GAP (AtlasRep) - Chapter 6: New GAP Objects and Utility Functions provided by the AtlasRep Package

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6 New GAP Objects and Utility Functions provided by the AtlasRep Package
6.1 Straight Line Decisions
6.1-1 IsStraightLineDecision
6.1-2 LinesOfStraightLineDecision
6.1-3 NrInputsOfStraightLineDecision
6.1-4 ScanStraightLineDecision
6.1-5 StraightLineDecision
6.1-6 ResultOfStraightLineDecision
6.1-7 Semi-Presentations and Presentations
6.1-8 AsStraightLineDecision
6.1-9 StraightLineProgramFromStraightLineDecision
6.2 Black Box Programs
6.2-1 IsBBoxProgram
6.2-2 ScanBBoxProgram
6.2-3 RunBBoxProgram
6.2-4 ResultOfBBoxProgram
6.2-5 AsBBoxProgram
6.2-6 AsStraightLineProgram
6.3 Representations of Minimal Degree
6.3-1 MinimalRepresentationInfo
6.3-2 MinimalRepresentationInfoData
6.3-3 SetMinimalRepresentationInfo
6.3-4 Criteria Used to Compute Minimality Information
6.4 A JSON Interface
6.4-1 Why JSON?
6.4-2 AGR.JsonText
6.4-3 AGR.GapObjectOfJsonText

### 6 New GAP Objects and Utility Functions provided by the AtlasRep Package

This chapter describes GAP objects and functions that are provided by the AtlasRep package but that might be of general interest.

The new objects are straight line decisions (see Section 6.1) and black box programs (see Section 6.2).

The new functions are concerned with representations of minimal degree, see Section 6.3, and a JSON interface, see Section 6.4.

#### 6.1 Straight Line Decisions

Straight line decisions are similar to straight line programs (see Section Reference: Straight Line Programs) but return `true` or `false`. A straight line decision checks whether its inputs have some property. An important example is to check whether a given list of group generators is in fact a list of standard generators (cf. Section3.3) for this group.

A straight line decision in GAP is represented by an object in the filter `IsStraightLineDecision` (6.1-1) that stores a list of "lines" each of which has one of the following three forms.

1. a nonempty dense list \(l\) of integers,

2. a pair \([ l, i ]\) where \(l\) is a list of form 1. and \(i\) is a positive integer,

3. a list \([\) `"Order"`\(, i, n ]\) where \(i\) and \(n\) are positive integers.

The first two forms have the same meaning as for straight line programs (see Section Reference: Straight Line Programs), the last form means a check whether the element stored at the \(i\)-th label has the order \(n\).

For the meaning of the list of lines, see `ResultOfStraightLineDecision` (6.1-6).

Straight line decisions can be constructed using `StraightLineDecision` (6.1-5), defining attributes for straight line decisions are `NrInputsOfStraightLineDecision` (6.1-3) and `LinesOfStraightLineDecision` (6.1-2), an operation for straight line decisions is `ResultOfStraightLineDecision` (6.1-6).

Special methods applicable to straight line decisions are installed for the operations `Display` (Reference: Display), `IsInternallyConsistent` (Reference: IsInternallyConsistent), `PrintObj` (Reference: PrintObj), and `ViewObj` (Reference: ViewObj).

For a straight line decision prog, the default `Display` (Reference: Display) method prints the interpretation of prog as a sequence of assignments of associative words and of order checks; a record with components `gensnames` (with value a list of strings) and `listname` (a string) may be entered as second argument of `Display` (Reference: Display), in this case these names are used, the default for `gensnames` is `[ g1, g2,` \(\ldots\)`]`, the default for listname is \(r\).

##### 6.1-1 IsStraightLineDecision

`‣ IsStraightLineDecision`( obj ) | ( category )

Each straight line decision in GAP lies in the filter `IsStraightLineDecision`.

##### 6.1-2 LinesOfStraightLineDecision

`‣ LinesOfStraightLineDecision`( prog ) | ( operation )

Returns: the list of lines that define the straight line decision.

This defining attribute for the straight line decision prog (see `IsStraightLineDecision` (6.1-1)) corresponds to `LinesOfStraightLineProgram` (Reference: LinesOfStraightLineProgram) for straight line programs.

    gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
    > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
    <straight line decision>
    gap> LinesOfStraightLineDecision( dec );
    [ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ],
      [ "Order", 3, 5 ] ]

##### 6.1-3 NrInputsOfStraightLineDecision

`‣ NrInputsOfStraightLineDecision`( prog ) | ( operation )

Returns: the number of inputs required for the straight line decision.

This defining attribute corresponds to `NrInputsOfStraightLineProgram` (Reference: NrInputsOfStraightLineProgram).

    gap> NrInputsOfStraightLineDecision( dec );
    2

##### 6.1-4 ScanStraightLineDecision

`‣ ScanStraightLineDecision`( string ) | ( function )

Returns: a record containing the straight line decision, or `fail`.

Let string be a string that encodes a straight line decision in the sense that it consists of the lines listed for `ScanStraightLineProgram` (7.4-1), except that `oup` lines are not allowed, and instead lines of the following form may occur.

`chor` \(a\) \(b\)

  means that it is checked whether the order of the element at label \(a\) is \(b\).

