GAP (FR) - Chapter 10: FR implementation details

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10 FR implementation details
10.1 The family of FR objects
10.1-1 FRMFamily
10.1-2 FREFamily
10.1-3 AlphabetOfFRObject
10.1-4 AsPermutation
10.1-5 AsTransformation
10.2 Filters for `FRObject`s
10.2-1 IsGroupFRMachine
10.2-2 IsFRMachineStrRep
10.2-3 IsMealyMachine
10.2-4 IsMealyElement
10.2-5 IsMealyMachineIntRep
10.2-6 IsMealyMachineDomainRep
10.2-7 IsVectorFRMachineRep
10.2-8 IsAlgebraFRMachineRep
10.2-9 IsLinearFRMachine
10.2-10 IsLinearFRElement
10.2-11 IsFRElement
10.2-12 IsFRMealyElement
10.2-13 IsFRObject
10.2-14 IsFRMachine
10.2-15 IsInvertible
10.2-16 IsFRGroup
10.2-17 IsFRAlgebra
10.3 Some of the algorithms implemented
10.3-1 FRMachineRWS
10.3-2 Order of FR elements
10.3-3 Membership in semigroups
10.3-4 The conjugacy problem
10.3-5 OrbitSignalizer
10.3-6 FRConjugacyAlgorithm
10.3-7 FRBranchGroupConjugacyData
10.3-8 Order of groups
10.3-9 Images and preimages of some groups in f.p. and l.p. groups
10.3-10 Comparison of FR, Mealy, vector, and algebra elements
10.3-11 Inverses of linear elements

### 10 FR implementation details

FR creates new categories for the various objects considered in the package. The first category is `FRObject`; all objects are in this category, and have an `Alphabet` method.

There are two categories below: `FRMachine` and `FRElement`. An `FRMachine` must have a `StateSet`, and methods for `Output` and a `Transition`. An `FRElement` must have an underlying `FRMachine` and `InitialState`, and `Output` and a `Transition` that use the initial state.

A self-similar group is simply a collections category of FR elements which is also a group.

#### 10.1 The family of FR objects

All FR objects have an associated `AlphabetOfFRObject` (10.1-3).

##### 10.1-1 FRMFamily

`‣ FRMFamily`( obj ) | ( operation )

Returns: the family of FR machines on alphabet obj.

The family of an FR object is the arity of the tree on which elements cat act; in other words, there is one family for each alphabet.

##### 10.1-2 FREFamily

`‣ FREFamily`( obj ) | ( operation )

Returns: the family of FR elements on alphabet obj.

The family of an FR object is the arity of the tree on which elements cat act; in other words, there is one family for each alphabet.

The argument may be an FR machine, an alphabet, or a family of FR machines.

##### 10.1-3 AlphabetOfFRObject

`‣ AlphabetOfFRObject`( obj ) | ( operation )

`‣ AlphabetOfFRAlgebra`( obj ) | ( operation )

`‣ AlphabetOfFRSemigroup`( obj ) | ( operation )

`‣ Alphabet`( obj ) | ( operation )

Returns: the alphabet associated with obj.

This command applies to the family of any FR object, or to the object themselves. Alphabets are returned as lists, and in pratice are generally of the form `[1..n]`.

##### 10.1-4 AsPermutation

`‣ AsPermutation`( o ) | ( method )

This method takes as argument an FR object o: machine, element, or group, and produces an equivalent object whose outputs are permutations. In particular, it converts Mealy machines from domain representation to int representation.

If this is not possible, the method returns `fail`.

##### 10.1-5 AsTransformation

`‣ AsTransformation`( o ) | ( method )

This method takes as argument an FR object o: machine, element, or group, and produces an equivalent object whose outputs are transformations. In particular, it converts Mealy machines from domain representation to int representation.

Since transformations can never be inverted by GAP, even when they are invertible, this function returns a monoid when applied to a full SC group.

#### 10.2 Filters for `FRObject`s

##### 10.2-1 IsGroupFRMachine

`‣ IsGroupFRMachine`( obj ) | ( property )

`‣ IsMonoidFRMachine`( obj ) | ( property )

`‣ IsSemigroupFRMachine`( obj ) | ( property )

Returns: `true` if obj is an FR machine whose stateset is a free group/monoid/semigroup.

This function is the acceptor for those functionally recursive machines whose stateset (accessible via `StateSet` (3.4-1)) is a free group, monoid or semigroup. The generating set of its stateset is accessible via `GeneratorsOfFRMachine` (3.4-2).

