smt(:ConstraintBody) since release 4.2
The arithmetic, membership, and propositional constraints described
earlier are transformed at compile time to conjunctions of library
constraints. Although linear in the size of the source code, the
expansion of a propositional formula over reifiable constraints to
library goals can have time and memory overheads, and propagates
disjunctions very weakly. Temporary variables holding intermediate
values may have to be introduced, and the grain size of the constraint
solver invocations can be rather small. As an alternative to the
default propagation of such constraint formulas, this constraint is a
front-end to the case/[3,4]
propagator, which treats such a
formula globally. The pruning can be stronger and it can run faster
than the default propagator, but this is not necessarily the case.
Bounds consistency is not guaranteed.
ConstraintBody should be of one of the following forms, or a propositional combination of such forms. See Syntax of Indexicals for the exact definition:
in
ConstantRange
element(var,CList,var)
table([VList],CTable)
X
stands for X#=1
}
count(+Val,+List,+RelOp,?Count) since release 4.0.5,deprecated
where Val is an integer, List is a list of domain variables,
Count a domain variable, and RelOp is a relational symbol as
in Arithmetic Constraints. True if N is the number of
elements of List that are equal to Val and N RelOp
Count. Implemented by decomposition into one sum/3
constraint,
one arithmetic comparison, and several reified equalities.
Corresponds to count_*/3
, exactly/3
in MiniZinc.
count/4
maintains domain consistency, but in practice, the
following constraint is a better alternative.
global_cardinality(+Xs,+Vals)
global_cardinality(+Xs,+Vals,+Options)
where Xs = [X1,…,Xd] is a list of domain variables, and Vals = [K1-V1,…,Kn-Vn] is a list of pairs where each key Ki is a unique integer and Vi is a domain variable. True if every element of Xs is equal to some key and for each pair Ki-Vi, exactly Vi elements of Xs are equal to Ki.
If either Xs or Vals is ground, and in many other special
cases, global_cardinality/[2,3]
maintains domain consistency, but
generally, bounds consistency cannot be guaranteed. A domain
consistency algorithm [Regin 96] is used, roughly linear in the total
size of the domains.
Corresponds to global_cardinality*/*
and distribute/3
in MiniZinc.
Options is a list of zero or more of the following:
consistency(Cons)
Which filtering algorithm to use. One of the following:
domain
The constraint will use the algorithm mentioned above.
Implies on(dom)
. The default.
bounds
The constraint will use the algorithm mentioned above.
Implies on(minmax)
.
value
The constraint will use a simple algorithm, which prevents too few or
too many of the Xs from taking values among the Vals.
Implies on(val)
.
on(On)
How eagerly to wake up the constraint. One of the following:
dom
to wake up when the domain of a variable is changed (the default);
minmax
to wake up when a bound of a variable is changed;
val
to wake up when a variable is fixed.
cost(Cost,Matrix)
Overrides any consistency/1
option value. A cost is associated
with the constraint and reflected into the domain variable Cost.
Matrix should be a d*n matrix of integers, represented as a
list of d lists, each of length n. Assume that each
Xi equals K(pi). The cost of the constraint is then
Matrix[1,p1]+…+Matrix[d,pd].
With this option, a domain consistency algorithm [Regin 99] is used, the complexity of which is roughly O(d(m + n log n)) where m is the total size of the domains.
all_different(+Variables)
all_different(+Variables, +Options)
all_distinct(+Variables)
all_distinct(+Variables, +Options)
where Variables is a list of domain variables. Each variable is constrained to take a value that is unique among the variables. Declaratively, this is equivalent to an inequality constraint for each pair of variables.
Corresponds to alldifferent/1
in MiniZinc.
Options is a list of zero or more of the following:
L #= R since release 4.3
where R should be an integer, and R an expressions on one of the following forms, where X1, …, Xj occur among Variables:
X1 + ... + Xj
X1*X1 + ... + Xj*Xj
X1 * ... * Xj
The given equation is a side constraint for the constraint to hold. A special bounds consistency algorithm is used, which considers the main constraint and the side constraints globally. This option is only valid if the domains of X1, …, Xj consist of integers strictly greater than zero.
consistency(Cons)
Which algorithm to use, one of the following:
domain
The default for all_distinct/[1,2]
and assignment/[2,3]
.
A domain consistency algorithm [Regin 94] is used, roughly linear in
the total size of the domains. Implies on(dom)
.
bounds
A bounds consistency algorithm [Lopez-Ortiz 03] is used, which runs
in O(n log n) time for n variables. Implies
on(minmax)
.
value
The default for all_different/[1,2]
. An algorithm achieving
exactly the same pruning as a set of pairwise inequality constraints is
used, roughly linear in the number of variables. Implies
on(val)
.
on(On)
How eagerly to wake up the constraint. One of the following:
dom
(the default for all_distinct/[1,2]
and assignment/[2,3]
),
to wake up when the domain of a variable is changed;
min
to wake up when the lower bound of a domain is changed;
max
to wake up when the upper bound of a domain is changed;
minmax
to wake up when some bound of a domain is changed;
val
(the default for all_different/[1,2]
), to wake up when a variable
is fixed.
nvalue(?N, +Variables)
where Variables is a list of domain variables with finite bounds,
and N is a domain variable. True if N is the number of
distinct values taken by Variables. Approximates
bounds consistency in N and domain consistency in Variables.
