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# construct_group(+Array, ++VarDimNames, ++ValueSpec, ++SymSpecs, ++GroupName, -MaxPoints, -ToPointPred, -FromPointPred)

Constructs a GAP group based on the provided symmetry specification.

Array
  The array of search variables
VarDimNames
  A list of symbolic names for each dimension in Array
ValueSpec
  The symbolic name for values and their range
SymSpecs
  A list of symmetry specifiers
GroupName
  The GAP name to give the constructed group (string)
MaxPoints
  The number of GAP points operated on by group
ToPointPred
  A predicate for converting a variable/value assignment to a GAP point
FromPointPred
  A predicate for converting a GAP point to a variable/value assignment

## Description

This predicate is intended for use in conjunction with a GAP/ECLiPSe-based symmetry breaking method. Given an array of search variables (Array) and a description of the symmetries, this predicate constructs the corresponding group in GAP (GroupName) and returns a pair of `to point' and `from point' predicates suitable for use in conjunction with it.

In order to facilitate the expression of the symmetries, a list of symbolic names (VarDimNames) is required, to specify a symbolic name for each of the dimensions of Array. Similarly, some way is needed to refer to the value `dimension'; this is provided by ValueSpec, which is of the form ValueName:ValueRange. ValueRange specifies the range of values the variables may take (this is just used for constructing the group; it is not imposed on the variables) and must be of the form Lo..Hi.

The symmetries (SymSpecs) are then specified by a list with entries of the form symmetry(Symmetry, SymmetryDimensions, OtherDimensions). Roughly speaking, Symmetry specifies the base symmetry, SymmetryDimensions specifies which dimensions it applies to, and OtherDimensions specifies which other dimensions should be treated specially with respect to this symmetry (if any). The options for Symmetry are:

s_n
  (1 dimension)
  Full permutation of the indices of the specified dimension.
cycle
  (1 dimension)
  Indices may be cycled (i.e. all shifted k positions modulo the size of the dimension).
r_4
  (2 dimensions)
  Rotational symmetry of the square. The two dimensions specified must be of equal size.
d_4
  (2 dimensions)
  Full symmetry of the square (i.e. including reflections). The two dimensions specified must be of equal size.
reverse
  (1 dimension)
  Indices may be reversed.
gap_group(GroupFunc)
  (no fixed number of dimensions)
  The symmetries described by the group returned by the GAP function GroupFunc. GroupFunc should accept a single argument, which is a list of the sizes of the dimensions that the symmetry is expected to act upon. The mapping between GAP points and the elements of the subarray that this symmetry acts upon is obtained by numbering the indices of the elements in lexicographic order (first dimension most significant). For example, if the symmetry subarray is of size MxN, then the element with index [I,J] is mapped to point (I-1)*N + J, so that [1,1]->1, [1,N]->N, [2,1]->N+1, etc.
function(Func)
  (no fixed number of dimensions)
  Allows a generator to be specified via an ECLiPSe function Func, a la classic SBDS symmetry functions. Func should accept three (extra) arguments: DimSizes, SrcIdx and DestIdx. The first, DimSizes, is a list of the sizes of the dimensions that the symmetry is expected to act upon. The second, SrcIdx, provides the index of an element of the subarray that the symmetry acts upon. The third, DestIdx, should then be unified with the index of the element that SrcIdx is mapped to by this function. For example, if the generator is intended to exchange the dimensions of a square array (i.e. reflect along the leading diagonal), if `swap_dim' is passed as Func, then the definition of `swap_dim' might be

      swap_dim([N, N], [I, J], [J, I]).

  Note that for 1-dimensional arrays, DestIdx may be returned as just an integer rather than a length-1 list containing the integer, if desired.
table(Generator)
  (no fixed number of dimensions)
  Allows a generator to be specified via a lookup table. Generator must be an array of same size as the subarray the symmetry acts upon, with each element containing the index of the element that that element should be mapped to.
  Note that for 1-dimensional arrays, an index may be specified using just an integer rather than a length-1 list containing the integer, if desired.

SymmetryDimensions must be a list with one entry for each dimension of the base symmetry. Each entry must either be the name of a dimension, or of the form Dimension:IdxSpec, where Dimension is the name of a dimension and IdxSpec specifies a subset of the indices of that dimension. IdxSpec may be an integer, an integer range L..H, or a list of these. Providing such a qualification means that the symmetry is only applied to the specified indices of the dimension, not the dimension as a whole.

By default, a symmetry affects the array as a whole; that is, for example, a symmetry applied to columns will affect all rows, and in particular all rows in the same way (it won't, say, swap columns 1 and 3 in row 1, but swap 5 and 6 in row 2; the columns will be rearranged in the same way in every row). If one or more indices for a dimension should be affected independently of the rest of the indices for that dimension, then this can be specified in the OtherDimensions field, which is a list of terms of the form Dimension:IdxSpec. This specifies that the symmetry should only affect the given indices of the given dimension, independent of the other indices (the specified indices are still affected synchronously). This allows one to express, for example, that the elements of a given row can be permuted, independent of the other rows. If OtherDimensions is just the empty list, then it may be omitted.

To do:
  Support multiple value dimensions
  Support multiple matrices
  More error checking... :)

### Modules

This predicate is sensitive to its module context (tool predicate, see @/2).

## Examples

Balanced Incomplete Block Design (BIBD)
  Standard model, with a 2-D array of booleans.
  Full row and column permutation.

dim(Array, [NRows, NCols]),
...
construct_group(Array, [rows, cols], values:0..1, [

  symmetry(s_n, [rows], []),
  symmetry(s_n, [cols], [])

  ], "bibd_group", MaxPoints, ToPointPred, FromPointPred)

Social golfer problem
  Boolean model, with one boolean variable for each round/group/player combination.
  Full permutation of the rounds, full permutation of the groups in any round, and full permutation of the players.

dim(Array, [NRounds, NGroups, NPlayers]),
...
construct_group(Array, [rounds, groups, players], values:0..1, [

  symmetry(s_n, [rounds], []),
  symmetry(s_n, [groups], [rounds:1]),
  symmetry(s_n, [players], [])

  ], "golf_group", MaxPoints, ToPointPred, FromPointPred)

  Note that the permutation of the groups within a round does not need to be specified for every round; the fact that the rounds are interchangeable means that it need only be specified for one round.

N Queens problem
  Standard row model, with one integer for each row specifying which column the queen appears in.
  The symmetries of the square, including reflection, applied to the row/column combination.

dim(Array, [NQueens]),
...
construct_group(Array, [rows], cols:1..NQueens, [

  symmetry(d_4, [rows, cols], [])

  ], "queens_group", MaxPoints, ToPointPred, FromPointPred)

N Queens problem
  Boolean model, with one boolean for each location on the board, specifying whether a queen appears there or not.
  The symmetries of the square, including reflection.

dim(Array, [NQueens, NQueens]),
...
construct_group(Array, [rows, cols], values:0..1, [

  symmetry(d_4, [rows, cols], [])

  ], "queens_group", MaxPoints, ToPointPred, FromPointPred)

  Note that the symmetry expression for this model is exactly the same as for the standard row model; whether the columns are values or a dimension of the variable array is irrelevant.