  
  [1X4 [33X[0;0YMethod Selection[133X[101X
  
  [33X[0;0YThe  setup  described  in  this chapter is intended for situations, in which
  lots  of  different  methods are available to fulfill a certain task, but in
  which  it  is  not  possible  in  the beginning to decide, which one to use.
  Therefore  this setup regulates, rather than just which method to choose, in
  which  order  the  various  methods are tried. The methods themselves return
  whether they were successful and, if not, whether it is sensible to try them
  again at a later stage.[133X
  
  [33X[0;0YThe  design is intentionally kept as simple as possible and at the same time
  as  versatile  as  possible,  thereby  providing a useful framework for many
  situations as described above.[133X
  
  [33X[0;0YNote the differences to the [5XGAP[105X method selection, which is designed with the
  idea  in  mind  that it will be quite clear in most situations, which one is
  [21Xthe  best[121X  method  for a given set of input data, and that we do not want to
  try  different  things. On the other hand, the [5XGAP[105X method selection is quite
  complicated,  which  is  to some extend necessary to make sure, that lots of
  different  information  about  the objects in question can be used to really
  find the best method.[133X
  
  [33X[0;0YOur  setup  here  in  particular has to fulfill the requirement, that in the
  end,  with  lots  of  methods installed, one still has to be able to have an
  overview and to [21Xprove[121X, that the whole system always does the right thing.[133X
  
  
  [1X4.1 [33X[0;0YWhat are methods?[133X[101X
  
  [33X[0;0YA  method  is  just  a  [5XGAP[105X  function  together with an agreement about what
  arguments  it  takes  and  what  result  it returns. The agreement about the
  arguments of course has to be made for every situation in which this generic
  method  selection  code  is  used,  and the user is completely free there. A
  method can (and has to) return one of the following four values:[133X
  
  [8X[10XSuccess[110X[8X[108X
        [33X[0;6Ymeans  that  the  method was successful and no more methods have to be
        tried.[133X
  
  [8X[10XNeverApplicable[110X[8X[108X
        [33X[0;6Ymeans that the method was not successful and that there is no point to
        call the method again in this situation whatsoever.[133X
  
  [8X[10XTemporaryFailure[110X[8X[108X
        [33X[0;6Ymeans  that  the  method  temporarily failed, that it however could be
        sensible  to  call  it  again in this situation at a later stage. This
        value  is  typical for a Las Vegas algorithm using randomised methods,
        which has failed, but which may succeed when called again.[133X
  
  [8X[10XNotEnoughInformation[110X[8X[108X
        [33X[0;6Ymeans that the method for some reason refused to do its work. However,
        it is possible that it will become applicable later such that it makes
        sense to call it again, may when more information is available.[133X
  
  [33X[0;0YFor  administration  in  the  method  selection,  a method is described by a
  record with the following components bound:[133X
  
  [8X[10Xmethod[110X[8X[108X
        [33X[0;6Yholds the function itself.[133X
  
  [8X[10Xrank[110X[8X[108X
        [33X[0;6Yholds an integer used to sort the various methods. Higher numbers mean
        that  the method is tried earlier. The numbering scheme is left to the
        user.[133X
  
  [8X[10Xstamp[110X[8X[108X
        [33X[0;6Yholds  a string value that uniquely describes the method. This is used
        for bookkeeping and to keep track of what has to be tried how often.[133X
  
  [8X[10Xcomment[110X[8X[108X
        [33X[0;6Ya string valued comment. This field is optional and can be left out.[133X
  
  [33X[0;0YThe  different  methods for a certain task are collected in so-called [21Xmethod
  databases[121X.  A  [13Xmethod  database[113X is just a list of records, each describing a
  method in the format described above. Usually, the ranks will be descending,
  but that is not necessary.[133X
  
  [33X[0;0YThere  is  one  convenience  function  to  put  a  new  method into a method
  database:[133X
  
  [1X4.1-1 AddMethod[101X
  
  [33X[1;0Y[29X[2XAddMethod[102X( [3Xdb[103X, [3Xmeth[103X, [3Xrank[103X, [3Xstamp[103X[, [3Xcomment[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0Y[3Xdb[103X must be a method database (list of records, see above) with non-ascending
  rank  values.  [3Xmeth[103X is the method function, [3Xrank[103X the rank and [3Xstamp[103X a string
  valued  stamp.  The  optional  argument [3Xcomment[103X can be a string comment. The
  record describing the method is created and inserted at the correct position
  in the method database. Nothing is returned.[133X
  
  
  [1X4.2 [33X[0;0YHow methods are called[133X[101X
  
  [33X[0;0YWhenever  the  method  selection  shall  be  used,  one  calls the following
  function:[133X
  
