In the preceding chapters, we have seen how to obtain information about mathematical objects in GAP: We have to pass the object as an argument to a function. For example, if G is a group one can call Size( G ), and the function will return a value, in our example an integer which is the size of G. Computing the size of a group generally requires a substantial amount of work, therefore it seems desirable to store the size somewhere once it has been calculated. You should imagine that GAP stores the size in some place associated with the object G when Size( G ) is executed for the first time, and if this function call is executed again later, the size is simply looked up and returned, without further computation.
This means that the behavior of the function Size (Reference: Size) has to depend on whether the size for the argument G is already known, and if not, that the size must be stored after it has been calculated. These two extra tasks are done by two other functions that accompany Size( G ), namely the tester HasSize( G ) and the setter SetSize( G, size ). The tester returns true or false according to whether G has already stored its size, and the setter puts size into a place from where G can directly look it up. The function Size (Reference: Size) itself is called the getter, and from the preceding discussion we see that there must really be at least two methods for the getter: One method is used when the tester returns false; it is the method which first does the real computation and then executes the setter with the computed value. A second method is used when the tester returns true; it simply returns the stored value. This second method is also called the system getter. GAP functions for which several methods can be available are called operations, so Size (Reference: Size) is an example of an operation.
gap> G := Group(List([1..3], i-> Random(SymmetricGroup(53))));; gap> Size( G ); time > 0; # the time may of course vary on your machine 4274883284060025564298013753389399649690343788366813724672000000000000 true gap> Size( G ); time; 4274883284060025564298013753389399649690343788366813724672000000000000 0
The convenient thing for the user is that GAP automatically chooses the right method for the getter, i.e., it calls a real-work getter at most once and the system getter in all subsequent occurrences. At most once because the value of a function call like Size( G ) can also be set for G before the getter is called at all; for example, one can call the setter directly if one knows the size.
The size of a group is an example of a class of things which in GAP are called attributes. Every attribute in GAP is represented by a triple of a getter, a setter and a tester. When a new attribute is declared, all three functions are created together and the getter contains references to the other two. This is necessary because when the getter is called, it must first consult the tester, and perhaps execute the setter in the end. Therefore the getter could be implemented as follows:
getter := function( obj )
local value;
if tester( obj ) then
value := system_getter( obj );
else
value := real_work_getter( obj );
setter( obj, value );
fi;
return value;
end;
The only function which depends on the mathematical nature of the attribute is the real-work getter, and this is of course what the programmer of an attribute has to install. In both cases, the getter returns the same value, which we also call the value of the attribute (properly: the value of the attribute for the object obj). By the way: The names for setter and tester of an attribute are always composed from the prefix Set resp. Has and the name of the getter.
As a (not typical) example, note that the GAP function Random (Reference: Random), although it takes only one argument, is of course not an attribute, because otherwise the first random element of a group would be stored by the setter and returned over and over again by the system getter every time Random (Reference: Random) is called in the sequel.
There is a general important rule about attributes: Once the value of an attribute for an object has been set, it cannot be reset, i.e., it cannot be changed any more. This is achieved by having two methods not only for the getter but also for the setter: If an object already has an attribute value stored, i.e., if the tester returns true, the setter simply does nothing.
gap> G := SymmetricGroup(8);; Size(G); 40320 gap> SetSize( G, 0 ); Size( G ); 40320
Summary. In this section we have introduced attributes as triples of getter, setter and tester and we have explained how these three functions work together behind the scene to provide automatic storage and look-up of values that have once been calculated. We have seen that there can be several methods for the same function among which GAP automatically selects an appropriate one.
Certain attributes, like IsAbelian (Reference: IsAbelian), are boolean-valued. Such attributes are known to GAP as properties, because their values are stored in a slightly different way. A property also has a getter, a setter and a tester, but in this case, the getter as well as the tester returns a boolean value. Therefore GAP stores both values in the same way, namely as bits in a boolean list, thereby treating property getters and all testers (of attributes or properties) uniformly. These boolean-valued functions are called filters. You can imagine a filter as a switch which is set either to true or to false. For every GAP object there is a boolean list which has reserved a bit for every filter GAP knows about. Strictly speaking, there is one bit for every simple filter, and these simple filters can be combined with and to form other filters (which are then true if and only if all the corresponding bits are set to true). For example, the filter IsPermGroup and IsSolvableGroup is made up from several simple filters.