`ScanStraightLineDecision` returns a record containing as the value of its component `program` the corresponding GAP straight line decision (see `IsStraightLineDecision` (6.1-1)) if the input string satisfies the syntax rules stated above, and returns `fail` otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of `InfoCMeatAxe` (7.1-2) is at least \(1\).

    gap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";;
    gap> prg:= ScanStraightLineDecision( str );
    rec( program := <straight line decision> )
    gap> prg:= prg.program;;
    gap> Display( prg );
    # input:
    r:= [ g1, g2 ];
    # program:
    if Order( r[1] ) <> 2 then  return false;  fi;
    if Order( r[2] ) <> 3 then  return false;  fi;
    r[3]:= r[1]*r[2];
    if Order( r[3] ) <> 5 then  return false;  fi;
    # return value:
    true

##### 6.1-5 StraightLineDecision

`‣ StraightLineDecision`( lines[, nrgens] ) | ( function )

`‣ StraightLineDecisionNC`( lines[, nrgens] ) | ( function )

Returns: the straight line decision given by the list of lines.

Let lines be a list of lists that defines a unique straight line decision (see `IsStraightLineDecision` (6.1-1)); in this case `StraightLineDecision` returns this program, otherwise an error is signalled. The optional argument nrgens specifies the number of input generators of the program; if a list of integers (a line of form 1. in the definition above) occurs in lines then this number is not determined by lines and therefore must be specified by the argument nrgens; if not then `StraightLineDecision` returns `fail`.

`StraightLineDecisionNC` does the same as `StraightLineDecision`, except that the internal consistency of the program is not checked.

##### 6.1-6 ResultOfStraightLineDecision

`‣ ResultOfStraightLineDecision`( prog, gens[, orderfunc] ) | ( operation )

Returns: `true` if all checks succeed, otherwise `false`.

`ResultOfStraightLineDecision` evaluates the straight line decision (see `IsStraightLineDecision` (6.1-1)) prog at the group elements in the list gens.

The function for computing the order of a group element can be given as the optional argument orderfunc. For example, this may be a function that gives up at a certain limit if one has to be aware of extremely huge orders in failure cases.

The result of a straight line decision with lines \(p_1, p_2, \ldots, p_k\) when applied to gens is defined as follows.

(a)

  First a list \(r\) of intermediate values is initialized with a shallow copy of gens.

(b)

  For \(i \leq k\), before the \(i\)-th step, let \(r\) be of length \(n\). If \(p_i\) is the external representation of an associative word in the first \(n\) generators then the image of this word under the homomorphism that is given by mapping \(r\) to these first \(n\) generators is added to \(r\). If \(p_i\) is a pair \([ l, j ]\), for a list \(l\), then the same element is computed, but instead of being added to \(r\), it replaces the \(j\)-th entry of \(r\). If \(p_i\) is a triple \([ \)`"Order"`\(, i, n ]\) then it is checked whether the order of \(r[i]\) is \(n\); if not then `false` is returned immediately.

(c)

  If all \(k\) lines have been processed and no order check has failed then `true` is returned.

Here are some examples.

    gap> dec:= StraightLineDecision( [ ], 1 );
    <straight line decision>
    gap> ResultOfStraightLineDecision( dec, [ () ] );
    true

The above straight line decision `dec` returns `true` –for any input of the right length.

    gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
    >       [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
    <straight line decision>
    gap> LinesOfStraightLineDecision( dec );
    [ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ],
      [ "Order", 3, 5 ] ]
    gap> ResultOfStraightLineDecision( dec, [ (), () ] );
    false
    gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,4,5) ] );
    true

The above straight line decision admits two inputs; it tests whether the orders of the inputs are \(2\) and \(3\), and the order of their product is \(5\).

##### 6.1-7 Semi-Presentations and Presentations

We can associate a finitely presented group \(F / R\) to each straight line decision dec, say, as follows. The free generators of the free group \(F\) are in bijection with the inputs, and the defining relators generating \(R\) as a normal subgroup of \(F\) are given by those words \(w^k\) for which dec contains a check whether the order of \(w\) equals \(k\).

So if dec returns `true` for the input list \([ g_1, g_2, \ldots, g_n ]\) then mapping the free generators of \(F\) to the inputs defines an epimorphism \(\Phi\) from \(F\) to the group \(G\), say, that is generated by these inputs, such that \(R\) is contained in the kernel of \(\Phi\).

(Note that "satisfying dec" is a stronger property than "satisfying a presentation". For example, \(\langle x \mid x^2 = x^3 = 1 \rangle\) is a presentation for the trivial group, but the straight line decision that checks whether the order of \(x\) is both \(2\) and \(3\) clearly always returns `false`.)

AtlasRep supports the following two kinds of straight line decisions.

- A presentation is a straight line decision dec that is defined for a set of standard generators of a group \(G\) and that returns `true` if and only if the list of inputs is in fact a sequence of such standard generators for \(G\). In other words, the relators derived from the order checks in the way described above are defining relators for \(G\), and moreover these relators are words in terms of standard generators. (In particular the kernel of the map \(\Phi\) equals \(R\) whenever dec returns `true`.)

- A semi-presentation is a straight line decision dec that is defined for a set of standard generators of a group \(G\) and that returns `true` for a list of inputs that is known to generate a group isomorphic with \(G\) if and only if these inputs form in fact a sequence of standard generators for \(G\). In other words, the relators derived from the order checks in the way described above are not necessarily defining relators for \(G\), but if we assume that the \(g_i\) generate \(G\) then they are standard generators. (In particular, \(F / R\) may be a larger group than \(G\) but in this case \(\Phi\) maps the free generators of \(F\) to standard generators of \(G\).)

