GAP (http://www.gap-system.org/) is an open-source system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. It is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more.
GAP includes a number of data libraries which are listed at the page http://www.gap-system.org/Datalib/datalib.html. Some of them are part of the core GAP system, while some others belong to GAP packages. Their exact format may vary, but in all cases there are some text files with data and there is certain code responsible for processing particular pieces of information from that files. In some cases, the data library may only consist of the GAP code which will construct GAP objects on demand. Documentation is contained in the manual of the GAP system or relevant packages; however, it may not contain technical details which in the best case will be placed in README files or as comments in the code. Usually once produced, the data libraries are only changed when new data are added to them. Existing data may be altered only in case of discovered errors.
Apart from the knowledge that is stored in data libraries as explained above, there is a wealth of knowledge about properties of algebraic objects, or how to compute them encoded in method installation and code. This knowledge can often not easily be extracted from the system.
The GAP type system has a number of advantages over a "standard" object-oriented model for algebraic computation. Among the most important are:
- Method selection based equally on the types of all arguments. Thus, in implementing an extension field K of an existing field L, new methods for multiplying kl and lk can be added without any special support. Similarly, inheritance applies to all arguments equally.
- Method selection can take account of information accumulated during the lifetime of an object. For instance, as soon as a group is found to be abelian, special methods for abelian groups will be applied to it. Similarly, when the size of a group has been determined once, not only is it remembered in case it is needed again, but different methods for other computations may be selected to take account of this information.
A central idea in the design of GAP is that as much of possible of the core functionality should be polymorphic, so that it can be applied to any mathematical object with appropriate properties, without knowing the underlying representation. Thus if you create some new kind of GAP object, supply a method for multiplying such objects, and claim that it is associative, then you should be able to make semigroups from your objects. With additional methods and some additional claims of algebraic properties, you can make groups, rings or algebras.
GAP has a kernel written in C. It implements:
- the GAP language,
- an interactive environment for developing and using GAP programs,
- memory management, and
- fast versions of time critical operations for various data types.
All the rest of the library of functions is written in the GAP language. Packages (user contributed extensions) are mainly written in the GAP language, but some also involve standalone executables. Some packages, for example, extend mathematical functionality of the system or add data libraries, while some others add infrastructural capabilities or links to other systems.