text="properties" Sect="Properties"
BookName="Tutorial"/></Alt>
<Alt Not="HTML"><E>properties</E></Alt>
and <Alt Only="HTML"><Ref Text="attributes" Sect="attributes"
BookName="Tutorial"/></Alt><Alt Not="HTML"><E>attributes</E></Alt>, as
often encountered in algebra and geometry. &GAP4; was thus designed to
address the needs of <E>mathematical</E> object-oriented programming
more than any other language designed by computer scientists. This was
primarily achieved by the advanced <Alt Only="HTML"><Ref Text="method
selection" Sect="method selection" BookName="Tutorial"/>
</Alt> <Alt Not="HTML"><E>method selection</E></Alt> techniques that
very much resemble the mathematical way of thinking. <P/>
Unlike the common object-oriented programming languages, methods in
&GAP4; are not bound to objects but
to <Alt Only="HTML"><Ref Text="operations"
Sect="Operations and Methods"
BookName="Tutorial"/></Alt><Alt Not="HTML">operations</Alt>.
In particular, one can also install methods that depend on two or more
arguments. The index of a subgroup is an easy example of an operation
illustrating this. While it would be sufficient to bind a method for
computing the order of a group to the object representing the group,
it is not clear what to do with the index, since its definition
involves two objects: a group <M>G</M> and a subgroup <M>U</M>. Note
that the index of <M>U</M> in a subgroup of <M>G</M> containing
<M>U</M> might also be of interest. Things become even more
complicated when the arguments of the operation are unrelated
objects. Moreover, binding methods to operations makes it possible for
the programming language to support the installation of one or more
methods for the same operation, depending on already known properties
or attributes of the involved objects. <P/>
Moreover &GAP4; supports so-called
<Alt Only="HTML"><Ref Text="immediate and true methods"
Sect="Immediate and True Methods"
BookName="Tutorial"/></Alt><Alt Not="HTML"><E>immediate and true
methods</E></Alt>. This considerably simplifies teaching theorems to
the computer. For example it takes one line of code to teach &GAP4;
that a reflexive left module over a ring with left global dimension
less or equal to two is projective. These logical implications are
installed globally and &GAP4; immediately uses them as soon as the
respective assumptions are fulfilled. This mechanism enables &GAP4; to
draw arbitrary long lines of conclusions. The more one knows about the
objects involved in the computation the more specialized efficient
algorithms can be utilized, while other computations can be completely
avoided. &homalg; is equipped with plenty of logical implications for
rings, matrices, modules, morphisms, and complexes. <P/>
When all these features become relevant to what you want to do, there
is hardly an alternative to &GAP4;.
</Subsection>
<Subsection Label="BigGAP">
<Heading>&GAP4; packages are easily extendible</Heading>
Being able to install several methods for a single operation (&see;
<Ref Label="Objectify"/>) has the additional advantage of making
&GAP4; packages easily extendible. If you have an algorithm that, in a
special case, performs better than existing algorithms you can install
it as a method which gets triggered when the special case occurs. You
don't need to break existing code to insert an additional elif
section contributing to an increasing unreadability of the code. Even
better, you don't even need to know anything about the code of
other existing methods. In addition to that, you can add (maybe
missing) properties and attributes (along with methods to compute
them) to existing objects.
</Subsection>
</Section>
<Section Label="WhyNotSage">
<Heading>Why not <Alt Only="Text">&Sage;</Alt><Alt Not="Text"><URL Text="Sage">http://www.sagemath.org/</URL></Alt>?</Heading>
Although the &python;-based &Sage; fulfills most of the above
requirements, it was primarily the points expressed in
<Ref Sect="Objectify"/> that finally favored &GAP4; over &Sage;: The
object-orientedness of &python;, although very modern, does not cover
the needs of the &homalg; package. At this place I would like to
thank <Alt Only="Text">William Stein</Alt>
<Alt Not="Text"><URL Text="William
Stein">http://modular.math.washington.edu/ for the helpful
discussion about &Sage; during the early stage of developing &homalg;,
and to Max Neunhöffer who explained me the advantages of the
object-oriented programming in &GAP4;.
</Section>
<Section Label="Sage">
<Heading>How does &homalg; compare to &Sage;?</Heading>
In what follows &homalg; often refers to the whole &homalg;
project. <P/>
<Subsection Label="homalg-Sage-objectives">
<Heading>They differ in objectives and scale</Heading>
First of all, &Sage; is a huge project, that, among other things, is
intended to replace commercial, general purpose computer algebra
systems like &Maple; and &Mathematica;. So while &Sage; targets (a
growing number of) different fields of computer algebra, &homalg; only
focuses on homological, and hopefully in the near future also
homotopical techniques (applicable to some of these different
fields). The two projects simply follow different goals and are
different in scale.
</Subsection>
<Subsection Label="homalg-Sage-language">
<Heading>They differ in the programming language</Heading>
&Sage; is based on &python; and the &C;-extension &cython; while
&homalg; is based on &GAP4;. Quoting from an email response William
Stein sent me on the 25. of February, 2008: <Q>Sage *is* Python + a
library</Q>. Although I seriously considered developing &homalg; as
part of &Sage;, for the reason mentioned in <Ref Sect="Objectify"/> I
finally decided to use &GAP4; as the programming language.
</Subsection>
<Subsection Label="homalg-Sage-communicate">
<Heading>They differ in the way they communicate with the outer world</Heading>
Both &Sage; and &homalg; rely for many things on external computer
algebra systems. But although one can simply invoke a &GAP; shell or a
&Singular; shell from within &Sage;, &Sage; normally runs the external
computer algebra systems in the background and tries to understand the
internals of the objects residing in them. An object in the external
computer algebra system is wrapped by an object in &Sage; and
supporting this external object involves understanding its details in
the external system. &homalg; follows a different strategy: The only
external objects &homalg; needs (beside rings) are non-empty
matrices. And being zero or not is basically the only thing &homalg;
wants to know about a matrix after knowing its dimension. I myself was
stunned by this insight, which culminated in <Alt Only="HTML"><Ref
Sect="the principle of least communication"
BookName="Modules"/></Alt><Alt Not="HTML"><E>the principle of least
communication</E> (&see; <Ref Label="the principle of least
communication" BookName="Modules"/>).
In particular, &Sage; can make use of all of &homalg;, but for in
order to make full use, &Sage; needs to understand the internals of
the &homalg; objects. On the contrary, &homalg; can only make limited
use of &Sage; (or of virtually any computer algebra system that
supports rings in a sufficient way (&see; <Ref
Sect="Rings supported in a sufficient way"
BookName="Modules"/>)), but without the need to delve into the inner
life of the &Sage; objects. <P/>
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