  More about semi-presentations can be found in [NW05].

Available presentations and semi-presentations are listed by `DisplayAtlasInfo` (3.5-1), they can be accessed via `AtlasProgram` (3.5-4). (Clearly each presentation is also a semi-presentation. So a semi-presentation for some standard generators of a group is regarded as available whenever a presentation for these standard generators and this group is available.)

Note that different groups can have the same semi-presentation. We illustrate this with an example that is mentioned in [NW05]. The groups \(L_2(7) \cong L_3(2)\) and \(L_2(8)\) are generated by elements of the orders \(2\) and \(3\) such that their product has order \(7\), and no further conditions are necessary to define standard generators.

    gap> check:= AtlasProgram( "L2(8)", "check" );
    rec( groupname := "L2(8)",
      identifier := [ "L2(8)", "L28G1-check1", 1, 1 ],
      program := <straight line decision>, standardization := 1,
      version := "1" )
    gap> gens:= AtlasGenerators( "L2(8)", 1 );
    rec( charactername := "1a+8a", constituents := [ 1, 6 ],
      contents := "core",
      generators := [ (1,2)(3,4)(6,7)(8,9), (1,3,2)(4,5,6)(7,8,9) ],
      groupname := "L2(8)", id := "",
      identifier := [ "L2(8)", [ "L28G1-p9B0.m1", "L28G1-p9B0.m2" ], 1, 9
         ], isPrimitive := true, maxnr := 1, p := 9, rankAction := 2,
      repname := "L28G1-p9B0", repnr := 1, size := 504,
      stabilizer := "2^3:7", standardization := 1, transitivity := 3,
      type := "perm" )
    gap> ResultOfStraightLineDecision( check.program, gens.generators );
    true
    gap> gens:= AtlasGenerators( "L3(2)", 1 );
    rec( contents := "core", generators := [ (2,4)(3,5), (1,2,3)(5,6,7) ],
      groupname := "L3(2)", id := "a",
      identifier := [ "L3(2)", [ "L27G1-p7aB0.m1", "L27G1-p7aB0.m2" ], 1,
          7 ], isPrimitive := true, maxnr := 1, p := 7, rankAction := 2,
      repname := "L27G1-p7aB0", repnr := 1, size := 168,
      stabilizer := "S4", standardization := 1, transitivity := 2,
      type := "perm" )
    gap> ResultOfStraightLineDecision( check.program, gens.generators );
    true

##### 6.1-8 AsStraightLineDecision

`‣ AsStraightLineDecision`( bbox ) | ( attribute )

Returns: an equivalent straight line decision for the given black box program, or `fail`.

For a black box program (see `IsBBoxProgram` (6.2-1)) bbox, `AsStraightLineDecision` returns a straight line decision (see `IsStraightLineDecision` (6.1-1)) with the same output as bbox, in the sense of `AsBBoxProgram` (6.2-5), if such a straight line decision exists, and `fail` otherwise.

    gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],
    >              [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;
    gap> dec:= StraightLineDecision( lines, 2 );
    <straight line decision>
    gap> bboxdec:= AsBBoxProgram( dec );
    <black box program>
    gap> asdec:= AsStraightLineDecision( bboxdec );
    <straight line decision>
    gap> LinesOfStraightLineDecision( asdec );
    [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ [ 1, 1, 2, 1 ], 3 ],
      [ "Order", 3, 5 ] ]

##### 6.1-9 StraightLineProgramFromStraightLineDecision

`‣ StraightLineProgramFromStraightLineDecision`( dec ) | ( operation )

Returns: the straight line program associated to the given straight line decision.

For a straight line decision dec (see `IsStraightLineDecision` (6.1-1), `StraightLineProgramFromStraightLineDecision` returns the straight line program (see `IsStraightLineProgram` (Reference: IsStraightLineProgram) obtained by replacing each line of type 3. (i.e, each order check) by an assignment of the power in question to a new slot, and by declaring the list of these elements as the return value.

This means that the return value describes exactly the defining relators of the presentation that is associated to the straight line decision, see 6.1-7.

For example, one can use the return value for printing the relators with `StringOfResultOfStraightLineProgram` (Reference: StringOfResultOfStraightLineProgram), or for explicitly constructing the relators as words in terms of free generators, by applying `ResultOfStraightLineProgram` (Reference: ResultOfStraightLineProgram) to the program and to these generators.

    gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
    > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
    <straight line decision>
    gap> prog:= StraightLineProgramFromStraightLineDecision( dec );
    <straight line program>
    gap> Display( prog );
    # input:
    r:= [ g1, g2 ];
    # program:
    r[3]:= r[1]*r[2];
    r[4]:= r[1]^2;
    r[5]:= r[2]^3;
    r[6]:= r[3]^5;
    # return values:
    [ r[4], r[5], r[6] ]
    gap> StringOfResultOfStraightLineProgram( prog, [ "a", "b" ] );
    "[ a^2, b^3, (ab)^5 ]"
    gap> gens:= GeneratorsOfGroup( FreeGroup( "a", "b" ) );
    [ a, b ]
    gap> ResultOfStraightLineProgram( prog, gens );
    [ a^2, b^3, (a*b)^5 ]

#### 6.2 Black Box Programs

Black box programs formalize the idea that one takes some group elements, forms arithmetic expressions in terms of them, tests properties of these expressions, executes conditional statements (including jumps inside the program) depending on the results of these tests, and eventually returns some result.

