/* CLP(ℤ): Constraint Logic Programming over Integers. Author: Markus Triska E-mail: triska@metalevel.at WWW: https://www.metalevel.at Copyright (C): 2016-2024 Markus Triska This library provides CLP(ℤ): Constraint Logic Programming over Integers ========================================== Highlights: -) DECLARATIVE implementation of integer arithmetic. -) Fully relational, MONOTONIC execution mode. -) Always TERMINATING labeling. Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */ :- module(clpz, [ 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 \/ op(150, fx, #), (#>)/2, (#<)/2, (#>=)/2, (#=<)/2, (#=)/2, (#\=)/2, (#\)/1, (#<==>)/2, (#==>)/2, (#<==)/2, (#\/)/2, (#\)/2, (#/\)/2, (in)/2, (ins)/2, all_different/1, all_distinct/1, nvalue/2, 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, zcompare/3, chain/2, fd_var/1, fd_inf/2, fd_sup/2, fd_size/2, fd_dom/2, % for use in predicates from library(reif) clpz_t/2, (#=)/3, (#<)/3 % called from goal_expansion % clpz_equal/2, % clpz_geq/2 ]). :- use_module(library(assoc)). :- use_module(library(pairs)). :- use_module(library(between)). :- use_module(library(lists)). :- use_module(library(atts)). :- use_module(library(iso_ext)). :- use_module(library(dcgs)). :- use_module(library(terms)). :- use_module(library(error), [domain_error/3, type_error/3, can_be/2]). :- use_module(library(si)). :- use_module(library(freeze)). :- use_module(library(arithmetic)). :- use_module(library(debug)). :- use_module(library(format)). % :- use_module(library(types)). :- attribute clpz/1, clpz_aux/1, clpz_relation/1, edges/1, flow/1, parent/1, free/1, g0_edges/1, used/1, lowlink/1, value/1, visited/1, index/1, in_stack/1, clpz_gcc_vs/1, clpz_gcc_num/1, clpz_gcc_occurred/1, queue/2, disabled/0. :- dynamic(monotonic/0). :- dynamic(clpz_equal_/2). :- dynamic(clpz_geq_/2). :- dynamic(clpz_neq/2). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Compatibility predicates. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ cyclic_term(T) :- \+ acyclic_term(T). must_be(What, Term) :- must_be(What, unknown(Term)-1, Term). must_be(ground, _, Term) :- !, ( ground(Term) -> true ; instantiation_error(Term) ). must_be(acyclic, Where, Term) :- !, ( acyclic_term(Term) -> true ; domain_error(acyclic_term, Term, Where) ). must_be(list, Where, Term) :- !, ( list_si(Term) -> true ; type_error(list, Term, Where) ). must_be(list(What), Where, Term) :- !, must_be(list, Where, Term), maplist(must_be(What, Where), Term). must_be(Type, _, Term) :- error:must_be(Type, Term). instantiation_error(Term) :- instantiation_error(Term, unknown(Term)-1). instantiation_error(_, Goal-Arg) :- throw(error(instantiation_error, instantiation_error(Goal, Arg))). domain_error(Expectation, Term) :- domain_error(Expectation, Term, unknown(Term)-1). type_error(Expectation, Term) :- type_error(Expectation, Term, unknown(Term)-1). :- meta_predicate(partition(1, ?, ?, ?)). partition(Pred, Ls0, As, Bs) :- include(Pred, Ls0, As), exclude(Pred, Ls0, Bs). partition(Pred, Ls0, Ls, Es, Gs) :- partition_(Ls0, Pred, Ls, Es, Gs). partition_([], _, [], [], []). partition_([X|Xs], Pred, Ls0, Es0, Gs0) :- call(Pred, X, Cmp), ( Cmp = (<) -> Ls0 = [X|Rest], partition_(Xs, Pred, Rest, Es0, Gs0) ; Cmp = (=) -> Es0 = [X|Rest], partition_(Xs, Pred, Ls0, Rest, Gs0) ; Cmp = (>) -> Gs0 = [X|Rest], partition_(Xs, Pred, Ls0, Es0, Rest) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - include/3 and exclude/3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- meta_predicate(include(1, ?, ?)). include(_, [], []). include(Goal, [L|Ls0], Ls) :- ( call(Goal, L) -> Ls = [L|Rest] ; Ls = Rest ), include(Goal, Ls0, Rest). :- meta_predicate(exclude(1, ?, ?)). exclude(_, [], []). exclude(Goal, [L|Ls0], Ls) :- ( call(Goal, L) -> Ls = Rest ; Ls = [L|Rest] ), exclude(Goal, Ls0, Rest). %:- discontiguous clpz:goal_expansion/5. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Public operators. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- 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 \/ :- op(150, fx, #). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Privately needed operators. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- op(700, xfx, cis). :- op(700, xfx, cis_geq). :- op(700, xfx, cis_gt). :- op(700, xfx, cis_leq). :- op(700, xfx, cis_lt). :- op(1200, xfx, ++>). /** Constraint Logic Programming over Integers ## Introduction This library provides CLP(ℤ): Constraint Logic Programming over Integers. CLP(ℤ) is an instance of the general CLP(_X_) scheme, extending logic programming with reasoning over specialised domains. CLP(ℤ) lets us reason about *integers* in a way that honors the relational nature of Prolog. There are two major use cases of CLP(ℤ) constraints: 1. [*declarative integer arithmetic*](#clpz-integer-arith) 2. solving *combinatorial problems* such as planning, scheduling and allocation tasks. The predicates of this library can be classified as: * _arithmetic_ constraints like `(#=)/2`, `(#>)/2` and `(#\=)/2` * the _membership_ constraints `(in)/2` and `(ins)/2` * the _enumeration_ predicates `indomain/1`, `label/1` and `labeling/2` * _combinatorial_ constraints like `all_distinct/1` and `global_cardinality/2` * _reification_ predicates such as `(#<==>)/2` * _reflection_ predicates such as `fd_dom/2` In most cases, [_arithmetic constraints_](#clpz-arith-constraints) are the only predicates you will ever need from this library. When reasoning over integers, simply replace low-level arithmetic predicates like `(is)/2` and `(>)/2` by the corresponding CLP(ℤ) constraints like `(#=)/2` and `(#>)/2` to honor and preserve declarative properties of your programs. For satisfactory performance, arithmetic constraints are implicitly rewritten at compilation time so that low-level fallback predicates are automatically used whenever possible. Almost all Prolog programs also reason about integers. Therefore, it is highly advisable that you make CLP(ℤ) constraints available in all your programs. One way to do this is to put the following directive in your `~/.scryerrc` initialisation file: ``` :- use_module(library(clpz)). ``` All example programs that appear in the CLP(ℤ) documentation assume that you have done this. Important concepts and principles of this library are illustrated by means of usage examples that are available in a public git repository: [*https://github.com/triska/clpz*](https://github.com/triska/clpz) If you are used to the complicated operational considerations that low-level arithmetic primitives necessitate, then moving to CLP(ℤ) 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 all popular Prolog systems, and they are designed to help you eliminate and avoid the use of low-level and less general primitives by providing declarative alternatives that are meant to be used instead. When teaching Prolog, CLP(ℤ) constraints should be introduced _before_ explaining low-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 impure limitations that are better deferred to more advanced lectures. More information about CLP(ℤ) constraints and their implementation is contained in: [*metalevel.at/drt.pdf*](https://www.metalevel.at/drt.pdf) The best way to discuss applying, improving and extending CLP(ℤ) constraints is to use the dedicated `clpz` tag on [stackoverflow.com](http://stackoverflow.com). Several of the world's foremost CLP(ℤ) experts regularly participate in these discussions and will help you for free on this platform. {#clpz-arith-constraints} ## Arithmetic constraints In modern Prolog systems, *arithmetic constraints* subsume and supersede low-level predicates over integers. The main advantage of arithmetic constraints is that they are true _relations_ and can be used in all directions. For most programs, arithmetic constraints are the only predicates you will ever need from this library. The most important arithmetic constraint is `(#=)/2`, which subsumes both `(is)/2` and `(=:=)/2` over integers. Use `(#=)/2` to make your programs more general. In total, the arithmetic constraints are: | Expr1 `#=` Expr2 | Expr1 equals Expr2 | | Expr1 `#\=` Expr2 | Expr1 is not equal to Expr2 | | Expr1 `#>=` Expr2 | Expr1 is greater than or equal to Expr2 | | Expr1 `#=<` Expr2 | Expr1 is less than or equal to Expr2 | | Expr1 `#>` Expr2 | Expr1 is greater than Expr2 | | Expr1 `#<` Expr2 | Expr1 is less than Expr2 | `Expr1` and `Expr2` denote *arithmetic expressions*, which are: | _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 | | sign(Expr) | Sign (-1, 0, 1) of Expr | | Expr // Expr | Truncated integer division | | Expr div Expr | Floored integer division | where `Expr` again denotes an arithmetic expression. The bitwise operations `(\)/1`, `(/\)/2`, `(\/)/2`, `(>>)/2`, `(<<)/2`, `lsb/1`, `msb/1`, `popcount/1` and `(xor)/2` are also supported. {#clpz-integer-arith} ## Declarative integer arithmetic The [_arithmetic constraints_](#clpz-arith-constraints) `(#=)/2`, `(#>)/2` etc. are meant to be used _instead_ of the primitives `(is)/2`, `(=:=)/2`, `(>)/2` etc. over integers. Almost all Prolog programs also reason about integers. Therefore, it is recommended that you put the following directive in your `~/.scryerrc` initialisation file to make CLP(ℤ) constraints available in all your programs: ``` :- use_module(library(clpz)). ``` Throughout the following, it is assumed that you have done this. The most basic use of CLP(ℤ) constraints is _evaluation_ of arithmetic expressions involving integers. For example: ``` ?- X #= 1+2. X = 3. ``` This could in principle also be achieved with the lower-level predicate `(is)/2`. However, an important advantage of arithmetic constraints is their purely relational nature: Constraints can be used in _all directions_, also if one or more of their arguments are only partially instantiated. For example: ``` ?- 3 #= Y+2. Y = 1. ``` This relational nature makes CLP(ℤ) constraints easy to explain and use, and well suited for beginners and experienced Prolog programmers alike. In contrast, when using low-level integer arithmetic, we get: ``` ?- 3 is Y+2. error(instantiation_error,(is)/2). ?