## A.7 library(clpfd): Constraint Logic Programming over Finite Domains

author
  Markus Triska

#### A.7.1 Introduction

Constraint programming is a declarative formalism that lets you describe conditions a solution must satisfy. This library provides CLP(FD), Constraint Logic Programming over Finite Domains. It can be used to model and solve various combinatorial problems such as planning, scheduling and allocation tasks.

Most predicates of this library are finite domain constraints, which are relations over integers. They generalise arithmetic evaluation of integer expressions in that propagation can proceed in all directions. This library also provides enumeration predicates, which let you systematically search for solutions on variables whose domains have become finite.

You can cite this library in your publications as:

    @inproceedings{Triska12,
      author    = {Markus Triska},
      title     = {The Finite Domain Constraint Solver of {SWI-Prolog}},
      booktitle = {FLOPS},
      series    = {LNCS},
      volume    = {7294},
      year      = {2012},
      pages     = {307-316}
    }

#### A.7.2 Arithmetic constraints

A finite domain arithmetic expression is one of:

> integer | Given value
> variable | Unknown integer
> ?(variable) | Unknown integer
> -Expr | Unary minus
> Expr + Expr | Addition
> Expr * Expr | Multiplication
> Expr - Expr | Subtraction
> Expr `^` Expr | Exponentiation
> `min(Expr,Expr)` | Minimum of two expressions
> `max(Expr,Expr)` | Maximum of two expressions
> Expr mod Expr | Modulo induced by floored division
> Expr rem Expr | Modulo induced by truncated division
> `abs(Expr)` | Absolute value
> Expr / Expr | Truncated integer division

Arithmetic constraints are relations between arithmetic expressions.

The most important arithmetic constraints are:

> Expr1 `#>=` Expr2 | Expr1 is greater than or equal to Expr2
> Expr1 `#=<` Expr2 | Expr1 is less than or equal to Expr2
> Expr1 `#=` Expr2 | Expr1 equals Expr2
> Expr1 `#\=` Expr2 | Expr1 is not equal to Expr2
> Expr1 `#>` Expr2 | Expr1 is greater than Expr2
> Expr1 `#<` Expr2 | Expr1 is less than Expr2

#### A.7.3 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 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.

#### A.7.4 Examples

Here is an example session with a few queries and their answers:

    ?- use_module(library(clpfd)).
    % library(clpfd) compiled into clpfd 0.06 sec, 633,732 bytes
    true.

    ?- X #> 3.
    X in 4..sup.

    ?- X #\= 20.
    X in inf..19\/21..sup.

    ?- 2*X #= 10.
    X = 5.

    ?- X*X #= 144.
    X in -12\/12.

    ?- 4*X + 2*Y #= 24, X + Y #= 9, [X,Y] ins 0..sup.
    X = 3,
    Y = 6.

    ?- Vs = [X,Y,Z], Vs ins 1..3, all_different(Vs), X = 1, Y #\= 2.
    Vs = [1, 3, 2],
    X = 1,
    Y = 3,
    Z = 2.

    ?- X #= Y #<==> B, X in 0..3, Y in 4..5.
    B = 0,
    X in 0..3,
    Y in 4..5.

In each case, and as for all pure programs, the answer is declaratively equivalent to the original query, and in many cases the constraint solver has deduced additional domain restrictions.

#### A.7.5 Search

A common usage of this library is to first post the desired constraints among the variables of a model, and then to use enumeration predicates to search for solutions. As an example of a constraint satisfaction problem, consider the cryptoarithmetic puzzle SEND + MORE = MONEY, where different letters denote distinct integers between 0 and 9. It can be modeled in CLP(FD) as follows:

    :- use_module(library(clpfd)).

    puzzle([S,E,N,D] + [M,O,R,E] = [M,O,N,E,Y]) :-
            Vars = [S,E,N,D,M,O,R,Y],
            Vars ins 0..9,
            all_different(Vars),
                      S*1000 + E*100 + N*10 + D +
                      M*1000 + O*100 + R*10 + E #=
            M*10000 + O*1000 + N*100 + E*10 + Y,
            M #\= 0, S #\= 0.

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]),
    1000*9+91*_A2+ -90*_A3+_A4+ -9000*1+ -900*0+10*_B3+ -1*_C5#=0,
    _A3 in 5..8,
    _A4 in 2..8,
    _B3 in 2..8,
    _C5 in 2..8.