A specification of the language can be found in [Nic06], see also

http://atlas.math.rwth-aachen.de/Atlas/info/blackbox.html.

The inputs of a black box program may be explicit group elements, and the program may also ask for random elements from a given group. The program steps form products, inverses, conjugates, commutators, etc. of known elements, tests concern essentially the orders of elements, and the result is a list of group elements or `true` or `false` or `fail`.

Examples that can be modeled by black box programs are

straight line programs,

  which require a fixed number of input elements and form arithmetic expressions of elements but do not use random elements, tests, conditional statements and jumps; the return value is always a list of elements; these programs are described in Section Reference: Straight Line Programs.

straight line decisions,

  which differ from straight line programs only in the sense that also order tests are admissible, and that the return value is `true` if all these tests are satisfied, and `false` as soon as the first such test fails; they are described in Section 6.1.

scripts for finding standard generators,

  which take a group and a function to generate a random element in this group but no explicit input elements, admit all control structures, and return either a list of standard generators or `fail`; see `ResultOfBBoxProgram` (6.2-4) for examples.

In the case of general black box programs, currently GAP provides only the possibility to read an existing program via `ScanBBoxProgram` (6.2-2), and to run the program using `RunBBoxProgram` (6.2-3). It is not our aim to write such programs in GAP.

The special case of the "find" scripts mentioned above is also admissible as an argument of `ResultOfBBoxProgram` (6.2-4), which returns either the set of found generators or `fail`.

Contrary to the general situation, more support is provided for straight line programs and straight line decisions in GAP, see Section Reference: Straight Line Programs for functions that manipulate them (compose, restrict etc.).

The functions `AsStraightLineProgram` (6.2-6) and `AsStraightLineDecision` (6.1-8) can be used to transform a general black box program object into a straight line program or a straight line decision if this is possible.

Conversely, one can create an equivalent general black box program from a straight line program or from a straight line decision with `AsBBoxProgram` (6.2-5).

Computing a straight line program related to a given straight line decision is supported in the sense of `StraightLineProgramFromStraightLineDecision` (6.1-9).

Note that none of these three kinds of objects is a special case of another: Running a black box program with `RunBBoxProgram` (6.2-3) yields a record, running a straight line program with `ResultOfStraightLineProgram` (Reference: ResultOfStraightLineProgram) yields a list of elements, and running a straight line decision with `ResultOfStraightLineDecision` (6.1-6) yields `true` or `false`.

##### 6.2-1 IsBBoxProgram

`‣ IsBBoxProgram`( obj ) | ( category )

Each black box program in GAP lies in the filter `IsBBoxProgram`.

##### 6.2-2 ScanBBoxProgram

`‣ ScanBBoxProgram`( string ) | ( function )

Returns: a record containing the black box program encoded by the input string, or `fail`.

For a string string that describes a black box program, e.g., the return value of `StringFile` (GAPDoc: StringFile), `ScanBBoxProgram` computes this black box program. If this is successful then the return value is a record containing as the value of its component `program` the corresponding GAP object that represents the program, otherwise `fail` is returned.

As the first example, we construct a black box program that tries to find standard generators for the alternating group \(A_5\); these standard generators are any pair of elements of the orders \(2\) and \(3\), respectively, such that their product has order \(5\).

    gap> findstr:= "\
    >   set V 0\n\
    > lbl START1\n\
    >   rand 1\n\
    >   ord 1 A\n\
    >   incr V\n\
    >   if V gt 100 then timeout\n\
    >   if A notin 1 2 3 5 then fail\n\
    >   if A noteq 2 then jmp START1\n\
    > lbl START2\n\
    >   rand 2\n\
    >   ord 2 B\n\
    >   incr V\n\
    >   if V gt 100 then timeout\n\
    >   if B notin 1 2 3 5 then fail\n\
    >   if B noteq 3 then jmp START2\n\
    >   # The elements 1 and 2 have the orders 2 and 3, respectively.\n\
    >   set X 0\n\
    > lbl CONJ\n\
    >   incr X\n\
    >   if X gt 100 then timeout\n\
    >   rand 3\n\
    >   cjr 2 3\n\
    >   mu 1 2 4   # ab\n\
    >   ord 4 C\n\
    >   if C notin 2 3 5 then fail\n\
    >   if C noteq 5 then jmp CONJ\n\
    >   oup 2 1 2";;
    gap> find:= ScanBBoxProgram( findstr );
    rec( program := <black box program> )

The second example is a black box program that checks whether its two inputs are standard generators for \(A_5\).

    gap> checkstr:= "\
    > chor 1 2\n\
    > chor 2 3\n\
    > mu 1 2 3\n\
    > chor 3 5";;
    gap> check:= ScanBBoxProgram( checkstr );
    rec( program := <black box program> )

##### 6.2-3 RunBBoxProgram

`‣ RunBBoxProgram`( prog, G, input, options ) | ( function )

Returns: a record describing the result and the statistics of running the black box program prog, or `fail`, or the string `"timeout"`.

For a black box program prog, a group G, a list input of group elements, and a record options, `RunBBoxProgram` applies prog to input, where G is used only to compute random elements.

The return value is `fail` if a syntax error or an explicit `fail` statement is reached at runtime, and the string `"timeout"` if a `timeout` statement is reached. (The latter might mean that the random choices were unlucky.) Otherwise a record with the following components is returned.

