- Documentation
- Reference manual
- The SWI-Prolog library
- library(aggregate): Aggregation operators on backtrackable predicates
- library(apply): Apply predicates on a list
- library(assoc): Association lists
- library(broadcast): Broadcast and receive event notifications
- library(charsio): I/O on Lists of Character Codes
- library(check): Consistency checking
- library(clpb): Constraint Logic Programming over Boolean Variables
- library(clpfd): Constraint Logic Programming over Finite Domains
- library(clpqr): Constraint Logic Programming over Rationals and Reals
- library(csv): Process CSV (Comma-Separated Values) data
- library(debug): Print debug messages and test assertions
- library(gensym): Generate unique identifiers
- library(iostream): Utilities to deal with streams
- library(lists): List Manipulation
- library(nb_set): Non-backtrackable set
- library(www_browser): Activating your Web-browser
- library(option): Option list processing
- library(optparse): command line parsing
- library(ordsets): Ordered set manipulation
- library(pairs): Operations on key-value lists
- library(persistency): Provide persistent dynamic predicates
- library(pio): Pure I/O
- library(predicate_options): Declare option-processing of predicates
- library(prolog_pack): A package manager for Prolog
- library(prolog_xref): Cross-reference data collection library
- library(quasi_quotations): Define Quasi Quotation syntax
- library(random): Random numbers
- library(readutil): Reading lines, streams and files
- library(record): Access named fields in a term
- library(registry): Manipulating the Windows registry
- library(simplex): Solve linear programming problems
- library(solution_sequences): Modify solution sequences
- library(thread_pool): Resource bounded thread management
- library(ugraphs): Unweighted Graphs
- library(url): Analysing and constructing URL
- library(varnumbers): Utilities for numbered terms
- library(yall): Lambda expressions

- The SWI-Prolog library
- Packages

- Reference manual

- author
- Markus Triska

This library provides CLP(FD): Constraint Logic Programming over Finite Domains.

CLP(FD) is an instance of the general CLP(.) scheme, extending logic
programming with reasoning over specialised domains. CLP(FD) lets you
reason about sets of *integers*.

There are two major use cases of this library:

- CLP(FD) constraints provide
**declarative integer arithmetic**: They implement pure*relations*between integer expressions and can be used in all directions, also if parts of expressions are variables. - In connection with enumeration predicates and more complex
constraints, CLP(FD) is often used to model and solve
**combinatorial problems**such as planning, scheduling and allocation tasks.

When teaching Prolog, we *strongly recommend* that you introduce
CLP(FD) constraints *before* explaining lower-level arithmetic
predicates and their procedural idiosyncrasies. This is because
constraints are easy to explain, understand and use due to their purely
relational nature. In contrast, the modedness and directionality of
low-level arithmetic primitives are non-declarative limitations that are
better deferred to more advanced lectures.

If you are used to the complicated operational considerations that
low-level arithmetic primitives necessitate, then moving to CLP(FD)
constraints may, due to their power and convenience, at first feel to
you excessive and almost like cheating. It *isn't*. Constraints are
an integral part of many Prolog systems and are available to help you
eliminate and avoid, as far as possible, the use of lower-level and less
general primitives by providing declarative alternatives that are meant
to be used instead.

For satisfactory performance, arithmetic constraints are implicitly rewritten at compilation time so that lower-level fallback predicates are automatically used whenever possible.

We recommend the following reference to cite this library in scientific publications:

@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} }

and the following URL to link to its documentation:

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

Usage examples of this library are available in a public git repository: https://github.com/triska/clpfd

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`

ExprModulo induced by floored division Expr `rem`

ExprModulo induced by truncated division `abs(Expr)`

Absolute value Expr `//`

ExprTruncated 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

CLP(FD) constraints let you declaratively express integer arithmetic. The CLP(FD) constraints #=/2, #>/2 etc. are meant to be used instead of the corresponding primitives is/2, =:=/2, >/2 etc. over integers.

An important advantage of arithmetic constraints is their purely relational nature. They are therefore easy to explain and use, and well suited for beginners and experienced Prolog programmers alike.

Consider for example the query:

?- X #> 3, X #= 5 + 2. X = 7.

