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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.