`gens`

  a list of group elements, bound if an `oup` statement was reached,

`result`

  `true` if a `true` statement was reached, `false` if either a `false` statement or a failed order check was reached,

The other components serve as statistical information about the numbers of the various operations (`multiply`, `invert`, `power`, `order`, `random`, `conjugate`, `conjugateinplace`, `commutator`), and the runtime in milliseconds (`timetaken`).

The following components of options are supported.

`randomfunction`

  the function called with argument G in order to compute a random element of G (default `PseudoRandom` (Reference: PseudoRandom))

`orderfunction`

  the function for computing element orders (default `Order` (Reference: Order)),

`quiet`

  if `true` then ignore `echo` statements (default `false`),

`verbose`

  if `true` then print information about the line that is currently processed, and about order checks (default `false`),

`allowbreaks`

  if `true` then call `Error` (Reference: Error) when a `break` statement is reached, otherwise ignore `break` statements (default `true`).

As an example, we run the black box programs constructed in the example for `ScanBBoxProgram` (6.2-2).

    gap> g:= AlternatingGroup( 5 );;
    gap> res:= RunBBoxProgram( find.program, g, [], rec() );;
    gap> IsBound( res.gens );  IsBound( res.result );
    true
    false
    gap> List( res.gens, Order );
    [ 2, 3 ]
    gap> Order( Product( res.gens ) );
    5
    gap> res:= RunBBoxProgram( check.program, "dummy", res.gens, rec() );;
    gap> IsBound( res.gens );  IsBound( res.result );
    false
    true
    gap> res.result;
    true
    gap> othergens:= GeneratorsOfGroup( g );;
    gap> res:= RunBBoxProgram( check.program, "dummy", othergens, rec() );;
    gap> res.result;
    false

##### 6.2-4 ResultOfBBoxProgram

`‣ ResultOfBBoxProgram`( prog, G[, options] ) | ( function )

Returns: a list of group elements or `true`, `false`, `fail`, or the string `"timeout"`.

This function calls `RunBBoxProgram` (6.2-3) with the black box program prog and second argument either a group or a list of group elements; if options is not given then the default options of `RunBBoxProgram` (6.2-3) are assumed. The return value is `fail` if this call yields `fail`, otherwise the `gens` component of the result, if bound, or the `result` component if not.

Note that a group G is used as the second argument in the call of `RunBBoxProgram` (6.2-3) (the source for random elements), whereas a list G is used as the third argument (the inputs).

As an example, we run the black box programs constructed in the example for `ScanBBoxProgram` (6.2-2).

    gap> g:= AlternatingGroup( 5 );;
    gap> res:= ResultOfBBoxProgram( find.program, g );;
    gap> List( res, Order );
    [ 2, 3 ]
    gap> Order( Product( res ) );
    5
    gap> res:= ResultOfBBoxProgram( check.program, res );
    true
    gap> othergens:= GeneratorsOfGroup( g );;
    gap> res:= ResultOfBBoxProgram( check.program, othergens );
    false

##### 6.2-5 AsBBoxProgram

`‣ AsBBoxProgram`( slp ) | ( attribute )

Returns: an equivalent black box program for the given straight line program or straight line decision.

Let slp be a straight line program (see `IsStraightLineProgram` (Reference: IsStraightLineProgram)) or a straight line decision (see `IsStraightLineDecision` (6.1-1)). Then `AsBBoxProgram` returns a black box program bbox (see `IsBBoxProgram` (6.2-1)) with the "same" output as slp, in the sense that `ResultOfBBoxProgram` (6.2-4) yields the same result for bbox as `ResultOfStraightLineProgram` (Reference: ResultOfStraightLineProgram) or `ResultOfStraightLineDecision` (6.1-6), respectively, for slp.

    gap> f:= FreeGroup( "x", "y" );;  gens:= GeneratorsOfGroup( f );;
    gap> slp:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 );
    <straight line program>
    gap> ResultOfStraightLineProgram( slp, gens );
    y^-3*x^-2
    gap> bboxslp:= AsBBoxProgram( slp );
    <black box program>
    gap> ResultOfBBoxProgram( bboxslp, gens );
    [ y^-3*x^-2 ]
    gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],
    >              [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;
    gap> dec:= StraightLineDecision( lines, 2 );
    <straight line decision>
    gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,5) ] );
    true
    gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,4) ] );
    false
    gap> bboxdec:= AsBBoxProgram( dec );
    <black box program>
    gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,5) ] );
    true
    gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,4) ] );
    false

##### 6.2-6 AsStraightLineProgram

`‣ AsStraightLineProgram`( bbox ) | ( attribute )

Returns: an equivalent straight line program for the given black box program, or `fail`.

For a black box program (see `AsBBoxProgram` (6.2-5)) bbox, `AsStraightLineProgram` returns a straight line program (see `IsStraightLineProgram` (Reference: IsStraightLineProgram)) with the same output as bbox if such a straight line program exists, and `fail` otherwise.

    gap> Display( AsStraightLineProgram( bboxslp ) );
    # input:
    r:= [ g1, g2 ];
    # program:
    r[3]:= r[1]^2;
    r[4]:= r[2]^3;
    r[5]:= r[3]*r[4];
    r[3]:= r[5]^-1;
    # return values:
    [ r[3] ]
    gap> AsStraightLineProgram( bboxdec );
    fail

#### 6.3 Representations of Minimal Degree

This section deals with minimal degrees of permutation and matrix representations. We do not provide an algorithm that computes these degrees for an arbitrary group, we only provide some tools for evaluating known databases, mainly concerning "bicyclic extensions" (see [CCN+85, Section 6.5]) of simple groups, in order to derive the minimal degrees, see Section 6.3-4.