Since they allow only two values, the bits which represent filters can be compared very quickly, and the scheme by which GAP chooses the method, e.g., for a getter or a setter (as we have seen in the previous section), is mostly based on the examination of filters, not on the examination of other attribute values. Details of this method selection are described in chapter Reference: Method Selection.
We only present the following rule of thumb here: Each installed method for an attribute, say Size (Reference: Size), has a "required filter", which is made up from certain simple filters which must yield true for the argument obj for this method to be applicable. To execute a call of Size( obj ), GAP selects among all applicable methods the one whose required filter combines the most simple filters; the idea behind is that the more an algorithm requires of obj, the more efficient it is expected to be. For example, if obj is a permutation group that is not (known to be) solvable, a method with required filter IsPermGroup and IsSolvableGroup is not applicable, whereas a method with required filter IsPermGroup (Reference: IsPermGroup) can be chosen. On the other hand, if obj was known to be solvable, the method with required filter IsPermGroup and IsSolvableGroup would be preferred to the one with required filter IsPermGroup (Reference: IsPermGroup).
It may happen that a method is applicable for a given argument but cannot compute the desired value. In such cases, the method will execute the statement TryNextMethod();, and GAP calls the next applicable method. For example, [Sim90] describes an algorithm to compute the size of a solvable permutation group, which can be used also to decide whether or not a permutation group is solvable. Suppose that the function size_solvable implements this algorithm, and that is returns the order of the group if it is solvable and fail otherwise. Then we can install the following method for Size (Reference: Size) with required filter IsPermGroup (Reference: IsPermGroup).
function( G )
local value;
value := size_solvable( G );
if value <> fail then return value;
else TryNextMethod(); fi;
end;
This method can then be tried on every permutation group (whether known to be solvable or not), and it would include a mandatory solvability test.
If no applicable method (or no next applicable method) is found, GAP stops with an error message of the form
Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `Size' on 1 arguments called from ... lines deleted here ...
You would get an error message as above if you asked for Size( 1 ). The message simply says that there is no method installed for calculating the size of 1. Section Reference: Recovery from NoMethodFound-Errors contains more information on how to deal with these messages.
Summary. In this section we have introduced properties as special attributes, and filters as the general concept behind property getters and attribute testers. The values of the filters of an object govern how the object is treated in the selection of methods for operations.
In the example in Section 8.2, we have mentioned that the operation Size (Reference: Size) has a method for solvable permutation groups that is so far superior to the method for general permutation groups that it seems worthwhile to try it even if nothing is known about solvability of the group of which the Size (Reference: Size) is to be calculated. There are other examples where certain methods are even "cheaper" to execute. For example, if the size of a group is known it is easy to check whether it is odd, and if so, the Feit-Thompson theorem allows us to set IsSolvableGroup (Reference: IsSolvableGroup) to true for this group. GAP utilizes this celebrated theorem by having an immediate method for IsSolvableGroup (Reference: IsSolvableGroup) with required filter HasSize which checks parity of the size and either sets IsSolvableGroup (Reference: IsSolvableGroup) or does nothing, i.e., calls TryNextMethod(). These immediate methods are executed automatically for an object whenever the value of a filter changes, so solvability of a group will automatically be detected when an odd size has been calculated for it (and therefore the value of HasSize for that group has changed to true).
Some methods are even more immediate, because they do not require any calculation at all: They allow a filter to be set if another filter is also set. In other words, they model a mathematical implication like IsGroup and IsCyclic implies IsSolvableGroup and such implications can be installed in GAP as true methods. To execute true methods, GAP only needs to do some bookkeeping with its filters, therefore true methods are much faster than immediate methods.