- 3 =:= Y+2. error(instantiation_error,(is)/2). ``` 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(ℤ) constraints are drop-in replacements of these low-level arithmetic predicates, often yielding more general programs. See [`n_factorial/2`](#clpz-factorial) for an example. This library uses `goal_expansion/2` to automatically rewrite constraints at compilation time so that low-level arithmetic predicates are _automatically_ used whenever possible. For example, the predicate: ``` positive_integer(N) :- N #>= 1. ``` is executed as if it were written as: ``` positive_integer(N) :- ( integer(N) -> N >= 1 ; N #>= 1 ). ``` This illustrates why the performance of CLP(ℤ) constraints is almost always completely satisfactory when they are used in modes that can be handled by low-level arithmetic. To disable the automatic rewriting, set the Prolog flag `clpz_goal_expansion` to `false`. If you are used to the complicated operational considerations that low-level arithmetic primitives necessitate, then moving to CLP(ℤ) 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 all popular Prolog systems, and they are designed to help you eliminate and avoid the use of low-level and less general primitives by providing declarative alternatives that are meant to be used instead. {#clpz-factorial} ## Example: Factorial relation We illustrate the benefit of using `(#=)/2` for more generality with a simple example. Consider first a rather conventional definition of `n_factorial/2`, relating each natural number _N_ to its factorial _F_: ``` n_factorial(0, 1). n_factorial(N, F) :- N #> 0, N1 #= N - 1, n_factorial(N1, F1), F #= N * F1. ``` This program uses CLP(ℤ) constraints _instead_ of low-level arithmetic throughout, and everything that _would have worked_ with low-level arithmetic _also_ works with CLP(ℤ) constraints, retaining roughly the same performance. For example: ``` ?- n_factorial(47, F). F = 258623241511168180642964355153611979969197632389120000000000 ; false. ``` Now the point: Due to the increased flexibility and generality of CLP(ℤ) constraints, we are free to _reorder_ the goals as follows: ``` n_factorial(0, 1). n_factorial(N, F) :- N #> 0, N1 #= N - 1, F #= N * F1, n_factorial(N1, F1). ``` In this concrete case, _termination_ properties of the predicate are improved. For example, the following queries now both terminate: ``` ?- 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. The value of CLP(ℤ) constraints does _not_ lie in completely freeing us from _all_ procedural phenomena. For example, the two programs do not even have the same _termination properties_ in all cases. Instead, the primary benefit of CLP(ℤ) constraints is that they allow you to try different execution orders and apply [*declarative debugging*](https://www.metalevel.at/prolog/debugging) techniques _at all_! Reordering goals (and clauses) can significantly impact the performance of Prolog programs, and you are free to try different variants if you use declarative approaches. Moreover, since all CLP(ℤ) constraints _always terminate_, placing them earlier can at most _improve_, never worsen, the termination properties of your programs. An additional benefit of CLP(ℤ) constraints is that they eliminate the complexity of introducing `(is)/2` and `(=:=)/2` to beginners, since _both_ predicates are subsumed by `(#=)/2` when reasoning over integers. {#clpz-combinatorial} ## Combinatorial constraints In addition to subsuming and replacing low-level arithmetic predicates, CLP(ℤ) constraints are often used to solve combinatorial problems such as planning, scheduling and allocation tasks. Among the most frequently used *combinatorial constraints* are `all_distinct/1`, `global_cardinality/2` and `cumulative/2`. This library also provides several other constraints like `disjoint2/1` and `automaton/8`, which are useful in more specialized applications. {#clpz-domains} ## Domains Each CLP(ℤ) variable has an associated set of admissible integers, which we call the variable's *domain*. Initially, the domain of each CLP(ℤ) variable is the set of _all_ integers. CLP(ℤ) constraints like `(#=)/2`, `(#>)/2` and `(#\=)/2` can at most reduce, and never extend, the domains of their arguments. The constraints `(in)/2` and `(ins)/2` let us explicitly state domains of CLP(ℤ) variables. The process of determining and adjusting domains of variables is called constraint *propagation*, and it is performed automatically by this library. When the domain of a variable contains only one element, then the variable is automatically unified to that element. Domains are taken into account when further constraints are stated, and by enumeration predicates like labeling/2. {#clpz-sudoku} ## Example: Sudoku As another example, consider _Sudoku_: It is a popular puzzle over integers that can be easily solved with CLP(ℤ) constraints. ``` sudoku(Rows) :- length(Rows, 9), maplist(same_length(Rows), Rows), append(Rows, Vs), Vs ins 1..9, maplist(all_distinct, Rows), transpose(Rows, Columns), maplist(all_distinct, Columns), Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is], blocks(As, Bs, Cs), blocks(Ds, Es, Fs), blocks(Gs, Hs, Is). blocks([], [], []). blocks([N1,N2,N3|Ns1], [N4,N5,N6|Ns2], [N7,N8,N9|Ns3]) :- all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]), blocks(Ns1, Ns2, Ns3). problem(1, [[_,_,_,_,_,_,_,_,_], [_,_,_,_,_,3,_,8,5], [_,_,1,_,2,_,_,_,_], [_,_,_,5,_,7,_,_,_], [_,_,4,_,_,_,1,_,_], [_,9,_,_,_,_,_,_,_], [5,_,_,_,_,_,_,7,3], [_,_,2,_,1,_,_,_,_], [_,_,_,_,4,_,_,_,9]]). ``` Sample query: ``` ?- problem(1, Rows), sudoku(Rows), maplist(portray_clause, Rows). [9,8,7,6,5,4,3,2,1]. [2,4,6,1,7,3,9,8,5]. [3,5,1,9,2,8,7,4,6]. [1,2,8,5,3,7,6,9,4]. [6,3,4,8,9,2,1,5,7]. [7,9,5,4,6,1,8,3,2]. [5,1,9,2,8,6,4,7,3]. [4,7,2,3,1,9,5,6,8]. [8,6,3,7,4,5,2,1,9]. Rows = [[9,8,7,6,5,4,3,2,1]|...]. ``` In this concrete case, the constraint solver is strong enough to find the unique solution without any search. {#clpz-residual-goals} ## Residual goals Here is an example session with a few queries and their answers: ``` ?- X #> 3. clpz:(X in 4..sup). ?- X #\= 20. clpz:(X in inf..19\/21..sup). ?- 2*X #= 10. X = 5. ?- X*X #= 144. clpz:(X in-12\/12) ; false. ?- 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, clpz:(X in 0..3), clpz:(Y in 4..5). ``` The answers emitted by the toplevel are called _residual programs_, and the goals that comprise each answer are called *residual goals*. In each case above, and as for all pure programs, the residual program is declaratively equivalent to the original query. From the residual goals, it is clear that the constraint solver has deduced additional domain restrictions in many cases. To inspect residual goals, it is best to let the toplevel display them for us. Wrap the call of your predicate into `call_residue_vars/2` to make sure that all constrained variables are displayed. To make the constraints a variable is involved in available as a Prolog term for further reasoning within your program, use `copy_term/3`. For example: ``` ?- X #= Y + Z, X in 0..5, copy_term([X,Y,Z], [X,Y,Z], Gs). Gs = [clpz: (X in 0..5), clpz: (Y+Z#=X)], X in 0..5, Y+Z#=X. ``` This library also provides _reflection_ predicates (like `fd_dom/2`, `fd_size/2` etc.) with which we can inspect a variable's current domain. These predicates can be useful if you want to implement your own labeling strategies. {#clpz-search} ## Core relations and search Using CLP(ℤ) 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 called the *core relation*, separate from the actual search for solutions. This lets us observe termination and determinism properties of the core relation 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(ℤ) as follows: ``` 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 this core relation _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. {#clpz-n-queens} ## Example: Eight queens puzzle We illustrate the concepts of the preceding sections by means of the so-called _eight queens puzzle_. The task is to place 8 queens on an 8x8 chessboard such that none of the queens is under attack. This means that no two queens share the same row, column or diagonal. To express this puzzle via CLP(ℤ) constraints, we must first pick a suitable representation. Since CLP(ℤ) constraints reason over _integers_, we must find a way to map the positions of queens to integers. Several such mappings are conceivable, and it is not immediately obvious which we should use. On top of that, different constraints can be used to express the desired relations. For such reasons, _modeling_ combinatorial problems via CLP(ℤ) constraints often necessitates some creativity and has been described as more of an art than a science. In our concrete case, we observe that there must be exactly one queen per column. The following representation therefore suggests itself: We are looking for 8 integers, one for each column, where each integer denotes the _row_ of the queen that is placed in the respective column, and which are subject to certain constraints. In fact, let us now generalize the task to the so-called _N queens puzzle_, which is obtained by replacing 8 by _N_ everywhere it occurs in the above description. We implement the above considerations in the *core relation* `n_queens/2`, where the first argument is the number of queens (which is identical to the number of rows and columns of the generalized chessboard), and the second argument is a list of _N_ integers that represents a solution in the form described above. ``` n_queens(N, Qs) :- length(Qs, N), Qs ins 1..N, safe_queens(Qs). safe_queens([]). safe_queens([Q|Qs]) :- safe_queens(Qs, Q, 1), safe_queens(Qs). safe_queens([], _, _). safe_queens([Q|Qs], Q0, D0) :- Q0 #\= Q, abs(Q0 - Q) #\= D0, D1 #= D0 + 1, safe_queens(Qs, Q0, D1). ``` Note that all these predicates can be used in _all directions_: We can use them to _find_ solutions, _test_ solutions and _complete_ partially instantiated solutions. The original task can be readily solved with the following query: ``` ?- n_queens(8, Qs), label(Qs). Qs = [1,5,8,6,3,7,2,4] ; ... . ``` Using suitable labeling strategies, we can easily find solutions with 80 queens and more: ``` ?- n_queens(80, Qs), labeling([ff], Qs). Qs = [1,3,5,44,42,4,50,7,68,57,76,61,6,39,30,40,8,54,36,41|...] ; ... . ?- time((n_queens(90, Qs), labeling([ff], Qs))). % CPU time: 2.382s Qs = [1,3,5,50,42,4,49,7,59,48,46,63,6,55,47,64,8,70,58,67|...] ; ... . ``` Experimenting with different search strategies is easy because we have separated the core relation from the actual search. {#clpz-optimisation} ## Optimisation We can use `labeling/2` to minimize or maximize the value of a CLP(ℤ) expression, and generate solutions in increasing or decreasing order of the value. See the labeling options `min(Expr)` and `max(Expr)`, respectively. Again, to easily try different labeling options in connection with optimisation, we recommend to introduce a dedicated predicate for posting constraints, and to use `labeling/2` in a separate goal. This way, we can observe properties of the core relation in isolation, and try different labeling options without recompiling our code. If necessary, we can use `once/1` to commit to the first optimal solution. However, it is often very valuable to see alternative solutions that are _also_ optimal, so that we can choose among optimal solutions by other criteria. For the sake of [*purity*](https://www.metalevel.at/prolog/purity) and completeness, we recommend to avoid `once/1` and other constructs that lead to impurities in CLP(ℤ) programs. Related to optimisation with CLP(ℤ) constraints are `library(simplex)` and CLP(Q) which reason about _linear_ constraints over rational numbers. {#clpz-reification} ## Reification The constraints `(in)/2`, `(#=)/2`, `(#\=)/2`, `(#<)/2`, `(#>)/2`, `(#=<)/2`, and `(#>=)/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. When reasoning over Boolean variables, also consider using CLP(B) constraints as provided by `library(clpb)`. {#clpz-monotonicity} ## Enabling monotonic CLP(ℤ) In the default execution mode, CLP(ℤ) constraints still exhibit some non-relational properties. For example, _adding_ constraints can yield new solutions: ``` ?