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<!-- %W  opers.tex                 GAP documentation            Heiko Theißen -->
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<!-- %H  @(#)<M>Id: opers.tex,v 4.23 2006/03/07 15:30:55 jjm Exp </M> -->
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<!-- %Y  Copyright 1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,   Germany -->
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<!-- %%  This file contains a tutorial introduction to operations. -->
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<P/>

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<Chapter Label="Operations and Methods">
<Heading>Operations and Methods</Heading>

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<Section Label="Attributes">
<Heading>Attributes</Heading>

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 <A>G</A> is a group one can call
<C>Size( <A>G</A> )</C>, and the function will return a value,
in our example an integer  which is the  size of <A>G</A>. 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  <A>G</A> when <C>Size( <A>G</A>  )</C>
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.
<P/>
<Index Subkey="of an attribute">getter</Index>
<Index Subkey="of an attribute">setter</Index>
<Index Subkey="of an attribute">tester</Index>
<Index>methods</Index>
This means that the behavior of the function
<Ref Func="Size" BookName="ref"/> has to  depend on
whether the size for the argument <A>G</A> 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 <C>Size( <A>G</A> )</C>,
namely the <E>tester</E> <C>HasSize( <A>G</A> )</C>
and the <E>setter</E> <C>SetSize( <A>G</A>, <A>size</A> )</C>.
The tester returns <K>true</K> or <K>false</K> according to
whether <A>G</A> has already stored its size,
and the setter puts <A>size</A> into
a place from where <A>G</A> can directly look it up.
The function <Ref Func="Size" BookName="ref"/> itself is called the
<E>getter</E>,
and from the preceding discussion we see
that there must really be at least two <E>methods</E> for the getter:
One method is used when the tester returns <K>false</K>;
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 <K>true</K>;
it simply returns the stored value.
This second method is also called the <E>system getter</E>.
&GAP; functions for which several methods can be available
are called <E>operations</E>,
so <Ref Func="Size" BookName="ref"/> is an example of an operation.
<P/>
<Log><![CDATA[
gap> G := Group(List([1..3], i-> Random(SymmetricGroup(53))));;
gap> Size( G ); time; # the time may of course vary on your machine
4274883284060025564298013753389399649690343788366813724672000000000000
196
gap> Size( G ); time;
4274883284060025564298013753389399649690343788366813724672000000000000
0
]]></Log>
<P/>
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. <E>At most
once</E> because the value of a function call like <C>Size( <A>G</A> )</C>
can also be set for <A>G</A> before the getter is called at all;
for example, one can call the setter directly if one knows the size.
<P/>
The size of a group is an example of a class of things  which in &GAP;
are called <E>attributes</E>.
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:
<Log><![CDATA[
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;
]]></Log>
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 <C>obj</C>).
By the way:
The names for setter and tester of an attribute are always composed from
the prefix <C>Set</C> resp.&nbsp;<C>Has</C> and the name of the getter.
<P/>
As a (not typical) example, note that the &GAP; function
<Ref Func="Random" BookName="ref"/>,
although it takes only one argument, is of course <E>not</E> 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 <Ref Func="Random" BookName="ref"/> is called in the sequel.
<P/>
There is a general important rule about attributes: <E>Once the value of an
attribute for an object has been set, it cannot be reset, i.e., it cannot
be changed any more.</E> 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 <K>true</K>, the setter simply does
nothing.
<P/>
<Example><![CDATA[
gap> G := SymmetricGroup(8);; Size(G);
40320
gap> SetSize( G, 0 ); Size( G );
40320
]]></Example>
<P/>
<E>Summary.</E> 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.