In the AtlasRep package, this information can be used for prescribing "minimality conditions" in `DisplayAtlasInfo` (3.5-1), `OneAtlasGeneratingSetInfo` (3.5-6), and `AllAtlasGeneratingSetInfos` (3.5-7). An overview of the stored minimal degrees can be shown with `BrowseMinimalDegrees` (3.6-1).

##### 6.3-1 MinimalRepresentationInfo

`‣ MinimalRepresentationInfo`( grpname, conditions ) | ( function )

Returns: a record with the components `value` and `source`, or `fail`

Let grpname be the GAP name of a group \(G\), say. If the information described by conditions about minimal representations of this group can be computed or is stored then `MinimalRepresentationInfo` returns a record with the components `value` and `source`, otherwise `fail` is returned.

The following values for conditions are supported.

- If conditions is `NrMovedPoints` (Reference: NrMovedPoints for a permutation) then `value`, if known, is the degree of a minimal faithful (not necessarily transitive) permutation representation for \(G\).

- If conditions consists of `Characteristic` (Reference: Characteristic) and a prime integer p then `value`, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation in characteristic p for \(G\).

- If conditions consists of `Size` (Reference: Size) and a prime power q then `value`, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation over the field of size q for \(G\).

In all cases, the value of the component `source` is a list of strings that describe sources of the information, which can be the ordinary or modular character table of \(G\) (see [CCN+85], [JLPW95], [HL89]), the table of marks of \(G\), or [Jan05]. For an overview of minimal degrees of faithful matrix representations for sporadic simple groups and their covering groups, see also

http://www.math.rwth-aachen.de/~MOC/mindeg/.

Note that `MinimalRepresentationInfo` cannot provide any information about minimal representations over prescribed fields in characteristic zero.

Information about groups that occur in the AtlasRep package is precomputed in `MinimalRepresentationInfoData` (6.3-2), so the packages CTblLib and TomLib are not needed when `MinimalRepresentationInfo` is called for these groups. (The only case that is not covered by this list is that one asks for the minimal degree of matrix representations over a prescribed field in characteristic coprime to the group order.)

One of the following strings can be given as an additional last argument.

`"cache"`

  means that the function tries to compute (and then store) values that are not stored in `MinimalRepresentationInfoData` (6.3-2), but stored values are preferred; this is also the default.

`"lookup"`

  means that stored values are returned but the function does not attempt to compute values that are not stored in `MinimalRepresentationInfoData` (6.3-2).

`"recompute"`

  means that the function always tries to compute the desired value, and checks the result against stored values.

    gap> MinimalRepresentationInfo( "A5", NrMovedPoints );
    rec(
      source := [ "computed (alternating group)",
          "computed (char. table)", "computed (subgroup tables)",
          "computed (subgroup tables, known repres.)",
          "computed (table of marks)" ], value := 5 )
    gap> MinimalRepresentationInfo( "A5", Characteristic, 2 );
    rec( source := [ "computed (char. table)" ], value := 2 )
    gap> MinimalRepresentationInfo( "A5", Size, 2 );
    rec( source := [ "computed (char. table)" ], value := 4 )

##### 6.3-2 MinimalRepresentationInfoData

`‣ MinimalRepresentationInfoData` | ( global variable )

This is a record whose components are GAP names of groups for which information about minimal permutation and matrix representations were known in advance or have been computed in the current GAP session. The value for the group \(G\), say, is a record with the following components.

`NrMovedPoints`

  a record with the components `value` (the degree of a smallest faithful permutation representation of \(G\)) and `source` (a string describing the source of this information).

`Characteristic`

  a record whose components are at most `0` and strings corresponding to prime integers, each bound to a record with the components `value` (the degree of a smallest faithful matrix representation of \(G\) in this characteristic) and `source` (a string describing the source of this information).

`CharacteristicAndSize`

  a record whose components are strings corresponding to prime integers p, each bound to a record with the components `sizes` (a list of powers q of p), `dimensions` (the corresponding list of minimal dimensions of faithful matrix representations of \(G\) over a field of size q), `sources` (the corresponding list of strings describing the source of this information), and `complete` (a record with the components `val` (`true` if the minimal dimension over any finite field in characteristic p can be derived from the values in the record, and `false` otherwise) and `source` (a string describing the source of this information)).

The values are set by `SetMinimalRepresentationInfo` (6.3-3).

##### 6.3-3 SetMinimalRepresentationInfo

`‣ SetMinimalRepresentationInfo`( grpname, op, value, source ) | ( function )

Returns: `true` if the values were successfully set, `false` if stored values contradict the given ones.

This function sets an entry in `MinimalRepresentationInfoData` (6.3-2) for the group \(G\), say, with GAP name grpname.