##### 10.2-2 IsFRMachineStrRep

`‣ IsFRMachineStrRep`( obj ) | ( filter )

Returns: `true` if obj is a standard (group,monoid,semigroup) FR machine.

There is a free object `free`, of rank \(N\), a list `transitions` of length \(N\), each entry a list, indexed by the alphabet, of elements of `free`, and a list `output` of length `N` of transformations or permutations of the alphabet.

##### 10.2-3 IsMealyMachine

`‣ IsMealyMachine`( obj ) | ( filter )

Returns: `true` if obj is a Mealy machine.

This function is the acceptor for the Mealy machine subcategory of FR machines.

##### 10.2-4 IsMealyElement

`‣ IsMealyElement`( obj ) | ( filter )

Returns: `true` if obj is a Mealy element.

This function is the acceptor for the Mealy element subcategory of FR elements.

##### 10.2-5 IsMealyMachineIntRep

`‣ IsMealyMachineIntRep`( obj ) | ( filter )

Returns: `true` if obj is a Mealy machine in integer representation.

A Mealy machine in integer representation has components `nrstates`, `transitions`, `output` and optionally `initial`.

Its stateset is `[1..nrstates]`, its transitions is a matrix with `transitions[s][x]` the transition from state `s` with input `x`, its output is a list of transformations or permutations, and its initial state is an integer.

##### 10.2-6 IsMealyMachineDomainRep

`‣ IsMealyMachineDomainRep`( obj ) | ( filter )

Returns: `true` if obj is a Mealy machine in domain representation.

A Mealy machine in domain representation has components `states`, `transitions`, `output` and optionally `initial`.

Its states is a domain, its transitions is a function with `transitions(s,x)` the transition from state `s` with input `x`, its output is a function with `output(s,x)` the output from input `x` in state `s`, and its initial state is an elemnent of `states`.

##### 10.2-7 IsVectorFRMachineRep

`‣ IsVectorFRMachineRep`( obj ) | ( filter )

Returns: `true` if obj is a vector machine

A vector machine is a representation of a linear machine by a finite-dimensional vector space (implicit in the structure), a transition tensor (represented as a matrix of matrices), and an output vector (represented as a list).

##### 10.2-8 IsAlgebraFRMachineRep

`‣ IsAlgebraFRMachineRep`( obj ) | ( filter )

Returns: `true` if obj is an algebra machine

An algebra machine is a representation of a linear machine by a finitely generated free algebra, a tensor of transitions, indexed by generator index and two alphabet indices, and an output vector, indexed by a generator index.

The transition tensor's last two entries are the 0 and 1 matrix over the free algebra, and the output tensor's last two entries are the 0 and 1 elements of the left acting domain.

##### 10.2-9 IsLinearFRMachine

`‣ IsLinearFRMachine`( obj ) | ( filter )

Returns: `true` if obj is a linear machine.

This function is the acceptor for the linear machine subcategory of FR machines.

##### 10.2-10 IsLinearFRElement

`‣ IsLinearFRElement`( obj ) | ( filter )

Returns: `true` if obj is a linear element.

This function is the acceptor for the linear element subcategory of FR elements.

##### 10.2-11 IsFRElement

`‣ IsFRElement`( obj ) | ( filter )

`‣ IsSemigroupFRElement`( obj ) | ( filter )

`‣ IsMonoidFRElement`( obj ) | ( filter )

`‣ IsGroupFRElement`( obj ) | ( filter )

Returns: `true` if obj is an FR element.

This filter is the acceptor for the functionally recursive element category.

It implies that obj has an underlying FR machine, may act on sequences, and has a recursive `DecompositionOfFRElement` (4.2-6).

The next filters specify the type of free object the stateset of obj is modelled on.

##### 10.2-12 IsFRMealyElement

`‣ IsFRMealyElement`( obj ) | ( filter )

`‣ IsSemigroupFRMealyElement`( obj ) | ( filter )

`‣ IsMonoidFRMealyElement`( obj ) | ( filter )

`‣ IsGroupFRMealyElement`( obj ) | ( filter )

`‣ UnderlyingMealyElement`( obj ) | ( attribute )

Returns: `true` if obj is an FR element.

This filter is the acceptor for the functionally recursive element category, with an additional Mealy element stored as attribute for faster calculations. It defines a subcategory of `IsFRElement` (10.2-11). This additional Mealy element may be obtained as `UnderlyingMealyElement(obj)`.

The next filters specify the type of free object the stateset of obj is modelled on.

