GAP (recog) - Chapter 9: How to write a recognition method

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9 How to write a recognition method
9.1 Leaf methods
9.2 Elements with memory
9.3 Splitting methods

### 9 How to write a recognition method

This chapter explains how to integrate a newly developed group recognition method into the framework provided by the recog package.

TODO: Refer to Chapter 4 for an explanation of methods. There are leaf methods and split methods. The next two sections describe how to implement leaf and split methods respectively, and include example code.

#### 9.1 Leaf methods

A leaf method must at the very least do the following, examples will be provided below (TODO: add a reference):

- Provide the order of the recognized group via `SetSize(ri, NNN)`.

- Provide a set of SLPs which map the original generators \(X\) to the nice generators \(Y\), as entry for the attribute `slptonice`.

- Provide a function which maps any element \(g\in G\) to a corresponding SLP in terms of the nice generators \(Y\), as entry for the attribute `slpforelement`.

- Call `SetFilterObj(ri, IsLeaf);` to mark the node as a leaf node.

There are further values that can be provided, in particular to speed up computations; we'll come back to that later. Let's first look at an example: The following method is used by recog to recognize trivial groups, as a base case for the recursive group recognition algorithm. It works for arbitrary groups.

TODO: AutoDoc inserts extra paragraph commands here:

    BindRecogMethod(FindHomMethodsGeneric, "TrivialGroup",
    "go through generators and compare to the identity",
    function(ri, G)
      local gens;
      # get the generators of the group
      gens := GeneratorsOfGroup(G);
    <P/>
      # check whether all generators are trivial
      # ri!.isone is explained below
      if not ForAll(gens, ri!.isone) then
        # NeverApplicable because it makes
        # no sense to call this method again
        return NeverApplicable;
      fi;
    <P/>
      # The group is trivial! Provide required information:
    <P/>
      # size of the group
      SetSize(ri, 1);
    <P/>
      # explained below
      Setslpforelement(ri, SLPforElementFuncsGeneric.TrivialGroup);
    <P/>
      # SLP from given generators to nice generators
      Setslptonice(ri, StraightLineProgramNC([[[1,0]]],
                       Length(gens)));
    <P/>
      # We have reached a leaf node.
      SetFilterObj(ri, IsLeaf);
      return Success;
    end);

The input is in the format described above (TODO), and the return value is "Success".

Two more comments:

- When we check whether all generators are the identity, we call `ri!.isone`, instead of `IsOne`. The reason for this is the need to support projective groups. For permutation groups and matrix groups, `ri!.isone` is simply defined to be `IsOne`. For projective groups, it is set to `IsOneProjective`, which can be read as "is one modulo scalars".

- The function `SLPforElementFuncs.TrivialGroup` takes `ri` as well as an element `g` as input. If \(g \in G\), then it is supposed to return an SLP for \(g\) in terms of the nice gens \(Y\). Otherwise it returns `fail`. Here is the concrete implementation:

    SLPforElementFuncsGeneric.TrivialGroup := function(ri,g)
        if not ri!.isone(g) then
            return fail;
        fi;
        return StraightLineProgramNC( [ [1,0] ], 1 );
    end;

Finally, we need to let recog know about this new recognition method. This is done via the `AddMethod` function. Another example!

    AddMethod(FindHomDbPerm, FindHomMethodsGeneric.TrivialGroup, 300);

TODO: refer to the `AddMethod` documentation instead. Also this is outdated now. The function `AddMethod` takes four mandatory arguments `db`, `meth`, `rank`, `stamp`, and an optional fifth argument `comment`. Their meaning is as follows:

- `db` is the "method database", and determines to which type of groups the methods should be applied. Allowed values are:

  - `FindHomDbPerm`

  - `FindHomDbMatrix`

  - `FindHomDbProj`

- `meth` is the recognition method we have defined. In our example this is `FindHomMethodsGeneric.TrivialGroup`.

- `rank` is the relative rank of the recognition method, given as an integer. The idea is that methods with a high rank get called before methods with a low rank, so [recog] tries recognition methods starting from the highest rank. What the "right" rank for a given method is depends on which other methods exist and what their ranks are. As a rule of thumb, methods which are either very fast or very likely to be applicable should be tried before slower methods, or methods which are less likely to be relevant.

- `stamp` holds a string value that uniquely describes the method. This is used for bookkeeping. It is also used in the manual, for printing the recognition tree, and for debugging purposes.

- `comment` is a string valued comment which in the example above has been used to explain what the method does. This argument is optional and can be left out.

Note that above, we only installed our method into `FindHomDbPerm`. But in recog, it is actually also installed for matrix and projective groups. We reproduce the corresponding `AddMethod` calls here. Note that the ranks differ, so the same method can be called with varying priority depending on the type of group.

    AddMethod(FindHomDbMatrix, FindHomMethodsGeneric.TrivialGroup, 3100);

    AddMethod(FindHomDbProjective, FindHomMethodsGeneric.TrivialGroup, 3000);

- TODO: more advanced example?