Supported values for op are

- `"NrMovedPoints"` (see `NrMovedPoints` (Reference: NrMovedPoints for a permutation)), which means that value is the degree of minimal faithful (not necessarily transitive) permutation representations of \(G\),

- a list of length two with first entry `"Characteristic"` (see `Characteristic` (Reference: Characteristic)) and second entry char either zero or a prime integer, which means that value is the dimension of minimal faithful (not necessarily irreducible) matrix representations of \(G\) in characteristic char,

- a list of length two with first entry `"Size"` (see `Size` (Reference: Size)) and second entry a prime power q, which means that value is the dimension of minimal faithful (not necessarily irreducible) matrix representations of \(G\) over the field with q elements, and

- a list of length three with first entry `"Characteristic"` (see `Characteristic` (Reference: Characteristic)), second entry a prime integer p, and third entry the string `"complete"`, which means that the information stored for characteristic p is complete in the sense that for any given power \(q\) of p, the minimal faithful degree over the field with \(q\) elements equals that for the largest stored field size of which \(q\) is a power.

In each case, source is a string describing the source of the data; computed values are detected from the prefix `"comp"` of source.

If the intended value is already stored and differs from value then an error message is printed.

    gap> SetMinimalRepresentationInfo( "A5", "NrMovedPoints", 5,
    >      "computed (alternating group)" );
    true
    gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 0 ], 3,
    >      "computed (char. table)" );
    true
    gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 2 ], 2,
    >      "computed (char. table)" );
    true
    gap> SetMinimalRepresentationInfo( "A5", [ "Size", 2 ], 4,
    >      "computed (char. table)" );
    true
    gap> SetMinimalRepresentationInfo( "A5", [ "Size", 4 ], 2,
    >      "computed (char. table)" );
    true
    gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 3 ], 3,
    >      "computed (char. table)" );
    true

##### 6.3-4 Criteria Used to Compute Minimality Information

The information about the minimal degree of a faithful matrix representation of \(G\) in a given characteristic or over a given field in positive characteristic is derived from the relevant (ordinary or modular) character table of \(G\), except in a few cases where this table itself is not known but enough information about the degrees is available in [HL89] and [Jan05].

The following criteria are used for deriving the minimal degree of a faithful permutation representation of \(G\) from the information in the GAP libraries of character tables and of tables of marks.

- If the name of \(G\) has the form `"A`\(n\)`"` or `"A`\(n\)`.2"` (denoting alternating and symmetric groups, respectively) then the minimal degree is \(n\), except if \(n\) is smaller than \(3\) or \(2\), respectively.

- If the name of \(G\) has the form `"L2(`\(q\)`)"` (denoting projective special linear groups in dimension two) then the minimal degree is \(q + 1\), except if \(q \in \{ 2, 3, 5, 7, 9, 11 \}\), see [Hup67, Satz II.8.28].

- If the largest maximal subgroup of \(G\) is core-free then the index of this subgroup is the minimal degree. (This is used when the two character tables in question and the class fusion are available in GAP's Character Table Library [Bre22]; this happens for many character tables of simple groups.)

- If \(G\) has a unique minimal normal subgroup then each minimal faithful permutation representation is transitive. (Note that the core of each point stabilizer is either trivial or contains the unique minimal normal subgroup.)

  In this case, the minimal degree can be computed directly from the information in the table of marks of \(G\) if this is available in GAP's Library of Tables of Marks [NMP18].

  Suppose that the largest maximal subgroup of \(G\) is not core-free but simple and normal in \(G\), and that the other maximal subgroups of \(G\) are core-free. In this case, we take the minimum of the indices of the core-free maximal subgroups and of the product of index and minimal degree of the normal maximal subgroup. (This suffices since no core-free subgroup of the whole group can contain a nontrivial normal subgroup of a normal maximal subgroup.)

  Let \(N\) be the unique minimal normal subgroup of \(G\), and assume that \(G/N\) is simple and has minimal degree \(n\), say. If there is a subgroup \(U\) of index \(n \cdot |N|\) in \(G\) that intersects \(N\) trivially then the minimal degree of \(G\) is \(n \cdot |N|\). (This is used for the case that \(N\) is central in \(G\) and \(N \times U\) occurs as a subgroup of \(G\).)

- If we know a subgroup of \(G\) whose minimal degree is \(n\), say, and if we know either (a class fusion from) a core-free subgroup of index \(n\) in \(G\) or a faithful permutation representation of degree \(n\) for \(G\) then \(n\) is the minimal degree for \(G\). (This happens often for tables of almost simple groups.)

#### 6.4 A JSON Interface

We define a mapping between certain GAP objects and JSON (JavaScript Object Notation) texts (see [JSO14]), as follows.

- The three GAP values `true`, `false`, and `fail` correspond to the JSON texts `true`, `false`, and `null`, respectively.

- GAP strings correspond to JSON strings; special characters in a GAP string (control characters ASCII \(0\) to \(31\), backslash and double quote) are mapped as defined in JSON's specification, and other ASCII characters are kept as they are; if a GAP string contains non-ASCII characters, it is assumed that it is UTF-8 encoded, and one may choose either to keep non-ASCII characters as they are, or to create an ASCII only JSON string, using JSON's syntax for Unicode code points ("`\uXXXX`"); in the other direction, JSON strings are assumed to be UTF-8 encoded, and are mapped to UTF-8 encoded GAP strings, by keeping the non-ASCII characters and converting substrings of the form `\uXXXX` accordingly.

- GAP integers (in the sense of `IsInt` (Reference: IsInt)) are mapped to JSON numbers that consist of digits and optionally a leading sign character `-`; in the other direction, JSON numbers of this form and also JSON numbers that involve no decimal dots and have no negative exponent (for example `"2e3"`) are mapped to GAP integers.