- X #= 2, X = 1+1. false. ?- X = 1+1, X #= 2, X = 1+1. X = 1+1. ``` This behaviour is highly problematic from a logical point of view, and it may render declarative debugging techniques inapplicable. Assert `clpz:monotonic` to make CLP(ℤ) *monotonic*: This means that _adding_ new constraints _cannot_ yield new solutions. When this flag is `true`, we must wrap variables that occur in arithmetic expressions with the functor `(?)/1` or `(#)/1`. For example: ``` ?- assertz(clpz:monotonic). true. ?- #X #= #Y + #Z. clpz:(#Y+ #Z#= #X). ?- X #= 2, X = 1+1. error(instantiation_error,instantiation_error(unknown(_408),1)). ``` The wrapper can be omitted for variables that are already constrained to integers. {#clpz-custom-constraints} ## Custom constraints We can 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: ``` :- multifile clpz:run_propagator/2. oneground(X, Y, Z) :- clpz:make_propagator(oneground(X, Y, Z), Prop), clpz:init_propagator(X, Prop), clpz:init_propagator(Y, Prop), clpz:trigger_once(Prop). clpz:run_propagator(oneground(X, Y, Z), MState) :- ( integer(X) -> clpz:kill(MState), Z = 1 ; integer(Y) -> clpz:kill(MState), Z = 1 ; true ). ``` First, `clpz:make_propagator/2` is used to transform a user-defined representation of the new constraint to an internal form. With `clpz: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, `clpz:trigger_once/1` is used to give the propagator its first chance for propagation even though the variables' domains have not yet changed. Finally, `clpz: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 `clpz: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 `clpz: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](https://www.metalevel.at) */ /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Duo DCGs ======== A Duo DCG is like a DCG, except that it describes *two* lists at the same time. A Duo DCG rule has the form Head ++> Body. The language construct As+Bs is used within Duo DCGs to describe that the elements in As occur in the first list, and the elements in Bs occur in the second list. Duo DCGs are compiled to Prolog code via term expansion. The interface predicates are: *) duophrase(NT, As, Bs) *) duophrase(NT, As0, As, Bs0, Bs) (difference list version). Duo DCGs could be used to efficiently describe scheduled propagators, taking into account the two possible propagator priorities. However, it turns out that passing around a single argument is more efficient than passing around multiple arguments, and therefore regular DCGs are used for propagator scheduling. Still, everything is completely pure: No global data structures are needed to schedule the propagators! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- meta_predicate(duophrase(4, ?, ?)). :- meta_predicate(duophrase(4, ?, ?, ?, ?)). duophrase(NT, As, Bs) :- duophrase(NT, As, [], Bs, []). duophrase(NT, As0, As, Bs0, Bs) :- call(NT, As0, As, Bs0, Bs). %:- multifile user:term_expansion/6. term_expansion(Term0, Term) :- nonvar(Term0), Term0 = (Head0 ++> Body0), Term = (Head :- Body), duodcg_head(Head0, Head, As0, As, Bs0, Bs), once(duodcg_body(Body0, Body, As0, As, Bs0, Bs)). duodcg_body([], (As0=As,Bs0=Bs), As0, As, Bs0, Bs). duodcg_body(Xs+Ys, (phrase(seq(Xs), As0, As), phrase(seq(Ys), Bs0, Bs)), As0, As, Bs0, Bs). duodcg_body({Goal}, call(Goal), As, As, Bs, Bs). duodcg_body((A0,B0), (A,B), As0, As, Bs0, Bs) :- duodcg_body(A0, A, As0, As1, Bs0, Bs1), duodcg_body(B0, B, As1, As, Bs1, Bs). duodcg_body((A0->B0;C0), (A->B;C), As0, As, Bs0, Bs) :- duodcg_body(A0, A, As0, As1, Bs0, Bs1), duodcg_body(B0, B, As1, As, Bs1, Bs), duodcg_body(C0, C, As0, As, Bs0, Bs). duodcg_body((A->B), Body, As0, As, Bs0, Bs) :- duodcg_body((A->B;false), Body, As0, As, Bs0, Bs). duodcg_body((A0;B0), (A;B), As0, As, Bs0, Bs) :- duodcg_body(A0, A, As0, As, Bs0, Bs), duodcg_body(B0, B, As0, As, Bs0, Bs). duodcg_body(NT0, NT, As0, As, Bs0, Bs) :- duodcg_head(NT0, NT, As0, As, Bs0, Bs). duodcg_head(Head0, Head, As0, As, Bs0, Bs) :- Head0 =.. [F|Args0], append(Args0, [As0,As,Bs0,Bs], Args), Head =.. [F|Args]. goal_expansion(get_attr(Var, Module, Value), (var(Var),get_atts(Var, Access))) :- Access =.. [Module,Value]. goal_expansion(put_attr(Var, Module, Value), put_atts(Var, Access)) :- Access =.. [Module,Value]. goal_expansion(del_attr(Var, Module), (var(Var) -> put_atts(Var, -Access);true)) :- Access =.. [Module,_]. 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). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ % 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(clpz_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). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% in(?Var, +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 :- clpz_in(Var, Dom). clpz_in(V, D) :- fd_variable(V), drep_to_domain(D, Dom), domain(V, Dom). fd_variable(V) :- can_be(integer, V). %% ins(+Vars, +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) ). fd_must_be_list(Ls, Where) :- ( fd_var(Ls) -> type_error(list, Ls, Where) ; must_be(list, Where, 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)). %% 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, labeling(Options, Vars)-1, Options), fd_must_be_list(Vars, labeling(Options, Vars)-2), maplist(finite_domain(labeling(Options, Vars), 2), Vars), label(Options, Options, default(leftmost), default(up), default(step), [], upto_ground, Vars). finite_domain(Goal, Arg, Var) :- ( fd_get(Var, Dom, _) -> ( domain_infimum(Dom, n(_)), domain_supremum(Dom, n(_)) -> true ; instantiation_error(Dom) ) ; integer(Var) -> true ; must_be(integer, Goal-Arg, Var) ). finite_domain(Var) :- finite_domain(finite_domain(Var), 1, 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), call_cleanup(optimise(Vars, [S,O,C], SVs), retractall(extremum(_))) ). % 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), 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). choice(step). choice(enum). choice(bisect). order(up). order(down). 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). 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, contracting(Vs)-1, Vs), maplist(finite_domain(contracting(Vs), 1), 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 the clause database to save the computed extremum over backtracking. Failure is used to get rid of copies of attributed variables that are created in intermediate steps. Example: ?- X in 1..3, call_residue_vars(labeling([min(X)], [X]), Vs). %@ X = 1, Vs = [] %@ ; X = 2, Vs = [] %@ ; X = 3, Vs = [] %@ ; false. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ :- dynamic(extremum/1). optimise(Vars, Options, Whats) :- Whats = [What|WhatsRest], asserta(extremum(mark)), ( catch(store_extremum(Vars, Options, What), time_limit_exceeded, false) ; once(extremum(Val0)), retract_until_mark, Val0 = n(Val), arg(1, What, Expr), append(WhatsRest, Options, Options1), ( Expr #= Val, labeling(Options1, Vars) ; Expr #\= Val, optimise(Vars, Options, Whats) ) ). retract_until_mark :- ( retract(extremum(E)), E == mark -> true ; retract_until_mark ). store_extremum(Vars, Options, What) :- 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). optimise(Direction, Options, Vars, Expr0, Expr) :- must_be(ground, Expr0), update_extremum(Expr0), catch((tighten(Direction, Expr, Expr0), labeling(Options, Vars), throw(v(Expr))), v(Expr1), true), optimise(Direction, Options, Vars, Expr1, Expr). update_extremum(Expr) :- ( once(extremum(Prev)), Prev = n(_) -> once(retract(extremum(_))) ; true ), asserta(extremum(n(Expr))). tighten(min, E, V) :- E #< V. tighten(max, E, V) :- E #> V. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% all_different(+Vars) % % Like all_distinct/1, but with weaker propagation. all_different(Ls) :- fd_must_be_list(Ls, all_different(Ls)-1), maplist(fd_variable, Ls), Orig = original_goal(_, all_different(Ls)), new_queue(Q0), phrase((all_different(Ls, [], Orig),do_queue), [Q0], _). 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(+Vars). % % True iff Vars are pairwise distinct. 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, all_distinct(Ls)-1), maplist(fd_variable, Ls), make_propagator(pdistinct(Ls), Prop), new_queue(Q0), phrase((distinct_attach(Ls, Prop, []),trigger_prop(Prop),do_queue), [Q0], _), variables_same_queue(Ls). %% nvalue(?N, +Vars). % % True if N is the number of distinct values taken by Vars. Vars is a % list of domain variables, and N is a domain variable. Can be % thought of as a relaxed version of all_distinct/1. nvalue(N, Vars) :- fd_must_be_list(Vars), maplist(fd_variable, Vars), length(Vars, Len), N in 0..Len, zero_or_more(Vars, N), propagator_init_trigger(Vars, pnvalue(N, Vars)). zero_or_more([], 0). zero_or_more([_|_], N) :- N #> 0. %% 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(ground, 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], Inf0, Sup0, Inf, Sup) ++> { fd_get(V, _, Inf1, Sup1, _) }, ( Inf1 = n(NInf) -> ( C < 0 -> Sup2 is Sup0 + C*NInf ; Inf2 is Inf0 + C*NInf ) ; ( C < 0 -> Sup2 = Sup0, []+[C*V] ; Inf2 = Inf0, [C*V]+[] ) ), ( Sup1 = n(NSup) -> ( C < 0 -> Inf2 is Inf0 + C*NSup ; Sup2 is Sup0 + C*NSup ) ; ( C < 0 -> Inf2 = Inf0, [C*V]+[] ; Sup2 = Sup0, []+[C*V] ) ), sum_finite_domains(Cs, Vs, 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). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Parsing a CLP(ℤ) 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_clpz(E, R, [g(cyclic_term(E)) => [g(domain_error(clpz_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)], #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(pplus(A,R,0))], 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(ptimes(R, B, A))], m(A//B) => [g(B #\= 0), p(ptzdiv(A, B, R))], m(A div B) => [g(#R #= (A - (A mod B)) // B)], m(A^B) => [p(pexp(A, B, R))], m(sign(A)) => [g(R in -1..1), p(psign(A, R))], % bitwise operations m(\A) => [p(pfunction(\, A, R))], m(msb(A)) => [p(pfunction(msb, A, R))], m(lsb(A)) => [p(pfunction(lsb, A, R))], m(popcount(A)) => [p(ppopcount(A, R))], m(A< [p(pfunction(<<, A, B, R))], m(A>>B) => [p(pfunction(>>, A, B, R))], m(A/\B) => [p(pfunction(/\, A, B, R))], m(A\/B) => [p(pfunction(\/, A, B, R))], m(xor(A, B)) => [p(pxor(A, B, R))], g(true) => [g(domain_error(clpz_expression, E))] ]). non_monotonic(X) :- ( \+ fd_var(X), monotonic -> instantiation_error(X) ; true ). % Here, we compile the committed choice language to a single % predicate, parse_clpz/2. make_parse_clpz(Clauses) :- parse_clpz_clauses(Clauses0), maplist(goals_goal, Clauses0, Clauses). goals_goal((Head :- Goals), (Head :- Body)) :- list_goal(Goals, Body). parse_clpz_clauses(Clauses) :- parse_clpz(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_clpz(Head, R) :- Goals). parse_condition(g(Goal), E, E) --> [Goal, !]