##### 10.2-13 IsFRObject

`‣ IsFRObject`( obj ) | ( filter )

Returns: `true` if obj is an FR machine or element.

This function is the acceptor for the most general FR category (which splits up as `IsFRMachine` (10.2-14) and `IsFRElement` (10.2-11)).

It implies that obj has an attribute `AlphabetOfFRObject` (10.1-3).

##### 10.2-14 IsFRMachine

`‣ IsFRMachine`( obj ) | ( filter )

Returns: `true` if obj is an FR machine.

This function is the acceptor for the functionally recursive machine category. It splits up as `IsGroupFRMachine` (10.2-1), `IsSemigroupFRMachine` (10.2-1), `IsMonoidFRMachine` (10.2-1) and `IsMealyMachine` (10.2-3)).

It implies that obj has attributes `StateSet` (3.4-1), `GeneratorsOfFRMachine` (3.4-2), and `WreathRecursion` (3.4-6); the last two are usually not used for Mealy machines.

##### 10.2-15 IsInvertible

`‣ IsInvertible`( m ) | ( property )

Returns: `true` if m is an invertible FR machine.

This function accepts invertible FR machines, i.e. machines m such that \((m,q)\) is an invertible transformation of the alphabet for all \(q\) in the stateset of m.

    gap> m := FRMachine([[[],[]]],[(1,2)]);
    <FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>
    gap> IsInvertible(m);
    true
    gap> m := FRMachine([[[],[]]],[[1,1]]);
    <FR machine with alphabet [ 1, 2 ] on Monoid( [ m1 ], ... )>
    gap> IsInvertible(m);
    false

##### 10.2-16 IsFRGroup

`‣ IsFRGroup`( obj ) | ( filter )

`‣ IsFRMonoid`( obj ) | ( filter )

`‣ IsFRSemigroup`( obj ) | ( filter )

Returns: `true` if obj is a FR group/monoid/semigroup.

These functions accept self-similar groups/monoids/semigroups, i.e. groups/monoids/semigroups whose elements are FR elements.

##### 10.2-17 IsFRAlgebra

`‣ IsFRAlgebra`( obj ) | ( filter )

`‣ IsFRAlgebraWithOne`( obj ) | ( filter )

Returns: `true` if obj is a FR algebra [with one].

These functions accept self-similar algebras [with one], i.e. algebras whose elements are linear FR elements.

#### 10.3 Some of the algorithms implemented

Few calculations with infinite groups can be guaranteed to terminate --- and especially to terminate within reasonable time. This section describes some of the algorithms implemented in FR.

##### 10.3-1 FRMachineRWS

`‣ FRMachineRWS`( m ) | ( attribute )

Returns: A record containing a rewriting system for m.

Elements of an FR machine are compared using a rewriting system, which records all known relations among states of the machine.

One may specify via an optional argument `:fr_maxlen:=n`, the maximal length of rules to be added. By default, this maximum length is 5.

    gap> n := FRMachine(["a","b"],[[[],[2]],[[],[1]]],[(1,2),()]);
    <FR machine with alphabet [ 1, 2 ] on Group( [ a, b ] )>
    gap> FRMachineRWS(n);
    rec( rws := Knuth Bendix Rewriting System for Monoid( [ a^-1, a, b^-1, b
         ], ... ) with rules
        [ [ a^-1*a, <identity ...> ], [ a*a^-1, <identity ...> ],
          [ b^-1*b, <identity ...> ], [ b*b^-1, <identity ...> ] ],
      tzrules := [ [ [ 1, 2 ], [  ] ], [ [ 2, 1 ], [  ] ], [ [ 3, 4 ], [  ] ],
          [ [ 4, 3 ], [  ] ] ], letterrep := function( w ) ... end,
      pi := function( w ) ... end, reduce := function( w ) ... end,
      addgprule := function( w ) ... end, commit := function(  ) ... end,
      restart := function(  ) ... end )

##### 10.3-2 Order of FR elements

The order of an FR element `e` is computed as follows: the tree is traversed recursively, filling it as follows. For each cycle of `e` on the first level, the product of the states on that cycle are computed. The method continues recursively with that product, remembering the order of the cycle. Once a state reappears in the traversal, FR determines if one instance of the state is in the subtree of the other, and if so whether the top one was raised to a non-trivial power to yield the second one as a state. If this happens, then `e` has infinite order. Otherwise, the least common multiple of the powers that appeared in the traversal is returned.

