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A global constraint expresses a relation that involves many variables at once. The most frequently used global constraints of this library are the combinatorial constraints all_distinct/1, global_cardinality/2 and cumulative/2.
?- maplist(in, Vs, [1\/3..4, 1..2\/4, 1..2\/4, 1..3, 1..3, 1..6]), all_distinct(Vs). false.
\=, #<, #>,
#=<or #>=. For example:
?- [A,B,C] ins 0..sup, sum([A,B,C], #=, 100). A in 0..100, A+B+C#=100, B in 0..100, C in 0..100.
\=, #<, #>,
?- tuples_in([[X,Y]], [[1,2],[1,5],[4,0],[4,3]]), X = 4. X = 4, Y in 0\/3.
As another example, consider a train schedule represented as a list of quadruples, denoting departure and arrival places and times for each train. In the following program, Ps is a feasible journey of length 3 from A to D via trains that are part of the given schedule.
trains([[1,2,0,1], [2,3,4,5], [2,3,0,1], [3,4,5,6], [3,4,2,3], [3,4,8,9]]). threepath(A, D, Ps) :- Ps = [[A,B,_T0,T1],[B,C,T2,T3],[C,D,T4,_T5]], T2 #> T1, T4 #> T3, trains(Ts), tuples_in(Ps, Ts).
In this example, the unique solution is found without labeling:
?- threepath(1, 4, Ps). Ps = [[1, 2, 0, 1], [2, 3, 4, 5], [3, 4, 8, 9]].
=<S_j or S_j + D_j
=<S_i for all 1
=<i < j
?- length(Vs, 3), Vs ins 0..3, serialized(Vs, [1,2,3]), label(Vs). Vs = [0, 1, 3] ; Vs = [2, 0, 3] ; false.
global_cardinality(Vs, Pairs, ). See global_cardinality/3.
?- Vs = [_,_,_], global_cardinality(Vs, [1-2,3-_]), label(Vs). Vs = [1, 1, 3] ; Vs = [1, 3, 1] ; Vs = [3, 1, 1].
?- length(Vs, _), circuit(Vs), label(Vs). Vs =  ; Vs =  ; Vs = [2, 1] ; Vs = [2, 3, 1] ; Vs = [3, 1, 2] ; Vs = [2, 3, 4, 1] .
cumulative(Tasks, [limit(1)]). See cumulative/2.
task(S_i, D_i, E_i, C_i, T_i). S_i denotes the start time, D_i the positive duration, E_i the end time, C_i the non-negative resource consumption, and T_i the task identifier. Each of these arguments must be a finite domain variable with bounded domain, or an integer. The constraint holds iff at each time slot during the start and end of each task, the total resource consumption of all tasks running at that time does not exceed the global resource limit. Options is a list of options. Currently, the only supported option is:
For example, given the following predicate that relates three tasks of durations 2 and 3 to a list containing their starting times:
tasks_starts(Tasks, [S1,S2,S3]) :- Tasks = [task(S1,3,_,1,_), task(S2,2,_,1,_), task(S3,2,_,1,_)].
We can use cumulative/2 as follows, and obtain a schedule:
?- tasks_starts(Tasks, Starts), Starts ins 0..10, cumulative(Tasks, [limit(2)]), label(Starts). Tasks = [task(0, 3, 3, 1, _G36), task(0, 2, 2, 1, _G45), ...], Starts = [0, 0, 2] .
automaton(Vs, _, Vs, Nodes, Arcs, , , _), a common use case of automaton/8. In the following example, a list of binary finite domain variables is constrained to contain at least two consecutive ones:
two_consecutive_ones(Vs) :- automaton(Vs, [source(a),sink(c)], [arc(a,0,a), arc(a,1,b), arc(b,0,a), arc(b,1,c), arc(c,0,c), arc(c,1,c)]).
?- length(Vs, 3), two_consecutive_ones(Vs), label(Vs). Vs = [0, 1, 1] ; Vs = [1, 1, 0] ; Vs = [1, 1, 1].
sink(Node)terms. Arcs is a list of
arc(Node,Integer,Node,Exprs)terms that denote the automaton's transitions. Each node is represented by an arbitrary term. Transitions that are not mentioned go to an implicit failure node. Exprs is a list of arithmetic expressions, of the same length as Counters. In each expression, variables occurring in Counters symbolically refer to previous counter values, and variables occurring in Template refer to the current element of Sequence. When a transition containing arithmetic expressions is taken, each counter is updated according to the result of the corresponding expression. When a transition without arithmetic expressions is taken, all counters remain unchanged. Counters is a list of variables. Initials is a list of finite domain variables or integers denoting, in the same order, the initial value of each counter. These values are related to Finals according to the arithmetic expressions of the taken transitions.
The following example is taken from Beldiceanu, Carlsson, Debruyne and Petit: "Reformulation of Global Constraints Based on Constraints Checkers", Constraints 10(4), pp 339-362 (2005). It relates a sequence of integers and finite domain variables to its number of inflexions, which are switches between strictly ascending and strictly descending subsequences:
sequence_inflexions(Vs, N) :- variables_signature(Vs, Sigs), automaton(Sigs, _, Sigs, [source(s),sink(i),sink(j),sink(s)], [arc(s,0,s), arc(s,1,j), arc(s,2,i), arc(i,0,i), arc(i,1,j,[C+1]), arc(i,2,i), arc(j,0,j), arc(j,1,j), arc(j,2,i,[C+1])], [C], , [N]). variables_signature(, ). variables_signature([V|Vs], Sigs) :- variables_signature_(Vs, V, Sigs). variables_signature_(, _, ). variables_signature_([V|Vs], Prev, [S|Sigs]) :- V #= Prev #<==> S #= 0, Prev #< V #<==> S #= 1, Prev #> V #<==> S #= 2, variables_signature_(Vs, V, Sigs).
?- sequence_inflexions([1,2,3,3,2,1,3,0], N). N = 3. ?- length(Ls, 5), Ls ins 0..1, sequence_inflexions(Ls, 3), label(Ls). Ls = [0, 1, 0, 1, 0] ; Ls = [1, 0, 1, 0, 1].
#<or #>. For example:
?- chain([X,Y,Z], #>=). X#>=Y, Y#>=Z.