. parse_condition(?(E), _, ?(E)) --> [!]. 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_clpz(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(Prop0)) --> { term_variables(Prop0, Vs), morphing_propagator(Prop0, Prop, _) }, [make_propagator(Prop, P), new_queue(Q0), phrase(init_propagator_(Vs, P), [Q0], [Q]), variables_same_queue(Vs), trigger_once_(P, Q)]. morphing(pplus). morphing(ptimes). morphing(pexp). morphing(ptzdiv). morphing_propagator(P0, P, Target) :- P0 =.. [F|Args0], ( morphing(F) -> append(Args0, [Last], Args), Target = p(Last) ; Args = Args0, Target = none ), P =.. [F|Args]. morph_into_propagator(MState, Vs, P0, Morph) --> kill(MState), { morphing_propagator(P0, P, _), make_propagator(P, Morph) }, init_propagator_(Vs, Morph), trigger_prop(Morph). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ?- use_module(library(lists)), use_module(library(format)), clpz:parse_clpz_clauses(Clauses), maplist(portray_clause, Clauses). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% trigger_once(Prop) :- new_queue(Q), trigger_once_(Prop, Q). trigger_once_(Prop, Q) :- phrase((trigger_prop(Prop),do_queue), [Q], _). 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), new_queue(Q0), phrase(init_propagator_(Vs, Prop), [Q0], [Q]), variables_same_queue(Vs), trigger_once_(Prop, Q) }. variables_same_queue(Vs0) :- include(var, Vs0, Vs), ( Vs == [] -> true ; maplist(variable_queue, Vs, Qs0), sort(Qs0, [Q|Qs]), Q =.. [_|Args], append_queues_(Qs, Args, [append,append,append,ignore]), maplist(clear_queue, Qs), maplist(=(Q), Qs) ). append_queues_([], _, _). append_queues_([Q|Qs], Args0, Is) :- Q =.. [_|Args], maplist(append_queue, Is, Args0, Args), append_queues_(Qs, Args, Is). append_queue(ignore, _, _). append_queue(append, Q0, Q) :- ( get_atts(Q0, queue(Ls0,Ls)) -> ( get_atts(Q, queue(Ms0,Ms)) -> Ls = Ms0, put_atts(Q0, queue(Ls0,Ms)) ; true ) ; ( get_atts(Q, queue(Ms0,Ms)) -> put_atts(Q0, queue(Ms0,Ms)) ; true ) ). clear_queue(queue(Goals,Fast,Slow,Aux)) :- put_atts(Goals, -queue(_,_)), put_atts(Fast, -queue(_,_)), put_atts(Slow, -queue(_,_)), put_atts(Aux, -disabled). variable_queue(Var, Q) :- get_attr(Var, clpz, Attr), Attr = clpz_attr(_Left,_Right,_Spread,_Dom,_Ps,Q). 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) :- new_queue(Q), phrase((geq(A, B),do_queue), [Q], _). 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) ) ) ; { fd_get(B, BD, BPs) } -> { domain_remove_greater_than(BD, A, BD1) }, fd_put(B, BD1, BPs) ; 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_clpz(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(#>=, clpz_geq_). match_expand(#=, clpz_equal_). match_expand(#\=, clpz_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)) #>= any(Z)) => [d(X, X1), d(Y, Y1), d(Z, Z1), g((abs(#A)#= #B,Y1+A#=X1,Z1#== 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(Y)) => [p(pplus(X,Y,0))], m_c(any(X) #= any(Y), left_right_linsum_const(X, Y, Cs, Vs, S)) => [g(scalar_product_(#=, Cs, Vs, S))], m_c(var(X) #= abs(var(Y)) + any(V0), X == Y) => [d(V0,V),p(x_eq_abs_plus_v(X,V))], m_c(var(X) #= abs(var(Y)) - any(V0), X == Y) => [d(-V0,V),p(x_eq_abs_plus_v(X,V))], 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)) => [p(pneq(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(clpz_expression, X) ; cyclic_term(Y) -> domain_error(clpz_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) ; 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_clpz(X, Y)]. match_goal(g(Goal), _) --> [Goal]. match_goal(p(Prop0), _) --> { term_variables(Prop0, Vs), morphing_propagator(Prop0, Prop, _) }, [make_propagator(Prop, P), new_queue(Q0), phrase(init_propagator_(Vs, P), [Q0], [Q]), variables_same_queue(Vs), trigger_once_(P, Q)]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% #>=(?X, ?Y) % % Same as Y #=< X. When reasoning over integers, replace (>=)/2 by (#>=)/2 % to obtain more general relations. X #>= Y :- clpz_geq(X, Y). clpz_geq(X, Y) :- clpz_geq_(X, Y), reinforce(X), reinforce(Y). %% #=<(?X, ?Y) % % The arithmetic expression X is less than or equal to Y. When % reasoning over integers, replace (=<)/2 by (#=<)/2 to obtain more % general relations. X #=< Y :- Y #>= X. %% #=(?X, ?Y) % % The arithmetic expression X equals Y. When reasoning over integers, % replace `(is)/2` by `(#=)/2` to obtain more general relations. X #= Y :- clpz_equal(X, Y). clpz_equal(X, Y) :- clpz_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), !, \+ monotonic }. expr_conds(E, E) --> { integer(E) }. 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)]. % Bitwise operations, added to make CLP(ℤ) usable in more cases expr_conds(\ A0, \ A) --> expr_conds(A0, A). expr_conds(A0< 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). expr_conds(xor(A0,B0), xor(A,B)) --> expr_conds(A0, A), expr_conds(B0, B). % expr_conds(lsb(A0), lsb(A)) --> expr_conds(A0, A). % expr_conds(msb(A0), msb(A)) --> expr_conds(A0, A). expr_conds(popcount(A0), Count) --> expr_conds(A0, A), [I is A, arithmetic:popcount(I, Count)]. no_popcount_t(Gs, T) :- ( member(arithmetic:popcount(_, _), Gs) -> T = false ; T = true ). clpz_expandable(_ in _). clpz_expandable(_ #= _). clpz_expandable(_ #>= _). clpz_expandable(_ #=< _). clpz_expandable(_ #> _). clpz_expandable(_ #< _). clpz_expandable(_ #\= _). clpz_expandable(_ #<==> _). clpz_expansion(Var in Dom, In) :- ( ground(Dom), Dom = L..U, integer(L), integer(U) -> expansion_simpler( ( integer(Var) -> between:between(L, U, Var) ; clpz:clpz_in(Var, Dom) ), In) ; In = clpz:clpz_in(Var, Dom) ). clpz_expansion(A #<==> B, Reif) :- nonvar(A), A =.. [F0,X0,Y0], clpz_builtin(F0, F), phrase(expr_conds(X0, X), Cs0, Cs), phrase(expr_conds(Y0, Y), Cs), list_goal(Cs0, Cond), Expr =.. [F,X,Y], expansion_simpler(( Cond, ( var(B) ; integer(B), clpz:between(0, 1, B) ) -> ( Expr -> B = 1 ; B = 0 ) ; clpz:reify(A, RA), clpz:reify(B, RA) ), Reif). clpz_expansion(X0 #= Y0, Equal) :- phrase(expr_conds(X0, X), CsX), phrase(expr_conds(Y0, Y), CsY), list_goal(CsX, CondX), list_goal(CsY, CondY), no_popcount_t(CsY, YT), no_popcount_t(CsX, XT), expansion_simpler( ( CondX -> ( YT, var(Y) -> Y is X ; CondY -> X =:= Y ; T is X, clpz:clpz_equal(T, Y0) ) ; CondY -> ( XT, var(X) -> X is Y ; T is Y, clpz:clpz_equal(X0, T) ) ; clpz:clpz_equal(X0, Y0) ), Equal). clpz_expansion(X0 #>= Y0, Geq) :- 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, clpz:clpz_geq(T, Y0) ) ; CondY -> T is Y, clpz:clpz_geq(X0, T) ; clpz:clpz_geq(X0, Y0) ), Geq). clpz_expansion(X #=< Y, Leq) :- clpz_expansion(Y #>= X, Leq). clpz_expansion(X #> Y, Gt) :- clpz_expansion(X #>= Y+1, Gt). clpz_expansion(X #< Y, Lt) :- clpz_expansion(Y #> X, Lt). clpz_expansion(X0 #\= Y0, Neq) :- 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, clpz:clpz_neq(T, Y0) ) ; CondY -> T is Y, clpz:clpz_neq(X0, T) ; clpz:clpz_neq(X0, Y0) ), Neq). clpz_builtin(#=, =:=). clpz_builtin(#\=, =\=). clpz_builtin(#>, >). clpz_builtin(#<, <). clpz_builtin(#>=, >=). clpz_builtin(#=<, =<). 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((A0,B0), (A,B)) :- !, expansion_simpler(A0, A), expansion_simpler(B0, B). expansion_simpler(Var is Expr0, Goal) :- ground(Expr0), !, phrase(expr_conds(Expr0, Expr), Gs), ( maplist(call, Gs) -> Value is Expr, Goal = (Var = Value) ; Goal = false ). expansion_simpler(Var =:= Expr0, Goal) :- ground(Expr0), !, phrase(expr_conds(Expr0, Expr), Gs), ( maplist(call, Gs) -> Value is Expr, Goal = (Var =:= Value) ; Goal = false ). expansion_simpler(between: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)). % :- endif. is_false((A,B)) :- is_false(A) ; is_false(B). is_false(var(X)) :- nonvar(X). :- dynamic(goal_expansion/1). user:goal_expansion(Goal0, Goal) :- \+ goal_expansion(false), clpz_expandable(Goal0), clpz_expansion(Goal0, Goal). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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(#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) ). samsort(Ls0, Ls) :- maplist(as_key, Ls0, LKs0), keysort(LKs0, LKs), maplist(as_key, Ls, LKs). as_key(E, E-t). 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), samsort(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 ), ( K < 0 -> ( N =:= 1 -> R = 1 ; N =:= -1 -> odd(K), R = -1 ; false ) ; ( 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) ). floor_integer_log_b(N, B, Log0, Log) :- T is B^Log0, ( T > N -> Log is Log0 - 1 ; T =:= N -> Log = Log0 ; T < N, Log1 is Log0 + 1, floor_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) % % The arithmetic expressions X and Y evaluate to distinct integers. % When reasoning over integers, replace (=\=)/2 by (#\=)/2 to obtain more % general relations. X #\= Y :- clpz_neq(X, Y). % 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 ). neq_num(X, N) --> ( { fd_get(X, XD, XPs) } -> { domain_remove(XD, N, XD1) }, fd_put(X, XD1, XPs) ; X =\= N ). %% #>(?X, ?Y) % % Same as Y #< X. X #> Y :- X #>= Y + 1. %% #<(?X, ?Y) % % The arithmetic expression X is less than Y. When reasoning over % integers, replace `(<)/2` by `(#<)/2` to obtain more general relations. % % In addition to its regular use in tasks that require it, this % constraint can also be useful to eliminate uninteresting symmetries % from a problem. For example, all possible matches between pairs % built from four players in total: % % ``` % ?- Vs = [A,B,C,D], Vs ins 1..4, % all_different(Vs), % A #< B, C #< D, A #< C, % findall(pair(A,B)-pair(C,D), label(Vs), Ms). % Ms = [ pair(1, 2)-pair(3, 4), % pair(1, 3)-pair(2, 4), % pair(1, 4)-pair(2, 3)]. % ``` 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). %% #<==>(?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: % % ``` % 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). %% #==>(?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). 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(V in I, V) :- var(V), integer(I). 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(_ in I) --> [I]. 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_clpz/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, and ls(Ls) means the literals Ls occur in the described list. When a constraint becomes entailed or subexpressions become undefined, created auxiliary constraints are killed, and the "clpz" 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. We cannot use a skeleton propagator for (/)/2, since (/)/2 can fail in cases such as 0 #==> X #= 1/2, where we expect success. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ parse_reified(E, R, D, [g(cyclic_term(E)) => [g(domain_error(clpz_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)], #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(pplus(A,R,0)), 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)) => [d(D), g(#R#>=0), p(pabs(A, R)), a(A,R)], m(A^B) => [d(D1), p(preified_exp(A,B,D2,R)), p(reified_and(D1,[],D2,[],D)),a(D2),a(A,B,R)], m(A/B) => [d(D1), p(preified_slash(A,B,D2,R)), p(reified_and(D1,[],D2,[],D)),a(D2),a(A,B,R)], m(A div B) => [d(D1), g(phrase(parse_reified_clpz(((A-(A mod B)) // B), R, D2), Ps)), ls(Ps), p(reified_and(D1,[],D2,[],D)),a(D2),a(A,B,R)], m(A//B) => [skeleton(A,B,D,R,ptzdiv)], m(A mod B) => [skeleton(A,B,D,R,pmod)], m(A rem B) => [skeleton(A,B,D,R,prem)], % bitwise operations m(\A) => [function(D,\,A,R)], m(msb(A)) => [g(#A#>0) ,function(D,msb,A,R)], m(lsb(A)) => [g(#A#>0), function(D,lsb,A,R)], m(popcount(A)) => [d(D), p(ppopcount(A, R)), a(A,R)], m(sign(A)) => [d(D), p(psign(A, R)), a(A,R)], m(A< [function(D,<<,A,B,R)], m(A>>B) => [function(D,>>,A,B,R)], m(A/\B) => [function(D,/\,A,B,R)], m(A\/B) => [function(D,\/,A,B,R)], m(xor(A, B)) => [skeleton(A,B,D,R,pxor)], g(true) => [g(domain_error(clpz_expression, E))]] ). % Again, we compile this to a predicate, parse_reified_clpz//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_clpz(Head, R, D) --> Goals). reified_condition(g(Goal), E, E, []) --> [{Goal}, !]. reified_condition(?(E), _, ?(E), []) --> [!]. 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_clpz(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, Prop0), _) --> { morphing_propagator(Prop0, Prop, Target) }, [{make_propagator(Prop, P)}], target_propagator(Target), parse_init_dcg(Vs, P), [{variables_same_queue(Vs), 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(function(D,Op,A,B,R), Ds) --> reified_goals([d(D),p(pfunction(Op,A,B,R)),a(A,B,R)], Ds). reified_goal(function(D,Op,A,R), Ds) --> reified_goals([d(D),p(pfunction(Op,A,R)),a(A,R)], Ds). reified_goal(skeleton(A,B,D,R,F), Ds) --> { Prop0 =.. [F,X,Y,Z], morphing_propagator(Prop0, Prop, Target) }, target_propagator(Target), 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). 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]]. reified_goal(ls(Ls), _) --> [Ls]. target_propagator(p(Prop)) --> [[p(Prop)]]. target_propagator(none) --> []. parse_init_dcg([], _) --> []. parse_init_dcg([V|Vs], P) --> [{init_propagator(V, P)}], parse_init_dcg(Vs, P). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ?- use_module(library(lists)), use_module(library(format)), clpz: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(clpz_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(#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_(#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), seq(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_clpz(L, LR, LD), parse_reified_clpz(R, RR, RD)), Ps), Prop =.. [Functor,LD,LR,RD,RR,Ps,B] }, seq(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] }, seq(Ps1), seq(Ps2), propagator_init_trigger([LR,RR,B], Prop), a(LR, RR, B). 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, clpz_relation, Relation), make_propagator(reified_tuple_in(Tuple, R, B), Prop), new_queue(Q0), phrase((tuple_freeze_(Tuple, Prop), init_propagator_([B], Prop), do_queue), [Q0], _). 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) :- ( propagator_state(Prop, State), State == dead -> true ; 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(clpz_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), 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, clpz, Attr) -> Attr = clpz_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). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Constraint 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. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ fd_put(X, Dom, Ps) --> put_terminating(X, Dom, Ps). fd_put(X, Dom, Ps) :- new_queue(Q), phrase((put_terminating(X, Dom, Ps), % { portray_clause(done_terminating) }, do_queue), [Q], _). put_terminating(X, Dom, Ps) --> Dom \== empty, ( Dom = from_to(F, F) -> queue_goal(F = n(X)) ; ( { get_attr(X, clpz, Attr) } -> { Attr = clpz_attr(Left,Right,Spread,OldDom, _OldPs,Q), put_attr(X, clpz, clpz_attr(Left,Right,Spread,Dom,Ps,Q)) }, ( { OldDom == Dom } -> [] ; { ( Left == (.) -> Bounded = yes ; domain_infimum(Dom, Inf), domain_supremum(Dom, Sup), ( Inf = n(_), Sup = n(_) -> Bounded = yes ; Bounded = no ) ) }, ( { Bounded == yes } -> { put_attr(X, clpz, clpz_attr(.,.,.,Dom,Ps,Q)) }, 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, clpz, clpz_attr(LeftP,RightP,SpreadP,Dom,Ps,Q)) }, ( { RightP == yes, Right = yes } -> [] ; { LeftP == yes, Left = yes } -> [] ; { SpreadP == yes, Spread = yes } -> [] ; trigger_props(Ps, X, OldDom, Dom) ) ) ) ; { var(X) } -> { new_queue(Q), put_attr(X, clpz, clpz_attr(no,no,no,Dom,Ps,Q)) } ; [] ) ). new_queue(queue(_Goals,_Fast,_Slow,_Aux)). queue_goal(Goal) --> insert_queue(Goal, 1). queue_fast(Prop) --> insert_queue(Prop, 2). queue_slow(Prop) --> insert_queue(Prop, 3). insert_queue(Element, Which) --> state(Queue), { arg(Which, Queue, Arg), ( get_atts(Arg, queue(Head0,Tail0)) -> Head = Head0, Tail0 = [Element|Tail] ; Head = [Element|Tail] ), put_atts(Arg, queue(Head,Tail)) }. 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). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - All relevant constraints get a propagation opportunity whenever a new constraint is posted. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ reinforce(X) :- term_variables(X, Vs), maplist(reinforce_, Vs). 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, clpz, Attr) -> Attr = clpz_attr(_,_,_,OldDom, _OldPs,Q), put_attr(X, clpz, clpz_attr(no,no,no,Dom,Ps,Q)), %format("putting dom: ~w\n", [Dom]), ( OldDom == Dom -> true ; new_queue(Q), % TODO: queue? phrase((trigger_props(Ps, X, OldDom, Dom), do_queue), [Q], _) ) ; var(X) -> %format('\t~w in ~w .. ~w\n',[X,L,U]), new_queue(Q), put_attr(X, clpz, clpz_attr(no,no,no,Dom,Ps,Q)) ; 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 } -> [] ; trigger_props_(Bs) ) ; trigger_props_(Bs) ) ; [] ), trigger_props_(Os). trigger_props(fd_props(Gs,Bs,Os), X) --> trigger_props_(Os), trigger_props_(Bs), ( { ground(X) } -> trigger_props_(Gs) ; [] ). 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(P) :- trigger_once(P). trigger_prop(Propagator) --> { propagator_state(Propagator, State) }, ( { State == dead } -> [] ; { get_attr(State, clpz_aux, queued) } -> [] ; { bb_get('$clpz_current_propagator', C), C == State } -> [] ; % passive %{ format("triggering: ~w\n", [Propagator]) }, { put_attr(State, clpz_aux, queued) }, ( { arg(1, Propagator, C), functor(C, F, _), global_constraint(F) } -> queue_slow(Propagator) ; queue_fast(Propagator) ) ). all_propagators(fd_props(Gs,Bs,Os)) --> propagators_(Gs), propagators_(Bs), propagators_(Os). propagators_([]) --> []. propagators_([P|Ps]) --> propagator_(P), propagators_(Ps). propagator_(Propagator) --> { propagator_state(Propagator, State) }, ( { State == dead } -> [] ; { get_attr(State, clpz_aux, queued) } -> [] ; % passive % format("triggering: ~w\n", [Propagator]), [clpz:trigger_prop(Propagator)] ). % DCG variants kill(State) --> { kill(State) }. kill(State, Ps) --> { kill(State, Ps) }. T =.. Ls --> { T =.. Ls }. A = A --> []. A == B --> { A == B }. A \== B --> { A \== B }. integer(I) --> { integer(I) }. nonvar(X) --> { nonvar(X) }. var(V) --> { var(V) }. ground(T) --> { ground(T) }. true --> []. false --> { false }. X >= Y --> { X >= Y }. X =< Y --> { X =< Y }. X =:= Y --> { X =:= Y }. X =\= Y --> { X =\= Y }. X is E --> { X is E }. X > Y --> { X > Y }. X < Y --> { X < Y }. /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Duo DCG variants - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ A = B ++> { A = B }. A < B ++> { A < B }. A is B ++> { A is B }. kill(State) :- del_attr(State, clpz_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, clpz). kill_entailed(a(X,B)) :- ( X == B -> true ; del_attr(B, clpz) ). kill_entailed(a(X,Y,B)) :- ( X == B -> true ; Y == B -> true ; del_attr(B, clpz) ). no_reactivation(rel_tuple(_,_)). no_reactivation(pdistinct(_)). no_reactivation(pnvalue(_)). no_reactivation(pgcc(_,_,_)). no_reactivation(pgcc_single(_,_)). %no_reactivation(scalar_product(_,_,_,_)). activate_propagator(propagator(P,State)) --> % { portray_clause(running(P)) }, ( State == dead -> [] ; { del_attr(State, clpz_aux) }, ( { no_reactivation(P) } -> { bb_b_put('$clpz_current_propagator', State) }, run_propagator(P, State), { bb_b_put('$clpz_current_propagator', []) } ; run_propagator(P, State) ) ). %do_queue --> print_queue, false. do_queue --> ( queue_enabled -> ( queue_get_goal(Goal) -> { call(Goal) }, do_queue ; queue_get_fast(Fast) -> activate_propagator(Fast), do_queue ; queue_get_slow(Slow) -> activate_propagator(Slow), do_queue ; true ) ; true ). :- meta_predicate(ignore(0)). ignore(Goal) :- ( Goal -> true ; true ). print_queue --> state(queue(Goal,Fast,Slow,_)), { ignore(get_atts(Goal, queue(GHs,_))), ignore(get_atts(Fast, queue(FHs,_))), ignore(get_atts(Slow, queue(SHs,_))), format("Current queue:~n goal: ~q~n fast: ~q~n slow: ~q~n~n", [GHs,FHs,SHs]) }. queue_get_goal(Goal) --> queue_get_arg(1, Goal). queue_get_fast(Fast) --> queue_get_arg(2, Fast). queue_get_slow(Slow) --> queue_get_arg(3, Slow). queue_get_arg(Which, Element) --> state(Queue), { queue_get_arg_(Queue, Which, Element) }. queue_get_arg_(Queue, Which, Element) :- arg(Which, Queue, Arg), get_atts(Arg, queue([Element|Elements],Tail)), ( var(Elements) -> put_atts(Arg, -queue(_,_)) ; put_atts(Arg, queue(Elements,Tail)) ). queue_enabled --> state(queue(_,_,_,Aux)), { \+ get_atts(Aux, disabled) }. disable_queue --> state(queue(_,_,_,Aux)), { put_atts(Aux, disabled) }. enable_queue --> state(queue(_,_,_,Aux)), { put_atts(Aux, -disabled) }. portray_propagator(propagator(P,_), F) :- functor(P, F, _). init_propagator_([], _) --> []. init_propagator_([V|Vs], Prop) --> ( { fd_get(V, Dom, Ps0) } -> { insert_propagator(Prop, Ps0, Ps) }, fd_put(V, Dom, Ps) ; [] ), init_propagator_(Vs, Prop). 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_eq_abs_plus_v, 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(pnvalue). 