This method is guaranteed to succeed if `e` is a bounded element. To improve chances of success, FR first computes whether `e` acts by vertex transformations belonging to an abelian group; and if so, if `e` is conjugate to an adding machine. In that case too, `e` has infinite order.

##### 10.3-3 Membership in semigroups

The following algorithm is used to determine whether a Mealy element belongs to a self-similar group. The corresponding problem of membership of an FR element in a state-closed self-similar group can be much simpler, because an FR element has an associated FR machine, all of whose states belong to the group.

Assume the group is given by generators. FR attempts to express the given Mealy element as a product of generators. At the same time, it constructs epimorphisms to finite groups. It is hoped that one of these two processes will stop.

This amounts, in fact, to the following. Consider a group \(G\) acting on a tree. It has a natural, profinite closure \(\overline G\). The algorithm then attempts either to write an element \(x\) as a product of generators of G, or to show that \(x\) does not belong to \(\overline G\).

There are groups \(G\) such that \(\overline G\setminus G\) contains Mealy machines. For these, the above algorithm will not terminate.

An additional refinement is implemented for bounded groups (see `IsBoundedFRSemigroup` (7.2-14)). The `Germs` (5.2-24) of an element are computed, and compared to the germs of elements in the group.

Finally, for a group that possesses self-similar data (see Section 10.3-9), very fast methods are implemented to recognize and express an FR element as a product of generators.

##### 10.3-4 The conjugacy problem

The conjugacy problem for self-similar branch groups has been implemented by Thorsten Groth, as part of his Diploma Thesis. His code is integrated in FR.

Specialized algorithms are implemented for the Grigorchuk and Gupta-Sidki groups, and a generic algorithm is implemented, which is however not guaranteed to succeed. The implementation follows [BBSZ13].

The following extra attibutes are part of his implementation:

##### 10.3-5 OrbitSignalizer

`‣ OrbitSignalizer`( g ) | ( attribute )

Returns: The Orbit Signalizer of the group element g

This attribute computes the orbit signalizer of an element. This is the set \(OS(g) := \{g^{|Orb_g(v)|}@v \mid v \in X^*\}\) where \(X\) is the alphabet of the element g and \(Orb_g(v)\) is the orbit of \(v\) under \(\langle g \rangle\).

    gap> a := MealyElement([[2,2],[2,2]],[(1,2),()],1);
    <Mealy element on alphabet [ 1 .. 2 ] with 2 states>
    gap> OrbitSignalizer(a);
    [ <Mealy element on alphabet [ 1 .. 2 ] with 2 states>, <Trivial Mealy element on alphabet [ 1 .. 2 ]> ]

DeclareAttribute("OrbitSignalizer", IsFRElement);

##### 10.3-6 FRConjugacyAlgorithm

`‣ FRConjugacyAlgorithm`( G ) | ( attribute )

Returns: A function which solves the conjugacy problem for G

This attribute stores a function in three arguments which computes a representative conjugator if exists or fail otherwise.

This attribute is not meant to have a standard setter but to be set if a specialized conjugacy algorithm for a certain group is discovered.

    gap> f := FRConjugacyAlgorithm(GrigorchukGroup);
    function( G, g, h ) ... end
    gap> AssignGeneratorVariables(GrigorchukGroup);
    #I  Assigned the global variables [ "a", "b", "c", "d" ]
    gap> f(GrigorchukGroup,a,a^b);
    <Mealy element on alphabet [ 1 .. 2 ] with 5 states>

DeclareAttribute("FRConjugacyAlgorithm", IsFRGroup,2);

##### 10.3-7 FRBranchGroupConjugacyData

`‣ FRBranchGroupConjugacyData`( G ) | ( attribute )

Returns: The initial data for the branch algorithm for G

This attribute records the data for the branch algorithm. The record has the following components:

initial_conj_dic:

  Dictionary of already known conjugacy pairs with corresponding conjugator tuples. This has to cover at least the TorsionNucleus of G

Branchstructure

  Usally calculated by the function BranchStructure

RepSystem

  List of representatives of \(G/K\) where \(K\) is the branching subgroup of G

DeclareAttribute("FRBranchGroupConjugacyData", IsFRGroup,"mutable");

##### 10.3-8 Order of groups

The order of an FR group is computed as follows: if all generators are finitary, then enumeration will succeed in computing the order. If the action of the group is primitive, and it comes from a bireversible automaton, then the Thompson-Wielandt theorem is tested against. This theorem states that, in our context (a group acting on a rooted tree, coming from a larger group acting transitively), if the group is finite then the stabilizer of a sphere of radius 2 is a \(p\)-group; see [BM00a, Proposition 2.1.1]. Then, FR attempts to find whether the group is level-transitive (in which case it would be infinite). Finally, it attempts to enumerate the group's elements, testing at the same time whether these elements have infinite order.