- TODO: also explain how verification works

- TODO: we need something that demonstrates the other two return values (Oh yes, good point.)

#### 9.2 Elements with memory

When using the memory of group elements, one currently has to always access ri!.gensHmem instead of doing `GroupWithMemory(Grp(ri))`. Namely, many functions for objects with memory assume that, if the elements live in the same group, then their `!.slp` components are identical.

#### 9.3 Splitting methods

Recall that splitting recognition methods produce an epimorphism \(\phi:G\to H\) and then delegate the work to the image \(H\) and the kernel \(N:=\ker(\phi)\). This means that now \(N\) and \(H\) have to be constructively recognized. Such a splitting recognition method only needs to provide a homomorphism, by calling `SetHomom(ri, hom);`. However, in practice one will want to provide additional data.

We start with an example, similar to a method used in recog. This refers to permutation groups only!

    BindRecogMethod(FindHomMethodsPerm, "NonTransitive",
    "try to find non-transitivity and restrict to orbit",
    rec(validatesOrAlwaysValidInput := true),
    function(ri, G)
        local hom,la,o;
    <P/>
        # test whether we can do something:
        if IsTransitive(G) then
            return NeverApplicable;
        fi;
    <P/>
        # compute orbit of the largest moved point
        la := LargestMovedPoint(G);
        o := Orb(G,la,OnPoints);
        Enumerate(o);
        # compute homomorphism into Sym(o), i.e, restrict
        # the permutation action of G to the orbit o
        hom := OrbActionHomomorphism(G,o);
        # TODO: explanation
        Setvalidatehomominput(ri, {ri,p} -> ForAll(o, x -> (x^p in o)));
        # store the homomorphism into the recognition node
        SetHomom(ri,hom);
    <P/>
        # TODO: explanation
        Setimmediateverification(ri, true);
    <P/>
        # indicate success
        return Success;
    end);

TODO Alternatively use this:

    FindHomMethodsPerm.NonTransitive := function(ri, G)
      local hom, la, o;

      # test whether we can do something:
      if IsTransitive(G) then
        # the action is transitive, so we can't do
        # anything, and there is no point in calling us again.
        return NeverApplicable;
      fi;

      # compute orbit of the largest moved point
      la := LargestMovedPoint(G);
      o := Orbit(G, la, OnPoints);

      # compute homomorphism into Sym(o), i.e, restrict
      # the permutation action of G to the orbit o
      hom := ActionHomomorphism(G, o);

      # store the homomorphism into the recognition node
      SetHomom(ri, hom);

      # indicate success
      return Success;
    end;

    AddMethod(FindHomDbPerm, FindHomMethodsPerm.NonTransitive, 90);

TODO: More complex example:

    FindHomMethodsMatrix.BlockLowerTriangular := function(ri, G)
      # This is only used coming from a hint, we know what to do:
      # A base change was done to get block lower triangular shape.
      # We first do the diagonal blocks, then the lower p-part:
      local H, data, hom, newgens;

      # we need to construct a homomorphism, but to defined it,
      # we need the image, but of course the image is defined in
      # terms of the homomorphism... to break this cycle, we do
      # the following: we first map the input generators using
      # the helper function RECOG.HomOntoBlockDiagonal; this
      # function is later also used as the underlying mapping
      # of the homomorphism.
      data := rec( blocks := ri!.blocks );
      newgens := List(GeneratorsOfGroup(G),
                      x -> RECOG.HomOntoBlockDiagonal(data, x));
      Assert(0, not fail in newgens);

      # now that we have the images of the generators, we can
      # defined the image group
      H := Group(newgens);

      # finally, we define the homomorphism
      hom := GroupHomByFuncWithData(G, H, RECOG.HomOntoBlockDiagonal, data);

      # ... and store it in the recognition node
      SetHomom(ri, hom);

      # since we know exactly what kind of group we are looking
      # at, we don't want to run generic recognition on the
      # image group and the kernel. So we provide "hints" to
      # ensure more appropriate recognition methods are applied
      # first.

      # Give hint to image
      InitialDataForImageRecogNode(ri).blocks := ri!.blocks;
      Add(InitialDataForImageRecogNode(ri).hints,
          rec( method := FindHomMethodsMatrix.BlockDiagonal,
               rank := 2000,
               stamp := "BlockDiagonal" ) );

      # Tell recog that we have a better method for finding kernel
      findgensNmeth(ri).method := FindKernelLowerLeftPGroup;
      findgensNmeth(ri).args := [];

      # Give hint to kernel N
      Add(InitialDataForKernelRecogNode(ri).hints,
          rec( method := FindHomMethodsMatrix.LowerLeftPGroup,
               rank := 2000,
               stamp := “LowerLeftPGroup" ));
      InitialDataForKernelRecogNode(ri).blocks := ri!.blocks;

      # This function always succeeds, because it is only
      # called for inputs for which it is known to apply.
      return Success;
    end;

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