/* Part of SWI-Prolog Author: Markus Triska E-mail: triska@gmx.at WWW: http://www.swi-prolog.org Copyright (C): 2007-2015 Markus Triska This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this library; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA As a special exception, if you link this library with other files, compiled with a Free Software compiler, to produce an executable, this library does not by itself cause the resulting executable to be covered by the GNU General Public License. This exception does not however invalidate any other reasons why the executable file might be covered by the GNU General Public License. */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Thanks to Tom Schrijvers for his "bounds.pl", the first finite domain constraint solver included with SWI-Prolog. I've learned a lot from it and could even use some of the code for this solver. The propagation queue idea is taken from "prop.pl", a prototype solver also written by Tom. Highlights of the present solver: Symbolic constants for infinities --------------------------------- ?- X #>= 0, Y #=< 0. %@ X in 0..sup, %@ Y in inf..0. No artificial limits (using GMP) --------------------------------- ?- N #= 2^66, X #\= N. %@ N = 73786976294838206464, %@ X in inf..73786976294838206463\/73786976294838206465..sup. Often stronger propagation --------------------------------- ?- Y #= abs(X), Y #\= 3, Z * Z #= 4. %@ Y in 0..2\/4..sup, %@ Y#=abs(X), %@ X in inf.. -4\/ -2..2\/4..sup, %@ Z in -2\/2. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Many things can be improved; if you need any additional features or want to help, please e-mail me. A good starting point is taking a propagation algorithm from the literature and adding it. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- module(clpfd, [ op(760, yfx, #<==>), op(750, xfy, #==>), op(750, yfx, #<==), op(740, yfx, #\/), op(730, yfx, #\), op(720, yfx, #/\), op(710, fy, #\), op(700, xfx, #>), op(700, xfx, #<), op(700, xfx, #>=), op(700, xfx, #=<), op(700, xfx, #=), op(700, xfx, #\=), op(700, xfx, in), op(700, xfx, ins), op(450, xfx, ..), % should bind more tightly than \/ (#>)/2, (#<)/2, (#>=)/2, (#=<)/2, (#=)/2, (#\=)/2, (#\)/1, (#<==>)/2, (#==>)/2, (#<==)/2, (#\/)/2, (#\)/2, (#/\)/2, (in)/2, (ins)/2, all_different/1, all_distinct/1, sum/3, scalar_product/4, tuples_in/2, labeling/2, label/1, indomain/1, lex_chain/1, serialized/2, global_cardinality/2, global_cardinality/3, circuit/1, cumulative/1, cumulative/2, disjoint2/1, element/3, automaton/3, automaton/8, transpose/2, zcompare/3, chain/2, fd_var/1, fd_inf/2, fd_sup/2, fd_size/2, fd_dom/2 ]). :- public % called from goal_expansion clpfd_equal/2, clpfd_geq/2. :- use_module(library(apply)). :- use_module(library(apply_macros)). :- use_module(library(assoc)). :- use_module(library(error)). :- use_module(library(lists)). :- use_module(library(pairs)). :- op(700, xfx, cis). :- op(700, xfx, cis_geq). :- op(700, xfx, cis_gt). :- op(700, xfx, cis_leq). :- op(700, xfx, cis_lt). /** Constraint Logic Programming over Finite Domains ### Introduction {#clpfd-intro} Constraint programming is a declarative formalism that lets you state relations between terms. This library provides CLP(FD), Constraint Logic Programming over Finite Domains. There are two major use cases of this library: 1. CLP(FD) constraints provide _declarative integer arithmetic_: They implement pure _relations_ between integer expressions and can be used in all directions, also if parts of expressions are variables. 2. In connection with enumeration predicates and more complex constraints, CLP(FD) is often used to model and solve combinatorial problems such as planning, scheduling and allocation tasks. When teaching Prolog, we _strongly_ recommend that you introduce CLP(FD) constraints _before_ explaining lower-level arithmetic predicates and their procedural idiosyncrasies. This is because constraints are easy to explain, understand and use due to their purely relational nature. In contrast, the modedness and directionality of low-level arithmetic primitives are non-declarative limitations that are better deferred to more advanced lectures. If you are used to the complicated operational considerations that low-level arithmetic primitives necessitate, then moving to CLP(FD) constraints may, due to their power and convenience, at first feel to you excessive and almost like cheating. It _isn't_. Constraints are an integral part of many Prolog systems and are available to help you eliminate and avoid, as far as possible, the use of lower-level and less general primitives by providing declarative alternatives that are meant to be used instead. For satisfactory performance, arithmetic constraints are implicitly rewritten at compilation time so that lower-level fallback predicates are automatically used whenever possible. You can cite this library in your publications as: == @inproceedings{Triska12, author = {Markus Triska}, title = {The Finite Domain Constraint Solver of {SWI-Prolog}}, booktitle = {FLOPS}, series = {LNCS}, volume = {7294}, year = {2012}, pages = {307-316} } == ### Arithmetic constraints {#cplfd-arith-constraints} A finite domain _arithmetic expression_ is one of: | _integer_ | Given value | | _variable_ | Unknown integer | | ?(_variable_) | Unknown integer | | -Expr | Unary minus | | Expr + Expr | Addition | | Expr * Expr | Multiplication | | Expr - Expr | Subtraction | | Expr ^ Expr | Exponentiation | | min(Expr,Expr) | Minimum of two expressions | | max(Expr,Expr) | Maximum of two expressions | | Expr mod Expr | Modulo induced by floored division | | Expr rem Expr | Modulo induced by truncated division | | abs(Expr) | Absolute value | | Expr // Expr | Truncated integer division | Arithmetic _constraints_ are relations between arithmetic expressions. The most important arithmetic constraints are: | Expr1 #>= Expr2 | Expr1 is greater than or equal to Expr2 | | Expr1 #=< Expr2 | Expr1 is less than or equal to Expr2 | | Expr1 #= Expr2 | Expr1 equals Expr2 | | Expr1 #\= Expr2 | Expr1 is not equal to Expr2 | | Expr1 #> Expr2 | Expr1 is greater than Expr2 | | Expr1 #< Expr2 | Expr1 is less than Expr2 | ### Declarative integer arithmetic {#clpfd-integer-arith} CLP(FD) constraints let you declaratively express integer arithmetic. The CLP(FD) constraints #=/2, #>/2 etc. are meant to be used instead of the corresponding primitives is/2, =:=/2, >/2 etc. An important advantage of arithmetic constraints is their purely relational nature. They are therefore easy to explain and use, and well suited for beginners and experienced Prolog programmers alike. Consider for example the query: == ?- X #> 3, X #= 5 + 2. X = 7. == In contrast, when using low-level integer arithmetic, we get: == ?- X > 3, X is 5 + 2. ERROR: >/2: Arguments are not sufficiently instantiated == Due to the necessary operational considerations, the use of these low-level arithmetic predicates is considerably harder to understand and should therefore be deferred to more advanced lectures. For supported expressions, CLP(FD) constraints are drop-in replacements of these low-level arithmetic predicates, often yielding more general programs. Here is an example: == :- use_module(library(clpfd)). n_factorial(0, 1). n_factorial(N, F) :- N #> 0, N1 #= N - 1, F #= N * F1, n_factorial(N1, F1). == This predicate can be used in all directions. For example: == ?- n_factorial(47, F). F = 258623241511168180642964355153611979969197632389120000000000 ; false. ?- n_factorial(N, 1). N = 0 ; N = 1 ; false. ?- n_factorial(N, 3). false. == To make the predicate terminate if any argument is instantiated, add the (implied) constraint F #\= 0 before the recursive call. Otherwise, the query n_factorial(N, 0) is the only non-terminating case of this kind. This library uses goal_expansion/2 to automatically rewrite arithmetic constraints at compilation time. The expansion's aim is to bring the performance of arithmetic constraints close to that of lower-level arithmetic predicates whenever possible. To disable the expansion, set the flag `clpfd_goal_expansion` to `false`. ### Reification {#clpfd-reification} The constraints in/2, #=/2, #\=/2, #/2, #==/2 can be _reified_, which means reflecting their truth values into Boolean values represented by the integers 0 and 1. Let P and Q denote reifiable constraints or Boolean variables, then: | #\ Q | True iff Q is false | | P #\/ Q | True iff either P or Q | | P #/\ Q | True iff both P and Q | | P #\ Q | True iff either P or Q, but not both | | P #<==> Q | True iff P and Q are equivalent | | P #==> Q | True iff P implies Q | | P #<== Q | True iff Q implies P | The constraints of this table are reifiable as well. ### Domains {#clpfd-domains} Each CLP(FD) variable has an associated set of admissible integers which we call the variable's _domain_. Initially, the domain of each CLP(FD) variable is the set of all integers. The constraints in/2 and ins/2 are the primary means to specify tighter domains of variables. Here are example queries and the system's declaratively equivalent answers: == ?- X in 100..sup. X in 100..sup. ?- X in 1..5 \/ 3..12. X in 1..12. ?- [X,Y,Z] ins 0..3. X in 0..3, Y in 0..3, Z in 0..3. == Domains are taken into account when further constraints are stated, and by enumeration predicates like labeling/2. ### Examples {#clpfd-examples} Here is an example session with a few queries and their answers: == ?- use_module(library(clpfd)). % library(clpfd) compiled into clpfd 0.06 sec, 633,732 bytes true. ?- X #> 3. X in 4..sup. ?- X #\= 20. X in inf..19\/21..sup. ?- 2*X #= 10. X = 5. ?- X*X #= 144. X in -12\/12. ?- 4*X + 2*Y #= 24, X + Y #= 9, [X,Y] ins 0..sup. X = 3, Y = 6. ?- X #= Y #<==> B, X in 0..3, Y in 4..5. B = 0, X in 0..3, Y in 4..5. == In each case, and as for all pure programs, the answer is declaratively equivalent to the original query, and in many cases the constraint solver has deduced additional domain restrictions. ### Enumeration predicates and search {#clpfd-search} In addition to being declarative replacements for low-level arithmetic predicates, CLP(FD) constraints are also often used to solve combinatorial problems such as planning, scheduling and allocation tasks. To let you conveniently model and solve such problems, this library provides several constraints beyond typical integer arithmetic, such as all_distinct/1, global_cardinality/2 and cumulative/1. Using CLP(FD) constraints to solve combinatorial tasks typically consists of two phases: 1. First, all relevant constraints are stated. 2. Second, if the domain of each involved variable is _finite_, then _enumeration predicates_ can be used to search for concrete solutions. It is good practice to keep the modeling part, via a dedicated predicate, separate from the actual search for solutions. This lets you observe termination and determinism properties of the modeling part in isolation from the search, and more easily try different search strategies. As an example of a constraint satisfaction problem, consider the cryptoarithmetic puzzle SEND + MORE = MONEY, where different letters denote distinct integers between 0 and 9. It can be modeled in CLP(FD) as follows: == :- use_module(library(clpfd)). puzzle([S,E,N,D] + [M,O,R,E] = [M,O,N,E,Y]) :- Vars = [S,E,N,D,M,O,R,Y], Vars ins 0..9, all_different(Vars), S*1000 + E*100 + N*10 + D + M*1000 + O*100 + R*10 + E #= M*10000 + O*1000 + N*100 + E*10 + Y, M #\= 0, S #\= 0. == Notice that we are _not_ using labeling/2 in this predicate, so that we can first execute and observe the modeling part in isolation. Sample query and its result (actual variables replaced for readability): == ?- puzzle(As+Bs=Cs). As = [9, A2, A3, A4], Bs = [1, 0, B3, A2], Cs = [1, 0, A3, A2, C5], A2 in 4..7, all_different([9, A2, A3, A4, 1, 0, B3, C5]), 91*A2+A4+10*B3#=90*A3+C5, A3 in 5..8, A4 in 2..8, B3 in 2..8, C5 in 2..8. == From this answer, we see that the modeling part _terminates_ and is in fact _deterministic_. Moreover, we see from the residual goals that the constraint solver has deduced more stringent bounds for all variables. Such observations are only possible if modeling and search parts are cleanly separated. Labeling can then be used to search for solutions in a separate predicate or goal: == ?- puzzle(As+Bs=Cs), label(As). As = [9, 5, 6, 7], Bs = [1, 0, 8, 5], Cs = [1, 0, 6, 5, 2] ; false. == In this case, it suffices to label a subset of variables to find the puzzle's unique solution, since the constraint solver is strong enough to reduce the domains of remaining variables to singleton sets. In general though, it is necessary to label all variables to obtain ground solutions. ### Advanced topics {#clpfd-advanced-topics} If you set the flag `clpfd_monotonic` to `true`, then CLP(FD) is monotonic: Adding new constraints cannot yield new solutions. When this flag is `true`, you must wrap variables that occur in arithmetic expressions with the functor `(?)/1`. For example, `?(X) #= ?(Y) + ?(Z)`. The wrapper can be omitted for variables that are already constrained to integers. Use call_residue_vars/2 and copy_term/3 to inspect residual goals and the constraints in which a variable is involved. This library also provides _reflection_ predicates (like fd_dom/2, fd_size/2 etc.) with which you can inspect a variable's current domain. These predicates can be useful if you want to implement your own labeling strategies. You can also define custom constraints. The mechanism to do this is not yet finalised, and we welcome suggestions and descriptions of use cases that are important to you. As an example of how it can be done currently, let us define a new custom constraint `oneground(X,Y,Z)`, where Z shall be 1 if at least one of X and Y is instantiated: == :- use_module(library(clpfd)). :- multifile clpfd:run_propagator/2. oneground(X, Y, Z) :- clpfd:make_propagator(oneground(X, Y, Z), Prop), clpfd:init_propagator(X, Prop), clpfd:init_propagator(Y, Prop), clpfd:trigger_once(Prop). clpfd:run_propagator(oneground(X, Y, Z), MState) :- ( integer(X) -> clpfd:kill(MState), Z = 1 ; integer(Y) -> clpfd:kill(MState), Z = 1 ; true ). == First, clpfd:make_propagator/2 is used to transform a user-defined representation of the new constraint to an internal form. With clpfd:init_propagator/2, this internal form is then attached to X and Y. From now on, the propagator will be invoked whenever the domains of X or Y are changed. Then, clpfd:trigger_once/1 is used to give the propagator its first chance for propagation even though the variables' domains have not yet changed. Finally, clpfd:run_propagator/2 is extended to define the actual propagator. As explained, this predicate is automatically called by the constraint solver. The first argument is the user-defined representation of the constraint as used in clpfd:make_propagator/2, and the second argument is a mutable state that can be used to prevent further invocations of the propagator when the constraint has become entailed, by using clpfd:kill/1. An example of using the new constraint: == ?- oneground(X, Y, Z), Y = 5. Y = 5, Z = 1, X in inf..sup. == @author Markus Triska */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A bound is either: n(N): integer N inf: infimum of Z (= negative infinity) sup: supremum of Z (= positive infinity) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ is_bound(n(N)) :- integer(N). is_bound(inf). is_bound(sup). defaulty_to_bound(D, P) :- ( integer(D) -> P = n(D) ; P = D ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Compactified is/2 and predicates for several arithmetic expressions with infinities, tailored for the modes needed by this solver. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ goal_expansion(A cis B, Expansion) :- phrase(cis_goals(B, A), Goals), list_goal(Goals, Expansion). goal_expansion(A cis_lt B, B cis_gt A). goal_expansion(A cis_leq B, B cis_geq A). goal_expansion(A cis_geq B, cis_leq_numeric(B, N)) :- nonvar(A), A = n(N). goal_expansion(A cis_geq B, cis_geq_numeric(A, N)) :- nonvar(B), B = n(N). goal_expansion(A cis_gt B, cis_lt_numeric(B, N)) :- nonvar(A), A = n(N). goal_expansion(A cis_gt B, cis_gt_numeric(A, N)) :- nonvar(B), B = n(N). % cis_gt only works for terms of depth 0 on both sides cis_gt(sup, B0) :- B0 \== sup. cis_gt(n(N), B) :- cis_lt_numeric(B, N). cis_lt_numeric(inf, _). cis_lt_numeric(n(B), A) :- B < A. cis_gt_numeric(sup, _). cis_gt_numeric(n(B), A) :- B > A. cis_geq(inf, inf). cis_geq(sup, _). cis_geq(n(N), B) :- cis_leq_numeric(B, N). cis_leq_numeric(inf, _). cis_leq_numeric(n(B), A) :- B =< A. cis_geq_numeric(sup, _). cis_geq_numeric(n(B), A) :- B >= A. cis_min(inf, _, inf). cis_min(sup, B, B). cis_min(n(N), B, Min) :- cis_min_(B, N, Min). cis_min_(inf, _, inf). cis_min_(sup, N, n(N)). cis_min_(n(B), A, n(M)) :- M is min(A,B). cis_max(sup, _, sup). cis_max(inf, B, B). cis_max(n(N), B, Max) :- cis_max_(B, N, Max). cis_max_(inf, N, n(N)). cis_max_(sup, _, sup). cis_max_(n(B), A, n(M)) :- M is max(A,B). cis_plus(inf, _, inf). cis_plus(sup, _, sup). cis_plus(n(A), B, Plus) :- cis_plus_(B, A, Plus). cis_plus_(sup, _, sup). cis_plus_(inf, _, inf). cis_plus_(n(B), A, n(S)) :- S is A + B. cis_minus(inf, _, inf). cis_minus(sup, _, sup). cis_minus(n(A), B, M) :- cis_minus_(B, A, M). cis_minus_(inf, _, sup). cis_minus_(sup, _, inf). cis_minus_(n(B), A, n(M)) :- M is A - B. cis_uminus(inf, sup). cis_uminus(sup, inf). cis_uminus(n(A), n(B)) :- B is -A. cis_abs(inf, sup). cis_abs(sup, sup). cis_abs(n(A), n(B)) :- B is abs(A). cis_times(inf, B, P) :- ( B cis_lt n(0) -> P = sup ; B cis_gt n(0) -> P = inf ; P = n(0) ). cis_times(sup, B, P) :- ( B cis_gt n(0) -> P = sup ; B cis_lt n(0) -> P = inf ; P = n(0) ). cis_times(n(N), B, P) :- cis_times_(B, N, P). cis_times_(inf, A, P) :- cis_times(inf, n(A), P). cis_times_(sup, A, P) :- cis_times(sup, n(A), P). cis_times_(n(B), A, n(P)) :- P is A * B. cis_exp(inf, n(Y), R) :- ( even(Y) -> R = sup ; R = inf ). cis_exp(sup, _, sup). cis_exp(n(N), Y, R) :- cis_exp_(Y, N, R). cis_exp_(n(Y), N, n(R)) :- R is N^Y. cis_exp_(sup, _, sup). cis_exp_(inf, _, inf). cis_goals(V, V) --> { var(V) }, !. cis_goals(n(N), n(N)) --> []. cis_goals(inf, inf) --> []. cis_goals(sup, sup) --> []. cis_goals(sign(A0), R) --> cis_goals(A0, A), [cis_sign(A, R)]. cis_goals(abs(A0), R) --> cis_goals(A0, A), [cis_abs(A, R)]. cis_goals(-A0, R) --> cis_goals(A0, A), [cis_uminus(A, R)]. cis_goals(A0+B0, R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_plus(A, B, R)]. cis_goals(A0-B0, R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_minus(A, B, R)]. cis_goals(min(A0,B0), R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_min(A, B, R)]. cis_goals(max(A0,B0), R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_max(A, B, R)]. cis_goals(A0*B0, R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_times(A, B, R)]. cis_goals(div(A0,B0), R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_div(A, B, R)]. cis_goals(A0//B0, R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_slash(A, B, R)]. cis_goals(A0^B0, R) --> cis_goals(A0, A), cis_goals(B0, B), [cis_exp(A, B, R)]. list_goal([], true). list_goal([G|Gs], Goal) :- foldl(list_goal_, Gs, G, Goal). list_goal_(G, G0, (G0,G)). cis_sign(sup, n(1)). cis_sign(inf, n(-1)). cis_sign(n(N), n(S)) :- S is sign(N). cis_div(sup, Y, Z) :- ( Y cis_geq n(0) -> Z = sup ; Z = inf ). cis_div(inf, Y, Z) :- ( Y cis_geq n(0) -> Z = inf ; Z = sup ). cis_div(n(X), Y, Z) :- cis_div_(Y, X, Z). cis_div_(sup, _, n(0)). cis_div_(inf, _, n(0)). cis_div_(n(Y), X, Z) :- ( Y =:= 0 -> ( X >= 0 -> Z = sup ; Z = inf ) ; Z0 is X // Y, Z = n(Z0) ). cis_slash(sup, _, sup). cis_slash(inf, _, inf). cis_slash(n(N), B, S) :- cis_slash_(B, N, S). cis_slash_(sup, _, n(0)). cis_slash_(inf, _, n(0)). cis_slash_(n(B), A, n(S)) :- S is A // B. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A domain is a finite set of disjoint intervals. Internally, domains are represented as trees. Each node is one of: empty: empty domain. split(N, Left, Right) - split on integer N, with Left and Right domains whose elements are all less than and greater than N, respectively. The domain is the union of Left and Right, i.e., N is a hole. from_to(From, To) - interval (From-1, To+1); From and To are bounds Desiderata: rebalance domains; singleton intervals. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Type definition and inspection of domains. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ check_domain(D) :- ( var(D) -> instantiation_error(D) ; is_domain(D) -> true ; domain_error(clpfd_domain, D) ). is_domain(empty). is_domain(from_to(From,To)) :- is_bound(From), is_bound(To), From cis_leq To. is_domain(split(S, Left, Right)) :- integer(S), is_domain(Left), is_domain(Right), all_less_than(Left, S), all_greater_than(Right, S). all_less_than(empty, _). all_less_than(from_to(From,To), S) :- From cis_lt n(S), To cis_lt n(S). all_less_than(split(S0,Left,Right), S) :- S0 < S, all_less_than(Left, S), all_less_than(Right, S). all_greater_than(empty, _). all_greater_than(from_to(From,To), S) :- From cis_gt n(S), To cis_gt n(S). all_greater_than(split(S0,Left,Right), S) :- S0 > S, all_greater_than(Left, S), all_greater_than(Right, S). default_domain(from_to(inf,sup)). domain_infimum(from_to(I, _), I). domain_infimum(split(_, Left, _), I) :- domain_infimum(Left, I). domain_supremum(from_to(_, S), S). domain_supremum(split(_, _, Right), S) :- domain_supremum(Right, S). domain_num_elements(empty, n(0)). domain_num_elements(from_to(From,To), Num) :- Num cis To - From + n(1). domain_num_elements(split(_, Left, Right), Num) :- domain_num_elements(Left, NL), domain_num_elements(Right, NR), Num cis NL + NR. domain_direction_element(from_to(n(From), n(To)), Dir, E) :- ( Dir == up -> between(From, To, E) ; between(From, To, E0), E is To - (E0 - From) ). domain_direction_element(split(_, D1, D2), Dir, E) :- ( Dir == up -> ( domain_direction_element(D1, Dir, E) ; domain_direction_element(D2, Dir, E) ) ; ( domain_direction_element(D2, Dir, E) ; domain_direction_element(D1, Dir, E) ) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Test whether domain contains a given integer. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_contains(from_to(From,To), I) :- From cis_leq n(I), n(I) cis_leq To. domain_contains(split(S, Left, Right), I) :- ( I < S -> domain_contains(Left, I) ; I > S -> domain_contains(Right, I) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Test whether a domain contains another domain. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_subdomain(Dom, Sub) :- domain_subdomain(Dom, Dom, Sub). domain_subdomain(from_to(_,_), Dom, Sub) :- domain_subdomain_fromto(Sub, Dom). domain_subdomain(split(_, _, _), Dom, Sub) :- domain_subdomain_split(Sub, Dom, Sub). domain_subdomain_split(empty, _, _). domain_subdomain_split(from_to(From,To), split(S,Left0,Right0), Sub) :- ( To cis_lt n(S) -> domain_subdomain(Left0, Left0, Sub) ; From cis_gt n(S) -> domain_subdomain(Right0, Right0, Sub) ). domain_subdomain_split(split(_,Left,Right), Dom, _) :- domain_subdomain(Dom, Dom, Left), domain_subdomain(Dom, Dom, Right). domain_subdomain_fromto(empty, _). domain_subdomain_fromto(from_to(From,To), from_to(From0,To0)) :- From0 cis_leq From, To0 cis_geq To. domain_subdomain_fromto(split(_,Left,Right), Dom) :- domain_subdomain_fromto(Left, Dom), domain_subdomain_fromto(Right, Dom). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Remove an integer from a domain. The domain is traversed until an interval is reached from which the element can be removed, or until it is clear that no such interval exists. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_remove(empty, _, empty). domain_remove(from_to(L0, U0), X, D) :- domain_remove_(L0, U0, X, D). domain_remove(split(S, Left0, Right0), X, D) :- ( X =:= S -> D = split(S, Left0, Right0) ; X < S -> domain_remove(Left0, X, Left1), ( Left1 == empty -> D = Right0 ; D = split(S, Left1, Right0) ) ; domain_remove(Right0, X, Right1), ( Right1 == empty -> D = Left0 ; D = split(S, Left0, Right1) ) ). %?- domain_remove(from_to(n(0),n(5)), 3, D). domain_remove_(inf, U0, X, D) :- ( U0 == n(X) -> U1 is X - 1, D = from_to(inf, n(U1)) ; U0 cis_lt n(X) -> D = from_to(inf,U0) ; L1 is X + 1, U1 is X - 1, D = split(X, from_to(inf, n(U1)), from_to(n(L1),U0)) ). domain_remove_(n(N), U0, X, D) :- domain_remove_upper(U0, N, X, D). domain_remove_upper(sup, L0, X, D) :- ( L0 =:= X -> L1 is X + 1, D = from_to(n(L1),sup) ; L0 > X -> D = from_to(n(L0),sup) ; L1 is X + 1, U1 is X - 1, D = split(X, from_to(n(L0),n(U1)), from_to(n(L1),sup)) ). domain_remove_upper(n(U0), L0, X, D) :- ( L0 =:= U0, X =:= L0 -> D = empty ; L0 =:= X -> L1 is X + 1, D = from_to(n(L1), n(U0)) ; U0 =:= X -> U1 is X - 1, D = from_to(n(L0), n(U1)) ; between(L0, U0, X) -> U1 is X - 1, L1 is X + 1, D = split(X, from_to(n(L0), n(U1)), from_to(n(L1), n(U0))) ; D = from_to(n(L0),n(U0)) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Remove all elements greater than / less than a constant. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_remove_greater_than(empty, _, empty). domain_remove_greater_than(from_to(From0,To0), G, D) :- ( From0 cis_gt n(G) -> D = empty ; To cis min(To0,n(G)), D = from_to(From0,To) ). domain_remove_greater_than(split(S,Left0,Right0), G, D) :- ( S =< G -> domain_remove_greater_than(Right0, G, Right), ( Right == empty -> D = Left0 ; D = split(S, Left0, Right) ) ; domain_remove_greater_than(Left0, G, D) ). domain_remove_smaller_than(empty, _, empty). domain_remove_smaller_than(from_to(From0,To0), V, D) :- ( To0 cis_lt n(V) -> D = empty ; From cis max(From0,n(V)), D = from_to(From,To0) ). domain_remove_smaller_than(split(S,Left0,Right0), V, D) :- ( S >= V -> domain_remove_smaller_than(Left0, V, Left), ( Left == empty -> D = Right0 ; D = split(S, Left, Right0) ) ; domain_remove_smaller_than(Right0, V, D) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Remove a whole domain from another domain. (Set difference.) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_subtract(Dom0, Sub, Dom) :- domain_subtract(Dom0, Dom0, Sub, Dom). domain_subtract(empty, _, _, empty). domain_subtract(from_to(From0,To0), Dom, Sub, D) :- ( Sub == empty -> D = Dom ; Sub = from_to(From,To) -> ( From == To -> From = n(X), domain_remove(Dom, X, D) ; From cis_gt To0 -> D = Dom ; To cis_lt From0 -> D = Dom ; From cis_leq From0 -> ( To cis_geq To0 -> D = empty ; From1 cis To + n(1), D = from_to(From1, To0) ) ; To1 cis From - n(1), ( To cis_lt To0 -> From = n(S), From2 cis To + n(1), D = split(S,from_to(From0,To1),from_to(From2,To0)) ; D = from_to(From0,To1) ) ) ; Sub = split(S, Left, Right) -> ( n(S) cis_gt To0 -> domain_subtract(Dom, Dom, Left, D) ; n(S) cis_lt From0 -> domain_subtract(Dom, Dom, Right, D) ; domain_subtract(Dom, Dom, Left, D1), domain_subtract(D1, D1, Right, D) ) ). domain_subtract(split(S, Left0, Right0), _, Sub, D) :- domain_subtract(Left0, Left0, Sub, Left), domain_subtract(Right0, Right0, Sub, Right), ( Left == empty -> D = Right ; Right == empty -> D = Left ; D = split(S, Left, Right) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Complement of a domain - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_complement(D, C) :- default_domain(Default), domain_subtract(Default, D, C). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Convert domain to a list of disjoint intervals From-To. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_intervals(D, Is) :- phrase(domain_intervals(D), Is). domain_intervals(split(_, Left, Right)) --> domain_intervals(Left), domain_intervals(Right). domain_intervals(empty) --> []. domain_intervals(from_to(From,To)) --> [From-To]. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - To compute the intersection of two domains D1 and D2, we choose D1 as the reference domain. For each interval of D1, we compute how far and to which values D2 lets us extend it. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domains_intersection(D1, D2, Intersection) :- domains_intersection_(D1, D2, Intersection), Intersection \== empty. domains_intersection_(empty, _, empty). domains_intersection_(from_to(L0,U0), D2, Dom) :- narrow(D2, L0, U0, Dom). domains_intersection_(split(S,Left0,Right0), D2, Dom) :- domains_intersection_(Left0, D2, Left1), domains_intersection_(Right0, D2, Right1), ( Left1 == empty -> Dom = Right1 ; Right1 == empty -> Dom = Left1 ; Dom = split(S, Left1, Right1) ). narrow(empty, _, _, empty). narrow(from_to(L0,U0), From0, To0, Dom) :- From1 cis max(From0,L0), To1 cis min(To0,U0), ( From1 cis_gt To1 -> Dom = empty ; Dom = from_to(From1,To1) ). narrow(split(S, Left0, Right0), From0, To0, Dom) :- ( To0 cis_lt n(S) -> narrow(Left0, From0, To0, Dom) ; From0 cis_gt n(S) -> narrow(Right0, From0, To0, Dom) ; narrow(Left0, From0, To0, Left1), narrow(Right0, From0, To0, Right1), ( Left1 == empty -> Dom = Right1 ; Right1 == empty -> Dom = Left1 ; Dom = split(S, Left1, Right1) ) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Union of 2 domains. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domains_union(D1, D2, Union) :- domain_intervals(D1, Is1), domain_intervals(D2, Is2), append(Is1, Is2, IsU0), merge_intervals(IsU0, IsU1), intervals_to_domain(IsU1, Union). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Shift the domain by an offset. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_shift(empty, _, empty). domain_shift(from_to(From0,To0), O, from_to(From,To)) :- From cis From0 + n(O), To cis To0 + n(O). domain_shift(split(S0, Left0, Right0), O, split(S, Left, Right)) :- S is S0 + O, domain_shift(Left0, O, Left), domain_shift(Right0, O, Right). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The new domain contains all values of the old domain, multiplied by a constant multiplier. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_expand(D0, M, D) :- ( M < 0 -> domain_negate(D0, D1), M1 is abs(M), domain_expand_(D1, M1, D) ; M =:= 1 -> D = D0 ; domain_expand_(D0, M, D) ). domain_expand_(empty, _, empty). domain_expand_(from_to(From0, To0), M, from_to(From,To)) :- From cis From0*n(M), To cis To0*n(M). domain_expand_(split(S0, Left0, Right0), M, split(S, Left, Right)) :- S is M*S0, domain_expand_(Left0, M, Left), domain_expand_(Right0, M, Right). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - similar to domain_expand/3, tailored for truncated division: an interval [From,To] is extended to [From*M, ((To+1)*M - 1)], i.e., to all values that truncated integer-divided by M yield a value from interval. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_expand_more(D0, M, D) :- %format("expanding ~w by ~w\n", [D0,M]), ( M < 0 -> domain_negate(D0, D1), M1 is abs(M) ; D1 = D0, M1 = M ), domain_expand_more_(D1, M1, D). %format("yield: ~w\n", [D]). domain_expand_more_(empty, _, empty). domain_expand_more_(from_to(From0, To0), M, from_to(From,To)) :- ( From0 cis_leq n(0) -> From cis (From0-n(1))*n(M) + n(1) ; From cis From0*n(M) ), ( To0 cis_lt n(0) -> To cis To0*n(M) ; To cis (To0+n(1))*n(M) - n(1) ). domain_expand_more_(split(S0, Left0, Right0), M, split(S, Left, Right)) :- S is M*S0, domain_expand_more_(Left0, M, Left), domain_expand_more_(Right0, M, Right). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Scale a domain down by a constant multiplier. Assuming (//)/2. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_contract(D0, M, D) :- %format("contracting ~w by ~w\n", [D0,M]), ( M < 0 -> domain_negate(D0, D1), M1 is abs(M) ; D1 = D0, M1 = M ), domain_contract_(D1, M1, D). domain_contract_(empty, _, empty). domain_contract_(from_to(From0, To0), M, from_to(From,To)) :- ( From0 cis_geq n(0) -> From cis (From0 + n(M) - n(1)) // n(M) ; From cis From0 // n(M) ), ( To0 cis_geq n(0) -> To cis To0 // n(M) ; To cis (To0 - n(M) + n(1)) // n(M) ). domain_contract_(split(_,Left0,Right0), M, D) :- % Scaled down domains do not necessarily retain any holes of % the original domain. domain_contract_(Left0, M, Left), domain_contract_(Right0, M, Right), domains_union(Left, Right, D). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Similar to domain_contract, tailored for division, i.e., {21,23} contracted by 4 is 5. It contracts "less". - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_contract_less(D0, M, D) :- ( M < 0 -> domain_negate(D0, D1), M1 is abs(M) ; D1 = D0, M1 = M ), domain_contract_less_(D1, M1, D). domain_contract_less_(empty, _, empty). domain_contract_less_(from_to(From0, To0), M, from_to(From,To)) :- From cis From0 // n(M), To cis To0 // n(M). domain_contract_less_(split(_,Left0,Right0), M, D) :- % Scaled down domains do not necessarily retain any holes of % the original domain. domain_contract_less_(Left0, M, Left), domain_contract_less_(Right0, M, Right), domains_union(Left, Right, D). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Negate the domain. Left and Right sub-domains and bounds switch sides. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ domain_negate(empty, empty). domain_negate(from_to(From0, To0), from_to(From, To)) :- From cis -To0, To cis -From0. domain_negate(split(S0, Left0, Right0), split(S, Left, Right)) :- S is -S0, domain_negate(Left0, Right), domain_negate(Right0, Left). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Construct a domain from a list of integers. Try to balance it. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ list_to_disjoint_intervals([], []). list_to_disjoint_intervals([N|Ns], Is) :- list_to_disjoint_intervals(Ns, N, N, Is). list_to_disjoint_intervals([], M, N, [n(M)-n(N)]). list_to_disjoint_intervals([B|Bs], M, N, Is) :- ( B =:= N + 1 -> list_to_disjoint_intervals(Bs, M, B, Is) ; Is = [n(M)-n(N)|Rest], list_to_disjoint_intervals(Bs, B, B, Rest) ). list_to_domain(List0, D) :- ( List0 == [] -> D = empty ; sort(List0, List), list_to_disjoint_intervals(List, Is), intervals_to_domain(Is, D) ). intervals_to_domain([], empty) :- !. intervals_to_domain([M-N], from_to(M,N)) :- !. intervals_to_domain(Is, D) :- length(Is, L), FL is L // 2, length(Front, FL), append(Front, Tail, Is), Tail = [n(Start)-_|_], Hole is Start - 1, intervals_to_domain(Front, Left), intervals_to_domain(Tail, Right), D = split(Hole, Left, Right). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% ?Var in +Domain % % Var is an element of Domain. Domain is one of: % % * Integer % Singleton set consisting only of _Integer_. % * Lower..Upper % All integers _I_ such that _Lower_ =< _I_ =< _Upper_. % _Lower_ must be an integer or the atom *inf*, which % denotes negative infinity. _Upper_ must be an integer or % the atom *sup*, which denotes positive infinity. % * Domain1 \/ Domain2 % The union of Domain1 and Domain2. Var in Dom :- clpfd_in(Var, Dom). clpfd_in(V, D) :- fd_variable(V), drep_to_domain(D, Dom), domain(V, Dom). fd_variable(V) :- ( var(V) -> true ; integer(V) -> true ; type_error(integer, V) ). %% +Vars ins +Domain % % The variables in the list Vars are elements of Domain. Vs ins D :- fd_must_be_list(Vs), maplist(fd_variable, Vs), drep_to_domain(D, Dom), domains(Vs, Dom). fd_must_be_list(Ls) :- ( fd_var(Ls) -> type_error(list, Ls) ; must_be(list, Ls) ). %% indomain(?Var) % % Bind Var to all feasible values of its domain on backtracking. The % domain of Var must be finite. indomain(Var) :- label([Var]). order_dom_next(up, Dom, Next) :- domain_infimum(Dom, n(Next)). order_dom_next(down, Dom, Next) :- domain_supremum(Dom, n(Next)). order_dom_next(random_value(_), Dom, Next) :- domain_to_list(Dom, Ls), length(Ls, L), I is random(L), nth0(I, Ls, Next). %% label(+Vars) % % Equivalent to labeling([], Vars). label(Vs) :- labeling([], Vs). %% labeling(+Options, +Vars) % % Assign a value to each variable in Vars. Labeling means systematically % trying out values for the finite domain variables Vars until all of % them are ground. The domain of each variable in Vars must be finite. % Options is a list of options that let you exhibit some control over % the search process. Several categories of options exist: % % The variable selection strategy lets you specify which variable of % Vars is labeled next and is one of: % % * leftmost % Label the variables in the order they occur in Vars. This is the % default. % % * ff % _|First fail|_. Label the leftmost variable with smallest domain next, % in order to detect infeasibility early. This is often a good % strategy. % % * ffc % Of the variables with smallest domains, the leftmost one % participating in most constraints is labeled next. % % * min % Label the leftmost variable whose lower bound is the lowest next. % % * max % Label the leftmost variable whose upper bound is the highest next. % % The value order is one of: % % * up % Try the elements of the chosen variable's domain in ascending order. % This is the default. % % * down % Try the domain elements in descending order. % % The branching strategy is one of: % % * step % For each variable X, a choice is made between X = V and X #\= V, % where V is determined by the value ordering options. This is the % default. % % * enum % For each variable X, a choice is made between X = V_1, X = V_2 % etc., for all values V_i of the domain of X. The order is % determined by the value ordering options. % % * bisect % For each variable X, a choice is made between X #=< M and X #> M, % where M is the midpoint of the domain of X. % % At most one option of each category can be specified, and an option % must not occur repeatedly. % % The order of solutions can be influenced with: % % * min(Expr) % * max(Expr) % % This generates solutions in ascending/descending order with respect % to the evaluation of the arithmetic expression Expr. Labeling Vars % must make Expr ground. If several such options are specified, they % are interpreted from left to right, e.g.: % % == % ?- [X,Y] ins 10..20, labeling([max(X),min(Y)],[X,Y]). % == % % This generates solutions in descending order of X, and for each % binding of X, solutions are generated in ascending order of Y. To % obtain the incomplete behaviour that other systems exhibit with % "maximize(Expr)" and "minimize(Expr)", use once/1, e.g.: % % == % once(labeling([max(Expr)], Vars)) % == % % Labeling is always complete, always terminates, and yields no % redundant solutions. % labeling(Options, Vars) :- must_be(list, Options), fd_must_be_list(Vars), maplist(finite_domain, Vars), label(Options, Options, default(leftmost), default(up), default(step), [], upto_ground, Vars). finite_domain(Var) :- ( fd_get(Var, Dom, _) -> ( domain_infimum(Dom, n(_)), domain_supremum(Dom, n(_)) -> true ; instantiation_error(Var) ) ; integer(Var) -> true ; must_be(integer, Var) ). label([O|Os], Options, Selection, Order, Choice, Optim, Consistency, Vars) :- ( var(O)-> instantiation_error(O) ; override(selection, Selection, O, Options, S1) -> label(Os, Options, S1, Order, Choice, Optim, Consistency, Vars) ; override(order, Order, O, Options, O1) -> label(Os, Options, Selection, O1, Choice, Optim, Consistency, Vars) ; override(choice, Choice, O, Options, C1) -> label(Os, Options, Selection, Order, C1, Optim, Consistency, Vars) ; optimisation(O) -> label(Os, Options, Selection, Order, Choice, [O|Optim], Consistency, Vars) ; consistency(O, O1) -> label(Os, Options, Selection, Order, Choice, Optim, O1, Vars) ; domain_error(labeling_option, O) ). label([], _, Selection, Order, Choice, Optim0, Consistency, Vars) :- maplist(arg(1), [Selection,Order,Choice], [S,O,C]), ( Optim0 == [] -> label(Vars, S, O, C, Consistency) ; reverse(Optim0, Optim), exprs_singlevars(Optim, SVs), optimise(Vars, [S,O,C], SVs) ). % Introduce new variables for each min/max expression to avoid % reparsing expressions during optimisation. exprs_singlevars([], []). exprs_singlevars([E|Es], [SV|SVs]) :- E =.. [F,Expr], ?(Single) #= Expr, SV =.. [F,Single], exprs_singlevars(Es, SVs). all_dead(fd_props(Bs,Gs,Os)) :- all_dead_(Bs), all_dead_(Gs), all_dead_(Os). all_dead_([]). all_dead_([propagator(_, S)|Ps]) :- S == dead, all_dead_(Ps). label([], _, _, _, Consistency) :- !, ( Consistency = upto_in(I0,I) -> I0 = I ; true ). label(Vars, Selection, Order, Choice, Consistency) :- ( Vars = [V|Vs], nonvar(V) -> label(Vs, Selection, Order, Choice, Consistency) ; select_var(Selection, Vars, Var, RVars), ( var(Var) -> ( Consistency = upto_in(I0,I), fd_get(Var, _, Ps), all_dead(Ps) -> fd_size(Var, Size), I1 is I0*Size, label(RVars, Selection, Order, Choice, upto_in(I1,I)) ; Consistency = upto_in, fd_get(Var, _, Ps), all_dead(Ps) -> label(RVars, Selection, Order, Choice, Consistency) ; choice_order_variable(Choice, Order, Var, RVars, Vars, Selection, Consistency) ) ; label(RVars, Selection, Order, Choice, Consistency) ) ). choice_order_variable(step, Order, Var, Vars, Vars0, Selection, Consistency) :- fd_get(Var, Dom, _), order_dom_next(Order, Dom, Next), ( Var = Next, label(Vars, Selection, Order, step, Consistency) ; neq_num(Var, Next), do_queue, label(Vars0, Selection, Order, step, Consistency) ). choice_order_variable(enum, Order, Var, Vars, _, Selection, Consistency) :- fd_get(Var, Dom0, _), domain_direction_element(Dom0, Order, Var), label(Vars, Selection, Order, enum, Consistency). choice_order_variable(bisect, Order, Var, _, Vars0, Selection, Consistency) :- fd_get(Var, Dom, _), domain_infimum(Dom, n(I)), domain_supremum(Dom, n(S)), Mid0 is (I + S) // 2, ( Mid0 =:= S -> Mid is Mid0 - 1 ; Mid = Mid0 ), ( Order == up -> ( Var #=< Mid ; Var #> Mid ) ; Order == down -> ( Var #> Mid ; Var #=< Mid ) ; domain_error(bisect_up_or_down, Order) ), label(Vars0, Selection, Order, bisect, Consistency). override(What, Prev, Value, Options, Result) :- call(What, Value), override_(Prev, Value, Options, Result). override_(default(_), Value, _, user(Value)). override_(user(Prev), Value, Options, _) :- ( Value == Prev -> domain_error(nonrepeating_labeling_options, Options) ; domain_error(consistent_labeling_options, Options) ). selection(ff). selection(ffc). selection(min). selection(max). selection(leftmost). selection(random_variable(Seed)) :- must_be(integer, Seed), set_random(seed(Seed)). choice(step). choice(enum). choice(bisect). order(up). order(down). % TODO: random_variable and random_value currently both set the seed, % so exchanging the options can yield different results. order(random_value(Seed)) :- must_be(integer, Seed), set_random(seed(Seed)). consistency(upto_in(I), upto_in(1, I)). consistency(upto_in, upto_in). consistency(upto_ground, upto_ground). optimisation(min(_)). optimisation(max(_)). select_var(leftmost, [Var|Vars], Var, Vars). select_var(min, [V|Vs], Var, RVars) :- find_min(Vs, V, Var), delete_eq([V|Vs], Var, RVars). select_var(max, [V|Vs], Var, RVars) :- find_max(Vs, V, Var), delete_eq([V|Vs], Var, RVars). select_var(ff, [V|Vs], Var, RVars) :- fd_size_(V, n(S)), find_ff(Vs, V, S, Var), delete_eq([V|Vs], Var, RVars). select_var(ffc, [V|Vs], Var, RVars) :- find_ffc(Vs, V, Var), delete_eq([V|Vs], Var, RVars). select_var(random_variable(_), Vars0, Var, Vars) :- length(Vars0, L), I is random(L), nth0(I, Vars0, Var), delete_eq(Vars0, Var, Vars). find_min([], Var, Var). find_min([V|Vs], CM, Min) :- ( min_lt(V, CM) -> find_min(Vs, V, Min) ; find_min(Vs, CM, Min) ). find_max([], Var, Var). find_max([V|Vs], CM, Max) :- ( max_gt(V, CM) -> find_max(Vs, V, Max) ; find_max(Vs, CM, Max) ). find_ff([], Var, _, Var). find_ff([V|Vs], CM, S0, FF) :- ( nonvar(V) -> find_ff(Vs, CM, S0, FF) ; ( fd_size_(V, n(S1)), S1 < S0 -> find_ff(Vs, V, S1, FF) ; find_ff(Vs, CM, S0, FF) ) ). find_ffc([], Var, Var). find_ffc([V|Vs], Prev, FFC) :- ( ffc_lt(V, Prev) -> find_ffc(Vs, V, FFC) ; find_ffc(Vs, Prev, FFC) ). ffc_lt(X, Y) :- ( fd_get(X, XD, XPs) -> domain_num_elements(XD, n(NXD)) ; NXD = 1, XPs = [] ), ( fd_get(Y, YD, YPs) -> domain_num_elements(YD, n(NYD)) ; NYD = 1, YPs = [] ), ( NXD < NYD -> true ; NXD =:= NYD, props_number(XPs, NXPs), props_number(YPs, NYPs), NXPs > NYPs ). min_lt(X,Y) :- bounds(X,LX,_), bounds(Y,LY,_), LX < LY. max_gt(X,Y) :- bounds(X,_,UX), bounds(Y,_,UY), UX > UY. bounds(X, L, U) :- ( fd_get(X, Dom, _) -> domain_infimum(Dom, n(L)), domain_supremum(Dom, n(U)) ; L = X, U = L ). delete_eq([], _, []). delete_eq([X|Xs], Y, List) :- ( nonvar(X) -> delete_eq(Xs, Y, List) ; X == Y -> List = Xs ; List = [X|Tail], delete_eq(Xs, Y, Tail) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - contracting/1 -- subject to change This can remove additional domain elements from the boundaries. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ contracting(Vs) :- must_be(list, Vs), maplist(finite_domain, Vs), contracting(Vs, false, Vs). contracting([], Repeat, Vars) :- ( Repeat -> contracting(Vars, false, Vars) ; true ). contracting([V|Vs], Repeat, Vars) :- fd_inf(V, Min), ( \+ \+ (V = Min) -> fd_sup(V, Max), ( \+ \+ (V = Max) -> contracting(Vs, Repeat, Vars) ; V #\= Max, contracting(Vs, true, Vars) ) ; V #\= Min, contracting(Vs, true, Vars) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - fds_sespsize(Vs, S). S is an upper bound on the search space size with respect to finite domain variables Vs. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ fds_sespsize(Vs, S) :- must_be(list, Vs), maplist(fd_variable, Vs), fds_sespsize(Vs, n(1), S1), bound_portray(S1, S). fd_size_(V, S) :- ( fd_get(V, D, _) -> domain_num_elements(D, S) ; S = n(1) ). fds_sespsize([], S, S). fds_sespsize([V|Vs], S0, S) :- fd_size_(V, S1), S2 cis S0*S1, fds_sespsize(Vs, S2, S). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Optimisation uses destructive assignment to save the computed extremum over backtracking. Failure is used to get rid of copies of attributed variables that are created in intermediate steps. At least that's the intention - it currently doesn't work in SWI: %?- X in 0..3, call_residue_vars(labeling([min(X)], [X]), Vs). %@ X = 0, %@ Vs = [_G6174, _G6177], %@ _G6174 in 0..3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ optimise(Vars, Options, Whats) :- Whats = [What|WhatsRest], Extremum = extremum(none), ( catch(store_extremum(Vars, Options, What, Extremum), time_limit_exceeded, false) ; Extremum = extremum(n(Val)), arg(1, What, Expr), append(WhatsRest, Options, Options1), ( Expr #= Val, labeling(Options1, Vars) ; Expr #\= Val, optimise(Vars, Options, Whats) ) ). store_extremum(Vars, Options, What, Extremum) :- catch((labeling(Options, Vars), throw(w(What))), w(What1), true), functor(What, Direction, _), maplist(arg(1), [What,What1], [Expr,Expr1]), optimise(Direction, Options, Vars, Expr1, Expr, Extremum). optimise(Direction, Options, Vars, Expr0, Expr, Extremum) :- must_be(ground, Expr0), nb_setarg(1, Extremum, n(Expr0)), catch((tighten(Direction, Expr, Expr0), labeling(Options, Vars), throw(v(Expr))), v(Expr1), true), optimise(Direction, Options, Vars, Expr1, Expr, Extremum). tighten(min, E, V) :- E #< V. tighten(max, E, V) :- E #> V. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% all_different(+Vars) % % Vars are pairwise distinct. all_different(Ls) :- fd_must_be_list(Ls), maplist(fd_variable, Ls), put_attr(Orig, clpfd_original, all_different(Ls)), all_different(Ls, [], Orig), do_queue. all_different([], _, _). all_different([X|Right], Left, Orig) :- ( var(X) -> make_propagator(pdifferent(Left,Right,X,Orig), Prop), init_propagator(X, Prop), trigger_prop(Prop) ; exclude_fire(Left, Right, X) ), all_different(Right, [X|Left], Orig). %% all_distinct(+Ls). % % Like all_different/1, with stronger propagation. For example, % all_distinct/1 can detect that not all variables can assume distinct % values given the following domains: % % == % ?- maplist(in, Vs, % [1\/3..4, 1..2\/4, 1..2\/4, 1..3, 1..3, 1..6]), % all_distinct(Vs). % false. % == all_distinct(Ls) :- fd_must_be_list(Ls), maplist(fd_variable, Ls), make_propagator(pdistinct(Ls), Prop), distinct_attach(Ls, Prop, []), trigger_once(Prop). %% sum(+Vars, +Rel, ?Expr) % % The sum of elements of the list Vars is in relation Rel to Expr. % Rel is one of #=, #\=, #<, #>, #=< 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. % == sum(Vs, Op, Value) :- must_be(list, Vs), same_length(Vs, Ones), maplist(=(1), Ones), scalar_product(Ones, Vs, Op, Value). %% scalar_product(+Cs, +Vs, +Rel, ?Expr) % % True iff the scalar product of Cs and Vs is in relation Rel to Expr. % Cs is a list of integers, Vs is a list of variables and integers. % Rel is #=, #\=, #<, #>, #=< or #>=. scalar_product(Cs, Vs, Op, Value) :- must_be(list(integer), Cs), must_be(list, Vs), maplist(fd_variable, Vs), ( Op = (#=), single_value(Value, Right), ground(Vs) -> foldl(coeff_int_linsum, Cs, Vs, 0, Right) ; must_be(callable, Op), ( memberchk(Op, [#=,#\=,#<,#>,#=<,#>=]) -> true ; domain_error(scalar_product_relation, Op) ), must_be(acyclic, Value), foldl(coeff_var_plusterm, Cs, Vs, 0, Left), ( left_right_linsum_const(Left, Value, Cs1, Vs1, Const) -> scalar_product_(Op, Cs1, Vs1, Const) ; sum(Cs, Vs, 0, Op, Value) ) ). single_value(V, V) :- var(V), !, non_monotonic(V). single_value(V, V) :- integer(V). single_value(?(V), V) :- fd_variable(V). coeff_var_plusterm(C, V, T0, T0+(C* ?(V))). coeff_int_linsum(C, I, S0, S) :- S is S0 + C*I. sum([], _, Sum, Op, Value) :- call(Op, Sum, Value). sum([C|Cs], [X|Xs], Acc, Op, Value) :- ?(NAcc) #= Acc + C* ?(X), sum(Cs, Xs, NAcc, Op, Value). multiples([], [], _). multiples([C|Cs], [V|Vs], Left) :- ( ( Cs = [N|_] ; Left = [N|_] ) -> ( N =\= 1, gcd(C,N) =:= 1 -> gcd(Cs, N, GCD0), gcd(Left, GCD0, GCD), ( GCD > 1 -> ?(V) #= GCD * ?(_) ; true ) ; true ) ; true ), multiples(Cs, Vs, [C|Left]). abs(N, A) :- A is abs(N). divide(D, N, R) :- R is N // D. scalar_product_(#=, Cs0, Vs, S0) :- ( Cs0 = [C|Rest] -> gcd(Rest, C, GCD), S0 mod GCD =:= 0, maplist(divide(GCD), [S0|Cs0], [S|Cs]) ; S0 =:= 0, S = S0, Cs = Cs0 ), ( S0 =:= 0 -> maplist(abs, Cs, As), multiples(As, Vs, []) ; true ), propagator_init_trigger(Vs, scalar_product_eq(Cs, Vs, S)). scalar_product_(#\=, Cs, Vs, C) :- propagator_init_trigger(Vs, scalar_product_neq(Cs, Vs, C)). scalar_product_(#=<, Cs, Vs, C) :- propagator_init_trigger(Vs, scalar_product_leq(Cs, Vs, C)). scalar_product_(#<, Cs, Vs, C) :- C1 is C - 1, scalar_product_(#=<, Cs, Vs, C1). scalar_product_(#>, Cs, Vs, C) :- C1 is C + 1, scalar_product_(#>=, Cs, Vs, C1). scalar_product_(#>=, Cs, Vs, C) :- maplist(negative, Cs, Cs1), C1 is -C, scalar_product_(#=<, Cs1, Vs, C1). negative(X0, X) :- X is -X0. coeffs_variables_const([], [], [], [], I, I). coeffs_variables_const([C|Cs], [V|Vs], Cs1, Vs1, I0, I) :- ( var(V) -> Cs1 = [C|CRest], Vs1 = [V|VRest], I1 = I0 ; I1 is I0 + C*V, Cs1 = CRest, Vs1 = VRest ), coeffs_variables_const(Cs, Vs, CRest, VRest, I1, I). sum_finite_domains([], [], [], [], Inf, Sup, Inf, Sup). sum_finite_domains([C|Cs], [V|Vs], Infs, Sups, Inf0, Sup0, Inf, Sup) :- fd_get(V, _, Inf1, Sup1, _), ( Inf1 = n(NInf) -> ( C < 0 -> Sup2 is Sup0 + C*NInf ; Inf2 is Inf0 + C*NInf ), Sups = Sups1, Infs = Infs1 ; ( C < 0 -> Sup2 = Sup0, Sups = [C*V|Sups1], Infs = Infs1 ; Inf2 = Inf0, Infs = [C*V|Infs1], Sups = Sups1 ) ), ( Sup1 = n(NSup) -> ( C < 0 -> Inf2 is Inf0 + C*NSup ; Sup2 is Sup0 + C*NSup ), Sups1 = Sups2, Infs1 = Infs2 ; ( C < 0 -> Inf2 = Inf0, Infs1 = [C*V|Infs2], Sups1 = Sups2 ; Sup2 = Sup0, Sups1 = [C*V|Sups2], Infs1 = Infs2 ) ), sum_finite_domains(Cs, Vs, Infs2, Sups2, Inf2, Sup2, Inf, Sup). remove_dist_upper_lower([], _, _, _). remove_dist_upper_lower([C|Cs], [V|Vs], D1, D2) :- ( fd_get(V, VD, VPs) -> ( C < 0 -> domain_supremum(VD, n(Sup)), L is Sup + D1//C, domain_remove_smaller_than(VD, L, VD1), domain_infimum(VD1, n(Inf)), G is Inf - D2//C, domain_remove_greater_than(VD1, G, VD2) ; domain_infimum(VD, n(Inf)), G is Inf + D1//C, domain_remove_greater_than(VD, G, VD1), domain_supremum(VD1, n(Sup)), L is Sup - D2//C, domain_remove_smaller_than(VD1, L, VD2) ), fd_put(V, VD2, VPs) ; true ), remove_dist_upper_lower(Cs, Vs, D1, D2). remove_dist_upper_leq([], _, _). remove_dist_upper_leq([C|Cs], [V|Vs], D1) :- ( fd_get(V, VD, VPs) -> ( C < 0 -> domain_supremum(VD, n(Sup)), L is Sup + D1//C, domain_remove_smaller_than(VD, L, VD1) ; domain_infimum(VD, n(Inf)), G is Inf + D1//C, domain_remove_greater_than(VD, G, VD1) ), fd_put(V, VD1, VPs) ; true ), remove_dist_upper_leq(Cs, Vs, D1). remove_dist_upper([], _). remove_dist_upper([C*V|CVs], D) :- ( fd_get(V, VD, VPs) -> ( C < 0 -> ( domain_supremum(VD, n(Sup)) -> L is Sup + D//C, domain_remove_smaller_than(VD, L, VD1) ; VD1 = VD ) ; ( domain_infimum(VD, n(Inf)) -> G is Inf + D//C, domain_remove_greater_than(VD, G, VD1) ; VD1 = VD ) ), fd_put(V, VD1, VPs) ; true ), remove_dist_upper(CVs, D). remove_dist_lower([], _). remove_dist_lower([C*V|CVs], D) :- ( fd_get(V, VD, VPs) -> ( C < 0 -> ( domain_infimum(VD, n(Inf)) -> G is Inf - D//C, domain_remove_greater_than(VD, G, VD1) ; VD1 = VD ) ; ( domain_supremum(VD, n(Sup)) -> L is Sup - D//C, domain_remove_smaller_than(VD, L, VD1) ; VD1 = VD ) ), fd_put(V, VD1, VPs) ; true ), remove_dist_lower(CVs, D). remove_upper([], _). remove_upper([C*X|CXs], Max) :- ( fd_get(X, XD, XPs) -> D is Max//C, ( C < 0 -> domain_remove_smaller_than(XD, D, XD1) ; domain_remove_greater_than(XD, D, XD1) ), fd_put(X, XD1, XPs) ; true ), remove_upper(CXs, Max). remove_lower([], _). remove_lower([C*X|CXs], Min) :- ( fd_get(X, XD, XPs) -> D is -Min//C, ( C < 0 -> domain_remove_greater_than(XD, D, XD1) ; domain_remove_smaller_than(XD, D, XD1) ), fd_put(X, XD1, XPs) ; true ), remove_lower(CXs, Min). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Constraint propagation proceeds as follows: Each CLP(FD) variable has an attribute that stores its associated domain and constraints. Constraints are triggered when the event they are registered for occurs (for example: variable is instantiated, bounds change etc.). do_queue/0 works off all triggered constraints, possibly triggering new ones, until fixpoint. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ % FIFO queue make_queue :- nb_setval('$clpfd_queue', fast_slow([], [])). push_queue(E, Which) :- nb_getval('$clpfd_queue', Qs), arg(Which, Qs, Q), ( Q == [] -> setarg(Which, Qs, [E|T]-T) ; Q = H-[E|T], setarg(Which, Qs, H-T) ). pop_queue(E) :- nb_getval('$clpfd_queue', Qs), ( pop_queue(E, Qs, 1) -> true ; pop_queue(E, Qs, 2) ). pop_queue(E, Qs, Which) :- arg(Which, Qs, [E|NH]-T), ( var(NH) -> setarg(Which, Qs, []) ; setarg(Which, Qs, NH-T) ). fetch_propagator(Prop) :- pop_queue(P), ( propagator_state(P, S), S == dead -> fetch_propagator(Prop) ; Prop = P ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Parsing a CLP(FD) expression has two important side-effects: First, it constrains the variables occurring in the expression to integers. Second, it constrains some of them even more: For example, in X/Y and X mod Y, Y is constrained to be #\= 0. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ constrain_to_integer(Var) :- ( integer(Var) -> true ; fd_get(Var, D, Ps), fd_put(Var, D, Ps) ). power_var_num(P, X, N) :- ( var(P) -> X = P, N = 1 ; P = Left*Right, power_var_num(Left, XL, L), power_var_num(Right, XR, R), XL == XR, X = XL, N is L + R ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Given expression E, we obtain the finite domain variable R by interpreting a simple committed-choice language that is a list of conditions and bodies. In conditions, g(Goal) means literally Goal, and m(Match) means that E can be decomposed as stated. The variables are to be understood as the result of parsing the subexpressions recursively. In the body, g(Goal) means again Goal, and p(Propagator) means to attach and trigger once a propagator. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- op(800, xfx, =>). parse_clpfd(E, R, [g(cyclic_term(E)) => [g(domain_error(clpfd_expression, E))], g(var(E)) => [g(non_monotonic(E)), g(constrain_to_integer(E)), g(E = R)], g(integer(E)) => [g(R = E)], ?(E) => [g(must_be_fd_integer(E)), g(R = E)], m(A+B) => [p(pplus(A, B, R))], % power_var_num/3 must occur before */2 to be useful g(power_var_num(E, V, N)) => [p(pexp(V, N, R))], m(A*B) => [p(ptimes(A, B, R))], m(A-B) => [p(pplus(R,B,A))], m(-A) => [p(ptimes(-1,A,R))], m(max(A,B)) => [g(A #=< ?(R)), g(B #=< R), p(pmax(A, B, R))], m(min(A,B)) => [g(A #>= ?(R)), g(B #>= R), p(pmin(A, B, R))], m(A mod B) => [g(B #\= 0), p(pmod(A, B, R))], m(A rem B) => [g(B #\= 0), p(prem(A, B, R))], m(abs(A)) => [g(?(R) #>= 0), p(pabs(A, R))], m(A/B) => [g(B #\= 0), p(ptzdiv(A, B, R))], m(A//B) => [g(B #\= 0), p(ptzdiv(A, B, R))], m(A div B) => [g(?(R) #= (A - (A mod B)) // B)], m(A rdiv B) => [g(B #\= 0), p(prdiv(A, B, R))], m(A^B) => [p(pexp(A, B, R))], g(true) => [g(domain_error(clpfd_expression, E))] ]). non_monotonic(X) :- ( \+ fd_var(X), current_prolog_flag(clpfd_monotonic, true) -> instantiation_error(X) ; true ). % Here, we compile the committed choice language to a single % predicate, parse_clpfd/2. make_parse_clpfd(Clauses) :- parse_clpfd_clauses(Clauses0), maplist(goals_goal, Clauses0, Clauses). goals_goal((Head :- Goals), (Head :- Body)) :- list_goal(Goals, Body). parse_clpfd_clauses(Clauses) :- parse_clpfd(E, R, Matchers), maplist(parse_matcher(E, R), Matchers, Clauses). parse_matcher(E, R, Matcher, Clause) :- Matcher = (Condition0 => Goals0), phrase((parse_condition(Condition0, E, Head), parse_goals(Goals0)), Goals), Clause = (parse_clpfd(Head, R) :- Goals). parse_condition(g(Goal), E, E) --> [Goal, !]. parse_condition(?(E), _, ?(E)) --> [!]. parse_condition(m(Match), _, Match0) --> [!], { copy_term(Match, Match0), term_variables(Match0, Vs0), term_variables(Match, Vs) }, parse_match_variables(Vs0, Vs). parse_match_variables([], []) --> []. parse_match_variables([V0|Vs0], [V|Vs]) --> [parse_clpfd(V0, V)], parse_match_variables(Vs0, Vs). parse_goals([]) --> []. parse_goals([G|Gs]) --> parse_goal(G), parse_goals(Gs). parse_goal(g(Goal)) --> [Goal]. parse_goal(p(Prop)) --> [make_propagator(Prop, P)], { term_variables(Prop, Vs) }, parse_init(Vs, P), [trigger_once(P)]. parse_init([], _) --> []. parse_init([V|Vs], P) --> [init_propagator(V, P)], parse_init(Vs, P). %?- set_prolog_flag(answer_write_options, [portray(true)]), % clpfd:parse_clpfd_clauses(Clauses), maplist(portray_clause, Clauses). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% trigger_once(Prop) :- trigger_prop(Prop), do_queue. neq(A, B) :- propagator_init_trigger(pneq(A, B)). propagator_init_trigger(P) --> { term_variables(P, Vs) }, propagator_init_trigger(Vs, P). propagator_init_trigger(Vs, P) --> [p(Prop)], { make_propagator(P, Prop), maplist(prop_init(Prop), Vs), trigger_once(Prop) }. propagator_init_trigger(P) :- phrase(propagator_init_trigger(P), _). propagator_init_trigger(Vs, P) :- phrase(propagator_init_trigger(Vs, P), _). prop_init(Prop, V) :- init_propagator(V, Prop). geq(A, B) :- ( fd_get(A, AD, APs) -> domain_infimum(AD, AI), ( fd_get(B, BD, _) -> domain_supremum(BD, BS), ( AI cis_geq BS -> true ; propagator_init_trigger(pgeq(A,B)) ) ; ( AI cis_geq n(B) -> true ; domain_remove_smaller_than(AD, B, AD1), fd_put(A, AD1, APs), do_queue ) ) ; fd_get(B, BD, BPs) -> domain_remove_greater_than(BD, A, BD1), fd_put(B, BD1, BPs), do_queue ; A >= B ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Naive parsing of inequalities and disequalities can result in a lot of unnecessary work if expressions of non-trivial depth are involved: Auxiliary variables are introduced for sub-expressions, and propagation proceeds on them as if they were involved in a tighter constraint (like equality), whereas eventually only very little of the propagated information is actually used. For example, only extremal values are of interest in inequalities. Introducing auxiliary variables should be avoided when possible, and specialised propagators should be used for common constraints. We again use a simple committed-choice language for matching special cases of constraints. m_c(M,C) means that M matches and C holds. d(X, Y) means decomposition, i.e., it is short for g(parse_clpfd(X, Y)). r(X, Y) means to rematch with X and Y. Two things are important: First, although the actual constraint functors (#\=2, #=/2 etc.) are used in the description, they must expand to the respective auxiliary predicates (match_expand/2) because the actual constraints are subject to goal expansion. Second, when specialised constraints (like scalar product) post simpler constraints on their own, these simpler versions must be handled separately and must occur before. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ match_expand(#>=, clpfd_geq_). match_expand(#=, clpfd_equal_). match_expand(#\=, clpfd_neq). symmetric(#=). symmetric(#\=). matches([ m_c(any(X) #>= any(Y), left_right_linsum_const(X, Y, Cs, Vs, Const)) => [g(( Cs = [1], Vs = [A] -> geq(A, Const) ; Cs = [-1], Vs = [A] -> Const1 is -Const, geq(Const1, A) ; Cs = [1,1], Vs = [A,B] -> ?(A) + ?(B) #= ?(S), geq(S, Const) ; Cs = [1,-1], Vs = [A,B] -> ( Const =:= 0 -> geq(A, B) ; C1 is -Const, propagator_init_trigger(x_leq_y_plus_c(B, A, C1)) ) ; Cs = [-1,1], Vs = [A,B] -> ( Const =:= 0 -> geq(B, A) ; C1 is -Const, propagator_init_trigger(x_leq_y_plus_c(A, B, C1)) ) ; Cs = [-1,-1], Vs = [A,B] -> ?(A) + ?(B) #= ?(S), Const1 is -Const, geq(Const1, S) ; scalar_product_(#>=, Cs, Vs, Const) ))], m(any(X) - any(Y) #>= integer(C)) => [d(X, X1), d(Y, Y1), g(C1 is -C), p(x_leq_y_plus_c(Y1, X1, C1))], m(integer(X) #>= any(Z) + integer(A)) => [g(C is X - A), r(C, Z)], m(abs(any(X)-any(Y)) #>= integer(I)) => [d(X, X1), d(Y, Y1), p(absdiff_geq(X1, Y1, I))], m(abs(any(X)) #>= integer(I)) => [d(X, RX), g((I>0 -> I1 is -I, RX in inf..I1 \/ I..sup; true))], m(integer(I) #>= abs(any(X))) => [d(X, RX), g(I>=0), g(I1 is -I), g(RX in I1..I)], m(any(X) #>= any(Y)) => [d(X, RX), d(Y, RY), g(geq(RX, RY))], %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m(var(X) #= var(Y)) => [g(constrain_to_integer(X)), g(X=Y)], m(var(X) #= var(Y)+var(Z)) => [p(pplus(Y,Z,X))], m(var(X) #= var(Y)-var(Z)) => [p(pplus(X,Z,Y))], m(var(X) #= var(Y)*var(Z)) => [p(ptimes(Y,Z,X))], m(var(X) #= -var(Z)) => [p(ptimes(-1, Z, X))], m_c(any(X) #= any(Y), left_right_linsum_const(X, Y, Cs, Vs, S)) => [g(scalar_product_(#=, Cs, Vs, S))], m(var(X) #= any(Y)) => [d(Y,X)], m(any(X) #= any(Y)) => [d(X, RX), d(Y, RX)], %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m(var(X) #\= integer(Y)) => [g(neq_num(X, Y))], m(var(X) #\= var(Y)) => [g(neq(X,Y))], m(var(X) #\= var(Y) + var(Z)) => [p(x_neq_y_plus_z(X, Y, Z))], m(var(X) #\= var(Y) - var(Z)) => [p(x_neq_y_plus_z(Y, X, Z))], m(var(X) #\= var(Y)*var(Z)) => [p(ptimes(Y,Z,P)), g(neq(X,P))], m(integer(X) #\= abs(any(Y)-any(Z))) => [d(Y, Y1), d(Z, Z1), p(absdiff_neq(Y1, Z1, X))], m_c(any(X) #\= any(Y), left_right_linsum_const(X, Y, Cs, Vs, S)) => [g(scalar_product_(#\=, Cs, Vs, S))], m(any(X) #\= any(Y) + any(Z)) => [d(X, X1), d(Y, Y1), d(Z, Z1), p(x_neq_y_plus_z(X1, Y1, Z1))], m(any(X) #\= any(Y) - any(Z)) => [d(X, X1), d(Y, Y1), d(Z, Z1), p(x_neq_y_plus_z(Y1, X1, Z1))], m(any(X) #\= any(Y)) => [d(X, RX), d(Y, RY), g(neq(RX, RY))] ]). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - We again compile the committed-choice matching language to the intended auxiliary predicates. We now must take care not to unintentionally unify a variable with a complex term. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ make_matches(Clauses) :- matches(Ms), findall(F, (member(M=>_, Ms), arg(1, M, M1), functor(M1, F, _)), Fs0), sort(Fs0, Fs), maplist(prevent_cyclic_argument, Fs, PrevCyclicClauses), phrase(matchers(Ms), Clauses0), maplist(goals_goal, Clauses0, MatcherClauses), append(PrevCyclicClauses, MatcherClauses, Clauses1), sort_by_predicate(Clauses1, Clauses). sort_by_predicate(Clauses, ByPred) :- map_list_to_pairs(predname, Clauses, Keyed), keysort(Keyed, KeyedByPred), pairs_values(KeyedByPred, ByPred). predname((H:-_), Key) :- !, predname(H, Key). predname(M:H, M:Key) :- !, predname(H, Key). predname(H, Name/Arity) :- !, functor(H, Name, Arity). prevent_cyclic_argument(F0, Clause) :- match_expand(F0, F), Head =.. [F,X,Y], Clause = (Head :- ( cyclic_term(X) -> domain_error(clpfd_expression, X) ; cyclic_term(Y) -> domain_error(clpfd_expression, Y) ; false )). matchers([]) --> []. matchers([Condition => Goals|Ms]) --> matcher(Condition, Goals), matchers(Ms). matcher(m(M), Gs) --> matcher(m_c(M,true), Gs). matcher(m_c(Matcher,Cond), Gs) --> [(Head :- Goals0)], { Matcher =.. [F,A,B], match_expand(F, Expand), Head =.. [Expand,X,Y], phrase((match(A, X), match(B, Y)), Goals0, [Cond,!|Goals1]), phrase(match_goals(Gs, Expand), Goals1) }, ( { symmetric(F), \+ (subsumes_term(A, B), subsumes_term(B, A)) } -> { Head1 =.. [Expand,Y,X] }, [(Head1 :- Goals0)] ; [] ). match(any(A), T) --> [A = T]. match(var(V), T) --> [( nonvar(T), T = ?(Var) -> must_be_fd_integer(Var), V = Var ; v_or_i(T), V = T )]. match(integer(I), T) --> [integer(T), I = T]. match(-X, T) --> [nonvar(T), T = -A], match(X, A). match(abs(X), T) --> [nonvar(T), T = abs(A)], match(X, A). match(X+Y, T) --> [nonvar(T), T = A + B], match(X, A), match(Y, B). match(X-Y, T) --> [nonvar(T), T = A - B], match(X, A), match(Y, B). match(X*Y, T) --> [nonvar(T), T = A * B], match(X, A), match(Y, B). match_goals([], _) --> []. match_goals([G|Gs], F) --> match_goal(G, F), match_goals(Gs, F). match_goal(r(X,Y), F) --> { G =.. [F,X,Y] }, [G]. match_goal(d(X,Y), _) --> [parse_clpfd(X, Y)]. match_goal(g(Goal), _) --> [Goal]. match_goal(p(Prop), _) --> [make_propagator(Prop, P)], { term_variables(Prop, Vs) }, parse_init(Vs, P), [trigger_once(P)]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% ?X #>= ?Y % % X is greater than or equal to Y. X #>= Y :- clpfd_geq(X, Y). clpfd_geq(X, Y) :- clpfd_geq_(X, Y), reinforce(X), reinforce(Y). %% ?X #=< ?Y % % X is less than or equal to Y. X #=< Y :- Y #>= X. %% ?X #= ?Y % % X equals Y. X #= Y :- clpfd_equal(X, Y). clpfd_equal(X, Y) :- clpfd_equal_(X, Y), reinforce(X). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Conditions under which an equality can be compiled to built-in arithmetic. Their order is significant. (/)/2 becomes (//)/2. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ expr_conds(E, E) --> [integer(E)], { var(E), !, \+ current_prolog_flag(clpfd_monotonic, true) }. expr_conds(E, E) --> { integer(E) }. expr_conds(?(E), E) --> [integer(E)]. expr_conds(-E0, -E) --> expr_conds(E0, E). expr_conds(abs(E0), abs(E)) --> expr_conds(E0, E). expr_conds(A0+B0, A+B) --> expr_conds(A0, A), expr_conds(B0, B). expr_conds(A0*B0, A*B) --> expr_conds(A0, A), expr_conds(B0, B). expr_conds(A0-B0, A-B) --> expr_conds(A0, A), expr_conds(B0, B). expr_conds(A0//B0, A//B) --> expr_conds(A0, A), expr_conds(B0, B), [B =\= 0]. expr_conds(A0/B0, AB) --> expr_conds(A0//B0, AB). expr_conds(min(A0,B0), min(A,B)) --> expr_conds(A0, A), expr_conds(B0, B). expr_conds(max(A0,B0), max(A,B)) --> expr_conds(A0, A), expr_conds(B0, B). expr_conds(A0 mod B0, A mod B) --> expr_conds(A0, A), expr_conds(B0, B), [B =\= 0]. expr_conds(A0^B0, A^B) --> expr_conds(A0, A), expr_conds(B0, B), [(B >= 0 ; A =:= -1)]. :- multifile user:goal_expansion/2. :- dynamic user:goal_expansion/2. user:goal_expansion(Var in Dom, In) :- \+ current_prolog_flag(clpfd_goal_expansion, false), ( ground(Dom), Dom = L..U, integer(L), integer(U) -> expansion_simpler( ( integer(Var) -> between(L, U, Var) ; clpfd:clpfd_in(Var, Dom) ), In) ; In = clpfd:clpfd_in(Var, Dom) ). user:goal_expansion(X0 #= Y0, Equal) :- \+ current_prolog_flag(clpfd_goal_expansion, false), phrase(expr_conds(X0, X), CsX), phrase(expr_conds(Y0, Y), CsY), list_goal(CsX, CondX), list_goal(CsY, CondY), expansion_simpler( ( CondX -> ( var(Y) -> Y is X ; CondY -> X =:= Y ; T is X, clpfd:clpfd_equal(T, Y0) ) ; CondY -> ( var(X) -> X is Y ; T is Y, clpfd:clpfd_equal(X0, T) ) ; clpfd:clpfd_equal(X0, Y0) ), Equal). user:goal_expansion(X0 #>= Y0, Geq) :- \+ current_prolog_flag(clpfd_goal_expansion, false), phrase(expr_conds(X0, X), CsX), phrase(expr_conds(Y0, Y), CsY), list_goal(CsX, CondX), list_goal(CsY, CondY), expansion_simpler( ( CondX -> ( CondY -> X >= Y ; T is X, clpfd:clpfd_geq(T, Y0) ) ; CondY -> T is Y, clpfd:clpfd_geq(X0, T) ; clpfd:clpfd_geq(X0, Y0) ), Geq). user:goal_expansion(X #=< Y, Leq) :- user:goal_expansion(Y #>= X, Leq). user:goal_expansion(X #> Y, Gt) :- user:goal_expansion(X #>= Y+1, Gt). user:goal_expansion(X #< Y, Lt) :- user:goal_expansion(Y #> X, Lt). expansion_simpler((True->Then0;_), Then) :- is_true(True), !, expansion_simpler(Then0, Then). expansion_simpler((False->_;Else0), Else) :- is_false(False), !, expansion_simpler(Else0, Else). expansion_simpler((If->Then0;Else0), (If->Then;Else)) :- !, expansion_simpler(Then0, Then), expansion_simpler(Else0, Else). expansion_simpler((Var is Expr,Goal), Goal) :- ground(Expr), !, Var is Expr. expansion_simpler((Var is Expr,Goal), (Var = Expr,Goal)) :- var(Expr), !. expansion_simpler(Var is Expr, Var = Expr) :- var(Expr), !. expansion_simpler(between(L,U,V), Goal) :- maplist(integer, [L,U,V]), !, ( between(L,U,V) -> Goal = true ; Goal = false ). expansion_simpler(Goal, Goal). is_true(true). is_true(integer(I)) :- integer(I). :- if(current_predicate(var_property/2)). is_true(var(X)) :- var(X), var_property(X, fresh(true)). is_false(integer(X)) :- var(X), var_property(X, fresh(true)). is_false((A,B)) :- is_false(A) ; is_false(B). :- endif. is_false(var(X)) :- nonvar(X). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% linsum(X, S, S) --> { var(X), !, non_monotonic(X) }, [vn(X,1)]. linsum(I, S0, S) --> { integer(I), S is S0 + I }. linsum(?(X), S, S) --> { must_be_fd_integer(X) }, [vn(X,1)]. linsum(-A, S0, S) --> mulsum(A, -1, S0, S). linsum(N*A, S0, S) --> { integer(N) }, !, mulsum(A, N, S0, S). linsum(A*N, S0, S) --> { integer(N) }, !, mulsum(A, N, S0, S). linsum(A+B, S0, S) --> linsum(A, S0, S1), linsum(B, S1, S). linsum(A-B, S0, S) --> linsum(A, S0, S1), mulsum(B, -1, S1, S). mulsum(A, M, S0, S) --> { phrase(linsum(A, 0, CA), As), S is S0 + M*CA }, lin_mul(As, M). lin_mul([], _) --> []. lin_mul([vn(X,N0)|VNs], M) --> { N is N0*M }, [vn(X,N)], lin_mul(VNs, M). v_or_i(V) :- var(V), !, non_monotonic(V). v_or_i(I) :- integer(I). must_be_fd_integer(X) :- ( var(X) -> constrain_to_integer(X) ; must_be(integer, X) ). left_right_linsum_const(Left, Right, Cs, Vs, Const) :- phrase(linsum(Left, 0, CL), Lefts0, Rights), phrase(linsum(Right, 0, CR), Rights0), maplist(linterm_negate, Rights0, Rights), msort(Lefts0, Lefts), Lefts = [vn(First,N)|LeftsRest], vns_coeffs_variables(LeftsRest, N, First, Cs0, Vs0), filter_linsum(Cs0, Vs0, Cs, Vs), Const is CR - CL. %format("linear sum: ~w ~w ~w\n", [Cs,Vs,Const]). linterm_negate(vn(V,N0), vn(V,N)) :- N is -N0. vns_coeffs_variables([], N, V, [N], [V]). vns_coeffs_variables([vn(V,N)|VNs], N0, V0, Ns, Vs) :- ( V == V0 -> N1 is N0 + N, vns_coeffs_variables(VNs, N1, V0, Ns, Vs) ; Ns = [N0|NRest], Vs = [V0|VRest], vns_coeffs_variables(VNs, N, V, NRest, VRest) ). filter_linsum([], [], [], []). filter_linsum([C0|Cs0], [V0|Vs0], Cs, Vs) :- ( C0 =:= 0 -> constrain_to_integer(V0), filter_linsum(Cs0, Vs0, Cs, Vs) ; Cs = [C0|Cs1], Vs = [V0|Vs1], filter_linsum(Cs0, Vs0, Cs1, Vs1) ). gcd([], G, G). gcd([N|Ns], G0, G) :- G1 is gcd(N, G0), gcd(Ns, G1, G). even(N) :- N mod 2 =:= 0. odd(N) :- \+ even(N). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - k-th root of N, if N is a k-th power. TODO: Replace this when the GMP function becomes available. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ integer_kth_root(N, K, R) :- ( even(K) -> N >= 0 ; true ), ( N < 0 -> odd(K), integer_kroot(N, 0, N, K, R) ; integer_kroot(0, N, N, K, R) ). integer_kroot(L, U, N, K, R) :- ( L =:= U -> N =:= L^K, R = L ; L + 1 =:= U -> ( L^K =:= N -> R = L ; U^K =:= N -> R = U ; false ) ; Mid is (L + U)//2, ( Mid^K > N -> integer_kroot(L, Mid, N, K, R) ; integer_kroot(Mid, U, N, K, R) ) ). integer_log_b(N, B, Log0, Log) :- T is B^Log0, ( T =:= N -> Log = Log0 ; T < N, Log1 is Log0 + 1, integer_log_b(N, B, Log1, Log) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Largest R such that R^K =< N. TODO: Replace this when the GMP function becomes available. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ integer_kth_root_leq(N, K, R) :- ( even(K) -> N >= 0 ; true ), ( N < 0 -> odd(K), integer_kroot_leq(N, 0, N, K, R) ; integer_kroot_leq(0, N, N, K, R) ). integer_kroot_leq(L, U, N, K, R) :- ( L =:= U -> R = L ; L + 1 =:= U -> ( U^K =< N -> R = U ; R = L ) ; Mid is (L + U)//2, ( Mid^K > N -> integer_kroot_leq(L, Mid, N, K, R) ; integer_kroot_leq(Mid, U, N, K, R) ) ). %% ?X #\= ?Y % % X is not Y. X #\= Y :- clpfd_neq(X, Y), do_queue. % X #\= Y + Z x_neq_y_plus_z(X, Y, Z) :- propagator_init_trigger(x_neq_y_plus_z(X,Y,Z)). % X is distinct from the number N. This is used internally, and does % not reinforce other constraints. neq_num(X, N) :- ( fd_get(X, XD, XPs) -> domain_remove(XD, N, XD1), fd_put(X, XD1, XPs) ; X =\= N ). %% ?X #> ?Y % % X is greater than Y. X #> Y :- X #>= Y + 1. %% #<(?X, ?Y) % % X is less than Y. In addition to its regular use in problems 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)]. % == X #< Y :- Y #> X. %% #\ +Q % % The reifiable constraint Q does _not_ hold. For example, to obtain % the complement of a domain: % % == % ?- #\ X in -3..0\/10..80. % X in inf.. -4\/1..9\/81..sup. % == #\ Q :- reify(Q, 0), do_queue. %% ?P #<==> ?Q % % P and Q are equivalent. 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: % % == % :- use_module(library(clpfd)). % % 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. % == L #<==> R :- reify(L, B), reify(R, B), do_queue. %% ?P #==> ?Q % % P implies Q. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Implication is special in that created auxiliary constraints can be retracted when the implication becomes entailed, for example: %?- X + 1 #= Y #==> Z, Z #= 1. %@ Z = 1, %@ X in inf..sup, %@ Y in inf..sup. We cannot use propagator_init_trigger/1 here because the states of auxiliary propagators are themselves part of the propagator. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ L #==> R :- reify(L, LB, LPs), reify(R, RB, RPs), append(LPs, RPs, Ps), propagator_init_trigger([LB,RB], pimpl(LB,RB,Ps)). %% ?P #<== ?Q % % Q implies P. L #<== R :- R #==> L. %% ?P #/\ ?Q % % P and Q hold. L #/\ R :- reify(L, 1), reify(R, 1), do_queue. conjunctive_neqs_var_drep(Eqs, Var, Drep) :- conjunctive_neqs_var(Eqs, Var), phrase(conjunctive_neqs_vals(Eqs), Vals), list_to_domain(Vals, Dom), domain_complement(Dom, C), domain_to_drep(C, Drep). conjunctive_neqs_var(V, _) :- var(V), !, false. conjunctive_neqs_var(L #\= R, Var) :- ( var(L), integer(R) -> Var = L ; integer(L), var(R) -> Var = R ; false ). conjunctive_neqs_var(A #/\ B, VA) :- conjunctive_neqs_var(A, VA), conjunctive_neqs_var(B, VB), VA == VB. conjunctive_neqs_vals(L #\= R) --> ( { integer(L) } -> [L] ; [R] ). conjunctive_neqs_vals(A #/\ B) --> conjunctive_neqs_vals(A), conjunctive_neqs_vals(B). %% ?P #\/ ?Q % % P or Q holds. 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. % == L #\/ R :- ( disjunctive_eqs_var_drep(L #\/ R, Var, Drep) -> Var in Drep ; reify(L, X, Ps1), reify(R, Y, Ps2), propagator_init_trigger([X,Y], reified_or(X,Ps1,Y,Ps2,1)) ). disjunctive_eqs_var_drep(Eqs, Var, Drep) :- disjunctive_eqs_var(Eqs, Var), phrase(disjunctive_eqs_vals(Eqs), Vals), list_to_drep(Vals, Drep). disjunctive_eqs_var(V, _) :- var(V), !, false. disjunctive_eqs_var(L #= R, Var) :- ( var(L), integer(R) -> Var = L ; integer(L), var(R) -> Var = R ; false ). disjunctive_eqs_var(A #\/ B, VA) :- disjunctive_eqs_var(A, VA), disjunctive_eqs_var(B, VB), VA == VB. disjunctive_eqs_vals(L #= R) --> ( { integer(L) } -> [L] ; [R] ). disjunctive_eqs_vals(A #\/ B) --> disjunctive_eqs_vals(A), disjunctive_eqs_vals(B). %% ?P #\ ?Q % % Either P holds or Q holds, but not both. L #\ R :- (L #\/ R) #/\ #\ (L #/\ R). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A constraint that is being reified need not hold. Therefore, in X/Y, Y can as well be 0, for example. Note that it is OK to constrain the *result* of an expression (which does not appear explicitly in the expression and is not visible to the outside), but not the operands, except for requiring that they be integers. In contrast to parse_clpfd/2, the result of an expression can now also be undefined, in which case the constraint cannot hold. Therefore, the committed-choice language is extended by an element d(D) that states D is 1 iff all subexpressions are defined. a(V) means that V is an auxiliary variable that was introduced while parsing a compound expression. a(X,V) means V is auxiliary unless it is ==/2 X, and a(X,Y,V) means V is auxiliary unless it is ==/2 X or Y. l(L) means the literal L occurs in the described list. When a constraint becomes entailed or subexpressions become undefined, created auxiliary constraints are killed, and the "clpfd" attribute is removed from auxiliary variables. For (/)/2, mod/2 and rem/2, we create a skeleton propagator and remember it as an auxiliary constraint. The pskeleton propagator can use the skeleton when the constraint is defined. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ parse_reified(E, R, D, [g(cyclic_term(E)) => [g(domain_error(clpfd_expression, E))], g(var(E)) => [g(non_monotonic(E)), g(constrain_to_integer(E)), g(R = E), g(D=1)], g(integer(E)) => [g(R=E), g(D=1)], ?(E) => [g(must_be_fd_integer(E)), g(R=E), g(D=1)], m(A+B) => [d(D), p(pplus(A,B,R)), a(A,B,R)], m(A*B) => [d(D), p(ptimes(A,B,R)), a(A,B,R)], m(A-B) => [d(D), p(pplus(R,B,A)), a(A,B,R)], m(-A) => [d(D), p(ptimes(-1,A,R)), a(R)], m(max(A,B)) => [d(D), p(pgeq(R, A)), p(pgeq(R, B)), p(pmax(A,B,R)), a(A,B,R)], m(min(A,B)) => [d(D), p(pgeq(A, R)), p(pgeq(B, R)), p(pmin(A,B,R)), a(A,B,R)], m(abs(A)) => [g(?(R)#>=0), d(D), p(pabs(A, R)), a(A,R)], m(A/B) => [skeleton(A,B,D,R,ptzdiv)], m(A//B) => [skeleton(A,B,D,R,ptzdiv)], m(A div B) => [skeleton(A,B,D,R,pdiv)], m(A rdiv B) => [skeleton(A,B,D,R,prdiv)], m(A mod B) => [skeleton(A,B,D,R,pmod)], m(A rem B) => [skeleton(A,B,D,R,prem)], m(A^B) => [d(D), p(pexp(A,B,R)), a(A,B,R)], g(true) => [g(domain_error(clpfd_expression, E))]] ). % Again, we compile this to a predicate, parse_reified_clpfd//3. This % time, it is a DCG that describes the list of auxiliary variables and % propagators for the given expression, in addition to relating it to % its reified (Boolean) finite domain variable and its Boolean % definedness. make_parse_reified(Clauses) :- parse_reified_clauses(Clauses0), maplist(goals_goal_dcg, Clauses0, Clauses). goals_goal_dcg((Head --> Goals), Clause) :- list_goal(Goals, Body), expand_term((Head --> Body), Clause). parse_reified_clauses(Clauses) :- parse_reified(E, R, D, Matchers), maplist(parse_reified(E, R, D), Matchers, Clauses). parse_reified(E, R, D, Matcher, Clause) :- Matcher = (Condition0 => Goals0), phrase((reified_condition(Condition0, E, Head, Ds), reified_goals(Goals0, Ds)), Goals, [a(D)]), Clause = (parse_reified_clpfd(Head, R, D) --> Goals). reified_condition(g(Goal), E, E, []) --> [{Goal}, !]. reified_condition(?(E), _, ?(E), []) --> [!]. reified_condition(m(Match), _, Match0, Ds) --> [!], { copy_term(Match, Match0), term_variables(Match0, Vs0), term_variables(Match, Vs) }, reified_variables(Vs0, Vs, Ds). reified_variables([], [], []) --> []. reified_variables([V0|Vs0], [V|Vs], [D|Ds]) --> [parse_reified_clpfd(V0, V, D)], reified_variables(Vs0, Vs, Ds). reified_goals([], _) --> []. reified_goals([G|Gs], Ds) --> reified_goal(G, Ds), reified_goals(Gs, Ds). reified_goal(d(D), Ds) --> ( { Ds = [X] } -> [{D=X}] ; { Ds = [X,Y] } -> { phrase(reified_goal(p(reified_and(X,[],Y,[],D)), _), Gs), list_goal(Gs, Goal) }, [( {X==1, Y==1} -> {D = 1} ; Goal )] ; { domain_error(one_or_two_element_list, Ds) } ). reified_goal(g(Goal), _) --> [{Goal}]. reified_goal(p(Vs, Prop), _) --> [{make_propagator(Prop, P)}], parse_init_dcg(Vs, P), [{trigger_once(P)}], [( { propagator_state(P, S), S == dead } -> [] ; [p(P)])]. reified_goal(p(Prop), Ds) --> { term_variables(Prop, Vs) }, reified_goal(p(Vs,Prop), Ds). reified_goal(skeleton(A,B,D,R,F), Ds) --> { Prop =.. [F,X,Y,Z], phrase(reified_goals([d(D1),l(p(P)),g(make_propagator(Prop, P)), p([A,B,D2,R], pskeleton(A,B,D2,[X,Y,Z]-P,R,F)), p(reified_and(D1,[],D2,[],D)),a(D2),a(A,B,R)], Ds), Goals), list_goal(Goals, Goal) }, [Goal]. reified_goal(a(V), _) --> [a(V)]. reified_goal(a(X,V), _) --> [a(X,V)]. reified_goal(a(X,Y,V), _) --> [a(X,Y,V)]. reified_goal(l(L), _) --> [[L]]. parse_init_dcg([], _) --> []. parse_init_dcg([V|Vs], P) --> [{init_propagator(V, P)}], parse_init_dcg(Vs, P). %?- set_prolog_flag(answer_write_options, [portray(true)]), % clpfd:parse_reified_clauses(Cs), maplist(portray_clause, Cs). reify(E, B) :- reify(E, B, _). reify(Expr, B, Ps) :- ( acyclic_term(Expr), reifiable(Expr) -> phrase(reify(Expr, B), Ps) ; domain_error(clpfd_reifiable_expression, Expr) ). reifiable(E) :- var(E), non_monotonic(E). reifiable(E) :- integer(E), E in 0..1. reifiable(?(E)) :- must_be_fd_integer(E). reifiable(V in _) :- fd_variable(V). reifiable(Expr) :- Expr =.. [Op,Left,Right], ( memberchk(Op, [#>=,#>,#=<,#<,#=,#\=]) ; memberchk(Op, [#==>,#<==,#<==>,#/\,#\/,#\]), reifiable(Left), reifiable(Right) ). reifiable(#\ E) :- reifiable(E). reifiable(tuples_in(Tuples, Relation)) :- must_be(list(list), Tuples), maplist(maplist(fd_variable), Tuples), must_be(list(list(integer)), Relation). reifiable(finite_domain(V)) :- fd_variable(V). reify(E, B) --> { B in 0..1 }, reify_(E, B). reify_(E, B) --> { var(E), !, E = B }. reify_(E, B) --> { integer(E), E = B }. reify_(?(B), B) --> []. reify_(V in Drep, B) --> { drep_to_domain(Drep, Dom) }, propagator_init_trigger(reified_in(V,Dom,B)), a(B). reify_(tuples_in(Tuples, Relation), B) --> { maplist(relation_tuple_b_prop(Relation), Tuples, Bs, Ps), maplist(monotonic, Bs, Bs1), fold_statement(conjunction, Bs1, And), ?(B) #<==> And }, propagator_init_trigger([B], tuples_not_in(Tuples, Relation, B)), kill_reified_tuples(Bs, Ps, Bs), list(Ps), as([B|Bs]). reify_(finite_domain(V), B) --> propagator_init_trigger(reified_fd(V,B)), a(B). reify_(L #>= R, B) --> arithmetic(L, R, B, reified_geq). reify_(L #= R, B) --> arithmetic(L, R, B, reified_eq). reify_(L #\= R, B) --> arithmetic(L, R, B, reified_neq). reify_(L #> R, B) --> reify_(L #>= (R+1), B). reify_(L #=< R, B) --> reify_(R #>= L, B). reify_(L #< R, B) --> reify_(R #>= (L+1), B). reify_(L #==> R, B) --> reify_((#\ L) #\/ R, B). reify_(L #<== R, B) --> reify_(R #==> L, B). reify_(L #<==> R, B) --> reify_((L #==> R) #/\ (R #==> L), B). reify_(L #\ R, B) --> reify_((L #\/ R) #/\ #\ (L #/\ R), B). reify_(L #/\ R, B) --> ( { conjunctive_neqs_var_drep(L #/\ R, V, D) } -> reify_(V in D, B) ; boolean(L, R, B, reified_and) ). reify_(L #\/ R, B) --> ( { disjunctive_eqs_var_drep(L #\/ R, V, D) } -> reify_(V in D, B) ; boolean(L, R, B, reified_or) ). reify_(#\ Q, B) --> reify(Q, QR), propagator_init_trigger(reified_not(QR,B)), a(B). arithmetic(L, R, B, Functor) --> { phrase((parse_reified_clpfd(L, LR, LD), parse_reified_clpfd(R, RR, RD)), Ps), Prop =.. [Functor,LD,LR,RD,RR,Ps,B] }, list(Ps), propagator_init_trigger([LD,LR,RD,RR,B], Prop), a(B). boolean(L, R, B, Functor) --> { reify(L, LR, Ps1), reify(R, RR, Ps2), Prop =.. [Functor,LR,Ps1,RR,Ps2,B] }, list(Ps1), list(Ps2), propagator_init_trigger([LR,RR,B], Prop), a(LR, RR, B). list([]) --> []. list([L|Ls]) --> [L], list(Ls). a(X,Y,B) --> ( { nonvar(X) } -> a(Y, B) ; { nonvar(Y) } -> a(X, B) ; [a(X,Y,B)] ). a(X, B) --> ( { var(X) } -> [a(X, B)] ; a(B) ). a(B) --> ( { var(B) } -> [a(B)] ; [] ). as([]) --> []. as([B|Bs]) --> a(B), as(Bs). kill_reified_tuples([], _, _) --> []. kill_reified_tuples([B|Bs], Ps, All) --> propagator_init_trigger([B], kill_reified_tuples(B, Ps, All)), kill_reified_tuples(Bs, Ps, All). relation_tuple_b_prop(Relation, Tuple, B, p(Prop)) :- put_attr(R, clpfd_relation, Relation), make_propagator(reified_tuple_in(Tuple, R, B), Prop), tuple_freeze_(Tuple, Prop), init_propagator(B, Prop). tuples_in_conjunction(Tuples, Relation, Conj) :- maplist(tuple_in_disjunction(Relation), Tuples, Disjs), fold_statement(conjunction, Disjs, Conj). tuple_in_disjunction(Relation, Tuple, Disj) :- maplist(tuple_in_conjunction(Tuple), Relation, Conjs), fold_statement(disjunction, Conjs, Disj). tuple_in_conjunction(Tuple, Element, Conj) :- maplist(var_eq, Tuple, Element, Eqs), fold_statement(conjunction, Eqs, Conj). fold_statement(Operation, List, Statement) :- ( List = [] -> Statement = 1 ; List = [First|Rest], foldl(Operation, Rest, First, Statement) ). conjunction(E, Conj, Conj #/\ E). disjunction(E, Disj, Disj #\/ E). var_eq(V, N, ?(V) #= N). % Match variables to created skeleton. skeleton(Vs, Vs-Prop) :- maplist(prop_init(Prop), Vs), trigger_once(Prop). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A drep is a user-accessible and visible domain representation. N, N..M, and D1 \/ D2 are dreps, if D1 and D2 are dreps. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ is_drep(N) :- integer(N). is_drep(N..M) :- drep_bound(N), drep_bound(M), N \== sup, M \== inf. is_drep(D1\/D2) :- is_drep(D1), is_drep(D2). is_drep({AI}) :- is_and_integers(AI). is_drep(\D) :- is_drep(D). is_and_integers(I) :- integer(I). is_and_integers((A,B)) :- is_and_integers(A), is_and_integers(B). drep_bound(I) :- integer(I). drep_bound(sup). drep_bound(inf). drep_to_intervals(I) --> { integer(I) }, [n(I)-n(I)]. drep_to_intervals(N..M) --> ( { defaulty_to_bound(N, N1), defaulty_to_bound(M, M1), N1 cis_leq M1} -> [N1-M1] ; [] ). drep_to_intervals(D1 \/ D2) --> drep_to_intervals(D1), drep_to_intervals(D2). drep_to_intervals(\D0) --> { drep_to_domain(D0, D1), domain_complement(D1, D), domain_to_drep(D, Drep) }, drep_to_intervals(Drep). drep_to_intervals({AI}) --> and_integers_(AI). and_integers_(I) --> { integer(I) }, [n(I)-n(I)]. and_integers_((A,B)) --> and_integers_(A), and_integers_(B). drep_to_domain(DR, D) :- must_be(ground, DR), ( is_drep(DR) -> true ; domain_error(clpfd_domain, DR) ), phrase(drep_to_intervals(DR), Is0), merge_intervals(Is0, Is1), intervals_to_domain(Is1, D). merge_intervals(Is0, Is) :- keysort(Is0, Is1), merge_overlapping(Is1, Is). merge_overlapping([], []). merge_overlapping([A-B0|ABs0], [A-B|ABs]) :- merge_remaining(ABs0, B0, B, Rest), merge_overlapping(Rest, ABs). merge_remaining([], B, B, []). merge_remaining([N-M|NMs], B0, B, Rest) :- Next cis B0 + n(1), ( N cis_gt Next -> B = B0, Rest = [N-M|NMs] ; B1 cis max(B0,M), merge_remaining(NMs, B1, B, Rest) ). domain(V, Dom) :- ( fd_get(V, Dom0, VPs) -> domains_intersection(Dom, Dom0, Dom1), %format("intersected\n: ~w\n ~w\n==> ~w\n\n", [Dom,Dom0,Dom1]), fd_put(V, Dom1, VPs), do_queue, reinforce(V) ; domain_contains(Dom, V) ). domains([], _). domains([V|Vs], D) :- domain(V, D), domains(Vs, D). props_number(fd_props(Gs,Bs,Os), N) :- length(Gs, N1), length(Bs, N2), length(Os, N3), N is N1 + N2 + N3. fd_get(X, Dom, Ps) :- ( get_attr(X, clpfd, Attr) -> Attr = clpfd_attr(_,_,_,Dom,Ps) ; var(X) -> default_domain(Dom), Ps = fd_props([],[],[]) ). fd_get(X, Dom, Inf, Sup, Ps) :- fd_get(X, Dom, Ps), domain_infimum(Dom, Inf), domain_supremum(Dom, Sup). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - By default, propagation always terminates. Currently, this is ensured by allowing the left and right boundaries, as well as the distance between the smallest and largest number occurring in the domain representation to be changed at most once after a constraint is posted, unless the domain is bounded. Set the experimental Prolog flag 'clpfd_propagation' to 'full' to make the solver propagate as much as possible. This can make queries non-terminating, like: X #> abs(X), or: X #> Y, Y #> X, X #> 0. Importantly, it can also make labeling non-terminating, as in: ?