Here, the constraint solver has deduced more stringent bounds for all variables. It is good practice to keep the modeling part separate from the actual search. This lets you observe termination and determinism properties of the modeling part in isolation from the search. 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.

#### A.7.6 Declarative integer arithmetic

You can also use CLP(FD) constraints as a more declarative alternative for ordinary integer arithmetic with is/2, >/2 etc. For example:

    :- use_module(library(clpfd)).

    n_factorial(0, 1).
    n_factorial(N, F) :-
            N #> 0, N1 #= N - 1, F #= N * F1,
            n_factorial(N1, F1).

This predicate can be used in all directions. For example:

    ?- n_factorial(47, F).
    F = 258623241511168180642964355153611979969197632389120000000000 ;
    false.

    ?- n_factorial(N, 1).
    N = 0 ;
    N = 1 ;
    false.

    ?- n_factorial(N, 3).
    false.

To make the predicate terminate if any argument is instantiated, add the (implied) constraint F `#\=` 0 before the recursive call. Otherwise, the query `n_factorial(N, 0)` is the only non-terminating case of this kind.

<### Advanced topics {#clpfd-advanced-topics}

This library uses goal_expansion/2 to rewrite constraints at compilation time. The expansion's aim is to transparently bring the performance of CLP(FD) constraints close to that of conventional arithmetic predicates (</2, =:=/2, is/2 etc.) when the constraints are used in modes that can also be handled by built-in arithmetic. To disable the expansion, set the flag `clpfd_goal_expansion` to `false`.

If you set the flag `clpfd_monotonic` to `true`, then CLP(FD) is monotonic: Adding new constraints cannot yield new solutions. When this flag is `true`, you must wrap variables that occur in arithmetic expressions with the functor `(?)/1`. For example, `?(X) #= ?(Y) + ?(Z)`. The wrapper can be omitted for variables that are already constrained to integers.

Use call_residue_vars/2 and copy_term/3 to inspect residual goals and the constraints in which a variable is involved. This library also provides reflection predicates (like fd_dom/2, fd_size/2 etc.) with which you can inspect a variable's current domain. These predicates can be useful if you want to implement your own labeling strategies.

You can also define custom constraints. The mechanism to do this is not yet finalised, and we welcome suggestions and descriptions of use cases that are important to you. As an example of how it can be done currently, let us define a new custom constraint `oneground(X,Y,Z)`, where Z shall be 1 if at least one of X and Y is instantiated:

    :- use_module(library(clpfd)).

    :- multifile clpfd:run_propagator/2.

    oneground(X, Y, Z) :-
            clpfd:make_propagator(oneground(X, Y, Z), Prop),
            clpfd:init_propagator(X, Prop),
            clpfd:init_propagator(Y, Prop),
            clpfd:trigger_once(Prop).

    clpfd:run_propagator(oneground(X, Y, Z), MState) :-
            (   integer(X) -> clpfd:kill(MState), Z = 1
            ;   integer(Y) -> clpfd:kill(MState), Z = 1
            ;   true
            ).

First, clpfd:make_propagator/2 is used to transform a user-defined representation of the new constraint to an internal form. With clpfd:init_propagator/2, this internal form is then attached to X and Y. From now on, the propagator will be invoked whenever the domains of X or Y are changed. Then, clpfd:trigger_once/1 is used to give the propagator its first chance for propagation even though the variables' domains have not yet changed. Finally, clpfd:run_propagator/2 is extended to define the actual propagator. As explained, this predicate is automatically called by the constraint solver. The first argument is the user-defined representation of the constraint as used in clpfd:make_propagator/2, and the second argument is a mutable state that can be used to prevent further invocations of the propagator when the constraint has become entailed, by using clpfd:kill/1. An example of using the new constraint:

    ?- oneground(X, Y, Z), Y = 5.
    Y = 5,
    Z = 1,
    X in inf..sup.

?Var in +Domain
  Var is an element of Domain. Domain is one of:
  Integer
    Singleton set consisting only of Integer.
  Lower .. Upper
    All integers I such that Lower `=<` I `=<` Upper. Lower must be an integer or the atom inf, which denotes negative infinity. Upper must be an integer or the atom sup, which denotes positive infinity.
  Domain1 `\/` Domain2
    The union of Domain1 and Domain2.
+Vars ins +Domain
  The variables in the list Vars are elements of Domain.
indomain(?Var)
  Bind Var to all feasible values of its domain on backtracking. The domain of Var must be finite.
label(+Vars)
  Equivalent to `labeling([], Vars)`.
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.

all_different(+Vars)
  Vars are pairwise distinct.
all_distinct(+Ls)
  Like all_different/1, with stronger propagation. For example, all_distinct/1 can detect that not all variables can assume distinct values given the following domains:

      ?- maplist(in, Vs,
                 [1\/3..4, 1..2\/4, 1..2\/4, 1..3, 1..3, 1..6]),
         all_distinct(Vs).
      false.

