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- Markus Triska

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 finite domain *arithmetic expression* is one of:

integerGiven value variableUnknown integer ?( variable)Unknown integer -Expr Unary minus Expr + Expr Addition Expr * Expr Multiplication Expr - Expr Subtraction Expr `^`

ExprExponentiation `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 `#>=`

Expr2Expr1 is greater than or equal to Expr2 Expr1 `#=<`

Expr2Expr1 is less than or equal to Expr2 Expr1 `#=`

Expr2Expr1 equals Expr2 Expr1 `#\=`

Expr2Expr1 is not equal to Expr2 Expr1 `#>`

Expr2Expr1 is greater than Expr2 Expr1 `#<`

Expr2Expr1 is less than Expr2

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:

`#\`

QTrue iff Q is false P `#\/`

QTrue iff either P or Q P `#/\`

QTrue iff both P and Q P `#<==>`

QTrue iff P and Q are equivalent P `#==>`

QTrue iff P implies Q P `#<==`

QTrue iff Q implies P

The constraints of this table are reifiable as well.

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 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.

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.

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`must be an integer or the atom`Lower`**inf**, which denotes negative infinity.must be an integer or the atom`Upper`**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.

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