</Section>


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<Section Label="Properties and Filters">
<Heading>Properties and Filters</Heading>

<Index>filters</Index>
<Index Subkey="selection">methods</Index>
Certain attributes, like <Ref Func="IsAbelian" BookName="ref"/>,
are boolean-valued. Such attributes
are known to &GAP; as <E>properties</E>, 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  <E>filters</E>.  You can imagine  a filter  as a switch
which is  set either  to <K>true</K> or  to <K>false</K>.  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  <E>simple
filter</E>, and these  simple filters can be  combined with <C>and</C> to form
other filters (which are then <K>true</K> if and only if all the corresponding
bits are set to <K>true</K>).
For example, the filter <C>IsPermGroup and IsSolvableGroup</C> is made up from
several simple filters.
<P/>
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 <E>method selection</E>   are
described in chapter&nbsp;<Ref Chap="Method Selection" BookName="ref"/>.
<P/>
We  only present the following  rule  of  thumb here:
Each installed method for an attribute, say <Ref Func="Size" BookName="ref"/>,
has a <Q>required filter</Q>, which is made  up from certain simple filters
which must yield <K>true</K> for the argument <A>obj</A> for this method to be
applicable.
To execute a call of <C>Size( <A>obj</A> )</C>, &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 <A>obj</A>,
the more efficient it is expected to be.
For example, if <A>obj</A> is a permutation group that is not (known to be)
solvable,
a method  with required filter <C>IsPermGroup and IsSolvableGroup</C> is not
applicable, whereas a method with required filter
<Ref Func="IsPermGroup" BookName="ref"/> can be chosen.
On the other hand, if <A>obj</A> was  known to be solvable,
the method with required filter <C>IsPermGroup and IsSolvableGroup</C>
would be preferred to the one with required filter
<Ref Func="IsPermGroup" BookName="ref"/>.
<P/>
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 <C>TryNextMethod();</C>,
<Index Key="TryNextMethod"><C>TryNextMethod</C></Index>
and &GAP; calls the next applicable method.
For example, <Cite Key="Sims90b"/> 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 <C>size_solvable</C> implements this algorithm,
and that is returns the order of the group if it is solvable and
<K>fail</K> otherwise.
Then we can install the following method for
<Ref Func="Size" BookName="ref"/> with required
filter <Ref Func="IsPermGroup" BookName="ref"/>.
<Log><![CDATA[
function( G )
local  value;
    value := size_solvable( G );
    if value <> fail  then  return value;
                      else  TryNextMethod();  fi;
end;
]]></Log>
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.
<P/>
If no applicable method  (or no next applicable  method) is found, &GAP;
stops with an error message of the form 
<Log><![CDATA[
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 ...
]]></Log>
<P/>
You would get an error message as above if you asked for <C>Size( 1 )</C>. 
The message simply says that there is no method installed for calculating
the size of <C>1</C>.
Section <Ref Sect="Recovery from NoMethodFound-Errors" BookName="ref"/>
contains more information on how to deal with these messages.
<P/>
<E>Summary.</E> 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.

</Section>


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<Section Label="Immediate and True Methods">
<Heading>Immediate and True Methods</Heading>

<Index Subkey="immediate">methods</Index>
<Index Subkey="true">methods</Index>
In the example in Section&nbsp;<Ref Sect="Properties and Filters"/>,
we have mentioned that the operation <Ref Func="Size" BookName="ref"/> 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
<Ref Func="Size" BookName="ref"/> is to be calculated.
There are other examples where certain
methods are even <Q>cheaper</Q> 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
<Ref Func="IsSolvableGroup" BookName="ref"/> to <K>true</K> for
this group.   &GAP; utilizes   this  celebrated  theorem  by  having  an
<E>immediate  method</E> for <Ref Func="IsSolvableGroup" BookName="ref"/>
with required filter <C>HasSize</C>
which checks parity of the size and either sets
<Ref Func="IsSolvableGroup" BookName="ref"/> or does
nothing, i.e.,   calls <C>TryNextMethod()</C>.   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 <C>HasSize</C>
for that group has changed to <K>true</K>).
<P/>
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
<C>IsGroup and IsCyclic</C> implies <C>IsSolvableGroup</C>  and   such
implications  can be installed  in &GAP;  as  <E>true methods</E>. To execute
true methods, &GAP; only needs to do  some bookkeeping with its filters,
therefore true methods are much faster than immediate methods.
<P/>
How immediate and true methods are installed is described in 
<Ref Sect="Immediate Methods" BookName="ref"/> and
<Ref Sect="Logical Implications" BookName="ref"/>.