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.

scalar_product(+Cs, +Vs, +Rel, ?Expr)
  Cs is a list of integers, Vs is a list of variables and integers. True if the scalar product of Cs and Vs is in relation Rel to Expr, where Rel is #=, #`\`=, #<, #>, `#=<` or #>=.
?X #>= ?Y
  X is greater than or equal to Y.
?X #=< ?Y
  X is less than or equal to Y.
?X #= ?Y
  X equals Y.
?X #\= ?Y
  X is not Y.
?X #> ?Y
  X is greater than Y.
?X #< ?Y
  X is less than Y. In addition to its regular use in problems that require it, this constraint can also be useful to eliminate uninteresting symmetries from a problem. For example, all possible matches between pairs built from four players in total:

      ?- Vs = [A,B,C,D], Vs ins 1..4,
              all_different(Vs),
              A #< B, C #< D, A #< C,
         findall(pair(A,B)-pair(C,D), label(Vs), Ms).
      Ms = [ pair(1, 2)-pair(3, 4),
             pair(1, 3)-pair(2, 4),
             pair(1, 4)-pair(2, 3)].

#\ +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.

?P #<==> ?Q
  P and Q are equivalent. For example:

      ?- X #= 4 #<==> B, X #\= 4.
      B = 0,
      X in inf..3\/5..sup.

  The following example uses reified constraints to relate a list of finite domain variables to the number of occurrences of a given value:

      :- use_module(library(clpfd)).

      vs_n_num(Vs, N, Num) :-
              maplist(eq_b(N), Vs, Bs),
              sum(Bs, #=, Num).

      eq_b(X, Y, B) :- X #= Y #<==> B.

  Sample queries and their results:

      ?- Vs = [X,Y,Z], Vs ins 0..1, vs_n_num(Vs, 4, Num).
      Vs = [X, Y, Z],
      Num = 0,
      X in 0..1,
      Y in 0..1,
      Z in 0..1.

      ?- vs_n_num([X,Y,Z], 2, 3).
      X = 2,
      Y = 2,
      Z = 2.

?P #==> ?Q
  P implies Q.
?P #<== ?Q
  Q implies P.
?P #/\ ?Q
  P and Q hold.
?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.

lex_chain(+Lists)
  Lists are lexicographically non-decreasing.
tuples_in(+Tuples, +Relation)
  Relation must be a list of lists of integers. The elements of the list Tuples are constrained to be elements of Relation. Arbitrary finite relations, such as compatibility tables, can be modeled in this way. For example, if 1 is compatible with 2 and 5, and 4 is compatible with 0 and 3:

      ?- tuples_in([[X,Y]], [[1,2],[1,5],[4,0],[4,3]]), X = 4.
      X = 4,
      Y in 0\/3.

  As another example, consider a train schedule represented as a list of quadruples, denoting departure and arrival places and times for each train. In the following program, Ps is a feasible journey of length 3 from A to D via trains that are part of the given schedule.

      :- use_module(library(clpfd)).

      trains([[1,2,0,1],
              [2,3,4,5],
              [2,3,0,1],
              [3,4,5,6],
              [3,4,2,3],
              [3,4,8,9]]).

      threepath(A, D, Ps) :-
              Ps = [[A,B,_T0,T1],[B,C,T2,T3],[C,D,T4,_T5]],
              T2 #> T1,
              T4 #> T3,
              trains(Ts),
              tuples_in(Ps, Ts).

  In this example, the unique solution is found without labeling:

      ?- threepath(1, 4, Ps).
      Ps = [[1, 2, 0, 1], [2, 3, 4, 5], [3, 4, 8, 9]].

