- Documentation
- Reference manual
- The SWI-Prolog library
- library(clpfd): CLP(FD): Constraint Logic Programming over Finite Domains
- Introduction
- Arithmetic constraints
- Declarative integer arithmetic
- Example: Factorial relation
- Combinatorial constraints
- Domains
- Example: Sudoku
- Residual goals
- Core relations and search
- Example: Eight queens puzzle
- Optimisation
- Reification
- Enabling monotonic CLP(FD)
- Custom constraints
- Applications
- Acknowledgments
- CLP(FD) predicate index
- Closing and opening words about CLP(FD)

- library(clpfd): CLP(FD): Constraint Logic Programming over Finite Domains

- The SWI-Prolog library
- Packages

- Reference manual

### A.8.17 CLP(FD) predicate index

In the following, each CLP(FD) predicate is described in more detail.

We recommend the following link to refer to this manual:

http://eu.swi-prolog.org/man/clpfd.html

#### A.8.17.1 Arithmetic constraints

*Arithmetic* constraints are the most basic use of CLP(FD).
Every time you use `(is)/2`

or one of the low-level
arithmetic comparisons (`(<)/2`

, `(>)/2`

etc.) over integers, consider using CLP(FD) constraints *instead*.
This can at most *increase* the generality of your programs. See
declarative integer arithmetic (section
A.8.3).

`?X`**#=**`?Y`- The arithmetic expression
`X`equals`Y`. This is the most important arithmetic constraint (section A.8.2), subsuming and replacing both`(is)/2`

*and*`(=:=)/2`

over integers. See declarative integer arithmetic (section A.8.3). `?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. See declarative integer arithmetic (section A.8.3). `?X`**#>=**`?Y`- Same as
`Y``#=<`

`X`. When reasoning over integers, replace`(>=)/2`

by #>=/2 to obtain more general relations. See declarative integer arithmetic (section A.8.3). `?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. See declarative integer arithmetic (section A.8.3). `?X`**#>**`?Y`- Same as
`Y``#<`

`X`. When reasoning over integers, replace`(>)/2`

by #>/2 to obtain more general relations See declarative integer arithmetic (section A.8.3). `?X`**#<**`?Y`- The arithmetic expression
`X`is less than`Y`. When reasoning over integers, replace`(<)/2`

by #</2 to obtain more general relations. See declarative integer arithmetic (section A.8.3).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)].

#### A.8.17.2 Membership constraints

If you are using CLP(FD) to model and solve combinatorial tasks, then
you typically need to specify the admissible domains of variables. The *membership
constraints* in/2 and ins/2
are useful in such cases.

`?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`. See in/2 for the syntax of`Domain`.

#### A.8.17.3 Enumeration predicates

When modeling combinatorial tasks, the actual search for solutions is
typically performed by *enumeration predicates* like labeling/2.
See the the section about *core relations* and search for more
information.

**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)`

. See labeling/2. **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. See core relations and search (section A.8.9) for usage advice.

#### A.8.17.4 Global constraints

A *global constraint* expresses a relation that involves many
variables at once. The most frequently used global constraints of this
library are the combinatorial constraints all_distinct/1,
global_cardinality/2
and cumulative/2.

**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_different**(`+Vars`)- Like all_distinct/1, but with weaker propagation. Consider using all_distinct/1 instead, since all_distinct/1 is typically acceptably efficient and propagates much more strongly.
**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`)- 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 #`>`=. **lex_chain**(`+Lists`)`Lists`are lexicographically non-decreasing.**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]].

**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, [])`

. See global_cardinality/3.Example:

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

**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 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] .

**cumulative**(`+Tasks`)- Equivalent to
`cumulative(Tasks, [limit(1)])`

. See cumulative/2. **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] .

**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. **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].

**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].

**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.

#### A.8.17.5 Reification predicates

Many CLP(FD) constraints can be *reified*. This means that their
truth value is itself turned into a CLP(FD) variable, so that we can
explicitly reason about whether a constraint holds or not. See
reification (section A.8.12).

**#\**`+Q``Q`does*not*hold. See reification (section A.8.12).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. See reification (section A.8.12).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.

`?P`**#==>**`?Q``P`implies`Q`. See reification (section A.8.12).`?P`**#<==**`?Q``Q`implies`P`. See reification (section A.8.12).`?P`**#/\**`?Q``P`and`Q`hold. See reification (section A.8.12).`?P`**#\/**`?Q``P`or`Q`holds. See reification (section A.8.12).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.

`?P`**#\**`?Q`- Either
`P`holds or`Q`holds, but not both. See reification (section A.8.12). **zcompare**(`?Order, ?A, ?B`)- Analogous to compare/3,
with finite domain variables
`A`and`B`.Think of zcompare/3 as

*reifying*an arithmetic comparison of two integers. This means that we can explicitly reason about the different cases*within*our programs. As in compare/3, the atoms`<`

,`>`

and`=`

denote the different cases of the trichotomy. In contrast to compare/3 though, zcompare/3 works correctly for*all modes*, also if only a subset of the arguments is instantiated. This allows you to make several predicates over integers deterministic while preserving their generality and completeness. For example:n_factorial(N, F) :- zcompare(C, N, 0), n_factorial_(C, N, F). n_factorial_(=, _, 1). n_factorial_(>, N, F) :- F #= F0*N, N1 #= N - 1, n_factorial(N1, F0).

This version of n_factorial/2 is deterministic if the first argument is instantiated, because argument indexing can distinguish the different clauses that reflect the possible and admissible outcomes of a comparison of

`N`against 0. Example:?- n_factorial(30, F). F = 265252859812191058636308480000000.

Since there is no clause for

`<`

, the predicate automatically*fails*if`N`is less than 0. The predicate can still be used in all directions, including the most general query:?- n_factorial(N, F). N = 0, F = 1 ; N = F, F = 1 ; N = F, F = 2 .

In this case, all clauses are tried on backtracking, and zcompare/3 ensures that the respective ordering between N and 0 holds in each case.

The truth value of a comparison can also be reified with (

`#<==>`

)/2 in combination with one of the*arithmetic constraints*(section A.8.2). See reification (section A.8.12). However, zcompare/3 lets you more conveniently distinguish the cases.

#### A.8.17.6 Reflection predicates

Reflection predicates let us obtain, in a well-defined way, information that is normally internal to this library. In addition to the predicates explained below, also take a look at call_residue_vars/2 and copy_term/3 to reason about CLP(FD) constraints that arise in programs. This can be useful in program analyzers and declarative debuggers.

**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`)- Reflect the current size of a 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.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.