chap8.txt

  
  8 Operations and Methods
  
  
  8.1 Attributes
  
  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.
  
    Example  
    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:
  
    Example  
    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.
  
    Example  
    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.
  
  
  8.2 Properties and Filters
  
  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).
  
    Example  
    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
  
    Example  
    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.
  
  
  8.3 Immediate and True Methods
  
  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'.
  
  
  8.4 Operations and Method Selection
  
  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 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.
  
    Example  
    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 ).
  
    Example  
    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.