  [1X4.2-1 CallMethods[101X
  
  [33X[1;0Y[29X[2XCallMethods[102X( [3Xdb[103X, [3Xlimit[103X[, [3Xfurtherargs[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya record [10Xms[110X describing this method selection procedure.[133X
  
  [33X[0;0YThe argument [3Xdb[103X must be a method database in the sense of Section [14X4.1[114X. [3Xlimit[103X
  must  be  a non-negative integer. [3Xfurtherargs[103X stands for an arbitrary number
  of  additional  arguments,  which  are handed down to the called methods. Of
  course  they  must  fulfill  the  conventions defined for the methods in the
  database [3Xdb[103X.[133X
  
  [33X[0;0YThe  function  first  creates a [21Xmethod selection[121X record keeping track of the
  things  that happened during the method trying procedure, which is also used
  during  this  procedure.  Then it calls methods with the algorithm described
  below and in the end returns the method selection record in its final state.[133X
  
  [33X[0;0YThe method selection record has the following components:[133X
  
  [8X[10XinapplicableMethods[110X[8X[108X
        [33X[0;6Ya  record, in which for every method that returned [10XNeverApplicable[110X the
        value 1 is bound to the component with name the stamp of the method.[133X
  
  [8X[10XfailedMethods[110X[8X[108X
        [33X[0;6Ya  record,  in which for every time a method returned [10XTemporaryFailure[110X
        the  value bound to the component with name the stamp of the method is
        increased by 1 (not being bound means zero).[133X
  
  [8X[10XsuccessMethod[110X[8X[108X
        [33X[0;6Ythe  stamp of the method that succeeded, if one did. This component is
        only bound after successful completion.[133X
  
  [8X[10Xresult[110X[8X[108X
        [33X[0;6Ya  boolean value which is either [10XSuccess[110X or [10XTemporaryFailure[110X depending
        on  whether  a  successful  method  was found or the procedure gave up
        respectively.  This  component  is  only bound after completion of the
        method selection procedure.[133X
  
  [8X[10Xtolerance[110X[8X[108X
        [33X[0;6Ythe number of times all methods failed until one succeeded. See below.[133X
  
  [33X[0;0YThe  algorithm  used  by  [2XCallMethods[102X is extremely simple: It sets a counter
  [10Xtolerance[110X  to  zero.  The  main  loop  starts at the beginning of the method
  database and runs through the methods in turn. Provided a method did not yet
  return  [10XNeverApplicable[110X  and  did  not yet return [10XTemporaryFailure[110X more than
  [10Xtolerance[110X  times before, it is tried. According to the value returned by the
  method, the following happens:[133X
  
  [8X[10XNeverApplicable[110X[8X[108X
        [33X[0;6Ythis is marked in the method selection record and the main loop starts
        again at the beginning of the method database.[133X
  
  [8X[10XTemporaryFailure[110X[8X[108X
        [33X[0;6Ythis  is  counted  in  the  method  selection record and the main loop
        starts again at the beginning of the method database.[133X
  
  [8X[10XNotEnoughInformation[110X[8X[108X
        [33X[0;6Ythe main loop goes to the next method in the method database.[133X
  
  [8X[10Xtrue[110X[8X[108X
        [33X[0;6Ythis  is  marked  in  the  method  selection  record and the procedure
        returns successfully.[133X
  
  [33X[0;0YIf  the  main  loop reaches the end of the method database without calling a
  method (because all methods have already failed or are not applicable), then
  the  counter  [10Xtolerance[110X  is  increased by one and everything starts all over
  again.  This  is repeated until [10Xtolerance[110X is greater than the [10Xlimit[110X which is
  the  second argument of [2XCallMethods[102X. The last value of the [10Xtolerance[110X counter
  is returned in the component [10Xtolerance[110X of the method selection record.[133X
  
  [33X[0;0YNote that the main loop starts again at the beginning of the method database
  after  each  failed  method call! However, this does not lead to an infinite
  loop,  because  the  failure is recorded in the method selection record such
  that the method is skipped until the [10Xtolerance[110X increases. Once the [10Xtolerance[110X
  has  been  increased methods having returned [10XTemporaryFailure[110X will be called
  again. The idea behind this approach is that even failed methods can collect
  additional  information  about the arguments changing them accordingly. This
  might  give  methods that come earlier and were not applicable up to now the
  opportunity  to  begin  working. Therefore one can install very good methods
  that  depend  on  some  already  known knowledge which will only be acquired
  during the method selection procedure by other methods, with a high rank.[133X
  
