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Pack tor -- prolog/tor_clpfd.pl |
The following predicates are exported from this file while their implementation is defined in imported modules or non-module files loaded by this module.
(>)/2
by
#>/2 to obtain more general relations See declarative integer
arithmetic.(<)/2
by #</2 to obtain more general relations. See
declarative integer arithmetic.
In addition to its regular use in tasks that require it, this constraint can also be useful to eliminate uninteresting symmetries from a problem. For example, all possible matches between pairs built from four players in total:
?- Vs = [A,B,C,D], Vs ins 1..4, all_different(Vs), A #< B, C #< D, A #< C, findall(pair(A,B)-pair(C,D), label(Vs), Ms). Ms = [ pair(1, 2)-pair(3, 4), pair(1, 3)-pair(2, 4), pair(1, 4)-pair(2, 3)].
(>=)/2
by
#>=/2 to obtain more general relations. See declarative integer
arithmetic.(=<)/2
by #=</2 to obtain more
general relations. See declarative integer
arithmetic.(is)/2
and (=:=)/2
over integers. See
declarative integer arithmetic.(=\=)/2
by #\=/2 to obtain
more general relations. See declarative integer
arithmetic.For example, to obtain the complement of a domain:
?- #\ X in -3..0\/10..80. X in inf.. -4\/1..9\/81..sup.
For example:
?- X #= 4 #<==> B, X #\= 4. B = 0, X in inf..3\/5..sup.
The following example uses reified constraints to relate a list of finite domain variables to the number of occurrences of a given value:
vs_n_num(Vs, N, Num) :- maplist(eq_b(N), Vs, Bs), sum(Bs, #=, Num). eq_b(X, Y, B) :- X #= Y #<==> B.
Sample queries and their results:
?- Vs = [X,Y,Z], Vs ins 0..1, vs_n_num(Vs, 4, Num). Vs = [X, Y, Z], Num = 0, X in 0..1, Y in 0..1, Z in 0..1. ?- vs_n_num([X,Y,Z], 2, 3). X = 2, Y = 2, Z = 2.
For example, the sum of natural numbers below 1000 that are multiples of 3 or 5:
?- findall(N, (N mod 3 #= 0 #\/ N mod 5 #= 0, N in 0..999, indomain(N)), Ns), sum(Ns, #=, Sum). Ns = [0, 3, 5, 6, 9, 10, 12, 15, 18|...], Sum = 233168.
?- maplist(in, Vs, [1\/3..4, 1..2\/4, 1..2\/4, 1..3, 1..3, 1..6]), all_distinct(Vs). false.
?- [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]].
?- 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.
Example:
?- 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 = [1] ; 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)]).
Example query:
?- length(Vs, 3), two_consecutive_ones(Vs), label(Vs). Vs = [0, 1, 1] ; Vs = [1, 1, 0] ; Vs = [1, 1, 1].
source(Node)
and sink(Node)
terms. Arcs is a list of
arc(Node,Integer,Node)
and 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], [0], [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).
Example queries:
?- 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].
?- transpose([[1,2,3],[4,5,6],[7,8,9]], Ts). Ts = [[1, 4, 7], [2, 5, 8], [3, 6, 9]].
This predicate is useful in many constraint programs. Consider for instance Sudoku:
sudoku(Rows) :- length(Rows, 9), maplist(same_length(Rows), Rows), append(Rows, Vs), Vs ins 1..9, maplist(all_distinct, Rows), transpose(Rows, Columns), maplist(all_distinct, Columns), Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is], blocks(As, Bs, Cs), blocks(Ds, Es, Fs), blocks(Gs, Hs, Is). blocks([], [], []). blocks([N1,N2,N3|Ns1], [N4,N5,N6|Ns2], [N7,N8,N9|Ns3]) :- all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]), blocks(Ns1, Ns2, Ns3). problem(1, [[_,_,_,_,_,_,_,_,_], [_,_,_,_,_,3,_,8,5], [_,_,1,_,2,_,_,_,_], [_,_,_,5,_,7,_,_,_], [_,_,4,_,_,_,1,_,_], [_,9,_,_,_,_,_,_,_], [5,_,_,_,_,_,_,7,3], [_,_,2,_,1,_,_,_,_], [_,_,_,_,4,_,_,_,9]]).
Sample query:
?- problem(1, Rows), sudoku(Rows), maplist(portray_clause, Rows). [9, 8, 7, 6, 5, 4, 3, 2, 1]. [2, 4, 6, 1, 7, 3, 9, 8, 5]. [3, 5, 1, 9, 2, 8, 7, 4, 6]. [1, 2, 8, 5, 3, 7, 6, 9, 4]. [6, 3, 4, 8, 9, 2, 1, 5, 7]. [7, 9, 5, 4, 6, 1, 8, 3, 2]. [5, 1, 9, 2, 8, 6, 4, 7, 3]. [4, 7, 2, 3, 1, 9, 5, 6, 8]. [8, 6, 3, 7, 4, 5, 2, 1, 9]. Rows = [[9, 8, 7, 6, 5, 4, 3, 2|...], ... , [...|...]].
Think of zcompare/3 as reifying an arithmetic comparison of two
integers. This means that we can explicitly reason about the
different cases within our programs. As in compare/3, the atoms
<
, >
and =
denote the different cases of the
trichotomy. In contrast to compare/3 though, zcompare/3 works
correctly for all modes, also if only a subset of the arguments is
instantiated. This allows you to make several predicates over
integers deterministic while preserving their generality and
completeness. For example:
n_factorial(N, F) :- zcompare(C, N, 0), n_factorial_(C, N, F). n_factorial_(=, _, 1). n_factorial_(>, N, F) :- F #= F0*N, N1 #= N - 1, n_factorial(N1, F0).
This version of n_factorial/2 is deterministic if the first argument is instantiated, because argument indexing can distinguish the different clauses that reflect the possible and admissible outcomes of a comparison of N against 0. Example:
?- n_factorial(30, F). F = 265252859812191058636308480000000.
Since there is no clause for <
, the predicate automatically
fails if N is less than 0. The predicate can still be used in
all directions, including the most general query:
?- n_factorial(N, F). N = 0, F = 1 ; N = F, F = 1 ; N = F, F = 2 .
In this case, all clauses are tried on backtracking, and zcompare/3 ensures that the respective ordering between N and 0 holds in each case.
The truth value of a comparison can also be reified with (#<==>)/2 in combination with one of the arithmetic constraints. See reification. However, zcompare/3 lets you more conveniently distinguish the cases.
?- chain([X,Y,Z], #>=). X#>=Y, Y#>=Z.
For example, to implement a custom labeling strategy, you may need to inspect the current domain of a finite domain variable. With the following code, you can convert a finite domain to a list of integers:
dom_integers(D, Is) :- phrase(dom_integers_(D), Is). dom_integers_(I) --> { integer(I) }, [I]. dom_integers_(L..U) --> { numlist(L, U, Is) }, Is. dom_integers_(D1\/D2) --> dom_integers_(D1), dom_integers_(D2).
Example:
?- X in 1..5, X #\= 4, fd_dom(X, D), dom_integers(D, Is). D = 1..3\/5, Is = [1,2,3,5], X in 1..3\/5.
If Max is sup, then Rest is the empty FD set. Otherwise, if Rest is non-empty, all elements of Rest are greater than Max+1.
This predicate should only be called with either Set or all other arguments being ground.
Either Interval or Min and Max must be ground.
Either Set or Elt must be ground.