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), lex_chain(Lss)-1, 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. % % ``` % 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), new_queue(Q0), phrase(tuples_relation(Tuples, Relation), [Q0], [Q]), append(Tuples, Vs), variables_same_queue(Vs), phrase(do_queue, [Q], _). tuples_relation([], _) --> []. tuples_relation([Tuple|Tuples], Relation) --> { relation_unifiable(Relation, Tuple, Us, _, _) }, ( ground(Tuple) -> { memberchk(Tuple, Relation) } ; tuple_domain(Tuple, Us), ( Tuple = [_,_|_] -> tuple_freeze(Tuple, Us) ; [] ) ), tuples_relation(Tuples, Relation). list_first_rest([L|Ls], L, Ls). tuple_domain([], _) --> []. tuple_domain([T|Ts], Relation0) --> { maplist(list_first_rest, Relation0, Firsts, Relation1) }, ( Firsts = [Unique] -> T = Unique ; ( var(T) -> { list_to_domain(Firsts, FDom), fd_get(T, TDom, TPs), domains_intersection(TDom, FDom, TDom1) }, fd_put(T, TDom1, TPs) ; [] ) ), tuple_domain(Ts, Relation1). tuple_freeze(Tuple, Relation) --> ( ground(Tuple) -> { memberchk(Tuple, Relation) } ; { put_attr(R, clpz_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) ; [] ), 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). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %run_propagator(P, _) --> { portray_clause(run_propagator(P)), false }. % 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(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(pnvalue(N, Vars), _MState) --> { propagate_nvalue(N, Vars) }. 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) ; [] ) ; { 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(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) ; [] ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(rel_tuple(R, Tuple), MState) --> { get_attr(R, clpz_relation, Relation) }, ( { ground(Tuple) } -> kill(MState), { del_attr(R, clpz_relation), memberchk(Tuple, Relation) } ; { relation_unifiable(Relation, Tuple, Us, false, Changed), Us = [_|_] }, ( { Tuple = [First,Second], ( ground(First) ; ground(Second) ) } -> kill(MState) ; [] ), ( { Us = [Single] } -> kill(MState), { del_attr(R, clpz_relation) }, Single = Tuple ; { Changed } -> { put_attr(R, clpz_relation, Us) }, disable_queue, tuple_domain(Tuple, Us), enable_queue ; [] ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 -> [] ; S_J + D_J =< S_I -> [] ; 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) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % X #\= Y 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) ; [] ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 } ; X == Y -> kill(MState) ; { 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) ; [] ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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) ; [] ). % X #= abs(X) + V run_propagator(x_eq_abs_plus_v(X,V), MState) --> ( nonvar(V) -> ( V =:= 0 -> kill(MState), { X in 0..sup } ; V < 0 -> kill(MState), { X #= V / 2 } ; false % V > 0 ) ; nonvar(X) -> kill(MState), { V #= X - abs(X) } ; 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) ; [] ) ; nonvar(Y) -> ( nonvar(Z) -> kill(MState), YZ is Y + Z, neq_num(X, YZ) ; Y =:= 0 -> kill(MState), { neq(X, Z) } ; [] ) ; 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) ; [] ), ( { 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) ; [] ) ; [] ) ) ). 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 ; { duophrase(sum_finite_domains(Cs, Vs, 0, 0, Inf, Sup), Infs, Sups) }, 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) ; 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 } ; { duophrase(sum_finite_domains(Cs, Vs, 0, 0, Inf, Sup), Infs, Sups) }, % { 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,Morph), 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) ; [] ) ) ; nonvar(Y) -> run_propagator(pplus(Y,X,Z,Morph), 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) ; [] ) ) ; ( X == Y -> morph_into_propagator(MState, [X,Z], ptimes(2,X,Z), Morph) ; 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) ; [] ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(ptimes(X,Y,Z,Morph), 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,Morph), 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 ; [] ) ; neq_num(X, 0), neq_num(Y, 0) ) ) ; ( X == Y -> morph_into_propagator(MState, [X,Z], pexp(X,2,Z), Morph) ; { 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) } -> [] ; neq_num(X, 0), neq_num(Y, 0) ) ; [] ) ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % X // Y = Z (round towards zero) run_propagator(ptzdiv(X,Y,Z,Morph), 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 -> morph_into_propagator(MState, [X,Z], pplus(X,Z,0), Morph) ; { 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) ) ). %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Z = X mod Y run_propagator(pmod(X,Y,Z), MState) --> ( Y == 0 -> false ; Y == Z -> false ; X == Y -> kill(MState), queue_goal(Z = 0) ; true ), ( nonvar(X), nonvar(Y) -> kill(MState), Z is X mod Y ; nonvar(Y), nonvar(Z) -> ( Y > 0 -> Z >= 0, Z < Y ; Z =< 0, Z > Y % Y < 0 ), ( { fd_get(X, _, n(XL), _, _) } -> ( (XL - Z) mod Y =\= 0 -> XMin is Z + Y * ((XL - Z) div Y + 1) ; XMin is XL ), { fd_get(X, XD0, XPs), domain_remove_smaller_than(XD0, XMin, XD2) }, fd_put(X, XD2, XPs) % queue_goal(X #>= XMin) ; true ), ( { fd_get(X, _, _, n(XU), _) } -> XMax is Z + Y * ((XU - Z) div Y), { fd_get(X, XD1, XPs), domain_remove_greater_than(XD1, XMax, XD3) }, fd_put(X, XD3, XPs) % queue_goal(X #=< XMax) ; true ) ; nonvar(Z), nonvar(X) -> ( Z > 0 -> ( X < 0 -> true ; X >= Z, % due to X = Z+Y*_ and Y > Z ( X-Z > 0 -> X-Z > Z ; true ) ) ; Z < 0 -> ( X > 0 -> true ; X =< Z, % due to X = Z+Y*_ and Y < Z ( X-Z < 0 -> X-Z < Z ; true ) ) ; Z =:= 0 % Multiple solutions so do nothing special. ), ( { fd_get(Y, _, _, n(YU), _), YU < X, X =< 0 } -> kill(MState), Z =:= X ; { fd_get(Y, _, n(YL), _, _), YL > X, X >= 0 } -> kill(MState), Z =:= X ; ( Z > 0, X < 0 -> { fd_get(Y, YD, YPs), YMin is Z+1, YMax is Z-X, domain_remove_smaller_than(YD, YMin, YD1), domain_remove_greater_than(YD1, YMax, YD2) }, fd_put(Y, YD2, YPs) % queue_goal((Y #> Z, Y #=< Z-X)) ; Z < 0, X > 0 -> { fd_get(Y, YD, YPs), YMax is Z-1, YMin is Z-X, domain_remove_greater_than(YD, YMax, YD1), domain_remove_smaller_than(YD1, YMin, YD2) }, fd_put(Y, YD2, YPs) % queue_goal((Y #< Z, Y #>= Z-X)) ; Z > 0 -> { fd_get(Y, YD, YPs), YMin is Z + 1, domain_remove_smaller_than(YD, YMin, YD1) }, fd_put(Y, YD1, YPs) % queue_goal(Y #> Z) ; Z < 0 -> { fd_get(Y, YD, YPs), YMax is Z - 1, domain_remove_greater_than(YD, YMax, YD1) }, fd_put(Y, YD1, YPs) % queue_goal(Y #< Z) ; Z =:= 0, ( X =:= 0 -> kill(MState) % trivial ; % only 4 solutions {-abs(X),-1,1,abs(X)} { YL is -abs(X), YU is abs(X), fd_get(Y, YD0, YPs), domain_remove_smaller_than(YD0, YL, YD1), domain_remove_greater_than(YD1, YU, YD) }, fd_put(Y, YD, YPs) ) ) ) ; run_propagator(pmodz(X,Y,Z), MState), run_propagator(pmody(X,Y,Z), MState), true ). run_propagator(pmodz(X,Y,Z), MState) --> ( nonvar(Z) -> true % Nothing to do. ; nonvar(X) -> ( X =:= 0 -> kill(MState), queue_goal(Z = X) ; ( X > 0 -> ( { fd_get(Y, _, n(YL), _, _), YL > X } -> kill(MState), queue_goal(Z = X) ; { fd_get(Z, ZD0, ZPs), domain_remove_greater_than(ZD0, X, ZD2) }, fd_put(Z, ZD2, ZPs) % queue_goal(Z #=< X) ) ; X < 0, ( { fd_get(Y, _, _, n(YU), _), YU < X } -> kill(MState), queue_goal(Z = X) ; { fd_get(Z, ZD0, ZPs), domain_remove_smaller_than(ZD0, X, ZD2) }, fd_put(Z, ZD2, ZPs) % queue_goal(Z #>= X) ) ), ( { fd_get(Y, _, n(YL), n(YU), _), YL > 0 } -> ZMax is YU - 1, { fd_get(Z, ZD1, ZPs), domain_remove_smaller_than(ZD1, 0, ZD3), domain_remove_greater_than(ZD3, ZMax, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in 0..ZMax) ; { fd_get(Y, _, n(YL), n(YU), _), YU < 0 } -> ZMin is YL + 1, { fd_get(Z, ZD1, ZPs), domain_remove_greater_than(ZD1, 0, ZD3), domain_remove_smaller_than(ZD3, ZMin, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in ZMin..0) ; true ) ) ; nonvar(Y) -> ( abs(Y) =:= 1 -> kill(MState), queue_goal(Z = 0) ; Y < 0 -> ( { fd_get(X, _, n(XL), n(XU), _), XU =< 0, Y < XL } -> kill(MState), queue_goal(Z = X) ; ZMin is Y + 1, { fd_get(Z, ZD1, ZPs), domain_remove_greater_than(ZD1, 0, ZD3), domain_remove_smaller_than(ZD3, ZMin, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in ZMin..0) ) ; Y > 0, ( { fd_get(X, _, n(XL), n(XU), _), XL >= 0, Y > XU } -> kill(MState), queue_goal(Z = X) ; ZMax is Y - 1, { fd_get(Z, ZD1, ZPs), domain_remove_smaller_than(ZD1, 0, ZD3), domain_remove_greater_than(ZD3, ZMax, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in 0..ZMax) ) ) ; ( { fd_get(X, _, n(XL), n(XU), _), XL >= 0, fd_get(Y, _, n(YL), _, _), XU < YL } -> kill(MState), queue_goal(Z = X) ; { fd_get(X, _, n(XL), n(XU), _), XU =< 0, fd_get(Y, _, _, n(YU), _), XL > YU } -> kill(MState), queue_goal(Z = X) ; ( { fd_get(X, _, n(XL), n(XU), _), XL >= 0 } -> { fd_get(Z, ZD0, ZPs), domain_remove_greater_than(ZD0, XU, ZD2) }, fd_put(Z, ZD2, ZPs) % queue_goal(Z #=< XU) ; { fd_get(X, _, n(XL), n(XU), _), XU =< 0 } -> { fd_get(Z, ZD0, ZPs), domain_remove_smaller_than(ZD0, XL, ZD2) }, fd_put(Z, ZD2, ZPs) % queue_goal(Z #>= XL) ; true ), ( { fd_get(Y, _, n(YL), n(YU), _), YL > 0 } -> ZMax is YU - 1, { fd_get(Z, ZD1, ZPs), domain_remove_smaller_than(ZD1, 0, ZD3), domain_remove_greater_than(ZD3, ZMax, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in 0..ZMax) ; { fd_get(Y, _, n(YL), n(YU), _), YU < 0 } -> ZMin is YL + 1, { fd_get(Z, ZD1, ZPs), domain_remove_greater_than(ZD1, 0, ZD3), domain_remove_smaller_than(ZD3, ZMin, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in ZMin..0) ; { fd_get(Y, _, n(YL), n(YU), _), YL < 0, YU > 0 } -> ZMin is YL + 1, ZMax is YU - 1, { fd_get(Z, ZD1, ZPs), domain_remove_greater_than(ZD1, ZMax, ZD3), domain_remove_smaller_than(ZD3, ZMin, ZD5) }, fd_put(Z, ZD5, ZPs) % queue_goal(Z in ZMin..ZMax) ; { fd_get(Y, _, _, n(YU), _), YU > 0 } -> { fd_get(Z, ZD1, ZPs), ZMax is YU - 1, domain_remove_greater_than(ZD1, ZMax, ZD3) }, fd_put(Z, ZD3, ZPs) % queue_goal(Z #< YU) ; { fd_get(Y, _, n(YL), _, _), YL < 0 } -> { fd_get(Z, ZD1, ZPs), ZMin is YL + 1, domain_remove_smaller_than(ZD1, ZMin, ZD3) }, fd_put(Z, ZD3, ZPs) % queue_goal(Z #> YL) ; true ) ) ). run_propagator(pmody(_X,Y,Z), _MState) --> ( nonvar(Y) -> true % Nothing to do. % ; nonvar(X) -> true ; nonvar(Z) -> ( Z > 0 -> { fd_get(Y, YD, YPs), YMin is Z + 1, domain_remove_smaller_than(YD, YMin, YD1) }, fd_put(Y, YD1, YPs) % queue_goal(Y #> Z) ; Z < 0 -> { fd_get(Y, YD, YPs), YMax is Z - 1, domain_remove_greater_than(YD, YMax, YD1) }, fd_put(Y, YD1, YPs) % queue_goal(Y #< Z) ; Z =:= 0 % Multiple solutions so do nothing special. ) ; ( { fd_get(Z, _, n(ZL), _, _), ZL > 0 } -> { fd_get(Y, YD, YPs), YMin is ZL + 1, domain_remove_smaller_than(YD, YMin, YD1) }, fd_put(Y, YD1, YPs) % queue_goal(Y #> ZL) ; { fd_get(Z, _, _, n(ZU), _), ZU < 0 } -> { fd_get(Y, YD, YPs), YMax is ZU - 1, domain_remove_greater_than(YD, YMax, YD1) }, fd_put(Y, YD1, YPs) % queue_goal(Y #< ZU) ; true ) ). %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % 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), queue_goal(Z is max(X,Y)) ; nonvar(Z) -> ( Z =:= X -> kill(MState), queue_goal(X #>= Y) ; Z > X -> queue_goal(Z = Y) ; false % Z < X ) ; Y == Z -> kill(MState), queue_goal(Y #>= X) ; { fd_get(Y, _, YInf, YSup, _) }, ( { YInf cis_gt n(X) } -> queue_goal(Z = Y) ; { YSup cis_lt n(X) } -> queue_goal(Z = X) ; YSup = n(M) -> { fd_get(Z, ZD, ZPs), domain_remove_greater_than(ZD, M, ZD1) }, fd_put(Z, ZD1, ZPs) ; [] ) ) ; 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 XSup } -> kill(MState), queue_goal(Z = Y) ; { YSup cis_lt XInf } -> kill(MState), queue_goal(Z = X) ; { n(M) cis max(XSup, YSup) } -> { domain_remove_greater_than(ZD, M, ZD1) }, fd_put(Z, ZD1, ZPs) ; [] ) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 ) ; Y == Z -> kill(MState), queue_goal(Y #=< 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) ; [] ) ) ; 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 XInf } -> 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) ; [] ) ; [] ). %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Z = X ^ Y run_propagator(pexp(X,Y,Z,Morph), MState) --> ( X == 1 -> kill(MState), Z = 1 ; X == 0 -> queue_goal((Z in 0..1, Y #>= 0)), morph_into_propagator(MState, [Y,Z], reified_eq(1,Y,1,0,[],Z), Morph) ; Y == 0 -> kill(MState), Z = 1 ; Y == 1 -> kill(MState), Z = X ; Y == Z -> kill(MState), X = Y, queue_goal(X in -1\/1) ; nonvar(X) -> ( nonvar(Y) -> ( Y >= 0 -> true ; X =:= -1 ), kill(MState), Z is X^Y ; nonvar(Z) -> ( Z > 1 -> abs(X) > 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 ), ( { X > 0, fd_get(Z, _, _, n(ZMax), _), ZMax > 0 } -> { floor_integer_log_b(ZMax, X, 1, YCeil) }, queue_goal(Y in inf..YCeil) ; 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) } -> { fd_get(Z, ZD0, ZPs0), domain_remove_smaller_than(ZD0, 0, ZDG0) }, fd_put(Z, ZDG0, ZPs0) ; 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), Num1 >= 0 -> 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 ). %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Y = sign(X) run_propagator(psign(X,Y), MState) --> ( nonvar(X) -> kill(MState), queue_goal(Y is sign(X)) ; Y == -1 -> kill(MState), queue_goal(X #< 0) ; Y == 0 -> kill(MState), queue_goal(X = 0) ; Y == 1 -> kill(MState), queue_goal(X #> 0) ; { fd_get(X, _, XL, XU, _) }, ( { XL = n(L), L > 0 } -> kill(MState), queue_goal(Y = 1) ; { XU = n(U), U < 0 } -> kill(MState), queue_goal(Y = -1) ; true ) ). %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Y = popcount(X) run_propagator(ppopcount(X,Y), MState) --> ( nonvar(X) -> kill(MState), queue_goal(popcount(X, Y)) ; true ). %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Z = X xor Y run_propagator(pxor(X,Y,Z), MState) --> ( nonvar(X), nonvar(Y) -> kill(MState), Z is xor(X, Y) ; nonvar(Y), nonvar(Z) -> kill(MState), X is xor(Y, Z) ; nonvar(Z), nonvar(X) -> kill(MState), Y is xor(Z, X) ; X == Y -> kill(MState), queue_goal(Z = 0) ; Y == Z -> kill(MState), queue_goal(X = 0) ; Z == X -> kill(MState), queue_goal(Y = 0) ; X == 0 -> kill(MState), queue_goal(Y = Z) ; Y == 0 -> kill(MState), queue_goal(Z = X) ; Z == 0 -> kill(MState), queue_goal(X = Y) ; 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 = (>) ; [] ) ) ; nonvar(B) -> { fd_get(A, _, AL, AU, _) }, ( { AL cis_gt n(B) } -> kill(MState), Order = (>) ; { AU cis_lt n(B) } -> kill(MState), Order = (<) ; [] ) ; { fd_get(A, _, AL, AU, _), fd_get(B, _, BL, BU, _) }, ( { AL cis_gt BU } -> kill(MState), Order = (>) ; { AU cis_lt BL } -> kill(MState), Order = (<) ; [] ) ) ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 ; [] ) ; kill(MState), B = 0 ) ). run_propagator(reified_tuple_in(Tuple, R, B), MState) --> { get_attr(R, clpz_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 ; [] ) ) ). run_propagator(tuples_not_in(Tuples, Relation, B), MState) --> ( B == 0 -> kill(MState), { tuples_in_conjunction(Tuples, Relation, Conj), #\ Conj } ; [] ). run_propagator(kill_reified_tuples(B, Ps, Bs), _) --> ( B == 0 -> { maplist(kill_entailed, Ps), phrase(as(Bs), As), maplist(kill_entailed, As) } ; [] ). 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) } -> [] ; { fd_sup(V, sup) } ) ; [] ). % 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) } ; [] ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Propagators for arithmetic functions that only propagate functionally. These are currently the bitwise operations. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ run_propagator(pfunction(Op,A,B,R), MState) --> ( integer(A), integer(B) -> kill(MState), Expr =.. [Op,A,B], R is Expr ; [] ). run_propagator(pfunction(Op,A,R), MState) --> ( integer(A) -> kill(MState), ( Op == msb -> { msb(A, R) } ; Op == lsb -> { lsb(A, R) } ; Expr =.. [Op,A], R is Expr ) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 ; [] ) ) ; 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 ; [] ) ; 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 ; [] ) ) ; B =:= 0 -> { kill(MState), X #< Y } ; [] ) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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) } -> [] ; 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 ; [] ) ) ; B =:= 0 -> kill(MState), { X #\= Y } ; [] ) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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) } -> [] ; 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 ; [] ) ) ; B =:= 0 -> kill(MState), X = Y ; [] ) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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) } ) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(preified_slash(X, Y, D, R), MState) --> ( Y == 0 -> kill(MState), D = 0 ; Y == 1 -> kill(MState), D = 1, R = X ; nonvar(X), nonvar(Y) -> kill(MState), ( X mod Y =:= 0 -> D = 1, R is X // Y ; D = 0 ) ; D == 1 -> kill(MState), queue_goal(X/Y #= R) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run_propagator(preified_exp(X, Y, D, R), MState) --> ( X == 1 -> kill(MState), D = 1, R = 1 ; Y == 0 -> kill(MState), D = 1, R = 1 ; Y == 1 -> kill(MState), D = 1, R = X ; nonvar(X), nonvar(Y) -> kill(MState), ( ( abs(X) =:= 1 ; Y >= 0 ) -> D = 1, R is X^Y ; D = 0 ) ; D == 1 -> kill(MState), queue_goal(X^Y #= R) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% update_bounds(X, XD, XPs, XL, XU, NXL, NXU) --> ( NXL == XL, NXU == XU -> [] ; { 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) :- % use findall/3 to forget auxiliary constraints that are only % needed temporarily for reasoning about domain boundaries findall(Min-Max, min_max_factor_(L1, U1, L2, U2, L3, U3, Min, Max), [Min-Max]). 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) }, seq(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) ). writeln(T) :- write(T), nl. :- meta_predicate(must_succeed(0)). must_succeed(G) :- ( G -> true ; throw(failed-G) ). 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), must_succeed(append_edge(Y, edges, flow_from(F,V))), must_succeed(append_edge(V, edges, flow_to(F,Y))) }, enumerate(Ns, V). append_edge(V, Attr, E) :- ( get_attr_(Attr, V, Es) -> put_attr_(Attr, V, [E|Es]) ; put_attr_(Attr, V, [E]) ). get_attr_(edges, V, Es) :- get_attr(V, edges, Es). get_attr_(g0_edges, V, Es) :- get_attr(V, g0_edges, Es). put_attr_(edges, V, E) :- put_attr(V, edges, E). put_attr_(g0_edges, V, E) :- put_attr(V, g0_edges, 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). :- meta_predicate(with_local_attributes(?, 0, ?)). :- dynamic(nat_copy/1). with_local_attributes(Vars, Goal, Result) :- catch((Goal, % Create a copy where all attributes are removed. Only % the result and its relation to Vars matters. We throw % an exception to undo all modifications to attributes % we made during propagation, and unify the variables % in the thrown copy with Vars in order to get the % intended variables in Result. copy_term_nat(Vars-Result, Copy), throw(local_attributes(Copy))), local_attributes(Vars-Result), true). distinct(Vars) --> { with_local_attributes(Vars, ( 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, neq_nums(Gs), enable_queue. neq_nums([]) --> []. neq_nums([neq_num(V,N)|VNs]) --> % { portray_clause(neq_num(V, N)) }, neq_num(V, N), neq_nums(VNs). 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. For more information about this technique, see: https://www.metalevel.at/prolog/dcg =================================== A Prolog implementation of this algorithm is also available as a standalone library from: https://www.metalevel.at/scc.pl - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 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) }. node_lowlink(V, L) :- get_attr(V, lowlink, L). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - nvalue/2: A relaxed version of all_distinct/1. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ maximal_matching([]) --> []. maximal_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) }, [FL] ; [] ), maximal_matching(FLs). propagate_nvalue(N, Vars0) :- sort(Vars0, Vars), include(integer, Vars, Ints), length(Ints, Distinct), vars_num_infinite(Vars, NumInfinite), N #>= Distinct, with_local_attributes(Vars, ( difference_arcs(Vars, FreeLeft, FreeRight0), maplist(put_free, FreeRight0), phrase(maximal_matching(FreeLeft), MatchedLeft), length(MatchedLeft, MaxFurther) ), MaxFurther), N #=< NumInfinite + Distinct + MaxFurther. vars_num_infinite(Vars, Num) :- foldl(num_infinite, Vars, 0, Num). num_infinite(Var, N0, N) :- ( integer(Var) -> N = N0 ; fd_get(Var, Dom, _), ( domain_infimum(Dom, n(_)), domain_supremum(Dom, n(_)) -> N = N0 ; N #= N0 + 1 ) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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), Orig = original_goal(_, weak_arc_all_distinct(Ls)), all_distinct(Ls, [], Orig). 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). 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), Orig = original_goal(_, 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) ; [] ) ; [] ). 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) ; [] ), { EST is EST_J+D_J, domain_remove_smaller_than(DomI, EST, DomI1) }, fd_put(I, DomI1, Ps) ; [] ). 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) ; [] ), { LST is LST_J-D_I, domain_remove_greater_than(DomI, LST, DomI1) }, fd_put(I, DomI1, Ps) ; [] ). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% 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] % ; false. % ``` 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: clpz_gcc_num: equal Num0, the user-visible counter variable clpz_gcc_vs: the remaining variables in the constraint that can be equal Key. clpz_gcc_occurred: stores how often Key already occurred in vs. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ gcc_pairs([], _, []). gcc_pairs([Key-Num0|KNs], Vs, [Key-Num|Rest]) :- put_attr(Num, clpz_gcc_num, Num0), put_attr(Num, clpz_gcc_vs, Vs), put_attr(Num, clpz_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) --> % at this point, all elements of clpz_gcc_vs must be % variables, which a previously scheduled and called % gcc_check//1 ensures. Note that gcc_check//1 disables the % queue and accumulates constraints in the queue. Do we need % to insert a call of do_queue//0 here to reach a fixpoint? I % think not, because verify_attributes/3 gives each variable % that is involved in a unification an opportunity to schedule % its propagators, even if the unifications happen % simultaneously (such as [A,B] = [0,1], which can happen in % the propagator of tuples_in/2). Hence: We need this only if % an example shows it, ideally found by a systematic search % that can be used to test the implementation. { with_local_attributes(Vs, (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, neq_nums(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, clpz_gcc_vs, Vs) -> get_attr(Num0, clpz_gcc_num, Num), get_attr(Num0, clpz_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), samsort(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 clpz_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, clpz_gcc_vs), del_attr(Num, clpz_gcc_num), del_attr(Num, clpz_gcc_occurred). gcc_check_([]) --> []. gcc_check_([Key-Num0|KNs]) --> ( { get_attr(Num0, clpz_gcc_vs, Vs) } -> { get_attr(Num0, clpz_gcc_num, Num), get_attr(Num0, clpz_gcc_occurred, Occ0), vs_key_min_others(Vs, Key, 0, Min, Os), put_attr(Num0, clpz_gcc_vs, Os), put_attr(Num0, clpz_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). 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), new_queue(Q0), phrase((distinct_attach(Vs, Prop, []),trigger_prop(Prop),do_queue), [Q0], _) ). 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. Options is a list of options. Currently, the only supported % option is: % % * limit(L) % The integer L is the global resource limit. Default is 1. % % 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 ; fully_elastic_relaxation(Tasks, L), 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), MinStarts = [Min|Mins], foldl(min_, Mins, Min, Start), MaxEnds = [Max|Maxs], foldl(max_, Maxs, Max, End), resource_limit(Start, End, Tasks, Bss, L) ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Trivial lower and upper bounds, assuming no gaps and not necessarily retaining the rectangular shape of each task. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ fully_elastic_relaxation(Tasks, Limit) :- maplist(task_duration_consumption, Tasks, Ds, Cs), maplist(area, Ds, Cs, As), sum(As, #=, #Area), #MinTime #= (Area + Limit - 1) // Limit, tasks_minstart_maxend(Tasks, MinStart, MaxEnd), MaxEnd #>= MinStart + MinTime. task_duration_consumption(task(_,D,_,C,_), D, C). area(X, Y, Area) :- #Area #= #X * #Y. tasks_minstart_maxend(Tasks, Start, End) :- maplist(task_start_end, Tasks, [Start0|Starts], [End0|Ends]), foldl(min_, Starts, Start0, Start), foldl(max_, Ends, End0, End). max_(E, M0, M) :- #M #= max(E, M0). min_(E, M0, M) :- #M #= min(E, M0). task_start_end(task(Start,_,End,_,_), #Start, #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(+Vs, +Nodes, +Arcs) % % Describes a list of finite domain variables with a finite % automaton. Equivalent to automaton(Vs, _, Vs, Nodes, Arcs, % [], [], _), a common use case of automaton/8. In the following % example, a list of binary finite domain variables is constrained to % contain at least two consecutive ones: % % ``` % two_consecutive_ones(Vs) :- % automaton(Vs, [source(a),sink(c)], % [arc(a,0,a), arc(a,1,b), % arc(b,0,a), arc(b,1,c), % arc(c,0,c), arc(c,1,c)]). % ``` % % Example query: % % ``` % ?- length(Vs, 3), two_consecutive_ones(Vs), label(Vs). % Vs = [0,1,1] % ; Vs = [1,1,0] % ; Vs = [1,1,1] % ; false. % ``` 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 symbolically refer to previous counter % values, and variables occurring in Template refer to the current % element of Sequence. When a transition containing arithmetic % expressions is taken, each counter is updated according to the % result of the corresponding expression. When a transition without % arithmetic expressions is taken, all counters remain unchanged. % Counters is a list of variables. Initials is a list of finite % domain variables or integers denoting, in the same order, the % initial value of each counter. These values are related to Finals % according to the arithmetic expressions of the taken transitions. % % The following example is taken from Beldiceanu, Carlsson, Debruyne % and Petit: "Reformulation of Global Constraints Based on % Constraints Checkers", Constraints 10(4), pp 339-362 (2005). It % relates a sequence of integers and finite domain variables to its % number of inflexions, which are switches between strictly ascending % and strictly descending subsequences: % % ``` % sequence_inflexions(Vs, N) :- % variables_signature(Vs, Sigs), % automaton(Sigs, _, Sigs, % [source(s),sink(i),sink(j),sink(s)], % [arc(s,0,s), arc(s,1,j), arc(s,2,i), % arc(i,0,i), arc(i,1,j,[C+1]), arc(i,2,i), % arc(j,0,j), arc(j,1,j), % arc(j,2,i,[C+1])], % [C], [0], [N]). % % variables_signature([], []). % variables_signature([V|Vs], Sigs) :- % variables_signature_(Vs, V, Sigs). % % variables_signature_([], _, []). % variables_signature_([V|Vs], Prev, [S|Sigs]) :- % V #= Prev #<==> S #= 0, % Prev #< V #<==> S #= 1, % Prev #> V #<==> S #= 2, % variables_signature_(Vs, V, Sigs). % ``` % % Example queries: % % ``` % ?- sequence_inflexions([1,2,3,3,2,1,3,0], N). % N = 3. % % ?- length(Ls, 5), Ls ins 0..1, % sequence_inflexions(Ls, 3), label(Ls). % Ls = [0, 1, 0, 1, 0] ; % Ls = [1, 0, 1, 0, 1]. % ``` 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) -> ( monotonic -> 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)). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% zcompare(?Order, ?A, ?B) % % Analogous to compare/3, with finite domain variables A and B. % % This predicate allows you to make several predicates over integers % deterministic while preserving their generality and completeness. % For example: % % ``` % n_factorial(N, F) :- % zcompare(C, N, 0), % n_factorial_(C, N, F). % % n_factorial_(=, _, 1). % n_factorial_(>, N, F) :- % F #= F0*N, N1 #= N - 1, % n_factorial(N1, F0). % ``` % % This version is deterministic if the first argument is instantiated, % because first argument indexing can distinguish the two different % clauses: % % ``` % ?- n_factorial(30, F). % F = 265252859812191058636308480000000. % ``` % % The predicate can still be used in all directions, including the % most general query: % % ``` % ?- n_factorial(N, F). % N = 0, F = 1 % ; N = 1, F = 1 % ; N = 2, F = 2 % ; ... . % ``` zcompare(Order, A, B) :- ( nonvar(Order) -> zcompare_(Order, A, B) ; integer(A), integer(B) -> compare(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_(O, A, B) :- ( member(O, "<=>") -> true ; domain_error(order, O, zcompare/3) ), zcompare__(O, A, B). zcompare__(=, A, B) :- #A #= #B. zcompare__(<, A, B) :- #A #< #B. zcompare__(>, A, B) :- #A #> #B. %% chain(+Relation, +Zs) % % 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(Relation, Zs) :- 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(ℤ) variable. fd_var(X) :- get_attr(X, clpz, _). %% 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: %?- clpz:goals_entail([X#>2], X #> 3). %@ false. %?- clpz:goals_entail([X#>1, X#<3], X #= 2). %@ true. %?- clpz:goals_entail([X#=Y+1], X #= Y+1). %@ ERROR: Arguments are not sufficiently instantiated %@ Exception: (15) throw(error(instantiation_error, _G2680)) ? %?- clpz: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 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ verify_attributes(Var, Other, Gs) :- % portray_clause(Var = Other), ( get_atts(Var, clpz(CLPZ)) -> CLPZ = clpz_attr(_,_,_,Dom,Ps,Q), ( nonvar(Other) -> ( integer(Other) -> true ; type_error(integer, Other) ), domain_contains(Dom, Other), phrase(trigger_props(Ps), [Q], [_]), Gs = [phrase(do_queue, [Q], _)] ; ( get_atts(Other, clpz(clpz_attr(_,_,_,OD,OPs,_))) -> domains_intersection(OD, Dom, Dom1), append_propagators(Ps, OPs, Ps1), new_queue(Q0), variables_same_queue([Var,Other]), phrase((fd_put(Other,Dom1,Ps1), trigger_props(Ps1)), [Q0], _), Gs = [phrase(do_queue, [Q0], _)] ; put_atts(Other, clpz(CLPZ)), Gs = [] ) ) ; Gs = [] ). 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, clpz, Attr), format("A: ~w\n", [Attr]) }, { get_attr(X, clpz, clpz_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) } -> [] ; [clpz:(X in Drep)] ), attributes_goals(Ps), { del_attr(X, clpz) }. attributes_goals([]) --> []. attributes_goals([propagator(P, State)|As]) --> ( { ground(State) } -> [] ; { phrase(attribute_goal_(P), Gs) } -> { del_attr(State, clpz_aux), State = processed, ( monotonic -> maplist(unwrap_with(bare_integer), Gs, Gs1) ; maplist(unwrap_with(=), Gs, Gs1) ), maplist(with_clpz, Gs1, Gs2) }, seq(Gs2) ; [P] % possibly user-defined constraint ), attributes_goals(As). with_clpz(G, clpz: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_(x_eq_abs_plus_v(X,V)) --> [#X #= abs(#X) + #V]. 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_(pexp(X,Y,Z,_)) --> [#X ^ #Y #= #Z]. attribute_goal_(psign(X,Y)) --> [#Y #= sign(#X)]. 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_(pxor(X,Y,Z)) --> [#Z #= xor(#X, #Y)]. attribute_goal_(ppopcount(X,Y)) --> [#Y #= popcount(#X)]. 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_(pnvalue(N, Vs)) --> [nvalue(N, 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(Pairs)) --> { pairs_values(Pairs, Nums), maplist(gcc_done, Nums) }. attribute_goal_(pcircuit(Vs)) --> [circuit(Vs)]. attribute_goal_(pserialized(_,_,_,_,O)) --> original_goal(O). attribute_goal_(rel_tuple(R, Tuple)) --> { get_attr(R, clpz_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, clpz_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_(preified_slash(X, Y, _, R)) --> [#X/ #Y #= R]. attribute_goal_(preified_exp(X, Y, _, R)) --> [#X^ #Y #= R]. attribute_goal_(pimpl(X, Y, _)) --> [#X #==> #Y]. attribute_goal_(pfunction(Op, A, B, R)) --> { Expr =.. [Op,#A,#B] }, [#R #= Expr]. attribute_goal_(pfunction(Op, A, R)) --> { Expr =.. [Op,#A] }, [#R #= Expr]. 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(original_goal(State, Goal)) --> ( { var(State) } -> { State = processed }, [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 ). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Reified predicates for use with predicates from library(reif). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ clpz_t(Expr, T) :- Expr #<==> #B, zo_t(B, T). #=(X, Y, T) :- clpz_t(X #= Y, T). #<(X, Y, T) :- clpz_t(X #< Y, T). zo_t(0, false). zo_t(1, true). /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Generated predicates - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ term_expansion(make_parse_clpz, Clauses) :- make_parse_clpz(Clauses). term_expansion(make_parse_reified, Clauses) :- make_parse_reified(Clauses). term_expansion(make_matches, Clauses) :- make_matches(Clauses). make_parse_clpz. make_parse_reified. make_matches.