In contrast, when using low-level integer arithmetic, we get:

?- X > 3, X is 5 + 2. ERROR: >/2: Arguments are not sufficiently instantiated

Due to the necessary operational considerations, the use of these low-level arithmetic predicates is considerably harder to understand and should therefore be deferred to more advanced lectures.

For supported expressions, CLP(FD) constraints are drop-in replacements of these low-level arithmetic predicates, often yielding more general programs.

Here is an example:

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

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

?- n_factorial(47, F). F = 258623241511168180642964355153611979969197632389120000000000 ; false. ?- n_factorial(N, 1). N = 0 ; N = 1 ; false. ?- n_factorial(N, 3). false.

To make the predicate terminate if any argument is instantiated, add
the (implied) constraint F `#\=`

0 before the recursive call.
Otherwise, the query `n_factorial(N, 0)`

is the only
non-terminating case of this kind.

This library uses goal_expansion/2
to automatically rewrite arithmetic constraints at compilation time. The
expansion's aim is to bring the performance of arithmetic constraints
close to that of lower-level arithmetic predicates whenever possible. To
disable the expansion, set the flag `clpfd_goal_expansion`

to `false`

.

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 either P or Q, but not both 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.

When reasoning over Boolean variables, also consider using
`library(clpb)`

and its dedicated CLP(B) constraints.

Each CLP(FD) variable has an associated set of admissible integers
which we call the variable's *domain*. Initially, the domain of
each CLP(FD) variable is the set of all integers. The constraints in/2
and
ins/2 are the primary means
to specify tighter domains of variables.

Here are example queries and the system's declaratively equivalent answers:

?- X in 100..sup. X in 100..sup. ?- X in 1..5 \/ 3..12. X in 1..12. ?- [X,Y,Z] ins 0..3. X in 0..3, Y in 0..3, Z in 0..3.

Domains are taken into account when further constraints are stated, and by enumeration predicates like labeling/2.

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

?- use_module(library(clpfd)). % library(clpfd) compiled into clpfd 0.06 sec, 633,732 bytes true. ?- X #> 3. X in 4..sup. ?- X #\= 20. X in inf..19\/21..sup. ?- 2*X #= 10. X = 5. ?- X*X #= 144. X in -12\/12. ?- 4*X + 2*Y #= 24, X + Y #= 9, [X,Y] ins 0..sup. X = 3, Y = 6. ?- X #= Y #<==> B, X in 0..3, Y in 4..5. B = 0, X in 0..3, Y in 4..5.

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

In addition to being declarative replacements for low-level arithmetic predicates, CLP(FD) constraints are also often used to solve combinatorial problems such as planning, scheduling and allocation tasks. To let you conveniently model and solve such problems, this library provides several constraints beyond typical integer arithmetic, such as all_distinct/1, global_cardinality/2 and cumulative/1.

Using CLP(FD) constraints to solve combinatorial tasks typically consists of two phases:

- First, all relevant constraints are stated.
- 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 you 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(FD) as follows:

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

Notice that we are *not* using labeling/2
in this predicate, so that we can first execute and observe the modeling
part in isolation. Sample query and its result (actual variables
replaced for readability):

?- puzzle(As+Bs=Cs). As = [9, A2, A3, A4], Bs = [1, 0, B3, A2], Cs = [1, 0, A3, A2, C5], A2 in 4..7, all_different([9, A2, A3, A4, 1, 0, B3, C5]), 91*A2+A4+10*B3#=90*A3+C5, A3 in 5..8, A4 in 2..8, B3 in 2..8, C5 in 2..8.

From this answer, we see that 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.

You can use labeling/2
to minimize or maximize the value of a CLP(FD) 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, you can observe properties of the core relation in
isolation, and try different labeling options without recompiling your
code.

If necessary, you 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 you can choose among
optimal solutions by other criteria. For the sake of purity and
completeness, we recommend to avoid `once/1`

and other
constructs that lead to impurities in CLP(FD) programs.

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`)- 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 #`>`=. `?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.

`?P`**#\**`?Q`- Either
`P`holds or`Q`holds, but not both. **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.

:- 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 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)])`

. **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::- 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 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:

:- use_module(library(clpfd)). 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].

**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(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(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`)`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.