How immediate and true methods are installed is described in Reference: Immediate Methods and Reference: Logical Implications.
The method selection is not only used to select methods for attribute getters but also for arbitrary operations, which can have more than one argument. In this case, there is a required filter for each argument (which must yield true for the corresponding arguments).
Additionally, a method with at least two arguments may require a certain relation between the arguments, which is expressed in terms of the families of the arguments. For example, the methods for ConjugateGroup( grp, elm ) require that elm lies in the family of elements from which grp is made, i.e., that the family of elm equals the "elements family" of grp.
For permutation groups, the situation is quite easy: all permutations form one family, PermutationsFamily (Reference: PermutationsFamily), and each collection of permutations, for example each permutation group, each coset of a permutation group, or each dense list of permutations, lies in CollectionsFamily( PermutationsFamily ).
For other kinds of group elements, the situation can be different. Every call of FreeGroup (Reference: FreeGroup) constructs a new family of free group elements. GAP refuses to compute One( FreeGroup( 1 ) ) * One( FreeGroup( 1 ) ) because the two operands of the multiplication lie in different families and no method is installed for this case.
For further information on family relations, see Reference: Families.
If you want to know which properties are already known for an object obj, or which properties are known to be true, you can use the functions KnownPropertiesOfObject( obj ) resp. KnownTruePropertiesOfObject( obj ). This will print a list of names of properties. These names are also the identifiers of the property getters, by which you can retrieve the value of the properties (and confirm that they are really true). Analogously, there is the function KnownAttributesOfObject (Reference: KnownAttributesOfObject) which lists the names of the known attributes, leaving out the properties.
Since GAP lets you know what it already knows about an object, it is only natural that it also lets you know what methods it considers applicable for a certain method, and in what order it will try them (in case TryNextMethod() occurs). ApplicableMethod( opr, [ arg_1, arg_2, ... ] ) returns the first applicable method for the call opr( arg_1, arg_2, ... ). More generally, ApplicableMethod( opr, [ ... ], 0, nr ) returns the nrth applicable method (i.e., the one that would be chosen after \(\textit{nr}-1\) calls of TryNextMethod) and if nr = "all", the sorted list of all applicable methods is returned. For details, see Reference: Applicable Methods and Method Selection.
If you want to see which methods are chosen for certain operations while GAP code is being executed, you can call the function TraceMethods (Reference: TraceMethods for a list of operations) with a list of these operations as arguments.
gap> TraceMethods( [ Size ] ); gap> g:= Group( (1,2,3), (1,2) );; Size( g ); #I Size: for a permutation group #I Setter(Size): system setter #I Size: system getter #I Size: system getter 6
The system getter is called once to fetch the freshly computed value for returning to the user. The second call is triggered by an immediate method. To find out by which, we can trace the immediate methods by saying TraceImmediateMethods( true ).
gap> TraceImmediateMethods( true ); gap> g:= Group( (1,2,3), (1,2) );; #I immediate: Size #I immediate: IsCyclic #I immediate: IsCommutative #I immediate: IsTrivial gap> Size( g ); #I Size: for a permutation group #I immediate: IsNonTrivial #I immediate: Size #I immediate: IsFreeAbelian #I immediate: IsTorsionFree #I immediate: IsNonTrivial #I immediate: GeneralizedPcgs #I Setter(Size): system setter #I Size: system getter #I immediate: IsPerfectGroup #I Size: system getter #I immediate: IsEmpty 6 gap> TraceImmediateMethods( false ); gap> UntraceMethods( [ Size ] );
The last two lines switch off tracing again. We now see that the system getter was called by the immediate method for IsPerfectGroup (Reference: IsPerfectGroup). Also the above-mentioned immediate method for IsSolvableGroup (Reference: IsSolvableGroup) was not used because the solvability of g was already found out during the size calculation (cf. the example in Section 8.2).
Summary. In this section and the last we have looked some more behind the scenes and seen that GAP automatically executes immediate and true methods to deduce information about objects that is cheaply available. We have seen how this can be supervised by tracing the methods.
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