- B #==> X #> abs(X), indomain(B). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ fd_put(X, Dom, Ps) :- ( current_prolog_flag(clpfd_propagation, full) -> put_full(X, Dom, Ps) ; put_terminating(X, Dom, Ps) ). put_terminating(X, Dom, Ps) :- Dom \== empty, ( Dom = from_to(F, F) -> F = n(X) ; ( get_attr(X, clpfd, Attr) -> Attr = clpfd_attr(Left,Right,Spread,OldDom, _OldPs), put_attr(X, clpfd, clpfd_attr(Left,Right,Spread,Dom,Ps)), ( OldDom == Dom -> true ; ( Left == (.) -> Bounded = yes ; domain_infimum(Dom, Inf), domain_supremum(Dom, Sup), ( Inf = n(_), Sup = n(_) -> Bounded = yes ; Bounded = no ) ), ( Bounded == yes -> put_attr(X, clpfd, clpfd_attr(.,.,.,Dom,Ps)), trigger_props(Ps, X, OldDom, Dom) ; % infinite domain; consider border and spread changes domain_infimum(OldDom, OldInf), ( Inf == OldInf -> LeftP = Left ; LeftP = yes ), domain_supremum(OldDom, OldSup), ( Sup == OldSup -> RightP = Right ; RightP = yes ), domain_spread(OldDom, OldSpread), domain_spread(Dom, NewSpread), ( NewSpread == OldSpread -> SpreadP = Spread ; NewSpread cis_lt OldSpread -> SpreadP = no ; SpreadP = yes ), put_attr(X, clpfd, clpfd_attr(LeftP,RightP,SpreadP,Dom,Ps)), ( RightP == yes, Right = yes -> true ; LeftP == yes, Left = yes -> true ; SpreadP == yes, Spread = yes -> true ; trigger_props(Ps, X, OldDom, Dom) ) ) ) ; var(X) -> put_attr(X, clpfd, clpfd_attr(no,no,no,Dom, Ps)) ; true ) ). domain_spread(Dom, Spread) :- domain_smallest_finite(Dom, S), domain_largest_finite(Dom, L), Spread cis L - S. smallest_finite(inf, Y, Y). smallest_finite(n(N), _, n(N)). domain_smallest_finite(from_to(F,T), S) :- smallest_finite(F, T, S). domain_smallest_finite(split(_, L, _), S) :- domain_smallest_finite(L, S). largest_finite(sup, Y, Y). largest_finite(n(N), _, n(N)). domain_largest_finite(from_to(F,T), L) :- largest_finite(T, F, L). domain_largest_finite(split(_, _, R), L) :- domain_largest_finite(R, L). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - With terminating propagation, all relevant constraints get a propagation opportunity whenever a new constraint is posted. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ reinforce(X) :- ( current_prolog_flag(clpfd_propagation, full) -> % full propagation propagates everything in any case true ; term_variables(X, Vs), maplist(reinforce_, Vs), do_queue ). reinforce_(X) :- ( fd_var(X), fd_get(X, Dom, Ps) -> put_full(X, Dom, Ps) ; true ). put_full(X, Dom, Ps) :- Dom \== empty, ( Dom = from_to(F, F) -> F = n(X) ; ( get_attr(X, clpfd, Attr) -> Attr = clpfd_attr(_,_,_,OldDom, _OldPs), put_attr(X, clpfd, clpfd_attr(no,no,no,Dom, Ps)), %format("putting dom: ~w\n", [Dom]), ( OldDom == Dom -> true ; trigger_props(Ps, X, OldDom, Dom) ) ; var(X) -> %format('\t~w in ~w .. ~w\n',[X,L,U]), put_attr(X, clpfd, clpfd_attr(no,no,no,Dom, Ps)) ; true ) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A propagator is a term of the form propagator(C, State), where C represents a constraint, and State is a free variable that can be used to destructively change the state of the propagator via attributes. This can be used to avoid redundant invocation of the same propagator, or to disable the propagator. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ make_propagator(C, propagator(C, _)). propagator_state(propagator(_,S), S). trigger_props(fd_props(Gs,Bs,Os), X, D0, D) :- ( ground(X) -> trigger_props_(Gs), trigger_props_(Bs) ; Bs \== [] -> domain_infimum(D0, I0), domain_infimum(D, I), ( I == I0 -> domain_supremum(D0, S0), domain_supremum(D, S), ( S == S0 -> true ; trigger_props_(Bs) ) ; trigger_props_(Bs) ) ; true ), trigger_props_(Os). trigger_props(fd_props(Gs,Bs,Os), X) :- trigger_props_(Os), trigger_props_(Bs), ( ground(X) -> trigger_props_(Gs) ; true ). trigger_props(fd_props(Gs,Bs,Os)) :- trigger_props_(Gs), trigger_props_(Bs), trigger_props_(Os). trigger_props_([]). trigger_props_([P|Ps]) :- trigger_prop(P), trigger_props_(Ps). trigger_prop(Propagator) :- propagator_state(Propagator, State), ( State == dead -> true ; get_attr(State, clpfd_aux, queued) -> true ; b_getval('$clpfd_current_propagator', C), C == State -> true ; % passive % format("triggering: ~w\n", [Propagator]), put_attr(State, clpfd_aux, queued), ( arg(1, Propagator, C), functor(C, F, _), global_constraint(F) -> push_queue(Propagator, 2) ; push_queue(Propagator, 1) ) ). kill(State) :- del_attr(State, clpfd_aux), State = dead. kill(State, Ps) :- kill(State), maplist(kill_entailed, Ps). kill_entailed(p(Prop)) :- propagator_state(Prop, State), kill(State). kill_entailed(a(V)) :- del_attr(V, clpfd). kill_entailed(a(X,B)) :- ( X == B -> true ; del_attr(B, clpfd) ). kill_entailed(a(X,Y,B)) :- ( X == B -> true ; Y == B -> true ; del_attr(B, clpfd) ). no_reactivation(rel_tuple(_,_)). no_reactivation(pdistinct(_)). no_reactivation(pgcc(_,_,_)). no_reactivation(pgcc_single(_,_)). %no_reactivation(scalar_product(_,_,_,_)). activate_propagator(propagator(P,State)) :- % format("running: ~w\n", [P]), del_attr(State, clpfd_aux), ( no_reactivation(P) -> b_setval('$clpfd_current_propagator', State), run_propagator(P, State), b_setval('$clpfd_current_propagator', []) ; run_propagator(P, State) ). disable_queue :- b_setval('$clpfd_queue_status', disabled). enable_queue :- b_setval('$clpfd_queue_status', enabled). portray_propagator(propagator(P,_), F) :- functor(P, F, _). portray_queue(V, []) :- var(V), !. portray_queue([P|Ps], [F|Fs]) :- portray_propagator(P, F), portray_queue(Ps, Fs). do_queue :- % b_getval('$clpfd_queue', H-_), % portray_queue(H, Port), % format("queue: ~w\n", [Port]), ( b_getval('$clpfd_queue_status', enabled) -> ( fetch_propagator(Propagator) -> activate_propagator(Propagator), do_queue ; true ) ; true ). init_propagator(Var, Prop) :- ( fd_get(Var, Dom, Ps0) -> insert_propagator(Prop, Ps0, Ps), fd_put(Var, Dom, Ps) ; true ). constraint_wake(pneq, ground). constraint_wake(x_neq_y_plus_z, ground). constraint_wake(absdiff_neq, ground). constraint_wake(pdifferent, ground). constraint_wake(pexclude, ground). constraint_wake(scalar_product_neq, ground). constraint_wake(x_leq_y_plus_c, bounds). constraint_wake(scalar_product_eq, bounds). constraint_wake(scalar_product_leq, bounds). constraint_wake(pplus, bounds). constraint_wake(pgeq, bounds). constraint_wake(pgcc_single, bounds). constraint_wake(pgcc_check_single, bounds). global_constraint(pdistinct). global_constraint(pgcc). global_constraint(pgcc_single). global_constraint(pcircuit). %global_constraint(rel_tuple). %global_constraint(scalar_product_eq). insert_propagator(Prop, Ps0, Ps) :- Ps0 = fd_props(Gs,Bs,Os), arg(1, Prop, Constraint), functor(Constraint, F, _), ( constraint_wake(F, ground) -> Ps = fd_props([Prop|Gs], Bs, Os) ; constraint_wake(F, bounds) -> Ps = fd_props(Gs, [Prop|Bs], Os) ; Ps = fd_props(Gs, Bs, [Prop|Os]) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% lex_chain(+Lists) % % Lists are lexicographically non-decreasing. lex_chain(Lss) :- must_be(list(list), Lss), maplist(maplist(fd_variable), Lss), ( Lss == [] -> true ; Lss = [First|Rest], make_propagator(presidual(lex_chain(Lss)), Prop), foldl(lex_chain_(Prop), Rest, First, _) ). lex_chain_(Prop, Ls, Prev, Ls) :- maplist(prop_init(Prop), Ls), lex_le(Prev, Ls). lex_le([], []). lex_le([V1|V1s], [V2|V2s]) :- ?(V1) #=< ?(V2), ( integer(V1) -> ( integer(V2) -> ( V1 =:= V2 -> lex_le(V1s, V2s) ; true ) ; freeze(V2, lex_le([V1|V1s], [V2|V2s])) ) ; freeze(V1, lex_le([V1|V1s], [V2|V2s])) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% tuples_in(+Tuples, +Relation). % % True iff all Tuples are elements of Relation. Each element of the % list Tuples is a list of integers or finite domain variables. % Relation is a list of lists of integers. Arbitrary finite relations, % such as compatibility tables, can be modeled in this way. For % example, if 1 is compatible with 2 and 5, and 4 is compatible with 0 % and 3: % % == % ?- 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. % % == % :- use_module(library(clpfd)). % % 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]]. % == tuples_in(Tuples, Relation) :- must_be(list(list), Tuples), maplist(maplist(fd_variable), Tuples), must_be(list(list(integer)), Relation), maplist(relation_tuple(Relation), Tuples), do_queue. relation_tuple(Relation, Tuple) :- relation_unifiable(Relation, Tuple, Us, _, _), ( ground(Tuple) -> memberchk(Tuple, Relation) ; tuple_domain(Tuple, Us), ( Tuple = [_,_|_] -> tuple_freeze(Tuple, Us) ; true ) ). tuple_domain([], _). tuple_domain([T|Ts], Relation0) :- maplist(list_first_rest, Relation0, Firsts, Relation1), ( var(T) -> ( Firsts = [Unique] -> T = Unique ; list_to_domain(Firsts, FDom), fd_get(T, TDom, TPs), domains_intersection(TDom, FDom, TDom1), fd_put(T, TDom1, TPs) ) ; true ), tuple_domain(Ts, Relation1). tuple_freeze(Tuple, Relation) :- put_attr(R, clpfd_relation, Relation), make_propagator(rel_tuple(R, Tuple), Prop), tuple_freeze_(Tuple, Prop). tuple_freeze_([], _). tuple_freeze_([T|Ts], Prop) :- ( var(T) -> init_propagator(T, Prop), trigger_prop(Prop) ; true ), tuple_freeze_(Ts, Prop). relation_unifiable([], _, [], Changed, Changed). relation_unifiable([R|Rs], Tuple, Us, Changed0, Changed) :- ( all_in_domain(R, Tuple) -> Us = [R|Rest], relation_unifiable(Rs, Tuple, Rest, Changed0, Changed) ; relation_unifiable(Rs, Tuple, Us, true, Changed) ). all_in_domain([], []). all_in_domain([A|As], [T|Ts]) :- ( fd_get(T, Dom, _) -> domain_contains(Dom, A) ; T =:= A ), all_in_domain(As, Ts). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % trivial propagator, used only to remember pending constraints run_propagator(presidual(_), _). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pdifferent(Left,Right,X,_), MState) :- run_propagator(pexclude(Left,Right,X), MState). run_propagator(weak_distinct(Left,Right,X,_), _MState) :- ( ground(X) -> disable_queue, exclude_fire(Left, Right, X), enable_queue ; outof_reducer(Left, Right, X) %( var(X) -> kill_if_isolated(Left, Right, X, MState) %; true %) ). run_propagator(pexclude(Left,Right,X), _) :- ( ground(X) -> disable_queue, exclude_fire(Left, Right, X), enable_queue ; true ). run_propagator(pdistinct(Ls), _MState) :- distinct(Ls). run_propagator(check_distinct(Left,Right,X), _) :- \+ list_contains(Left, X), \+ list_contains(Right, X). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pelement(N, Is, V), MState) :- ( fd_get(N, NDom, _) -> ( fd_get(V, VDom, VPs) -> integers_remaining(Is, 1, NDom, empty, VDom1), domains_intersection(VDom, VDom1, VDom2), fd_put(V, VDom2, VPs) ; true ) ; kill(MState), nth1(N, Is, V) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pgcc_single(Vs, Pairs), _) :- gcc_global(Vs, Pairs). run_propagator(pgcc_check_single(Pairs), _) :- gcc_check(Pairs). run_propagator(pgcc_check(Pairs), _) :- gcc_check(Pairs). run_propagator(pgcc(Vs, _, Pairs), _) :- gcc_global(Vs, Pairs). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pcircuit(Vs), _MState) :- distinct(Vs), propagate_circuit(Vs). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pneq(A, B), MState) :- ( nonvar(A) -> ( nonvar(B) -> A =\= B, kill(MState) ; fd_get(B, BD0, BExp0), domain_remove(BD0, A, BD1), kill(MState), fd_put(B, BD1, BExp0) ) ; nonvar(B) -> run_propagator(pneq(B, A), MState) ; A \== B, fd_get(A, _, AI, AS, _), fd_get(B, _, BI, BS, _), ( AS cis_lt BI -> kill(MState) ; AI cis_gt BS -> kill(MState) ; true ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pgeq(A,B), MState) :- ( A == B -> kill(MState) ; nonvar(A) -> ( nonvar(B) -> kill(MState), A >= B ; fd_get(B, BD, BPs), domain_remove_greater_than(BD, A, BD1), kill(MState), fd_put(B, BD1, BPs) ) ; nonvar(B) -> fd_get(A, AD, APs), domain_remove_smaller_than(AD, B, AD1), kill(MState), fd_put(A, AD1, APs) ; fd_get(A, AD, AL, AU, APs), fd_get(B, _, BL, BU, _), AU cis_geq BL, ( AL cis_geq BU -> kill(MState) ; AU == BL -> kill(MState), A = B ; NAL cis max(AL,BL), domains_intersection(AD, from_to(NAL,AU), NAD), fd_put(A, NAD, APs), ( fd_get(B, BD2, BL2, BU2, BPs2) -> NBU cis min(BU2, AU), domains_intersection(BD2, from_to(BL2,NBU), NBD), fd_put(B, NBD, BPs2) ; true ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(rel_tuple(R, Tuple), MState) :- get_attr(R, clpfd_relation, Relation), ( ground(Tuple) -> kill(MState), memberchk(Tuple, Relation) ; relation_unifiable(Relation, Tuple, Us, false, Changed), Us = [_|_], ( Tuple = [First,Second], ( ground(First) ; ground(Second) ) -> kill(MState) ; true ), ( Us = [Single] -> kill(MState), Single = Tuple ; Changed -> put_attr(R, clpfd_relation, Us), disable_queue, tuple_domain(Tuple, Us), enable_queue ; true ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pserialized(S_I, D_I, S_J, D_J, _), MState) :- ( nonvar(S_I), nonvar(S_J) -> kill(MState), ( S_I + D_I =< S_J -> true ; S_J + D_J =< S_I -> true ; false ) ; serialize_lower_upper(S_I, D_I, S_J, D_J, MState), serialize_lower_upper(S_J, D_J, S_I, D_I, MState) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % abs(X-Y) #\= C run_propagator(absdiff_neq(X,Y,C), MState) :- ( C < 0 -> kill(MState) ; nonvar(X) -> kill(MState), ( nonvar(Y) -> abs(X - Y) =\= C ; V1 is X - C, neq_num(Y, V1), V2 is C + X, neq_num(Y, V2) ) ; nonvar(Y) -> kill(MState), V1 is C + Y, neq_num(X, V1), V2 is Y - C, neq_num(X, V2) ; true ). % abs(X-Y) #>= C run_propagator(absdiff_geq(X,Y,C), MState) :- ( C =< 0 -> kill(MState) ; nonvar(X) -> kill(MState), ( nonvar(Y) -> abs(X-Y) >= C ; P1 is X - C, P2 is X + C, Y in inf..P1 \/ P2..sup ) ; nonvar(Y) -> kill(MState), P1 is Y - C, P2 is Y + C, X in inf..P1 \/ P2..sup ; true ). % X #\= Y + Z run_propagator(x_neq_y_plus_z(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> ( nonvar(Z) -> kill(MState), X =\= Y + Z ; kill(MState), XY is X - Y, neq_num(Z, XY) ) ; nonvar(Z) -> kill(MState), XZ is X - Z, neq_num(Y, XZ) ; true ) ; nonvar(Y) -> ( nonvar(Z) -> kill(MState), YZ is Y + Z, neq_num(X, YZ) ; Y =:= 0 -> kill(MState), neq(X, Z) ; true ) ; Z == 0 -> kill(MState), neq(X, Y) ; true ). % X #=< Y + C run_propagator(x_leq_y_plus_c(X,Y,C), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), X =< Y + C ; kill(MState), R is X - C, fd_get(Y, YD, YPs), domain_remove_smaller_than(YD, R, YD1), fd_put(Y, YD1, YPs) ) ; nonvar(Y) -> kill(MState), R is Y + C, fd_get(X, XD, XPs), domain_remove_greater_than(XD, R, XD1), fd_put(X, XD1, XPs) ; ( X == Y -> C >= 0, kill(MState) ; fd_get(Y, YD, _), ( domain_supremum(YD, n(YSup)) -> YS1 is YSup + C, fd_get(X, XD, XPs), domain_remove_greater_than(XD, YS1, XD1), fd_put(X, XD1, XPs) ; true ), ( fd_get(X, XD2, _), domain_infimum(XD2, n(XInf)) -> XI1 is XInf - C, ( fd_get(Y, YD1, YPs1) -> domain_remove_smaller_than(YD1, XI1, YD2), ( domain_infimum(YD2, n(YInf)), domain_supremum(XD2, n(XSup)), XSup =< YInf + C -> kill(MState) ; true ), fd_put(Y, YD2, YPs1) ; true ) ; true ) ) ). run_propagator(scalar_product_neq(Cs0,Vs0,P0), MState) :- coeffs_variables_const(Cs0, Vs0, Cs, Vs, 0, I), P is P0 - I, ( Vs = [] -> kill(MState), P =\= 0 ; Vs = [V], Cs = [C] -> kill(MState), ( C =:= 1 -> neq_num(V, P) ; C*V #\= P ) ; Cs == [1,-1] -> kill(MState), Vs = [A,B], x_neq_y_plus_z(A, B, P) ; Cs == [-1,1] -> kill(MState), Vs = [A,B], x_neq_y_plus_z(B, A, P) ; P =:= 0, Cs = [1,1,-1] -> kill(MState), Vs = [A,B,C], x_neq_y_plus_z(C, A, B) ; P =:= 0, Cs = [1,-1,1] -> kill(MState), Vs = [A,B,C], x_neq_y_plus_z(B, A, C) ; P =:= 0, Cs = [-1,1,1] -> kill(MState), Vs = [A,B,C], x_neq_y_plus_z(A, B, C) ; true ). run_propagator(scalar_product_leq(Cs0,Vs0,P0), MState) :- coeffs_variables_const(Cs0, Vs0, Cs, Vs, 0, I), P is P0 - I, ( Vs = [] -> kill(MState), P >= 0 ; sum_finite_domains(Cs, Vs, Infs, Sups, 0, 0, Inf, Sup), D1 is P - Inf, disable_queue, ( Infs == [], Sups == [] -> Inf =< P, ( Sup =< P -> kill(MState) ; remove_dist_upper_leq(Cs, Vs, D1) ) ; Infs == [] -> Inf =< P, remove_dist_upper(Sups, D1) ; Sups = [_], Infs = [_] -> remove_upper(Infs, D1) ; Infs = [_] -> remove_upper(Infs, D1) ; true ), enable_queue ). run_propagator(scalar_product_eq(Cs0,Vs0,P0), MState) :- coeffs_variables_const(Cs0, Vs0, Cs, Vs, 0, I), P is P0 - I, ( Vs = [] -> kill(MState), P =:= 0 ; Vs = [V], Cs = [C] -> kill(MState), P mod C =:= 0, V is P // C ; Cs == [1,1] -> kill(MState), Vs = [A,B], A + B #= P ; Cs == [1,-1] -> kill(MState), Vs = [A,B], A #= P + B ; Cs == [-1,1] -> kill(MState), Vs = [A,B], B #= P + A ; Cs == [-1,-1] -> kill(MState), Vs = [A,B], P1 is -P, A + B #= P1 ; P =:= 0, Cs == [1,1,-1] -> kill(MState), Vs = [A,B,C], A + B #= C ; P =:= 0, Cs == [1,-1,1] -> kill(MState), Vs = [A,B,C], A + C #= B ; P =:= 0, Cs == [-1,1,1] -> kill(MState), Vs = [A,B,C], B + C #= A ; sum_finite_domains(Cs, Vs, Infs, Sups, 0, 0, Inf, Sup), % nl, writeln(Infs-Sups-Inf-Sup), D1 is P - Inf, D2 is Sup - P, disable_queue, ( Infs == [], Sups == [] -> between(Inf, Sup, P), remove_dist_upper_lower(Cs, Vs, D1, D2) ; Sups = [] -> P =< Sup, remove_dist_lower(Infs, D2) ; Infs = [] -> Inf =< P, remove_dist_upper(Sups, D1) ; Sups = [_], Infs = [_] -> remove_lower(Sups, D2), remove_upper(Infs, D1) ; Infs = [_] -> remove_upper(Infs, D1) ; Sups = [_] -> remove_lower(Sups, D2) ; true ), enable_queue ). % X + Y = Z run_propagator(pplus(X,Y,Z), MState) :- ( nonvar(X) -> ( X =:= 0 -> kill(MState), Y = Z ; Y == Z -> kill(MState), X =:= 0 ; nonvar(Y) -> kill(MState), Z is X + Y ; nonvar(Z) -> kill(MState), Y is Z - X ; fd_get(Z, ZD, ZPs), fd_get(Y, YD, _), domain_shift(YD, X, Shifted_YD), domains_intersection(ZD, Shifted_YD, ZD1), fd_put(Z, ZD1, ZPs), ( fd_get(Y, YD1, YPs) -> O is -X, domain_shift(ZD1, O, YD2), domains_intersection(YD1, YD2, YD3), fd_put(Y, YD3, YPs) ; true ) ) ; nonvar(Y) -> run_propagator(pplus(Y,X,Z), MState) ; nonvar(Z) -> ( X == Y -> kill(MState), even(Z), X is Z // 2 ; fd_get(X, XD, _), fd_get(Y, YD, YPs), domain_negate(XD, XDN), domain_shift(XDN, Z, YD1), domains_intersection(YD, YD1, YD2), fd_put(Y, YD2, YPs), ( fd_get(X, XD1, XPs) -> domain_negate(YD2, YD2N), domain_shift(YD2N, Z, XD2), domains_intersection(XD1, XD2, XD3), fd_put(X, XD3, XPs) ; true ) ) ; ( X == Y -> kill(MState), 2*X #= Z ; X == Z -> kill(MState), Y = 0 ; Y == Z -> kill(MState), X = 0 ; fd_get(X, XD, XL, XU, XPs), fd_get(Y, _, YL, YU, _), fd_get(Z, _, ZL, ZU, _), NXL cis max(XL, ZL-YU), NXU cis min(XU, ZU-YL), update_bounds(X, XD, XPs, XL, XU, NXL, NXU), ( fd_get(Y, YD2, YL2, YU2, YPs2) -> NYL cis max(YL2, ZL-NXU), NYU cis min(YU2, ZU-NXL), update_bounds(Y, YD2, YPs2, YL2, YU2, NYL, NYU) ; NYL = n(Y), NYU = n(Y) ), ( fd_get(Z, ZD2, ZL2, ZU2, ZPs2) -> NZL cis max(ZL2,NXL+NYL), NZU cis min(ZU2,NXU+NYU), update_bounds(Z, ZD2, ZPs2, ZL2, ZU2, NZL, NZU) ; true ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(ptimes(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), Z is X * Y ; X =:= 0 -> kill(MState), Z = 0 ; X =:= 1 -> kill(MState), Z = Y ; nonvar(Z) -> kill(MState), 0 =:= Z mod X, Y is Z // X ; ( Y == Z -> kill(MState), Y = 0 ; fd_get(Y, YD, _), fd_get(Z, ZD, ZPs), domain_expand(YD, X, Scaled_YD), domains_intersection(ZD, Scaled_YD, ZD1), fd_put(Z, ZD1, ZPs), ( fd_get(Y, YDom2, YPs2) -> domain_contract(ZD1, X, Contract), domains_intersection(YDom2, Contract, NYDom), fd_put(Y, NYDom, YPs2) ; kill(MState), Z is X * Y ) ) ) ; nonvar(Y) -> run_propagator(ptimes(Y,X,Z), MState) ; nonvar(Z) -> ( X == Y -> kill(MState), integer_kth_root(Z, 2, R), NR is -R, X in NR \/ R ; fd_get(X, XD, XL, XU, XPs), fd_get(Y, YD, YL, YU, _), min_max_factor(n(Z), n(Z), YL, YU, XL, XU, NXL, NXU), update_bounds(X, XD, XPs, XL, XU, NXL, NXU), ( fd_get(Y, YD2, YL2, YU2, YPs2) -> min_max_factor(n(Z), n(Z), NXL, NXU, YL2, YU2, NYL, NYU), update_bounds(Y, YD2, YPs2, YL2, YU2, NYL, NYU) ; ( Y =\= 0 -> 0 =:= Z mod Y, kill(MState), X is Z // Y ; kill(MState), Z = 0 ) ), ( Z =:= 0 -> ( \+ domain_contains(XD, 0) -> kill(MState), Y = 0 ; \+ domain_contains(YD, 0) -> kill(MState), X = 0 ; true ) ; neq_num(X, 0), neq_num(Y, 0) ) ) ; ( X == Y -> kill(MState), X^2 #= Z ; fd_get(X, XD, XL, XU, XPs), fd_get(Y, _, YL, YU, _), fd_get(Z, ZD, ZL, ZU, _), ( Y == Z, \+ domain_contains(ZD, 0) -> kill(MState), X = 1 ; X == Z, \+ domain_contains(ZD, 0) -> kill(MState), Y = 1 ; min_max_factor(ZL, ZU, YL, YU, XL, XU, NXL, NXU), update_bounds(X, XD, XPs, XL, XU, NXL, NXU), ( fd_get(Y, YD2, YL2, YU2, YPs2) -> min_max_factor(ZL, ZU, NXL, NXU, YL2, YU2, NYL, NYU), update_bounds(Y, YD2, YPs2, YL2, YU2, NYL, NYU) ; NYL = n(Y), NYU = n(Y) ), ( fd_get(Z, ZD2, ZL2, ZU2, ZPs2) -> min_product(NXL, NXU, NYL, NYU, NZL), max_product(NXL, NXU, NYL, NYU, NZU), ( NZL cis_leq ZL2, NZU cis_geq ZU2 -> ZD3 = ZD2 ; domains_intersection(ZD2, from_to(NZL,NZU), ZD3), fd_put(Z, ZD3, ZPs2) ), ( domain_contains(ZD3, 0) -> true ; neq_num(X, 0), neq_num(Y, 0) ) ; true ) ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % X div Y = Z run_propagator(pdiv(X,Y,Z), MState) :- kill(MState), Z #= (X-(X mod Y)) // Y. % X rdiv Y = Z run_propagator(prdiv(X,Y,Z), MState) :- kill(MState), Z*Y #= X. % X // Y = Z (round towards zero) run_propagator(ptzdiv(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), Y =\= 0, Z is X // Y ; fd_get(Y, YD, YL, YU, YPs), ( nonvar(Z) -> ( Z =:= 0 -> NYL is -abs(X) - 1, NYU is abs(X) + 1, domains_intersection(YD, split(0, from_to(inf,n(NYL)), from_to(n(NYU), sup)), NYD), fd_put(Y, NYD, YPs) ; ( sign(X) =:= sign(Z) -> NYL cis max(n(X) // (n(Z)+sign(n(Z))) + n(1), YL), NYU cis min(n(X) // n(Z), YU) ; NYL cis max(n(X) // n(Z), YL), NYU cis min(n(X) // (n(Z)+sign(n(Z))) - n(1), YU) ), update_bounds(Y, YD, YPs, YL, YU, NYL, NYU) ) ; fd_get(Z, ZD, ZL, ZU, ZPs), ( X >= 0, ( YL cis_gt n(0) ; YU cis_lt n(0) )-> NZL cis max(n(X)//YU, ZL), NZU cis min(n(X)//YL, ZU) ; X < 0, ( YL cis_gt n(0) ; YU cis_lt n(0) ) -> NZL cis max(n(X)//YL, ZL), NZU cis min(n(X)//YU, ZU) ; % TODO: more stringent bounds, cover Y NZL cis max(-abs(n(X)), ZL), NZU cis