Can be thought of as a relaxed version of all_distinct/2
.
Corresponds to nvalue/2
in MiniZinc.
The following is a constraint over two lists of length n of variables. Each variable is constrained to take a value in [1,n] that is unique for its list. Furthermore, the lists are dual in a sense described below.
assignment(+Xs, +Ys)
assignment(+Xs, +Ys, +Options)
where Xs = [X1,…,Xn] and Ys = [Y1,…,Yn] are lists of domain variables. True if all Xi, Yi in [1,n] and Xi=j iff Yj=i.
Corresponds to inverse/2
in MiniZinc.
Options is a list of zero or more of the following, where
Boolean must be true
or false
(false
is the
default):
on(On)
Same meaning as for all_different/2
.
consistency(Cons)
Same meaning as for all_different/2
.
circuit(Boolean)
If true
, circuit(Xs,Ys)
must hold for the
constraint to be true.
cost(Cost,Matrix)
A cost is associated with the constraint and reflected into the domain variable Cost. Matrix should be an n*n matrix of integers, represented as a list of lists. The cost of the constraint is Matrix[1,X1]+…+Matrix[n,Xn].
With this option, a domain consistency algorithm [Sellmann 02] is used, the complexity of which is roughly O(n(m + n log n)) where m is the total size of the domains.
The following constraint captures the relation between a list of values, a list of the values in ascending order, and their positions in the original list:
sorting(+Xs,+Ps,+Ys)
where Xs = [X1,…,Xn], Ps = [P1,…,Pn], and Ys = [Y1,…,Yn] are lists of domain variables. The constraint holds if the following are true:
In practice, the underlying algorithm [Mehlhorn 00] is likely to achieve bounds consistency, and is guaranteed to do so if Ps is ground or completely free.
Corresponds to sort/2
in MiniZinc.
The following constraint is a generalization of sorting/3
, namely:
keysorting(+Xs,+Ys) since release 4.3.1
keysorting(+Xs,+Ys,+Options)
where Xs and Ys are lists of tuples of domain variables. Both lists should be of the same length n, and all tuples should have the same length m. The constraint holds if the following are true:
The filtering algorithm is based on [Mehlhorn 00] and endeavors to achieve bounds consistency, but does not guarantee it.
Corresponds to Prolog’s keysort/2
. In particular, the sort is stable by definition.
Options is a list of zero or more of the following:
keys(Keys)
where Keys should be a positive integer, denoting the length of the key prefix. The default is 1.
permutation(Ps)
where Ps should be a list of length n of domain variables. Its meaning is described above.
The following constraints express the fact that several vectors of domain variables are in ascending lexicographic order:
lex_chain(+Vectors)
lex_chain(+Vectors,+Options)
where Vectors is a list of vectors (lists) of domain variables with finite bounds. The constraint holds if Vectors are in ascending lexicographic order.
Corresponds to *lex2/1
, lex_greater*/2
, lex_less*/2
in MiniZinc.
Options is a list of zero or more of the following:
op(Op)
If Op is the atom #=<
(the default), the constraints holds
if Vectors are in non-descending lexicographic order. If Op
is the atom #<
, the constraints holds if Vectors are in
strictly ascending lexicographic order.
increasing
This option imposes the additional constraint that each vector in Vectors be sorted in strictly ascending order.
among(Least,Most,Values)
If given, Least and Most should be integers such that 0 <= Least <= Most and Values should be a list of distinct integers. This option imposes the additional constraint on each vector in Vectors that at least Least and at most Most elements belong to Values.
global(Boolean) since release 4.2.1
if true
, a more expensive algorithm [Carlsson & Beldiceanu 02],
which guaranteed domain consistency unless the increasing/0
or
among/3
options are given, will be used.
In the following constraints, a literal is either a term X
or a term #\ X
, where X
is a 0/1 variable. They maintain
domain consistency:
bool_and(+Lits, +Lit) since release 4.3
where Lits is a list of literals [L0,...,Lj]
and Lit is a literal.
True if Lit equals the Boolean conjunction of Lits, and
usually more efficient than the equivalent L0#/\...#/\Lj #<=> Lit
.
bool_or(+Lits, +Lit) since release 4.3
where Lits is a list of literals [L0,...,Lj]
and Lit is a literal.
True if Lit equals the Boolean disjunction of Lits, and
usually more efficient than the equivalent L0#\/...#\/Lj #<=> Lit
.
bool_xor(+Lits, +Lit) since release 4.3
where Lits is a list of literals [L0,...,Lj]
and Lit is a literal.
True if Lit equals the Boolean exclusive or of Lits, and
usually more efficient than the equivalent L0#\...#\Lj #<=> Lit
.
bool_channel(+Lits, ?Y, +Relop, +Offset) since release 4.3
where Lits is a list of literals [L0,...,Lj]
, Y is a
domain variable, RelOp is an arithmetic comparison as in
Syntax of Arithmetic Expressions, and Offset is an integer.
Expresses the constraint Li #<=> (Y RelOp i+Offset) for
i in 0..j
. Usually more efficient than a bunch of reified
comparisons between a given variable and a sequence of integers.