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 also
    Dorndorf et al. 2000, "Constraint Propagation Techniques for the Disjunctive Scheduling Problem"
element(?N, +Vs, ?V)
  The N-th element of the list of finite domain variables Vs is V. Analogous to nth1/3.
global_cardinality(+Vs, +Pairs)
  Global Cardinality constraint. Equivalent to `global_cardinality(Vs, Pairs, [])`. Example:

      ?- Vs = [_,_,_], global_cardinality(Vs, [1-2,3-_]), label(Vs).
      Vs = [1, 1, 3] ;
      Vs = [1, 3, 1] ;
      Vs = [3, 1, 1].

global_cardinality(+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.
circuit(+Vs)
  True if 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] .

cumulative(+Tasks)
  Equivalent to `cumulative(Tasks, [limit(1)])`.
cumulative(+Tasks, +Options)
  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 if at any time during the start and end of each task, the total resource consumption of all tasks running at that time does not exceed the global resource limit (which is 1 by default). Options is a list of options. Currently, the only supported option is:
  limit(L)
    The integer L is the global resource limit.

  For example, given the following predicate that relates three tasks of durations 2 and 3 to a list containing their starting times:

      tasks_starts(Tasks, [S1,S2,S3]) :-
              Tasks = [task(S1,3,_,1,_),
                       task(S2,2,_,1,_),
                       task(S3,2,_,1,_)].

  We can use cumulative/2 as follows, and obtain a schedule:

      ?- tasks_starts(Tasks, Starts), Starts ins 0..10,
         cumulative(Tasks, [limit(2)]), label(Starts).
      Tasks = [task(0, 3, 3, 1, _G36), task(0, 2, 2, 1, _G45), ...],
      Starts = [0, 0, 2] .

automaton(+Signature, +Nodes, +Arcs)
  Describes a list of finite domain variables with a finite automaton. Equivalent to `automaton(_, _, Signature, Nodes, Arcs, [], [], _)`, a common use case of automaton/8. In the following example, a list of binary finite domain variables is constrained to contain at least two consecutive ones:

      :- use_module(library(clpfd)).

      two_consecutive_ones(Vs) :-
              automaton(Vs, [source(a),sink(c)],
                        [arc(a,0,a), arc(a,1,b),
                         arc(b,0,a), arc(b,1,c),
                         arc(c,0,c), arc(c,1,c)]).

  Example query:

      ?- length(Vs, 3), two_consecutive_ones(Vs), label(Vs).
      Vs = [0, 1, 1] ;
      Vs = [1, 1, 0] ;
      Vs = [1, 1, 1].

automaton(?Sequence, ?Template, +Signature, +Nodes, +Arcs, +Counters, +Initials, ?Finals)
  Describes a list of finite domain variables with a finite automaton. True if the finite automaton induced by Nodes and Arcs (extended with Counters) accepts Signature. Sequence is a list of terms, all of the same shape. Additional constraints must link Sequence to Signature, if necessary. Nodes is a list of `source(Node)` and `sink(Node)` terms. Arcs is a list of `arc(Node,Integer,Node)` and `arc(Node,Integer,Node,Exprs)` terms that denote the automaton's transitions. Each node is represented by an arbitrary term. Transitions that are not mentioned go to an implicit failure node. Exprs is a list of arithmetic expressions, of the same length as Counters. In each expression, variables occurring in Counters correspond to old counter values, and variables occurring in Template correspond to the current element of Sequence. When a transition containing expressions is taken, counters are updated as stated. By default, counters remain unchanged. Counters is a list of variables that must not occur anywhere outside of the constraint goal. Initials is a list of the same length as Counters. Counter arithmetic on the transitions relates the counter values in Initials to Finals.

  The following example is taken from Beldiceanu, Carlsson, Debruyne and Petit: "Reformulation of Global Constraints Based on Constraints Checkers", Constraints 10(4), pp 339-362 (2005). It relates a sequence of integers and finite domain variables to its number of inflexions, which are switches between strictly ascending and strictly descending subsequences:

      :- use_module(library(clpfd)).

      sequence_inflexions(Vs, N) :-
              variables_signature(Vs, Sigs),
              automaton(_, _, Sigs,
                        [source(s),sink(i),sink(j),sink(s)],
                        [arc(s,0,s), arc(s,1,j), arc(s,2,i),
                         arc(i,0,i), arc(i,1,j,[C+1]), arc(i,2,i),
                         arc(j,0,j), arc(j,1,j),
                         arc(j,2,i,[C+1])],
                        [C], [0], [N]).

      variables_signature([], []).
      variables_signature([V|Vs], Sigs) :-
              variables_signature_(Vs, V, Sigs).

      variables_signature_([], _, []).
      variables_signature_([V|Vs], Prev, [S|Sigs]) :-
              V #= Prev #<==> S #= 0,
              Prev #< V #<==> S #= 1,
              Prev #> V #<==> S #= 2,
              variables_signature_(Vs, V, Sigs).