min(abs(n(X)), ZU) ), update_bounds(Z, ZD, ZPs, ZL, ZU, NZL, NZU), ( X >= 0, NZL cis_gt n(0), fd_get(Y, YD1, YPs1) -> NYL cis n(X) // (NZU + n(1)) + n(1), NYU cis n(X) // NZL, domains_intersection(YD1, from_to(NYL, NYU), NYD1), fd_put(Y, NYD1, YPs1) ; true ) ) ) ; nonvar(Y) -> Y =\= 0, ( Y =:= 1 -> kill(MState), X = Z ; Y =:= -1 -> kill(MState), Z #= -X ; fd_get(X, XD, XL, XU, XPs), ( nonvar(Z) -> kill(MState), ( sign(Z) =:= sign(Y) -> NXL cis max(n(Z)*n(Y), XL), NXU cis min((abs(n(Z))+n(1))*abs(n(Y))-n(1), XU) ; Z =:= 0 -> NXL cis max(-abs(n(Y)) + n(1), XL), NXU cis min(abs(n(Y)) - n(1), XU) ; NXL cis max((n(Z)+sign(n(Z)))*n(Y)+n(1), XL), NXU cis min(n(Z)*n(Y), XU) ), update_bounds(X, XD, XPs, XL, XU, NXL, NXU) ; fd_get(Z, ZD, ZPs), domain_contract_less(XD, Y, Contracted), domains_intersection(ZD, Contracted, NZD), fd_put(Z, NZD, ZPs), ( fd_get(X, XD2, XPs2) -> domain_expand_more(NZD, Y, Expanded), domains_intersection(XD2, Expanded, NXD2), fd_put(X, NXD2, XPs2) ; true ) ) ) ; nonvar(Z) -> fd_get(X, XD, XL, XU, XPs), fd_get(Y, _, YL, YU, _), ( YL cis_geq n(0), XL cis_geq n(0) -> NXL cis max(YL*n(Z), XL), NXU cis min(YU*(n(Z)+n(1))-n(1), XU) ; %TODO: cover more cases NXL = XL, NXU = XU ), update_bounds(X, XD, XPs, XL, XU, NXL, NXU) ; ( X == Y -> kill(MState), Z = 1 ; fd_get(X, _, XL, XU, _), fd_get(Y, _, YL, _, _), fd_get(Z, ZD, ZPs), NZU cis max(abs(XL), XU), NZL cis -NZU, domains_intersection(ZD, from_to(NZL,NZU), NZD0), ( XL cis_geq n(0), YL cis_geq n(0) -> domain_remove_smaller_than(NZD0, 0, NZD1) ; % TODO: cover more cases NZD1 = NZD0 ), fd_put(Z, NZD1, ZPs) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Y = abs(X) run_propagator(pabs(X,Y), MState) :- ( nonvar(X) -> kill(MState), Y is abs(X) ; nonvar(Y) -> kill(MState), Y >= 0, YN is -Y, X in YN \/ Y ; fd_get(X, XD, XPs), fd_get(Y, YD, _), domain_negate(YD, YDNegative), domains_union(YD, YDNegative, XD1), domains_intersection(XD, XD1, XD2), fd_put(X, XD2, XPs), ( fd_get(Y, YD1, YPs1) -> domain_negate(XD2, XD2Neg), domains_union(XD2, XD2Neg, YD2), domain_remove_smaller_than(YD2, 0, YD3), domains_intersection(YD1, YD3, YD4), fd_put(Y, YD4, YPs1) ; true ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Z = X mod Y run_propagator(pmod(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), Y =\= 0, Z is X mod Y ; true ) ; nonvar(Y) -> Y =\= 0, ( abs(Y) =:= 1 -> kill(MState), Z = 0 ; var(Z) -> YP is abs(Y) - 1, ( Y > 0, fd_get(X, _, n(XL), n(XU), _) -> ( XL >= 0, XU < Y -> kill(MState), Z = X, ZL = XL, ZU = XU ; ZL = 0, ZU = YP ) ; Y > 0 -> ZL = 0, ZU = YP ; YN is -YP, ZL = YN, ZU = 0 ), ( fd_get(Z, ZD, ZPs) -> domains_intersection(ZD, from_to(n(ZL), n(ZU)), ZD1), domain_infimum(ZD1, n(ZMin)), domain_supremum(ZD1, n(ZMax)), fd_put(Z, ZD1, ZPs) ; ZMin = Z, ZMax = Z ), ( fd_get(X, XD, XPs), domain_infimum(XD, n(XMin)) -> Z1 is XMin mod Y, ( between(ZMin, ZMax, Z1) -> true ; Y > 0 -> Next is ((XMin - ZMin + Y - 1) div Y)*Y + ZMin, domain_remove_smaller_than(XD, Next, XD1), fd_put(X, XD1, XPs) ; neq_num(X, XMin) ) ; true ), ( fd_get(X, XD2, XPs2), domain_supremum(XD2, n(XMax)) -> Z2 is XMax mod Y, ( between(ZMin, ZMax, Z2) -> true ; Y > 0 -> Prev is ((XMax - ZMin) div Y)*Y + ZMax, domain_remove_greater_than(XD2, Prev, XD3), fd_put(X, XD3, XPs2) ; neq_num(X, XMax) ) ; true ) ; fd_get(X, XD, XPs), % if possible, propagate at the boundaries ( domain_infimum(XD, n(Min)) -> ( Min mod Y =:= Z -> true ; Y > 0 -> Next is ((Min - Z + Y - 1) div Y)*Y + Z, domain_remove_smaller_than(XD, Next, XD1), fd_put(X, XD1, XPs) ; neq_num(X, Min) ) ; true ), ( fd_get(X, XD2, XPs2) -> ( domain_supremum(XD2, n(Max)) -> ( Max mod Y =:= Z -> true ; Y > 0 -> Prev is ((Max - Z) div Y)*Y + Z, domain_remove_greater_than(XD2, Prev, XD3), fd_put(X, XD3, XPs2) ; neq_num(X, Max) ) ; true ) ; true ) ) ; X == Y -> kill(MState), Z = 0 ; true % TODO: propagate more ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Z = X rem Y run_propagator(prem(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), Y =\= 0, Z is X rem Y ; U is abs(X), fd_get(Y, YD, _), ( X >=0, domain_infimum(YD, n(Min)), Min >= 0 -> L = 0 ; L is -U ), Z in L..U ) ; nonvar(Y) -> Y =\= 0, ( abs(Y) =:= 1 -> kill(MState), Z = 0 ; var(Z) -> YP is abs(Y) - 1, YN is -YP, ( Y > 0, fd_get(X, _, n(XL), n(XU), _) -> ( abs(XL) < Y, XU < Y -> kill(MState), Z = X, ZL = XL ; XL < 0, abs(XL) < Y -> ZL = XL ; XL >= 0 -> ZL = 0 ; ZL = YN ), ( XU > 0, XU < Y -> ZU = XU ; XU < 0 -> ZU = 0 ; ZU = YP ) ; ZL = YN, ZU = YP ), ( fd_get(Z, ZD, ZPs) -> domains_intersection(ZD, from_to(n(ZL), n(ZU)), ZD1), fd_put(Z, ZD1, ZPs) ; ZD1 = from_to(n(Z), n(Z)) ), ( fd_get(X, XD, _), domain_infimum(XD, n(Min)) -> Z1 is Min rem Y, ( domain_contains(ZD1, Z1) -> true ; neq_num(X, Min) ) ; true ), ( fd_get(X, XD1, _), domain_supremum(XD1, n(Max)) -> Z2 is Max rem Y, ( domain_contains(ZD1, Z2) -> true ; neq_num(X, Max) ) ; true ) ; fd_get(X, XD1, XPs1), % if possible, propagate at the boundaries ( domain_infimum(XD1, n(Min)) -> ( Min rem Y =:= Z -> true ; Y > 0, Min > 0 -> Next is ((Min - Z + Y - 1) div Y)*Y + Z, domain_remove_smaller_than(XD1, Next, XD2), fd_put(X, XD2, XPs1) ; % TODO: bigger steps in other cases as well neq_num(X, Min) ) ; true ), ( fd_get(X, XD3, XPs3) -> ( domain_supremum(XD3, n(Max)) -> ( Max rem Y =:= Z -> true ; Y > 0, Max > 0 -> Prev is ((Max - Z) div Y)*Y + Z, domain_remove_greater_than(XD3, Prev, XD4), fd_put(X, XD4, XPs3) ; % TODO: bigger steps in other cases as well neq_num(X, Max) ) ; true ) ; true ) ) ; X == Y -> kill(MState), Z = 0 ; fd_get(Z, ZD, ZPs) -> fd_get(Y, _, YInf, YSup, _), fd_get(X, _, XInf, XSup, _), M cis max(abs(YInf),YSup), ( XInf cis_geq n(0) -> Inf0 = n(0) ; Inf0 = XInf ), ( XSup cis_leq n(0) -> Sup0 = n(0) ; Sup0 = XSup ), NInf cis max(max(Inf0, -M + n(1)), min(XInf,-XSup)), NSup cis min(min(Sup0, M - n(1)), max(abs(XInf),XSup)), domains_intersection(ZD, from_to(NInf,NSup), ZD1), fd_put(Z, ZD1, ZPs) ; true % TODO: propagate more ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Z = max(X,Y) run_propagator(pmax(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), Z is max(X,Y) ; nonvar(Z) -> ( Z =:= X -> kill(MState), X #>= Y ; Z > X -> Z = Y ; false % Z < X ) ; fd_get(Y, _, YInf, YSup, _), ( YInf cis_gt n(X) -> Z = Y ; YSup cis_lt n(X) -> Z = X ; YSup = n(M) -> fd_get(Z, ZD, ZPs), domain_remove_greater_than(ZD, M, ZD1), fd_put(Z, ZD1, ZPs) ; true ) ) ; nonvar(Y) -> run_propagator(pmax(Y,X,Z), MState) ; fd_get(Z, ZD, ZPs) -> fd_get(X, _, XInf, XSup, _), fd_get(Y, _, YInf, YSup, _), ( YInf cis_gt YSup -> kill(MState), Z = Y ; YSup cis_lt XInf -> kill(MState), Z = X ; n(M) cis max(XSup, YSup) -> domain_remove_greater_than(ZD, M, ZD1), fd_put(Z, ZD1, ZPs) ; true ) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Z = min(X,Y) run_propagator(pmin(X,Y,Z), MState) :- ( nonvar(X) -> ( nonvar(Y) -> kill(MState), Z is min(X,Y) ; nonvar(Z) -> ( Z =:= X -> kill(MState), X #=< Y ; Z < X -> Z = Y ; false % Z > X ) ; fd_get(Y, _, YInf, YSup, _), ( YSup cis_lt n(X) -> Z = Y ; YInf cis_gt n(X) -> Z = X ; YInf = n(M) -> fd_get(Z, ZD, ZPs), domain_remove_smaller_than(ZD, M, ZD1), fd_put(Z, ZD1, ZPs) ; true ) ) ; nonvar(Y) -> run_propagator(pmin(Y,X,Z), MState) ; fd_get(Z, ZD, ZPs) -> fd_get(X, _, XInf, XSup, _), fd_get(Y, _, YInf, YSup, _), ( YSup cis_lt YInf -> kill(MState), Z = Y ; YInf cis_gt XSup -> kill(MState), Z = X ; n(M) cis min(XInf, YInf) -> domain_remove_smaller_than(ZD, M, ZD1), fd_put(Z, ZD1, ZPs) ; true ) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Z = X ^ Y run_propagator(pexp(X,Y,Z), MState) :- ( X == 1 -> kill(MState), Z = 1 ; X == 0 -> kill(MState), Z in 0..1, Z #<==> Y #= 0 ; Y == 0 -> kill(MState), Z = 1 ; Y == 1 -> kill(MState), Z = X ; nonvar(X) -> ( nonvar(Y) -> ( Y >= 0 -> true ; X =:= -1 ), kill(MState), Z is X^Y ; nonvar(Z) -> ( Z > 1 -> kill(MState), integer_log_b(Z, X, 1, Y) ; true ) ; fd_get(Y, _, YL, YU, _), fd_get(Z, ZD, ZPs), ( X > 0, YL cis_geq n(0) -> NZL cis n(X)^YL, NZU cis n(X)^YU, domains_intersection(ZD, from_to(NZL,NZU), NZD), fd_put(Z, NZD, ZPs) ; true ) ) ; nonvar(Z) -> ( nonvar(Y) -> integer_kth_root(Z, Y, R), kill(MState), ( even(Y) -> N is -R, X in N \/ R ; X = R ) ; fd_get(X, _, n(NXL), _, _), NXL > 1 -> ( Z > 1, between(NXL, Z, Exp), NXL^Exp > Z -> Exp1 is Exp - 1, fd_get(Y, YD, YPs), domains_intersection(YD, from_to(n(1),n(Exp1)), YD1), fd_put(Y, YD1, YPs), ( fd_get(X, XD, XPs) -> domain_infimum(YD1, n(YL)), integer_kth_root_leq(Z, YL, RU), domains_intersection(XD, from_to(n(NXL),n(RU)), XD1), fd_put(X, XD1, XPs) ; true ) ; true ) ; true ) ; nonvar(Y), Y > 0 -> ( even(Y) -> geq(Z, 0) ; true ), ( fd_get(X, XD, XL, XU, _), fd_get(Z, ZD, ZL, ZU, ZPs) -> ( domain_contains(ZD, 0) -> XD1 = XD ; domain_remove(XD, 0, XD1) ), ( domain_contains(XD, 0) -> ZD1 = ZD ; domain_remove(ZD, 0, ZD1) ), ( even(Y) -> ( XL cis_geq n(0) -> NZL cis XL^n(Y) ; XU cis_leq n(0) -> NZL cis XU^n(Y) ; NZL = n(0) ), NZU cis max(abs(XL),abs(XU))^n(Y), domains_intersection(ZD1, from_to(NZL,NZU), ZD2) ; ( finite(XL) -> NZL cis XL^n(Y), NZU cis XU^n(Y), domains_intersection(ZD1, from_to(NZL,NZU), ZD2) ; ZD2 = ZD1 ) ), fd_put(Z, ZD2, ZPs), ( even(Y), ZU = n(Num) -> integer_kth_root_leq(Num, Y, RU), ( XL cis_geq n(0), ZL = n(Num1) -> integer_kth_root_leq(Num1, Y, RL0), ( RL0^Y < Num1 -> RL is RL0 + 1 ; RL = RL0 ) ; RL is -RU ), RL =< RU, NXD = from_to(n(RL),n(RU)) ; odd(Y), ZL cis_geq n(0), ZU = n(Num) -> integer_kth_root_leq(Num, Y, RU), ZL = n(Num1), integer_kth_root_leq(Num1, Y, RL0), ( RL0^Y < Num1 -> RL is RL0 + 1 ; RL = RL0 ), RL =< RU, NXD = from_to(n(RL),n(RU)) ; NXD = XD1 % TODO: propagate more ), ( fd_get(X, XD2, XPs) -> domains_intersection(XD2, XD1, XD3), domains_intersection(XD3, NXD, XD4), fd_put(X, XD4, XPs) ; true ) ; true ) ; fd_get(X, _, XL, _, _), XL cis_gt n(0), fd_get(Y, _, YL, _, _), YL cis_gt n(0), fd_get(Z, ZD, ZPs) -> n(NZL) cis XL^YL, domain_remove_smaller_than(ZD, NZL, ZD1), fd_put(Z, ZD1, ZPs) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pzcompare(Order, A, B), MState) :- ( A == B -> kill(MState), Order = (=) ; ( nonvar(A) -> ( nonvar(B) -> kill(MState), ( A > B -> Order = (>) ; Order = (<) ) ; fd_get(B, _, BL, BU, _), ( BL cis_gt n(A) -> kill(MState), Order = (<) ; BU cis_lt n(A) -> kill(MState), Order = (>) ; true ) ) ; nonvar(B) -> fd_get(A, _, AL, AU, _), ( AL cis_gt n(B) -> kill(MState), Order = (>) ; AU cis_lt n(B) -> kill(MState), Order = (<) ; true ) ; fd_get(A, _, AL, AU, _), fd_get(B, _, BL, BU, _), ( AL cis_gt BU -> kill(MState), Order = (>) ; AU cis_lt BL -> kill(MState), Order = (<) ; true ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % reified constraints %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_in(V,Dom,B), MState) :- ( integer(V) -> kill(MState), ( domain_contains(Dom, V) -> B = 1 ; B = 0 ) ; B == 1 -> kill(MState), domain(V, Dom) ; B == 0 -> kill(MState), domain_complement(Dom, C), domain(V, C) ; fd_get(V, VD, _), ( domains_intersection(VD, Dom, I) -> ( I == VD -> kill(MState), B = 1 ; true ) ; kill(MState), B = 0 ) ). run_propagator(reified_tuple_in(Tuple, R, B), MState) :- get_attr(R, clpfd_relation, Relation), ( B == 1 -> kill(MState), tuples_in([Tuple], Relation) ; ( ground(Tuple) -> kill(MState), ( memberchk(Tuple, Relation) -> B = 1 ; B = 0 ) ; relation_unifiable(Relation, Tuple, Us, _, _), ( Us = [] -> kill(MState), B = 0 ; true ) ) ). run_propagator(tuples_not_in(Tuples, Relation, B), MState) :- ( B == 0 -> kill(MState), tuples_in_conjunction(Tuples, Relation, Conj), #\ Conj ; true ). run_propagator(kill_reified_tuples(B, Ps, Bs), _) :- ( B == 0 -> maplist(kill_entailed, Ps), phrase(as(Bs), As), maplist(kill_entailed, As) ; true ). run_propagator(reified_fd(V,B), MState) :- ( fd_inf(V, I), I \== inf, fd_sup(V, S), S \== sup -> kill(MState), B = 1 ; B == 0 -> ( fd_inf(V, inf) -> true ; fd_sup(V, sup) -> true ; false ) ; true ). % The result of X/Y, X mod Y, and X rem Y is undefined iff Y is 0. run_propagator(pskeleton(X,Y,D,Skel,Z,_), MState) :- ( Y == 0 -> kill(MState), D = 0 ; D == 1 -> kill(MState), neq_num(Y, 0), skeleton([X,Y,Z], Skel) ; integer(Y), Y =\= 0 -> kill(MState), D = 1, skeleton([X,Y,Z], Skel) ; fd_get(Y, YD, _), \+ domain_contains(YD, 0) -> kill(MState), D = 1, skeleton([X,Y,Z], Skel) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_geq(DX,X,DY,Y,Ps,B), MState) :- ( DX == 0 -> kill(MState, Ps), B = 0 ; DY == 0 -> kill(MState, Ps), B = 0 ; B == 1 -> kill(MState), DX = 1, DY = 1, geq(X, Y) ; DX == 1, DY == 1 -> ( var(B) -> ( nonvar(X) -> ( nonvar(Y) -> kill(MState), ( X >= Y -> B = 1 ; B = 0 ) ; fd_get(Y, _, YL, YU, _), ( n(X) cis_geq YU -> kill(MState, Ps), B = 1 ; n(X) cis_lt YL -> kill(MState, Ps), B = 0 ; true ) ) ; nonvar(Y) -> fd_get(X, _, XL, XU, _), ( XL cis_geq n(Y) -> kill(MState, Ps), B = 1 ; XU cis_lt n(Y) -> kill(MState, Ps), B = 0 ; true ) ; X == Y -> kill(MState, Ps), B = 1 ; fd_get(X, _, XL, XU, _), fd_get(Y, _, YL, YU, _), ( XL cis_geq YU -> kill(MState, Ps), B = 1 ; XU cis_lt YL -> kill(MState, Ps), B = 0 ; true ) ) ; B =:= 0 -> kill(MState), X #< Y ; true ) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_eq(DX,X,DY,Y,Ps,B), MState) :- ( DX == 0 -> kill(MState, Ps), B = 0 ; DY == 0 -> kill(MState, Ps), B = 0 ; B == 1 -> kill(MState), DX = 1, DY = 1, X = Y ; DX == 1, DY == 1 -> ( var(B) -> ( nonvar(X) -> ( nonvar(Y) -> kill(MState), ( X =:= Y -> B = 1 ; B = 0) ; fd_get(Y, YD, _), ( domain_contains(YD, X) -> true ; kill(MState, Ps), B = 0 ) ) ; nonvar(Y) -> run_propagator(reified_eq(DY,Y,DX,X,Ps,B), MState) ; X == Y -> kill(MState), B = 1 ; fd_get(X, _, XL, XU, _), fd_get(Y, _, YL, YU, _), ( XL cis_gt YU -> kill(MState, Ps), B = 0 ; YL cis_gt XU -> kill(MState, Ps), B = 0 ; true ) ) ; B =:= 0 -> kill(MState), X #\= Y ; true ) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_neq(DX,X,DY,Y,Ps,B), MState) :- ( DX == 0 -> kill(MState, Ps), B = 0 ; DY == 0 -> kill(MState, Ps), B = 0 ; B == 1 -> kill(MState), DX = 1, DY = 1, X #\= Y ; DX == 1, DY == 1 -> ( var(B) -> ( nonvar(X) -> ( nonvar(Y) -> kill(MState), ( X =\= Y -> B = 1 ; B = 0) ; fd_get(Y, YD, _), ( domain_contains(YD, X) -> true ; kill(MState, Ps), B = 1 ) ) ; nonvar(Y) -> run_propagator(reified_neq(DY,Y,DX,X,Ps,B), MState) ; X == Y -> kill(MState), B = 0 ; fd_get(X, _, XL, XU, _), fd_get(Y, _, YL, YU, _), ( XL cis_gt YU -> kill(MState, Ps), B = 1 ; YL cis_gt XU -> kill(MState, Ps), B = 1 ; true ) ) ; B =:= 0 -> kill(MState), X = Y ; true ) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_and(X,Ps1,Y,Ps2,B), MState) :- ( nonvar(X) -> kill(MState), ( X =:= 0 -> maplist(kill_entailed, Ps2), B = 0 ; B = Y ) ; nonvar(Y) -> run_propagator(reified_and(Y,Ps2,X,Ps1,B), MState) ; B == 1 -> kill(MState), X = 1, Y = 1 ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_or(X,Ps1,Y,Ps2,B), MState) :- ( nonvar(X) -> kill(MState), ( X =:= 1 -> maplist(kill_entailed, Ps2), B = 1 ; B = Y ) ; nonvar(Y) -> run_propagator(reified_or(Y,Ps2,X,Ps1,B), MState) ; B == 0 -> kill(MState), X = 0, Y = 0 ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(reified_not(X,Y), MState) :- ( X == 0 -> kill(MState), Y = 1 ; X == 1 -> kill(MState), Y = 0 ; Y == 0 -> kill(MState), X = 1 ; Y == 1 -> kill(MState), X = 0 ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(pimpl(X, Y, Ps), MState) :- ( nonvar(X) -> kill(MState), ( X =:= 1 -> Y = 1 ; maplist(kill_entailed, Ps) ) ; nonvar(Y) -> kill(MState), ( Y =:= 0 -> X = 0 ; maplist(kill_entailed, Ps) ) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% update_bounds(X, XD, XPs, XL, XU, NXL, NXU) :- ( NXL == XL, NXU == XU -> true ; domains_intersection(XD, from_to(NXL, NXU), NXD), fd_put(X, NXD, XPs) ). min_product(L1, U1, L2, U2, Min) :- Min cis min(min(L1*L2,L1*U2),min(U1*L2,U1*U2)). max_product(L1, U1, L2, U2, Max) :- Max cis max(max(L1*L2,L1*U2),max(U1*L2,U1*U2)). finite(n(_)). in_(L, U, X) :- fd_get(X, XD, XPs), domains_intersection(XD, from_to(L,U), NXD), fd_put(X, NXD, XPs). min_max_factor(L1, U1, L2, U2, L3, U3, Min, Max) :- ( U1 cis_lt n(0), L2 cis_lt n(0), U2 cis_gt n(0), L3 cis_lt n(0), U3 cis_gt n(0) -> maplist(in_(L1,U1), [Z1,Z2]), in_(L2, n(-1), X1), in_(n(1), U3, Y1), ( X1*Y1 #= Z1 -> ( fd_get(Y1, _, Inf1, Sup1, _) -> true ; Inf1 = n(Y1), Sup1 = n(Y1) ) ; Inf1 = inf, Sup1 = n(-1) ), in_(n(1), U2, X2), in_(L3, n(-1), Y2), ( X2*Y2 #= Z2 -> ( fd_get(Y2, _, Inf2, Sup2, _) -> true ; Inf2 = n(Y2), Sup2 = n(Y2) ) ; Inf2 = n(1), Sup2 = sup ), Min cis max(min(Inf1,Inf2), L3), Max cis min(max(Sup1,Sup2), U3) ; L1 cis_gt n(0), L2 cis_lt n(0), U2 cis_gt n(0), L3 cis_lt n(0), U3 cis_gt n(0) -> maplist(in_(L1,U1), [Z1,Z2]), in_(L2, n(-1), X1), in_(L3, n(-1), Y1), ( X1*Y1 #= Z1 -> ( fd_get(Y1, _, Inf1, Sup1, _) -> true ; Inf1 = n(Y1), Sup1 = n(Y1) ) ; Inf1 = n(1), Sup1 = sup ), in_(n(1), U2, X2), in_(n(1), U3, Y2), ( X2*Y2 #= Z2 -> ( fd_get(Y2, _, Inf2, Sup2, _) -> true ; Inf2 = n(Y2), Sup2 = n(Y2) ) ; Inf2 = inf, Sup2 = n(-1) ), Min cis max(min(Inf1,Inf2), L3), Max cis min(max(Sup1,Sup2), U3) ; min_factor(L1, U1, L2, U2, Min0), Min cis max(L3,Min0), max_factor(L1, U1, L2, U2, Max0), Max cis min(U3,Max0) ). min_factor(L1, U1, L2, U2, Min) :- ( L1 cis_geq n(0), L2 cis_gt n(0), finite(U2) -> Min cis div(L1+U2-n(1),U2) ; L1 cis_gt n(0), U2 cis_lt n(0) -> Min cis div(U1,U2) ; L1 cis_gt n(0), L2 cis_geq n(0) -> Min = n(1) ; L1 cis_gt n(0) -> Min cis -U1 ; U1 cis_lt n(0), U2 cis_leq n(0) -> ( finite(L2) -> Min cis div(U1+L2+n(1),L2) ; Min = n(1) ) ; U1 cis_lt n(0), L2 cis_geq n(0) -> Min cis div(L1,L2) ; U1 cis_lt n(0) -> Min = L1 ; L2 cis_leq n(0), U2 cis_geq n(0) -> Min = inf ; Min cis min(min(div(L1,L2),div(L1,U2)),min(div(U1,L2),div(U1,U2))) ). max_factor(L1, U1, L2, U2, Max) :- ( L1 cis_geq n(0), L2 cis_geq n(0) -> Max cis div(U1,L2) ; L1 cis_gt n(0), U2 cis_leq n(0) -> ( finite(L2) -> Max cis div(L1-L2-n(1),L2) ; Max = n(-1) ) ; L1 cis_gt n(0) -> Max = U1 ; U1 cis_lt n(0), U2 cis_lt n(0) -> Max cis div(L1,U2) ; U1 cis_lt n(0), L2 cis_geq n(0) -> ( finite(U2) -> Max cis div(U1-U2+n(1),U2) ; Max = n(-1) ) ; U1 cis_lt n(0) -> Max cis -L1 ; L2 cis_leq n(0), U2 cis_geq n(0) -> Max = sup ; Max cis max(max(div(L1,L2),div(L1,U2)),max(div(U1,L2),div(U1,U2))) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - J-C. Régin: "A filtering algorithm for constraints of difference in CSPs", AAAI-94, Seattle, WA, USA, pp 362--367, 1994 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ distinct_attach([], _, _). distinct_attach([X|Xs], Prop, Right) :- ( var(X) -> init_propagator(X, Prop), make_propagator(pexclude(Xs,Right,X), P1), init_propagator(X, P1), trigger_prop(P1) ; exclude_fire(Xs, Right, X) ), distinct_attach(Xs, Prop, [X|Right]). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - For each integer of the union of domains, an attributed variable is introduced, to benefit from constant-time access. Attributes are: value ... integer corresponding to the node free ... whether this (right) node is still free edges ... [flow_from(F,From)] and [flow_to(F,To)] where F has an attribute "flow" that is either 0 or 1 and an attribute "used" if it is part of a maximum matching parent ... used in breadth-first search g0_edges ... [flow_to(F,To)] as above visited ... true if node was visited in DFS index, in_stack, lowlink ... used in Tarjan's SCC algorithm - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ difference_arcs(Vars, FreeLeft, FreeRight) :- empty_assoc(E), phrase(difference_arcs(Vars, FreeLeft), [E], [NumVar]), assoc_to_list(NumVar, LsNumVar), pairs_values(LsNumVar, FreeRight). domain_to_list(Domain, List) :- phrase(domain_to_list(Domain), List). domain_to_list(split(_, Left, Right)) --> domain_to_list(Left), domain_to_list(Right). domain_to_list(empty) --> []. domain_to_list(from_to(n(F),n(T))) --> { numlist(F, T, Ns) }, list(Ns). difference_arcs([], []) --> []. difference_arcs([V|Vs], FL0) --> ( { fd_get(V, Dom, _), domain_to_list(Dom, Ns) } -> { FL0 = [V|FL] }, enumerate(Ns, V), difference_arcs(Vs, FL) ; difference_arcs(Vs, FL0) ). enumerate([], _) --> []. enumerate([N|Ns], V) --> state(NumVar0, NumVar), { ( get_assoc(N, NumVar0, Y) -> NumVar0 = NumVar ; put_assoc(N, NumVar0, Y, NumVar), put_attr(Y, value, N) ), put_attr(F, flow, 0), append_edge(Y, edges, flow_from(F,V)), append_edge(V, edges, flow_to(F,Y)) }, enumerate(Ns, V). append_edge(V, Attr, E) :- ( get_attr(V, Attr, Es) -> put_attr(V, Attr, [E|Es]) ; put_attr(V, Attr, [E]) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Strategy: Breadth-first search until we find a free right vertex in the value graph, then find an augmenting path in reverse. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ clear_parent(V) :- del_attr(V, parent). maximum_matching([]). maximum_matching([FL|FLs]) :- augmenting_path_to([[FL]], Levels, To), phrase(augmenting_path(FL, To), Path), maplist(maplist(clear_parent), Levels), del_attr(To, free), adjust_alternate_1(Path), maximum_matching(FLs). reachables([]) --> []. reachables([V|Vs]) --> { get_attr(V, edges, Es) }, reachables_(Es, V), reachables(Vs). reachables_([], _) --> []. reachables_([E|Es], V) --> edge_reachable(E, V), reachables_(Es, V). edge_reachable(flow_to(F,To), V) --> ( { get_attr(F, flow, 0), \+ get_attr(To, parent, _) } -> { put_attr(To, parent, V-F) }, [To] ; [] ). edge_reachable(flow_from(F,From), V) --> ( { get_attr(F, flow, 1), \+ get_attr(From, parent, _) } -> { put_attr(From, parent, V-F) }, [From] ; [] ). augmenting_path_to(Levels0, Levels, Right) :- Levels0 = [Vs|_], Levels1 = [Tos|Levels0], phrase(reachables(Vs), Tos), Tos = [_|_], ( member(Right, Tos), get_attr(Right, free, true) -> Levels = Levels1 ; augmenting_path_to(Levels1, Levels, Right) ). augmenting_path(S, V) --> ( { V == S } -> [] ; { get_attr(V, parent, V1-Augment) }, [Augment], augmenting_path(S, V1) ). adjust_alternate_1([A|Arcs]) :- put_attr(A, flow, 1), adjust_alternate_0(Arcs). adjust_alternate_0([]). adjust_alternate_0([A|Arcs]) :- put_attr(A, flow, 0), adjust_alternate_1(Arcs). % Instead of applying Berge's property directly, we can translate the % problem in such a way, that we have to search for the so-called % strongly connected components of the graph. g_g0(V) :- get_attr(V, edges, Es), maplist(g_g0_(V), Es). g_g0_(V, flow_to(F,To)) :- ( get_attr(F, flow, 1) -> append_edge(V, g0_edges, flow_to(F,To)) ; append_edge(To, g0_edges, flow_to(F,V)) ). g0_successors(V, Tos) :- ( get_attr(V, g0_edges, Tos0) -> maplist(arg(2), Tos0, Tos) ; Tos = [] ). put_free(F) :- put_attr(F, free, true). free_node(F) :- get_attr(F, free, true). del_vars_attr(Vars, Attr) :- maplist(del_attr_(Attr), Vars). del_attr_(Attr, Var) :- del_attr(Var, Attr). with_local_attributes(Vars, Attrs, Goal, Result) :- catch((maplist(del_vars_attr(Vars), Attrs), Goal, maplist(del_attrs, Vars), % reset all attributes, only the result matters throw(local_attributes(Result,Vars))), local_attributes(Result,Vars), true). distinct(Vars) :- with_local_attributes(Vars, [edges,parent,g0_edges,index,visited], (difference_arcs(Vars, FreeLeft, FreeRight0), length(FreeLeft, LFL), length(FreeRight0, LFR), LFL =< LFR, maplist(put_free, FreeRight0), maximum_matching(FreeLeft), include(free_node, FreeRight0, FreeRight), maplist(g_g0, FreeLeft), scc(FreeLeft, g0_successors), maplist(dfs_used, FreeRight), phrase(distinct_goals(FreeLeft), Gs)), Gs), disable_queue, maplist(call, Gs), enable_queue. distinct_goals([]) --> []. distinct_goals([V|Vs]) --> { get_attr(V, edges, Es) }, distinct_goals_(Es, V), distinct_goals(Vs). distinct_goals_([], _) --> []. distinct_goals_([flow_to(F,To)|Es], V) --> ( { get_attr(F, flow, 0), \+ get_attr(F, used, true), get_attr(V, lowlink, L1), get_attr(To, lowlink, L2), L1 =\= L2 } -> { get_attr(To, value, N) }, [neq_num(V, N)] ; [] ), distinct_goals_(Es, V). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Mark used edges. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ dfs_used(V) :- ( get_attr(V, visited, true) -> true ; put_attr(V, visited, true), ( get_attr(V, g0_edges, Es) -> dfs_used_edges(Es) ; true ) ). dfs_used_edges([]). dfs_used_edges([flow_to(F,To)|Es]) :- put_attr(F, used, true), dfs_used(To), dfs_used_edges(Es). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Tarjan's strongly connected components algorithm. DCGs are used to implicitly pass around the global index, stack and the predicate relating a vertex to its successors. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ scc(Vs, Succ) :- phrase(scc(Vs), [s(0,[],Succ)], _). scc([]) --> []. scc([V|Vs]) --> ( vindex_defined(V) -> scc(Vs) ; scc_(V), scc(Vs) ). vindex_defined(V) --> { get_attr(V, index, _) }. vindex_is_index(V) --> state(s(Index,_,_)), { put_attr(V, index, Index) }. vlowlink_is_index(V) --> state(s(Index,_,_)), { put_attr(V, lowlink, Index) }. index_plus_one --> state(s(I,Stack,Succ), s(I1,Stack,Succ)), { I1 is I+1 }. s_push(V) --> state(s(I,Stack,Succ), s(I,[V|Stack],Succ)), { put_attr(V, in_stack, true) }. vlowlink_min_lowlink(V, VP) --> { get_attr(V, lowlink, VL), get_attr(VP, lowlink, VPL), VL1 is min(VL, VPL), put_attr(V, lowlink, VL1) }. successors(V, Tos) --> state(s(_,_,Succ)), { call(Succ, V, Tos) }. scc_(V) --> vindex_is_index(V), vlowlink_is_index(V), index_plus_one, s_push(V), successors(V, Tos), each_edge(Tos, V), ( { get_attr(V, index, VI), get_attr(V, lowlink, VI) } -> pop_stack_to(V, VI) ; [] ). pop_stack_to(V, N) --> state(s(I,[First|Stack],Succ), s(I,Stack,Succ)), { del_attr(First, in_stack) }, ( { First == V } -> [] ; { put_attr(First, lowlink, N) }, pop_stack_to(V, N) ). each_edge([], _) --> []. each_edge([VP|VPs], V) --> ( vindex_defined(VP) -> ( v_in_stack(VP) -> vlowlink_min_lowlink(V, VP) ; [] ) ; scc_(VP), vlowlink_min_lowlink(V, VP) ), each_edge(VPs, V). state(S), [S] --> [S]. state(S0, S), [S] --> [S0]. v_in_stack(V) --> { get_attr(V, in_stack, true) }. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Weak arc consistent constraint of difference, currently only available internally. Candidate for all_different/2 option. See Neng-Fa Zhou: "Programming Finite-Domain Constraint Propagators in Action Rules", Theory and Practice of Logic Programming, Vol.6, No.5, pp 483-508, 2006 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ weak_arc_all_distinct(Ls) :- must_be(list, Ls), put_attr(O, clpfd_original, weak_arc_all_distinct(Ls)), all_distinct(Ls, [], O), do_queue. all_distinct([], _, _). all_distinct([X|Right], Left, Orig) :- %\+ list_contains(Right, X), ( var(X) -> make_propagator(weak_distinct(Left,Right,X,Orig), Prop), init_propagator(X, Prop), trigger_prop(Prop) % make_propagator(check_distinct(Left,Right,X), Prop2), % init_propagator(X, Prop2), % trigger_prop(Prop2) ; exclude_fire(Left, Right, X) ), outof_reducer(Left, Right, X), all_distinct(Right, [X|Left], Orig). exclude_fire(Left, Right, E) :- all_neq(Left, E), all_neq(Right, E). list_contains([X|Xs], Y) :- ( X == Y -> true ; list_contains(Xs, Y) ). kill_if_isolated(Left, Right, X, MState) :- append(Left, Right, Others), fd_get(X, XDom, _), ( all_empty_intersection(Others, XDom) -> kill(MState) ; true ). all_empty_intersection([], _). all_empty_intersection([V|Vs], XDom) :- ( fd_get(V, VDom, _) -> domains_intersection_(VDom, XDom, empty), all_empty_intersection(Vs, XDom) ; all_empty_intersection(Vs, XDom) ). outof_reducer(Left, Right, Var) :- ( fd_get(Var, Dom, _) -> append(Left, Right, Others), domain_num_elements(Dom, N), num_subsets(Others, Dom, 0, Num, NonSubs), ( n(Num) cis_geq N -> false ; n(Num) cis N - n(1) -> reduce_from_others(NonSubs, Dom) ; true ) ; %\+ list_contains(Right, Var), %\+ list_contains(Left, Var) true ). reduce_from_others([], _). reduce_from_others([X|Xs], Dom) :- ( fd_get(X, XDom, XPs) -> domain_subtract(XDom, Dom, NXDom), fd_put(X, NXDom, XPs) ; true ), reduce_from_others(Xs, Dom). num_subsets([], _Dom, Num, Num, []). num_subsets([S|Ss], Dom, Num0, Num, NonSubs) :- ( fd_get(S, SDom, _) -> ( domain_subdomain(Dom, SDom) -> Num1 is Num0 + 1, num_subsets(Ss, Dom, Num1, Num, NonSubs) ; NonSubs = [S|Rest], num_subsets(Ss, Dom, Num0, Num, Rest) ) ; num_subsets(Ss, Dom, Num0, Num, NonSubs) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% serialized(+Starts, +Durations) % % Describes a set of non-overlapping tasks. % Starts = [S_1,...,S_n], is a list of variables or integers, % Durations = [D_1,...,D_n] is a list of non-negative integers. % Constrains Starts and Durations to denote a set of % non-overlapping tasks, i.e.: S_i + D_i =< S_j or S_j + D_j =< % S_i for all 1 =< i < j =< n. Example: % % == % ?- length(Vs, 3), % Vs ins 0..3, % serialized(Vs, [1,2,3]), % label(Vs). % Vs = [0, 1, 3] ; % Vs = [2, 0, 3] ; % false. % == % % @see Dorndorf et al. 2000, "Constraint Propagation Techniques for the % Disjunctive Scheduling Problem" serialized(Starts, Durations) :- must_be(list(integer), Durations), pairs_keys_values(SDs, Starts, Durations), put_attr(Orig, clpfd_original, serialized(Starts, Durations)), serialize(SDs, Orig). serialize([], _). serialize([S-D|SDs], Orig) :- D >= 0, serialize(SDs, S, D, Orig), serialize(SDs, Orig). serialize([], _, _, _). serialize([S-D|Rest], S0, D0, Orig) :- D >= 0, propagator_init_trigger([S0,S], pserialized(S,D,S0,D0,Orig)), serialize(Rest, S0, D0, Orig). % consistency check / propagation % Currently implements 2-b-consistency earliest_start_time(Start, EST) :- ( fd_get(Start, D, _) -> domain_infimum(D, EST) ; EST = n(Start) ). latest_start_time(Start, LST) :- ( fd_get(Start, D, _) -> domain_supremum(D, LST) ; LST = n(Start) ). serialize_lower_upper(S_I, D_I, S_J, D_J, MState) :- ( var(S_I) -> serialize_lower_bound(S_I, D_I, S_J, D_J, MState), ( var(S_I) -> serialize_upper_bound(S_I, D_I, S_J, D_J, MState) ; true ) ; true ). serialize_lower_bound(I, D_I, J, D_J, MState) :- fd_get(I, DomI, Ps), ( domain_infimum(DomI, n(EST_I)), latest_start_time(J, n(LST_J)), EST_I + D_I > LST_J, earliest_start_time(J, n(EST_J)) -> ( nonvar(J) -> kill(MState) ; true ), EST is EST_J+D_J, domain_remove_smaller_than(DomI, EST, DomI1), fd_put(I, DomI1, Ps) ; true ). serialize_upper_bound(I, D_I, J, D_J, MState) :- fd_get(I, DomI, Ps), ( domain_supremum(DomI, n(LST_I)), earliest_start_time(J, n(EST_J)), EST_J + D_J > LST_I, latest_start_time(J, n(LST_J)) -> ( nonvar(J) -> kill(MState) ; true ), LST is LST_J-D_I, domain_remove_greater_than(DomI, LST, DomI1), fd_put(I, DomI1, Ps) ; true ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% element(?N, +Vs, ?V) % % The N-th element of the list of finite domain variables Vs is V. % Analogous to nth1/3. element(N, Is, V) :- must_be(list, Is), length(Is, L), N in 1..L, element_(Is, 1, N, V), propagator_init_trigger([N|Is], pelement(N,Is,V)). element_domain(V, VD) :- ( fd_get(V, VD, _) -> true ; VD = from_to(n(V), n(V)) ). element_([], _, _, _). element_([I|Is], N0, N, V) :- ?(I) #\= ?(V) #==> ?(N) #\= N0, N1 is N0 + 1, element_(Is, N1, N, V). integers_remaining([], _, _, D, D). integers_remaining([V|Vs], N0, Dom, D0, D) :- ( domain_contains(Dom, N0) -> element_domain(V, VD), domains_union(D0, VD, D1) ; D1 = D0 ), N1 is N0 + 1, integers_remaining(Vs, N1, Dom, D1, D). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% global_cardinality(+Vs, +Pairs) % % Global Cardinality constraint. Equivalent to % global_cardinality(Vs, Pairs, []). Example: % % == % ?- Vs = [_,_,_], global_cardinality(Vs, [1-2,3-_]), label(Vs). % Vs = [1, 1, 3] ; % Vs = [1, 3, 1] ; % Vs = [3, 1, 1]. % == global_cardinality(Xs, Pairs) :- global_cardinality(Xs, Pairs, []). %% global_cardinality(+Vs, +Pairs, +Options) % % Global Cardinality constraint. Vs is a list of finite domain % variables, Pairs is a list of Key-Num pairs, where Key is an % integer and Num is a finite domain variable. The constraint holds % iff each V in Vs is equal to some key, and for each Key-Num pair % in Pairs, the number of occurrences of Key in Vs is Num. Options % is a list of options. Supported options are: % % * consistency(value) % A weaker form of consistency is used. % % * cost(Cost, Matrix) % Matrix is a list of rows, one for each variable, in the order % they occur in Vs. Each of these rows is a list of integers, one % for each key, in the order these keys occur in Pairs. When % variable v_i is assigned the value of key k_j, then the % associated cost is Matrix_{ij}. Cost is the sum of all costs. global_cardinality(Xs, Pairs, Options) :- must_be(list(list), [Xs,Pairs,Options]), maplist(fd_variable, Xs), maplist(gcc_pair, Pairs), pairs_keys_values(Pairs, Keys, Nums), ( sort(Keys, Keys1), same_length(Keys, Keys1) -> true ; domain_error(gcc_unique_key_pairs, Pairs) ), length(Xs, L), Nums ins 0..L, list_to_drep(Keys, Drep), Xs ins Drep, gcc_pairs(Pairs, Xs, Pairs1), % pgcc_check must be installed before triggering other % propagators propagator_init_trigger(Xs, pgcc_check(Pairs1)), propagator_init_trigger(Nums, pgcc_check_single(Pairs1)), ( member(OD, Options), OD == consistency(value) -> true ; propagator_init_trigger(Nums, pgcc_single(Xs, Pairs1)), propagator_init_trigger(Xs, pgcc(Xs, Pairs, Pairs1)) ), ( member(OC, Options), functor(OC, cost, 2) -> OC = cost(Cost, Matrix), must_be(list(list(integer)), Matrix), maplist(keys_costs(Keys), Xs, Matrix, Costs), sum(Costs, #=, Cost) ; true ). keys_costs(Keys, X, Row, C) :- element(N, Keys, X), element(N, Row, C). gcc_pair(Pair) :- ( Pair = Key-Val -> must_be(integer, Key), fd_variable(Val) ; domain_error(gcc_pair, Pair) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - For each Key-Num0 pair, we introduce an auxiliary variable Num and attach the following attributes to it: clpfd_gcc_num: equal Num0, the user-visible counter variable clpfd_gcc_vs: the remaining variables in the constraint that can be equal Key. clpfd_gcc_occurred: stores how often Key already occurred in vs. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ gcc_pairs([], _, []). gcc_pairs([Key-Num0|KNs], Vs, [Key-Num|Rest]) :- put_attr(Num, clpfd_gcc_num, Num0), put_attr(Num, clpfd_gcc_vs, Vs), put_attr(Num, clpfd_gcc_occurred, 0), gcc_pairs(KNs, Vs, Rest). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - J.-C. Régin: "Generalized Arc Consistency for Global Cardinality Constraint", AAAI-96 Portland, OR, USA, pp 209--215, 1996 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ gcc_global(Vs, KNs) :- gcc_check(KNs), % reach fix-point: all elements of clpfd_gcc_vs must be variables do_queue, with_local_attributes(Vs, [edges,parent,index], (gcc_arcs(KNs, S, Vals), variables_with_num_occurrences(Vs, VNs), maplist(target_to_v(T), VNs), ( get_attr(S, edges, Es) -> put_attr(S, parent, none), % Mark S as seen to avoid going back to S. feasible_flow(Es, S, T), % First construct a feasible flow (if any) maximum_flow(S, T), % only then, maximize it. gcc_consistent(T), scc(Vals, gcc_successors), phrase(gcc_goals(Vals), Gs) ; Gs = [] )), Gs), disable_queue, maplist(call, Gs), enable_queue. gcc_consistent(T) :- get_attr(T, edges, Es), maplist(saturated_arc, Es). saturated_arc(arc_from(_,U,_,Flow)) :- get_attr(Flow, flow, U). gcc_goals([]) --> []. gcc_goals([Val|Vals]) --> { get_attr(Val, edges, Es) }, gcc_edges_goals(Es, Val), gcc_goals(Vals). gcc_edges_goals([], _) --> []. gcc_edges_goals([E|Es], Val) --> gcc_edge_goal(E, Val), gcc_edges_goals(Es, Val). gcc_edge_goal(arc_from(_,_,_,_), _) --> []. gcc_edge_goal(arc_to(_,_,V,F), Val) --> ( { get_attr(F, flow, 0), get_attr(V, lowlink, L1), get_attr(Val, lowlink, L2), L1 =\= L2, get_attr(Val, value, Value) } -> [neq_num(V, Value)] ; [] ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Like in all_distinct/1, first use breadth-first search, then construct an augmenting path in reverse. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ maximum_flow(S, T) :- ( gcc_augmenting_path([[S]], Levels, T) -> phrase(augmenting_path(S, T), Path), Path = [augment(_,First,_)|Rest], path_minimum(Rest, First, Min), maplist(gcc_augment(Min), Path), maplist(maplist(clear_parent), Levels), maximum_flow(S, T) ; true ). feasible_flow([], _, _). feasible_flow([A|As], S, T) :- make_arc_feasible(A, S, T), feasible_flow(As, S, T). make_arc_feasible(A, S, T) :- A = arc_to(L,_,V,F), get_attr(F, flow, Flow), ( Flow >= L -> true ; Diff is L - Flow, put_attr(V, parent, S-augment(F,Diff,+)), gcc_augmenting_path([[V]], Levels, T), phrase(augmenting_path(S, T), Path), path_minimum(Path, Diff, Min), maplist(gcc_augment(Min), Path), maplist(maplist(clear_parent), Levels), make_arc_feasible(A, S, T) ). gcc_augmenting_path(Levels0, Levels, T) :- Levels0 = [Vs|_], Levels1 = [Tos|Levels0], phrase(gcc_reachables(Vs), Tos), Tos = [_|_], ( member(To, Tos), To == T -> Levels = Levels1 ; gcc_augmenting_path(Levels1, Levels, T) ). gcc_reachables([]) --> []. gcc_reachables([V|Vs]) --> { get_attr(V, edges, Es) }, gcc_reachables_(Es, V), gcc_reachables(Vs). gcc_reachables_([], _) --> []. gcc_reachables_([E|Es], V) --> gcc_reachable(E, V), gcc_reachables_(Es, V). gcc_reachable(arc_from(_,_,V,F), P) --> ( { \+ get_attr(V, parent, _), get_attr(F, flow, Flow), Flow > 0 } -> { put_attr(V, parent, P-augment(F,Flow,-)) }, [V] ; [] ). gcc_reachable(arc_to(_L,U,V,F), P) --> ( { \+ get_attr(V, parent, _), get_attr(F, flow, Flow), Flow < U } -> { Diff is U - Flow, put_attr(V, parent, P-augment(F,Diff,+)) }, [V] ; [] ). path_minimum([], Min, Min). path_minimum([augment(_,A,_)|As], Min0, Min) :- Min1 is min(Min0,A), path_minimum(As, Min1, Min). gcc_augment(Min, augment(F,_,Sign)) :- get_attr(F, flow, Flow0), gcc_flow_(Sign, Flow0, Min, Flow), put_attr(F, flow, Flow). gcc_flow_(+, F0, A, F) :- F is F0 + A. gcc_flow_(-, F0, A, F) :- F is F0 - A. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Build value network for global cardinality constraint. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ gcc_arcs([], _, []). gcc_arcs([Key-Num0|KNs], S, Vals) :- ( get_attr(Num0, clpfd_gcc_vs, Vs) -> get_attr(Num0, clpfd_gcc_num, Num), get_attr(Num0, clpfd_gcc_occurred, Occ), ( nonvar(Num) -> U is Num - Occ, U = L ; fd_get(Num, _, n(L0), n(U0), _), L is L0 - Occ, U is U0 - Occ ), put_attr(Val, value, Key), Vals = [Val|Rest], put_attr(F, flow, 0), append_edge(S, edges, arc_to(L, U, Val, F)), put_attr(Val, edges, [arc_from(L, U, S, F)]), variables_with_num_occurrences(Vs, VNs), maplist(val_to_v(Val), VNs) ; Vals = Rest ), gcc_arcs(KNs, S, Rest). variables_with_num_occurrences(Vs0, VNs) :- include(var, Vs0, Vs1), msort(Vs1, Vs), ( Vs == [] -> VNs = [] ; Vs = [V|Rest], variables_with_num_occurrences(Rest, V, 1, VNs) ). variables_with_num_occurrences([], Prev, Count, [Prev-Count]). variables_with_num_occurrences([V|Vs], Prev, Count0, VNs) :- ( V == Prev -> Count1 is Count0 + 1, variables_with_num_occurrences(Vs, Prev, Count1, VNs) ; VNs = [Prev-Count0|Rest], variables_with_num_occurrences(Vs, V, 1, Rest) ). target_to_v(T, V-Count) :- put_attr(F, flow, 0), append_edge(V, edges, arc_to(0, Count, T, F)), append_edge(T, edges, arc_from(0, Count, V, F)). val_to_v(Val, V-Count) :- put_attr(F, flow, 0), append_edge(V, edges, arc_from(0, Count, Val, F)), append_edge(Val, edges, arc_to(0, Count, V, F)). gcc_successors(V, Tos) :- get_attr(V, edges, Tos0), phrase(gcc_successors_(Tos0), Tos). gcc_successors_([]) --> []. gcc_successors_([E|Es]) --> gcc_succ_edge(E), gcc_successors_(Es). gcc_succ_edge(arc_to(_,U,V,F)) --> ( { get_attr(F, flow, Flow), Flow < U } -> [V] ; [] ). gcc_succ_edge(arc_from(_,_,V,F)) --> ( { get_attr(F, flow, Flow), Flow > 0 } -> [V] ; [] ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Simple consistency check, run before global propagation. Importantly, it removes all ground values from clpfd_gcc_vs. The pgcc_check/1 propagator in itself suffices to ensure consistency. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ gcc_check(Pairs) :- disable_queue, gcc_check_(Pairs), enable_queue. gcc_done(Num) :- del_attr(Num, clpfd_gcc_vs), del_attr(Num, clpfd_gcc_num), del_attr(Num, clpfd_gcc_occurred). gcc_check_([]). gcc_check_([Key-Num0|KNs]) :- ( get_attr(Num0, clpfd_gcc_vs, Vs) -> get_attr(Num0, clpfd_gcc_num, Num), get_attr(Num0, clpfd_gcc_occurred, Occ0), vs_key_min_others(Vs, Key, 0, Min, Os), put_attr(Num0, clpfd_gcc_vs, Os), put_attr(Num0, clpfd_gcc_occurred, Occ1), Occ1 is Occ0 + Min, geq(Num, Occ1), % The queue is disabled for efficiency here in any case. % If it were enabled, make sure to retain the invariant % that gcc_global is never triggered during an % inconsistent state (after gcc_done/1 but before all % relevant constraints are posted). ( Occ1 == Num -> all_neq(Os, Key), gcc_done(Num0) ; Os == [] -> gcc_done(Num0), Num = Occ1 ; length(Os, L), Max is Occ1 + L, geq(Max, Num), ( nonvar(Num) -> Diff is Num - Occ1 ; fd_get(Num, ND, _), domain_infimum(ND, n(NInf)), Diff is NInf - Occ1 ), L >= Diff, ( L =:= Diff -> Num is Occ1 + Diff, maplist(=(Key), Os), gcc_done(Num0) ; true ) ) ; true ), gcc_check_(KNs). vs_key_min_others([], _, Min, Min, []). vs_key_min_others([V|Vs], Key, Min0, Min, Others) :- ( fd_get(V, VD, _) -> ( domain_contains(VD, Key) -> Others = [V|Rest], vs_key_min_others(Vs, Key, Min0, Min, Rest) ; vs_key_min_others(Vs, Key, Min0, Min, Others) ) ; ( V =:= Key -> Min1 is Min0 + 1, vs_key_min_others(Vs, Key, Min1, Min, Others) ; vs_key_min_others(Vs, Key, Min0, Min, Others) ) ). all_neq([], _). all_neq([X|Xs], C) :- neq_num(X, C), all_neq(Xs, C). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% circuit(+Vs) % % True iff the list Vs of finite domain variables induces a % Hamiltonian circuit. The k-th element of Vs denotes the % successor of node k. Node indexing starts with 1. Examples: % % == % ?- 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] . % == circuit(Vs) :- must_be(list, Vs), maplist(fd_variable, Vs), length(Vs, L), Vs ins 1..L, ( L =:= 1 -> true ; neq_index(Vs, 1), make_propagator(pcircuit(Vs), Prop), distinct_attach(Vs, Prop, []), trigger_once(Prop) ). neq_index([], _). neq_index([X|Xs], N) :- neq_num(X, N), N1 is N + 1, neq_index(Xs, N1). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Necessary condition for existence of a Hamiltonian circuit: The graph has a single strongly connected component. If the list is ground, the condition is also sufficient. Ts are used as temporary variables to attach attributes: lowlink, index: used for SCC [arc_to(V)]: possible successors - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ propagate_circuit(Vs) :- with_local_attributes([], [], (same_length(Vs, Ts), circuit_graph(Vs, Ts, Ts), scc(Ts, circuit_successors), maplist(single_component, Ts)), _). single_component(V) :- get_attr(V, lowlink, 0). circuit_graph([], _, _). circuit_graph([V|Vs], Ts0, [T|Ts]) :- ( nonvar(V) -> Ns = [V] ; fd_get(V, Dom, _), domain_to_list(Dom, Ns) ), phrase(circuit_edges(Ns, Ts0), Es), put_attr(T, edges, Es), circuit_graph(Vs, Ts0, Ts). circuit_edges([], _) --> []. circuit_edges([N|Ns], Ts) --> { nth1(N, Ts, T) }, [arc_to(T)], circuit_edges(Ns, Ts). circuit_successors(V, Tos) :- get_attr(V, edges, Tos0), maplist(arg(1), Tos0, Tos). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% cumulative(+Tasks) % % Equivalent to cumulative(Tasks, [limit(1)]). cumulative(Tasks) :- cumulative(Tasks, [limit(1)]). %% cumulative(+Tasks, +Options) % % Schedule with a limited resource. Tasks is a list of tasks, each of % the form 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 (which is 1 by default). Options is a list of options. % Currently, the only supported option is: % % * limit(L) % The integer L is the global resource limit. % % 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] . % == cumulative(Tasks, Options) :- must_be(list(list), [Tasks,Options]), ( Options = [] -> L = 1 ; Options = [limit(L)] -> must_be(integer, L) ; domain_error(cumulative_options_empty_or_limit, Options) ), ( Tasks = [] -> true ; maplist(task_bs, Tasks, Bss), maplist(arg(1), Tasks, Starts), maplist(fd_inf, Starts, MinStarts), maplist(arg(3), Tasks, Ends), maplist(fd_sup, Ends, MaxEnds), min_list(MinStarts, Start), max_list(MaxEnds, End), min_end_time(Tasks, L, Start), resource_limit(Start, End, Tasks, Bss, L) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Naive lower bound for global end time, assuming no gaps. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ min_end_time(Tasks, Limit, Start) :- maplist(task_duration_consumption, Tasks, Ds, Cs), maplist(area, Ds, Cs, As), sum(As, #=, ?(Area)), ?(MinTime) #= (Area + Limit - 1) // Limit, tasks_all_done(Tasks, DoneTime), DoneTime #>= Start + MinTime. task_duration_consumption(task(_,D,_,C,_), D, C). area(X, Y, Area) :- ?(Area) #= ?(X) * ?(Y). tasks_all_done(Tasks, End) :- maplist(task_end, Tasks, [End0|Es]), foldl(max_, Es, End0, End). max_(E, M0, M) :- ?(M) #= max(E, M0). task_end(task(_,_,End,_,_), ?(End)). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - All time slots must respect the resource limit. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ resource_limit(T, T, _, _, _) :- !. resource_limit(T0, T, Tasks, Bss, L) :- maplist(contribution_at(T0), Tasks, Bss, Cs), sum(Cs, #=<, L), T1 is T0 + 1, resource_limit(T1, T, Tasks, Bss, L). task_bs(Task, InfStart-Bs) :- Task = task(Start,D,End,_,_Id), ?(D) #> 0, ?(End) #= ?(Start) + ?(D), maplist(finite_domain, [End,Start,D]), fd_inf(Start, InfStart), fd_sup(End, SupEnd), L is SupEnd - InfStart, length(Bs, L), task_running(Bs, Start, End, InfStart). task_running([], _, _, _). task_running([B|Bs], Start, End, T) :- ((T #>= Start) #/\ (T #< End)) #<==> ?(B), T1 is T + 1, task_running(Bs, Start, End, T1). contribution_at(T, Task, Offset-Bs, Contribution) :- Task = task(Start,_,End,C,_), ?(C) #>= 0, fd_inf(Start, InfStart), fd_sup(End, SupEnd), ( T < InfStart -> Contribution = 0 ; T >= SupEnd -> Contribution = 0 ; Index is T - Offset, nth0(Index, Bs, B), ?(Contribution) #= B*C ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% disjoint2(+Rectangles) % % True iff Rectangles are not overlapping. Rectangles is a list of % terms of the form F(X_i, W_i, Y_i, H_i), where F is any functor, % and the arguments are finite domain variables or integers that % denote, respectively, the X coordinate, width, Y coordinate and % height of each rectangle. disjoint2(Rs0) :- must_be(list, Rs0), maplist(=.., Rs0, Rs), non_overlapping(Rs). non_overlapping([]). non_overlapping([R|Rs]) :- maplist(non_overlapping_(R), Rs), non_overlapping(Rs). non_overlapping_(A, B) :- a_not_in_b(A, B), a_not_in_b(B, A). a_not_in_b([_,AX,AW,AY,AH], [_,BX,BW,BY,BH]) :- ?(AX) #=< ?(BX) #/\ ?(BX) #< ?(AX) + ?(AW) #==> ?(AY) + ?(AH) #=< ?(BY) #\/ ?(BY) + ?(BH) #=< ?(AY), ?(AY) #=< ?(BY) #/\ ?(BY) #< ?(AY) + ?(AH) #==> ?(AX) + ?(AW) #=< ?(BX) #\/ ?(BX) + ?(BW) #=< ?(AX). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% automaton(+Signature, +Nodes, +Arcs) % % Describes a list of finite domain variables with a finite % automaton. Equivalent to automaton(_, _, Signature, 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: % % == % :- use_module(library(clpfd)). % % 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]. % == automaton(Sigs, Ns, As) :- automaton(_, _, Sigs, Ns, As, [], [], _). %% automaton(?Sequence, ?Template, +Signature, +Nodes, +Arcs, +Counters, +Initials, ?Finals) % % Describes a list of finite domain variables with a finite % automaton. True iff the finite automaton induced by Nodes and Arcs % (extended with Counters) accepts Signature. Sequence is a list of % terms, all of the same shape. Additional constraints must link % Sequence to Signature, if necessary. Nodes is a list of % 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 correspond to old counter values, and % variables occurring in Template correspond to the current element % of Sequence. When a transition containing expressions is taken, % each counter is updated as stated by the result of the % corresponding arithmetic expression. By default, counters remain % unchanged. Counters is a list of variables that must not occur % anywhere outside of the constraint goal. Initials is a list of the % same length as Counters. Counter arithmetic on the transitions % relates the counter values in Initials to Finals. % % 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: % % == % :- use_module(library(clpfd)). % % sequence_inflexions(Vs, N) :- % variables_signature(Vs, Sigs), % automaton(_, _, 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]. % == template_var_path(V, Var, []) :- var(V), !, V == Var. template_var_path(T, Var, [N|Ns]) :- arg(N, T, Arg), template_var_path(Arg, Var, Ns). path_term_variable([], V, V). path_term_variable([P|Ps], T, V) :- arg(P, T, Arg), path_term_variable(Ps, Arg, V). initial_expr(_, []-1). automaton(Seqs, Template, Sigs, Ns, As0, Cs, Is, Fs) :- must_be(list(list), [Sigs,Ns,As0,Cs,Is]), ( var(Seqs) -> ( current_prolog_flag(clpfd_monotonic, true) -> instantiation_error(Seqs) ; Seqs = Sigs ) ; must_be(list, Seqs) ), maplist(monotonic, Cs, CsM), maplist(arc_normalized(CsM), As0, As), include_args1(sink, Ns, Sinks), include_args1(source, Ns, Sources), maplist(initial_expr, Cs, Exprs0), phrase((arcs_relation(As, Relation), nodes_nums(Sinks, SinkNums0), nodes_nums(Sources, SourceNums0)), [s([]-0, Exprs0)], [s(_,Exprs1)]), maplist(expr0_expr, Exprs1, Exprs), phrase(transitions(Seqs, Template, Sigs, Start, End, Exprs, Cs, Is, Fs), Tuples), list_to_drep(SourceNums0, SourceDrep), Start in SourceDrep, list_to_drep(SinkNums0, SinkDrep), End in SinkDrep, tuples_in(Tuples, Relation). expr0_expr(Es0-_, Es) :- pairs_keys(Es0, Es1), reverse(Es1, Es). transitions([], _, [], S, S, _, _, Cs, Cs) --> []. transitions([Seq|Seqs], Template, [Sig|Sigs], S0, S, Exprs, Counters, Cs0, Cs) --> [[S0,Sig,S1|Is]], { phrase(exprs_next(Exprs, Is, Cs1), [s(Seq,Template,Counters,Cs0)], _) }, transitions(Seqs, Template, Sigs, S1, S, Exprs, Counters, Cs1, Cs). exprs_next([], [], []) --> []. exprs_next([Es|Ess], [I|Is], [C|Cs]) --> exprs_values(Es, Vs), { element(I, Vs, C) }, exprs_next(Ess, Is, Cs). exprs_values([], []) --> []. exprs_values([E0|Es], [V|Vs]) --> { term_variables(E0, EVs0), copy_term(E0, E), term_variables(E, EVs), ?(V) #= E }, match_variables(EVs0, EVs), exprs_values(Es, Vs). match_variables([], _) --> []. match_variables([V0|Vs0], [V|Vs]) --> state(s(Seq,Template,Counters,Cs0)), { ( template_var_path(Template, V0, Ps) -> path_term_variable(Ps, Seq, V) ; template_var_path(Counters, V0, Ps) -> path_term_variable(Ps, Cs0, V) ; domain_error(variable_from_template_or_counters, V0) ) }, match_variables(Vs0, Vs). nodes_nums([], []) --> []. nodes_nums([Node|Nodes], [Num|Nums]) --> node_num(Node, Num), nodes_nums(Nodes, Nums). arcs_relation([], []) --> []. arcs_relation([arc(S0,L,S1,Es)|As], [[From,L,To|Ns]|Rs]) --> node_num(S0, From), node_num(S1, To), state(s(Nodes, Exprs0), s(Nodes, Exprs)), { exprs_nums(Es, Ns, Exprs0, Exprs) }, arcs_relation(As, Rs). exprs_nums([], [], [], []). exprs_nums([E|Es], [N|Ns], [Ex0-C0|Exs0], [Ex-C|Exs]) :- ( member(Exp-N, Ex0), Exp == E -> C = C0, Ex = Ex0 ; N = C0, C is C0 + 1, Ex = [E-C0|Ex0] ), exprs_nums(Es, Ns, Exs0, Exs). node_num(Node, Num) --> state(s(Nodes0-C0, Exprs), s(Nodes-C, Exprs)), { ( member(N-Num, Nodes0), N == Node -> C = C0, Nodes = Nodes0 ; Num = C0, C is C0 + 1, Nodes = [Node-C0|Nodes0] ) }. include_args1(Goal, Ls0, As) :- include(Goal, Ls0, Ls), maplist(arg(1), Ls, As). source(source(_)). sink(sink(_)). monotonic(Var, ?(Var)). arc_normalized(Cs, Arc0, Arc) :- arc_normalized_(Arc0, Cs, Arc). arc_normalized_(arc(S0,L,S,Cs), _, arc(S0,L,S,Cs)). arc_normalized_(arc(S0,L,S), Cs, arc(S0,L,S,Cs)). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% transpose(+Matrix, ?Transpose) % % Transpose a list of lists of the same length. Example: % % == % ?- 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: % % == % :- use_module(library(clpfd)). % % sudoku(Rows) :- % length(Rows, 9), maplist(length_list(9), Rows), % append(Rows, Vs), Vs ins 1..9, % maplist(all_distinct, Rows), % transpose(Rows, Columns), % maplist(all_distinct, Columns), % Rows = [A,B,C,D,E,F,G,H,I], % blocks(A, B, C), blocks(D, E, F), blocks(G, H, I). % % length_list(L, Ls) :- length(Ls, L). % % blocks([], [], []). % blocks([A,B,C|Bs1], [D,E,F|Bs2], [G,H,I|Bs3]) :- % all_distinct([A,B,C,D,E,F,G,H,I]), % blocks(Bs1, Bs2, Bs3). % % 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(writeln, 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|...], ... , [...|...]]. % == transpose(Ls, Ts) :- must_be(list(list), Ls), lists_transpose(Ls, Ts). lists_transpose([], []). lists_transpose([L|Ls], Ts) :- foldl(transpose_, L, Ts, [L|Ls], _). transpose_(_, Fs, Lists0, Lists) :- maplist(list_first_rest, Lists0, Fs, Lists). list_first_rest([L|Ls], L, Ls). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% zcompare(?Order, ?A, ?B) % % Analogous to compare/3, with finite domain variables A and B. % Example: % % == % :- use_module(library(clpfd)). % % 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 is deterministic if the first argument is instantiated: % % == % ?- n_factorial(30, F). % F = 265252859812191058636308480000000. % == zcompare(Order, A, B) :- ( nonvar(Order) -> zcompare_(Order, A, B) ; freeze(Order, zcompare_(Order, A, B)), fd_variable(A), fd_variable(B), propagator_init_trigger([A,B], pzcompare(Order, A, B)) ). zcompare_(=, A, B) :- ?(A) #= ?(B). zcompare_(<, A, B) :- ?(A) #< ?(B). zcompare_(>, A, B) :- ?(A) #> ?(B). %% chain(+Zs, +Relation) % % Zs form a chain with respect to Relation. Zs is a list of finite % domain variables that are a chain with respect to the partial order % Relation, in the order they appear in the list. Relation must be #=, % #=<, #>=, #< or #>. For example: % % == % ?- chain([X,Y,Z], #>=). % X#>=Y, % Y#>=Z. % == chain(Zs, Relation) :- must_be(list, Zs), maplist(fd_variable, Zs), must_be(ground, Relation), ( chain_relation(Relation) -> true ; domain_error(chain_relation, Relation) ), chain_(Zs, Relation). chain_([], _). chain_([X|Xs], Relation) :- foldl(chain(Relation), Xs, X, _). chain_relation(#=). chain_relation(#<). chain_relation(#=<). chain_relation(#>). chain_relation(#>=). chain(Relation, X, Prev, X) :- call(Relation, ?(Prev), ?(X)). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Reflection predicates - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ %% fd_var(+Var) % % True iff Var is a CLP(FD) variable. fd_var(X) :- get_attr(X, clpfd, _). %% fd_inf(+Var, -Inf) % % Inf is the infimum of the current domain of Var. fd_inf(X, Inf) :- ( fd_get(X, XD, _) -> domain_infimum(XD, Inf0), bound_portray(Inf0, Inf) ; must_be(integer, X), Inf = X ). %% fd_sup(+Var, -Sup) % % Sup is the supremum of the current domain of Var. fd_sup(X, Sup) :- ( fd_get(X, XD, _) -> domain_supremum(XD, Sup0), bound_portray(Sup0, Sup) ; must_be(integer, X), Sup = X ). %% fd_size(+Var, -Size) % % Size is the number of elements of the current domain of Var, or the % atom *sup* if the domain is unbounded. fd_size(X, S) :- ( fd_get(X, XD, _) -> domain_num_elements(XD, S0), bound_portray(S0, S) ; must_be(integer, X), S = 1 ). %% fd_dom(+Var, -Dom) % % Dom is the current domain (see in/2) of Var. This predicate is % useful if you want to reason about domains. It is _not_ needed if % you only want to display remaining domains; instead, separate your % model from the search part and let the toplevel display this % information via residual goals. % % 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. % == fd_dom(X, Drep) :- ( fd_get(X, XD, _) -> domain_to_drep(XD, Drep) ; must_be(integer, X), Drep = X..X ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Entailment detection. Subject to change. Currently, Goals entail E if posting ({#\ E} U Goals), then labeling all variables, fails. E must be reifiable. Examples: %?- clpfd:goals_entail([X#>2], X #> 3). %@ false. %?- clpfd:goals_entail([X#>1, X#<3], X #= 2). %@ true. %?- clpfd:goals_entail([X#=Y+1], X #= Y+1). %@ ERROR: Arguments are not sufficiently instantiated %@ Exception: (15) throw(error(instantiation_error, _G2680)) ? %?- clpfd:goals_entail([[X,Y] ins 0..10, X#=Y+1], X #= Y+1). %@ true. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ goals_entail(Goals, E) :- must_be(list, Goals), \+ ( maplist(call, Goals), #\ E, term_variables(Goals-E, Vs), label(Vs) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Unification hook and constraint projection - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ attr_unify_hook(clpfd_attr(_,_,_,Dom,Ps), Other) :- ( nonvar(Other) -> ( integer(Other) -> true ; type_error(integer, Other) ), domain_contains(Dom, Other), trigger_props(Ps), do_queue ; fd_get(Other, OD, OPs), domains_intersection(OD, Dom, Dom1), append_propagators(Ps, OPs, Ps1), fd_put(Other, Dom1, Ps1), trigger_props(Ps1), do_queue ). append_propagators(fd_props(Gs0,Bs0,Os0), fd_props(Gs1,Bs1,Os1), fd_props(Gs,Bs,Os)) :- maplist(append, [Gs0,Bs0,Os0], [Gs1,Bs1,Os1], [Gs,Bs,Os]). bound_portray(inf, inf). bound_portray(sup, sup). bound_portray(n(N), N). list_to_drep(List, Drep) :- list_to_domain(List, Dom), domain_to_drep(Dom, Drep). domain_to_drep(Dom, Drep) :- domain_intervals(Dom, [A0-B0|Rest]), bound_portray(A0, A), bound_portray(B0, B), ( A == B -> Drep0 = A ; Drep0 = A..B ), intervals_to_drep(Rest, Drep0, Drep). intervals_to_drep([], Drep, Drep). intervals_to_drep([A0-B0|Rest], Drep0, Drep) :- bound_portray(A0, A), bound_portray(B0, B), ( A == B -> D1 = A ; D1 = A..B ), intervals_to_drep(Rest, Drep0 \/ D1, Drep). attribute_goals(X) --> % { get_attr(X, clpfd, Attr), format("A: ~w\n", [Attr]) }, { get_attr(X, clpfd, clpfd_attr(_,_,_,Dom,fd_props(Gs,Bs,Os))), append(Gs, Bs, Ps0), append(Ps0, Os, Ps), domain_to_drep(Dom, Drep) }, ( { default_domain(Dom), \+ all_dead_(Ps) } -> [] ; [clpfd:(X in Drep)] ), attributes_goals(Ps). clpfd_aux:attribute_goals(_) --> []. clpfd_aux:attr_unify_hook(_,_) :- false. clpfd_gcc_vs:attribute_goals(_) --> []. clpfd_gcc_vs:attr_unify_hook(_,_) :- false. clpfd_gcc_num:attribute_goals(_) --> []. clpfd_gcc_num:attr_unify_hook(_,_) :- false. clpfd_gcc_occurred:attribute_goals(_) --> []. clpfd_gcc_occurred:attr_unify_hook(_,_) :- false. clpfd_relation:attribute_goals(_) --> []. clpfd_relation:attr_unify_hook(_,_) :- false. clpfd_original:attribute_goals(_) --> []. clpfd_original:attr_unify_hook(_,_) :- false. attributes_goals([]) --> []. attributes_goals([propagator(P, State)|As]) --> ( { ground(State) } -> [] ; { phrase(attribute_goal_(P), Gs) } -> { del_attr(State, clpfd_aux), State = processed, ( current_prolog_flag(clpfd_monotonic, true) -> maplist(unwrap_with(bare_integer), Gs, Gs1) ; maplist(unwrap_with(=), Gs, Gs1) ), maplist(with_clpfd, Gs1, Gs2) }, list(Gs2) ; [P] % possibly user-defined constraint ), attributes_goals(As). with_clpfd(G, clpfd:G). unwrap_with(_, V, V) :- var(V), !. unwrap_with(Goal, ?(V0), V) :- !, call(Goal, V0, V). unwrap_with(Goal, Term0, Term) :- Term0 =.. [F|Args0], maplist(unwrap_with(Goal), Args0, Args), Term =.. [F|Args]. bare_integer(V0, V) :- ( integer(V0) -> V = V0 ; V = ?(V0) ). attribute_goal_(presidual(Goal)) --> [Goal]. attribute_goal_(pgeq(A,B)) --> [?(A) #>= ?(B)]. attribute_goal_(pplus(X,Y,Z)) --> [?(X) + ?(Y) #= ?(Z)]. attribute_goal_(pneq(A,B)) --> [?(A) #\= ?(B)]. attribute_goal_(ptimes(X,Y,Z)) --> [?(X) * ?(Y) #= ?(Z)]. attribute_goal_(absdiff_neq(X,Y,C)) --> [abs(?(X) - ?(Y)) #\= C]. attribute_goal_(absdiff_geq(X,Y,C)) --> [abs(?(X) - ?(Y)) #>= C]. attribute_goal_(x_neq_y_plus_z(X,Y,Z)) --> [?(X) #\= ?(Y) + ?(Z)]. attribute_goal_(x_leq_y_plus_c(X,Y,C)) --> [?(X) #=< ?(Y) + C]. attribute_goal_(ptzdiv(X,Y,Z)) --> [?(X) // ?(Y) #= ?(Z)]. attribute_goal_(pdiv(X,Y,Z)) --> [?(X) div ?(Y) #= ?(Z)]. attribute_goal_(prdiv(X,Y,Z)) --> [?(X) rdiv ?(Y) #= ?(Z)]. attribute_goal_(pexp(X,Y,Z)) --> [?(X) ^ ?(Y) #= ?(Z)]. attribute_goal_(pabs(X,Y)) --> [?(Y) #= abs(?(X))]. attribute_goal_(pmod(X,M,K)) --> [?(X) mod ?(M) #= ?(K)]. attribute_goal_(prem(X,Y,Z)) --> [?(X) rem ?(Y) #= ?(Z)]. attribute_goal_(pmax(X,Y,Z)) --> [?(Z) #= max(?(X),?(Y))]. attribute_goal_(pmin(X,Y,Z)) --> [?(Z) #= min(?(X),?(Y))]. attribute_goal_(scalar_product_neq(Cs,Vs,C)) --> [Left #\= Right], { scalar_product_left_right([-1|Cs], [C|Vs], Left, Right) }. attribute_goal_(scalar_product_eq(Cs,Vs,C)) --> [Left #= Right], { scalar_product_left_right([-1|Cs], [C|Vs], Left, Right) }. attribute_goal_(scalar_product_leq(Cs,Vs,C)) --> [Left #=< Right], { scalar_product_left_right([-1|Cs], [C|Vs], Left, Right) }. attribute_goal_(pdifferent(_,_,_,O)) --> original_goal(O). attribute_goal_(weak_distinct(_,_,_,O)) --> original_goal(O). attribute_goal_(pdistinct(Vs)) --> [all_distinct(Vs)]. attribute_goal_(pexclude(_,_,_)) --> []. attribute_goal_(pelement(N,Is,V)) --> [element(N, Is, V)]. attribute_goal_(pgcc(Vs, Pairs, _)) --> [global_cardinality(Vs, Pairs)]. attribute_goal_(pgcc_single(_,_)) --> []. attribute_goal_(pgcc_check_single(_)) --> []. attribute_goal_(pgcc_check(_)) --> []. attribute_goal_(pcircuit(Vs)) --> [circuit(Vs)]. attribute_goal_(pserialized(_,_,_,_,O)) --> original_goal(O). attribute_goal_(rel_tuple(R, Tuple)) --> { get_attr(R, clpfd_relation, Rel) }, [tuples_in([Tuple], Rel)]. attribute_goal_(pzcompare(O,A,B)) --> [zcompare(O,A,B)]. % reified constraints attribute_goal_(reified_in(V, D, B)) --> [V in Drep #<==> ?(B)], { domain_to_drep(D, Drep) }. attribute_goal_(reified_tuple_in(Tuple, R, B)) --> { get_attr(R, clpfd_relation, Rel) }, [tuples_in([Tuple], Rel) #<==> ?(B)]. attribute_goal_(kill_reified_tuples(_,_,_)) --> []. attribute_goal_(tuples_not_in(_,_,_)) --> []. attribute_goal_(reified_fd(V,B)) --> [finite_domain(V) #<==> ?(B)]. attribute_goal_(pskeleton(X,Y,D,_,Z,F)) --> { Prop =.. [F,X,Y,Z], phrase(attribute_goal_(Prop), Goals), list_goal(Goals, Goal) }, [?(D) #= 1 #==> Goal, ?(Y) #\= 0 #==> ?(D) #= 1]. attribute_goal_(reified_neq(DX,X,DY,Y,_,B)) --> conjunction(DX, DY, ?(X) #\= ?(Y), B). attribute_goal_(reified_eq(DX,X,DY,Y,_,B)) --> conjunction(DX, DY, ?(X) #= ?(Y), B). attribute_goal_(reified_geq(DX,X,DY,Y,_,B)) --> conjunction(DX, DY, ?(X) #>= ?(Y), B). attribute_goal_(reified_and(X,_,Y,_,B)) --> [?(X) #/\ ?(Y) #<==> ?(B)]. attribute_goal_(reified_or(X, _, Y, _, B)) --> [?(X) #\/ ?(Y) #<==> ?(B)]. attribute_goal_(reified_not(X, Y)) --> [#\ ?(X) #<==> ?(Y)]. attribute_goal_(pimpl(X, Y, _)) --> [?(X) #==> ?(Y)]. conjunction(A, B, G, D) --> ( { A == 1, B == 1 } -> [G #<==> ?(D)] ; { A == 1 } -> [(?(B) #/\ G) #<==> ?(D)] ; { B == 1 } -> [(?(A) #/\ G) #<==> ?(D)] ; [(?(A) #/\ ?(B) #/\ G) #<==> ?(D)] ). original_goal(V) --> ( { get_attr(V, clpfd_original, Goal) } -> { del_attr(V, clpfd_original) }, [Goal] ; [] ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Projection of scalar product. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ scalar_product_left_right(Cs, Vs, Left, Right) :- pairs_keys_values(Pairs0, Cs, Vs), partition(ground, Pairs0, Grounds, Pairs), maplist(pair_product, Grounds, Prods), sum_list(Prods, Const), NConst is -Const, partition(compare_coeff0, Pairs, Negatives, _, Positives), maplist(negate_coeff, Negatives, Rights), scalar_plusterm(Rights, Right0), scalar_plusterm(Positives, Left0), ( Const =:= 0 -> Left = Left0, Right = Right0 ; Right0 == 0 -> Left = Left0, Right = NConst ; Left0 == 0 -> Left = Const, Right = Right0 ; ( Const < 0 -> Left = Left0, Right = Right0+NConst ; Left = Left0+Const, Right = Right0 ) ). negate_coeff(A0-B, A-B) :- A is -A0. pair_product(A-B, Prod) :- Prod is A*B. compare_coeff0(Coeff-_, Compare) :- compare(Compare, Coeff, 0). scalar_plusterm([], 0). scalar_plusterm([CV|CVs], T) :- coeff_var_term(CV, T0), foldl(plusterm_, CVs, T0, T). plusterm_(CV, T0, T0+T) :- coeff_var_term(CV, T). coeff_var_term(C-V, T) :- ( C =:= 1 -> T = ?(V) ; T = C * ?(V) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Generated predicates - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- discontiguous term_expansion/2. term_expansion(make_parse_clpfd, Clauses) :- make_parse_clpfd(Clauses). term_expansion(make_parse_reified, Clauses) :- make_parse_reified(Clauses). term_expansion(make_matches, Clauses) :- make_matches(Clauses). make_parse_clpfd. make_parse_reified. make_matches. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Global variables - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ make_clpfd_var('$clpfd_queue') :- make_queue. make_clpfd_var('$clpfd_current_propagator') :- nb_setval('$clpfd_current_propagator', []). make_clpfd_var('$clpfd_queue_status') :- nb_setval('$clpfd_queue_status', enabled). :- multifile user:exception/3. user:exception(undefined_global_variable, Name, retry) :- make_clpfd_var(Name), !. warn_if_bounded_arithmetic :- ( current_prolog_flag(bounded, true) -> print_message(warning, clpfd(bounded)) ; true ). :- initialization(warn_if_bounded_arithmetic). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Messages - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- multifile prolog:message//1. prolog:message(clpfd(bounded)) --> ['Using CLP(FD) with bounded arithmetic may yield wrong results.'-[]]. /******************************* * SANDBOX * *******************************/ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The clpfd library cannot be analysed completely by library(sandbox). However, the API does not provide any meta predicates. It provides some unification hooks, but put_attr/3 does not allow injecting in arbitrary attributes. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- multifile sandbox:safe_primitive/1. safe_api(Name/Arity, sandbox:safe_primitive(clpfd:Head)) :- functor(Head, Name, Arity). term_expansion(safe_api, Clauses) :- module_property(clpfd, exports(API)), maplist(safe_api, API, Clauses). safe_api. % Support clpfd goal expansion. sandbox:safe_primitive(clpfd:clpfd_equal(_,_)). sandbox:safe_primitive(clpfd:clpfd_geq(_,_)). sandbox:safe_primitive(clpfd:clpfd_in(_,_)).