- GAP rationals (in the sense of `IsRat` (Reference: IsRat)) which are not integers are represented by JSON floating point numbers; the JSON representation (and hence the precision) is given by first applying `Float` (Reference: Float) and then `String` (Reference: String).

- GAP floats (in the sense of Chapter Reference: Floats in the GAP Reference Manual) are mapped to JSON floating point numbers; the JSON representation (and hence the precision) is given by applying `String` (Reference: String); in the other direction, JSON numbers that involve a decimal dot or a negative exponent are mapped to GAP floats.

- (Nested and not self-referential) dense GAP lists of objects correspond to JSON arrays such that the list entries correspond to each other. (Note that JSON does not support non-dense arrays.)

- (Nested and not self-referential) GAP records correspond to JSON objects such that both labels (which are strings in GAP and JSON) and values correspond to each other.

The GAP functions `AGR.JsonText` (6.4-2) and `AGR.GapObjectOfJsonText` (6.4-3) can be used to create a JSON text from a suitable GAP object and the GAP object that corresponds to a given JSON text, respectively.

Note that the composition of the two functions is in general not the identity mapping, because `AGR.JsonText` (6.4-2) accepts non-integer rationals, whereas `AGR.GapObjectOfJsonText` (6.4-3) does not create such objects.

Note also that the results of `AGR.JsonText` (6.4-2) do not contain information about dependencies between common subobjects. This is another reason why applying first `AGR.JsonText` (6.4-2) and then `AGR.GapObjectOfJsonText` (6.4-3) may yield a GAP object with different behaviour.

Applying `AGR.JsonText` (6.4-2) to a self-referential object such as `[ ~ ]` will raise a "recursion depth trap" error.

##### 6.4-1 Why JSON?

The aim of this JSON interface is to read and write certain data files with GAP such that these files become easily accessible independent of GAP. The function `AGR.JsonText` (6.4-2) is intended just as a prototype, variants of this function are very likely to appear in other contexts, for example in order to force certain line formatting or ordering of record components.

It is not the aim of the JSON interface to provide self-contained descriptions of arbitrary GAP objects, in order to read them into a GAP session. Note that those GAP objects for which a JSON equivalent exists (and many more) can be easily written to files as they are, and GAP can read them efficiently. On the other hand, more complicated GAP objects can be written and read via the so-called pickling, for which a framework is provided by the GAP package IO [Neu14].

Here are a few situations which are handled well by pickling but which cannot be addressed with a JSON interface.

- Pickling and unpickling take care of common subobjects of the given GAP object. The following example shows that the applying first `AGR.JsonText` (6.4-2) and then `AGR.GapObjectOfJsonText` (6.4-3) may yield an object which behaves differently.

      gap> l:= [ [ 1 ] ];; l[2]:= l[1];;  l;
      [ [ 1 ], [ 1 ] ]
      gap> new:= AGR.GapObjectOfJsonText( AGR.JsonText( l ) ).value;
      [ [ 1 ], [ 1 ] ]
      gap> Add( l[1], 2 );  l;
      [ [ 1, 2 ], [ 1, 2 ] ]
      gap> Add( new[1], 2 );  new;
      [ [ 1, 2 ], [ 1 ] ]

- GAP admits self-referential objects, for example as follows.

      gap> l:= [];;  l[1]:= l;;

  Pickling and unpickling take care of self-referential objects, but `AGR.JsonText` (6.4-2) does not support the conversion of such objects.

##### 6.4-2 AGR.JsonText

`‣ AGR.JsonText`( obj[, mode] ) | ( function )

Returns: a new mutable string that describes obj as a JSON text, or `fail`.

If obj is a GAP object for which a corresponding JSON text exists, according to the mapping described above, then such a JSON text is returned. Otherwise, `fail` is returned.

If the optional argument mode is given and has the value `"ASCII"` then the result in an ASCII string, otherwise the encoding of strings that are involved in obj is kept.

    gap> AGR.JsonText( [] );
    "[]"
    gap> AGR.JsonText( "" );
    "\"\""
    gap> AGR.JsonText( "abc\ndef\cghi" );
    "\"abc\\ndef\\u0003ghi\""
    gap> AGR.JsonText( rec() );
    "{}"
    gap> AGR.JsonText( [ , 2 ] );
    fail
    gap> str:= [ '\303', '\266' ];;  # umlaut o
    gap> json:= AGR.JsonText( str );;  List( json, IntChar );
    [ 34, 195, 182, 34 ]
    gap> AGR.JsonText( str, "ASCII" );
    "\"\\u00F6\""

##### 6.4-3 AGR.GapObjectOfJsonText

`‣ AGR.GapObjectOfJsonText`( string ) | ( function )

Returns: a new mutable record whose `value` component, if bound, contains a mutable GAP object that represents the JSON text string.

If string is a string that represents a JSON text then the result is a record with the components `value` (the corresponding GAP object in the sense of the above interface) and `status` (value `true`). Otherwise, the result is a record with the components `status` (value `false`) and `errpos` (the position in string where the string turns out to be not valid JSON).

    gap> AGR.GapObjectOfJsonText( "{ \"a\": 1 }" );
    rec( status := true, value := rec( a := 1 ) )
    gap> AGR.GapObjectOfJsonText( "{ \"a\": x }" );
    rec( errpos := 8, status := false )

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