  Example queries:

      ?- sequence_inflexions([1,2,3,3,2,1,3,0], N).
      N = 3.

      ?- length(Ls, 5), Ls ins 0..1,
         sequence_inflexions(Ls, 3), label(Ls).
      Ls = [0, 1, 0, 1, 0] ;
      Ls = [1, 0, 1, 0, 1].

transpose(+Matrix, ?Transpose)
  Transpose a list of lists of the same length. Example:

      ?- transpose([[1,2,3],[4,5,6],[7,8,9]], Ts).
      Ts = [[1, 4, 7], [2, 5, 8], [3, 6, 9]].

  This predicate is useful in many constraint programs. Consider for instance Sudoku:

      :- use_module(library(clpfd)).

      sudoku(Rows) :-
              length(Rows, 9), maplist(length_(9), Rows),
              append(Rows, Vs), Vs ins 1..9,
              maplist(all_distinct, Rows),
              transpose(Rows, Columns),
              maplist(all_distinct, Columns),
              Rows = [A,B,C,D,E,F,G,H,I],
              blocks(A, B, C), blocks(D, E, F), blocks(G, H, I).

      length_(L, Ls) :- length(Ls, L).

      blocks([], [], []).
      blocks([A,B,C|Bs1], [D,E,F|Bs2], [G,H,I|Bs3]) :-
              all_distinct([A,B,C,D,E,F,G,H,I]),
              blocks(Bs1, Bs2, Bs3).

      problem(1, [[_,_,_,_,_,_,_,_,_],
                  [_,_,_,_,_,3,_,8,5],
                  [_,_,1,_,2,_,_,_,_],
                  [_,_,_,5,_,7,_,_,_],
                  [_,_,4,_,_,_,1,_,_],
                  [_,9,_,_,_,_,_,_,_],
                  [5,_,_,_,_,_,_,7,3],
                  [_,_,2,_,1,_,_,_,_],
                  [_,_,_,_,4,_,_,_,9]]).

  Sample query:

      ?- problem(1, Rows), sudoku(Rows), maplist(writeln, Rows).
      [9, 8, 7, 6, 5, 4, 3, 2, 1]
      [2, 4, 6, 1, 7, 3, 9, 8, 5]
      [3, 5, 1, 9, 2, 8, 7, 4, 6]
      [1, 2, 8, 5, 3, 7, 6, 9, 4]
      [6, 3, 4, 8, 9, 2, 1, 5, 7]
      [7, 9, 5, 4, 6, 1, 8, 3, 2]
      [5, 1, 9, 2, 8, 6, 4, 7, 3]
      [4, 7, 2, 3, 1, 9, 5, 6, 8]
      [8, 6, 3, 7, 4, 5, 2, 1, 9]
      Rows = [[9, 8, 7, 6, 5, 4, 3, 2|...], ... , [...|...]].

zcompare(?Order, ?A, ?B)
  Analogous to compare/3, with finite domain variables A and B. Example:

      :- use_module(library(clpfd)).

       n_factorial(N, F) :-
               zcompare(C, N, 0),
               n_factorial_(C, N, F).

       n_factorial_(=, _, 1).
       n_factorial_(>, N, F) :-
            F #= F0*N, N1 #= N - 1,
            n_factorial(N1, F0).

  This version is deterministic if the first argument is instantiated:

      ?- n_factorial(30, F).
      F = 265252859812191058636308480000000.

chain(+Zs, +Relation)
  Zs form a chain with respect to Relation. Zs is a list of finite domain variables that are a chain with respect to the partial order Relation, in the order they appear in the list. Relation must be #=, #=<, #>=, `#<` or #>. For example:

      ?- chain([X,Y,Z], #>=).
      X#>=Y,
      Y#>=Z.

fd_var(+Var)
  True iff Var is a CLP(FD) variable.
fd_inf(+Var, -Inf)
  Inf is the infimum of the current domain of Var.
fd_sup(+Var, -Sup)
  Sup is the supremum of the current domain of Var.
fd_size(+Var, -Size)
  Determine the size of a variable's domain. Size is the number of elements of the current domain of Var, or the atom sup if the domain is unbounded.
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.