Needless to say, none except the first few steps are guaranteed to succeed.

##### 10.3-9 Images and preimages of some groups in f.p. and l.p. groups

Contracting, branched groups admit finite L-presentations (see [Bar03a]), that is, presentations by finitely many generators, relators and endomorphisms; the (usual) relators are the images of the given relators under iteration by all endomorphisms.

Using the package NQL, it is possible to construct infinite nilpotent quotients of self-similar groups, and perform fast computations in them.

It is possible to construct, algorithmically, such an L-presentation from a self-similar groups; however, this algorithm has not been implemented yet, mainly because efficiency issues would make it usable only in very few cases.

For groups with an isomorphism to an L-presented group (constructed by `IsomorphismLpGroup` (7.2-31)), a fast method expresses group elements as words in the L-presented group's generators. It proceeds recursively on the decomposition of the element, mapping elements that are expressible by short words over the nucleus (usually length 1; length 3 is needed for the `BrunnerSidkiVieiraGroup` (9.1-13)) to their value in the L-presented group, and using the presentation's endomorphism to construct words with appropriate decompositions.

In particular, the algorithm will stop, returning `fail`, if during the recursion it reaches an element \(x\) such that \(x\) is a state of \(x\) but \(x\) does not belong to the nucleus.

##### 10.3-10 Comparison of FR, Mealy, vector, and algebra elements

FR and Mealy elements can be compared quite efficiently, as long as they are distinct. The algorithm runs as follows: let the two elements be \(x\) and \(y\). Considering both in turn, FR constructs the first entries of minimal Mealy elements expressing \(x\) and \(y\); as soon as an output entry is distinct for \(x\) and for \(y\), the status of \(x<y\) is determined; and similarly for transition entries. Finally, if either of \(x\) or \(y\) is finite-state and the entries were identical up to that step, then the element with smallest stateset is considered smaller.

In this way, FR and Mealy elements can efficiently be compared. For Mealy elements, it suffices to follow their internal data; while for FR elements, this amounts to constructing Mealy elements approximating them to a sufficient precision so that they can be compared as such.

The algorithm first tries to test its arguments for equality; this test is not guaranteed to succeed.

A similar algorithm applies for linear elements. Here, one constructs vector element approximations; and compares, for ever-increasing values of \(i\), first the output vectors of basis state \(i\); then the transitions from state \(i\) to state \(j\), for all \(j\in\{1,\ldots,i\}\); then the transitions from state \(j\) to state \(i\) for all \(j\in\{1,\ldots,i-1\}\).

##### 10.3-11 Inverses of linear elements

It is probably difficult to compute the inverse of a vector element. The following approach is used: to compute the inverse of \(x\), large (scalar) matrix approximations of \(x\) are computed; they are inverted using linear algebra; a vector element representing this inverse is guessed; and the guess is checked. As long as that check fails, larger approximations are computed.

Needless to say, this method need not succeed; for there are vector elements that are invertible, but whose inverse is not a vector element. A good test example appears in [Bac08]: consider the infinite matrix with 1's on the diagonal, and \(\omega\) below the diagonal. This element admits an inverse if and only if \(\omega\) is a root of unity. The complexity of the inverse grows as the degree of \(\omega\) grows. Here is an illustation:

    gap> bacher := function(n)
      local f;
      f := CyclotomicField(n);
      return VectorElement(f,One(f)*[[[[1,0],[0,0]],
    	[[0,0],[0,1]]],[[[0,1],[0,0]],[[1,0],[0,0]]]],[One(f),E(n)],[One(f),Zero(f)]);
    end;;
    gap> Inverse(bacher(3));
    <Linear element on alphabet CF(3)^2 with 4-dimensional stateset>
    6 gap> Inverse(bacher(5));
    <Linear element on alphabet CF(5)^2 with 6-dimensional stateset>

Table: Dimension of states of inverse
\(n\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
dimension |  | 2 | 4 | 4 | 6 | 3 | 5 | 5 | 8 | 5
\(n\) | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
dimension | ? | 5 | ? | 4 | 6 | 6 | ? | 7 | ? | 7
\(n\) | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40
dimension | ? | 6 | ? | 6 | ? | 7 | ? | ? | ? | ?

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