</Section>


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<Section Label="Operations and Method Selection">
<Heading>Operations and Method Selection</Heading>

<Index>operations</Index>
The method selection  is not only  used to  select  methods for attribute
getters but also for arbitrary <E>operations</E>, which can have more than one
argument.  In this case,   there is a   required filter for each argument
(which must yield <K>true</K>  for the corresponding  arguments).
<P/>
Additionally, a method with at least two arguments may require a certain
relation between the arguments,
which is expressed in terms of the <E>families</E> of the arguments.
For example, the methods for  <C>ConjugateGroup( <A>grp</A>, <A>elm</A> )</C>
require that <A>elm</A>  lies in the family   of elements from  which
<A>grp</A> is made, i.e., that  the family of  <A>elm</A> equals the
<Q>elements family</Q>  of <A>grp</A>.
<P/>
For permutation groups, the situation is quite easy:
all permutations form  one family,
<Ref Var="PermutationsFamily" BookName="ref"/>,
and each collection of permutations,
for example each permutation group, each coset of a permutation group,
or each dense list of permutations,
lies in <C>CollectionsFamily( PermutationsFamily )</C>.
<P/>
For other kinds of group elements, the situation can be different.
Every call of <Ref Func="FreeGroup" BookName="ref"/> constructs a new family
of free group elements.
&GAP; refuses to compute <C>One( FreeGroup( 1 ) ) * One( FreeGroup( 1 ) )</C>
because the two operands of the multiplication lie in different families
and no method is installed for this case.
<P/>
For further information on family relations,
see <Ref Sect="Families" BookName="ref"/>.
<P/>
<Index Key="KnownPropertiesOfObject"><C>KnownPropertiesOfObject</C></Index>
<Index Key="KnownTruePropertiesOfObject"><C>KnownTruePropertiesOfObject</C></Index>
<Index Key="KnownAttributesOfObject"><C>KnownAttributesOfObject</C></Index>
If you want to know  which properties are already known for an object
<A>obj</A>, or which properties are known to be true, you can use the
functions <C>KnownPropertiesOfObject( <A>obj</A> )</C> resp.
<C>KnownTruePropertiesOfObject( <A>obj</A> )</C>.
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 <K>true</K>). Analogously, there is the function
<Ref Func="KnownAttributesOfObject" BookName="ref"/> which lists the names
of the known attributes, leaving out the properties.
<P/>
<Index Key="ApplicableMethod"><C>ApplicableMethod</C></Index>
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 <C>TryNextMethod()</C>  occurs).
<C>ApplicableMethod( <A>opr</A>, [ arg_1, arg_2, ... ] )</C>
returns the  first  applicable method for the  call
<C><A>opr</A>( arg_1, arg_2, ... )</C>.
More  generally,
<C>ApplicableMethod( <A>opr</A>, [ ... ], 0, <A>nr</A> )</C>
returns the <A>nr</A>th applicable method (i.e.,
the one that would be chosen after <M><A>nr</A>-1</M> calls of
<C>TryNextMethod</C>) and if <A>nr</A><C> = "all"</C>,
the sorted list  of  all applicable methods is returned.
For  details,  see
<Ref Sect="Applicable Methods and Method Selection" BookName="ref"/>.
<P/>
<Index Key="TraceMethods"><C>TraceMethods</C></Index>
If you want to see which methods  are chosen for certain operations while
&GAP;  code is being executed, you can call the function 
<Ref Func="TraceMethods" BookName="ref" Label="for a list of operations"/>
with a list of these operations as arguments.
<P/>
<Log><![CDATA[
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
]]></Log>
<P/>
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 <C>TraceImmediateMethods( true )</C>.
<P/>
<Log><![CDATA[
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 ] );
]]></Log>
<P/>
The last two lines switch off tracing  again.
We now  see that the system getter was called by the immediate method for
<Ref Func="IsPerfectGroup" BookName="ref"/>. 
Also the above-mentioned immediate method for
<Ref Func="IsSolvableGroup" BookName="ref"/>
was not used because the solvability of <C>g</C> was already found out during
the size calculation
(cf.&nbsp;the example in Section&nbsp;<Ref Sect="Properties and Filters"/>).
<P/>
<E>Summary.</E>  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.

</Section>
</Chapter>


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