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

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- Introduction
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- Example: Factorial relation
- Combinatorial constraints
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- 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)

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- The SWI-Prolog library
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- author
- Markus Triska

**Development of this library has moved to SICStus Prolog.**

Please see
**CLP(Z)**
for more information.

This library provides CLP(FD): Constraint Logic Programming over
Finite Domains. This is an instance of the general CLP(*X*) scheme
(section 8), extending logic
programming with reasoning over specialised domains. CLP(FD) lets us
reason about **integers** in a way that honors the relational nature
of Prolog.

Read **The
Power of Prolog** to understand how this library is meant to be
used in practice.

There are two major use cases of CLP(FD) constraints:

**declarative integer arithmetic**(section A.8.3)- solving
**combinatorial problems**such as planning, scheduling and allocation tasks.

The predicates of this library can be classified as:

*arithmetic*constraints like #=/2, #>/2 and #\=/2 (section A.8.17.1)- the
*membership*constraints in/2 and ins/2 (section A.8.17.2) - the
*enumeration*predicates indomain/1, label/1 and labeling/2 (section A.8.17.3) *combinatorial*constraints like all_distinct/1 and global_cardinality/2 (section A.8.17.4)*reification*predicates such as #<==>/2 (section A.8.17.5)*reflection*predicates such as fd_dom/2 (section A.8.17.6)

In most cases, *arithmetic constraints* (section
A.8.2) are the only predicates you will ever need from this library.
When reasoning over integers, simply replace low-level arithmetic
predicates like `(is)/2`

and `(>)/2`

by the
corresponding CLP(FD) constraints like #=/2
and #>/2 to honor and
preserve declarative properties of your programs. For satisfactory
performance, arithmetic constraints are implicitly rewritten at
compilation time so that low-level fallback predicates are automatically
used whenever possible.

Almost all Prolog programs also reason about integers. Therefore, it
is highly advisable that you make CLP(FD) constraints available in all
your programs. One way to do this is to put the following directive in
your `~/.swiplrc`

initialisation file:

:- use_module(library(clpfd)).

All example programs that appear in the CLP(FD) documentation assume that you have done this.

Important concepts and principles of this library are illustrated by
means of usage examples that are available in a public git repository:
**github.com/triska/clpfd**

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 all popular Prolog systems, and they are designed to
help you eliminate and avoid the use of low-level and less general
primitives by providing declarative alternatives that are meant to be
used instead.

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

We recommend the following reference (PDF: metalevel.at/swiclpfd.pdf) for citing 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} }

More information about CLP(FD) constraints and their implementation
is contained in: **metalevel.at/drt.pdf**

The best way to discuss applying, improving and extending CLP(FD)
constraints is to use the dedicated `clpfd`

tag on
stackoverflow.com.
Several of the world's foremost CLP(FD) experts regularly participate in
these discussions and will help you for free on this platform.

In modern Prolog systems, **arithmetic constraints** subsume and
supersede low-level predicates over integers. The main advantage of
arithmetic constraints is that they are true *relations* and can be
used in all directions. For most programs, arithmetic constraints are
the only predicates you will ever need from this library.

The most important arithmetic constraint is #=/2,
which subsumes both
`(is)/2`

and `(=:=)/2`

over integers. Use #=/2
to make your programs more general. See declarative integer arithmetic (section
A.8.3).

In total, the arithmetic constraints are:

Expr1 `#=`

Expr2Expr1 equals Expr2 Expr1 `#\=`

Expr2Expr1 is not equal to Expr2 Expr1 `#>=`

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

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

Expr2Expr1 is greater than Expr2 Expr1 `#<`

Expr2Expr1 is less than Expr2

`Expr1` and `Expr2` denote **arithmetic
expressions**, which are:

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 Expr div Expr Floored integer division

where `Expr` again denotes an arithmetic expression.

The bitwise operations `(\)/1`

, `(/\)/2`

, `(\/)/2`

, `(>>)/2`

,
`(<<)/2`

, `lsb/1`

, `msb/1`

, `popcount/1`

and `(xor)/2`

are also supported.

The *arithmetic constraints* (section
A.8.2) #=/2, #>/2
etc. are meant to be used *instead* of the primitives `(is)/2`

,
`(=:=)/2`

, `(>)/2`

etc. over integers. Almost
all Prolog programs also reason about integers. Therefore, it is
recommended that you put the following directive in your `~/.swiplrc`

initialisation file to make CLP(FD) constraints available in all your
programs:

:- use_module(library(clpfd)).

Throughout the following, it is assumed that you have done this.

The most basic use of CLP(FD) constraints is *evaluation* of
arithmetic expressions involving integers. For example:

?- X #= 1+2. X = 3.

This could in principle also be achieved with the lower-level
predicate `(is)/2`

. However, an important advantage of
arithmetic constraints is their purely relational nature: Constraints
can be used in *all directions*, also if one or more of their
arguments are only partially instantiated. For example:

?- 3 #= Y+2. Y = 1.

This relational nature makes CLP(FD) constraints easy to explain and use, and well suited for beginners and experienced Prolog programmers alike. In contrast, when using low-level integer arithmetic, we get:

?- 3 is Y+2. ERROR: is/2: Arguments are not sufficiently instantiated ?- 3 =:= Y+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. See `n_factorial/2`

(section
A.8.4) for an example.

This library uses goal_expansion/2
to automatically rewrite constraints at compilation time so that
low-level arithmetic predicates are *automatically* used whenever
possible. For example, the predicate:

positive_integer(N) :- N #>= 1.

is executed as if it were written as:

positive_integer(N) :- ( integer(N) -> N >= 1 ; N #>= 1 ).

This illustrates why the performance of CLP(FD) constraints is almost
always completely satisfactory when they are used in modes that can be
handled by low-level arithmetic. To disable the automatic rewriting, set
the Prolog flag `clpfd_goal_expansion`

to `false`

.

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 all popular Prolog systems, and they are designed to
help you eliminate and avoid the use of low-level and less general
primitives by providing declarative alternatives that are meant to be
used instead.

We illustrate the benefit of using #=/2 for more generality with a simple example.

Consider first a rather conventional definition of `n_factorial/2`

,
relating each natural number *N* to its factorial *F*:

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

This program uses CLP(FD) constraints *instead* of low-level
arithmetic throughout, and everything that *would have worked* with
low-level arithmetic *also* works with CLP(FD) constraints,
retaining roughly the same performance. For example:

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

Now the point: Due to the increased flexibility and generality of
CLP(FD) constraints, we are free to *reorder* the goals as follows:

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

In this concrete case, *termination* properties of the predicate
are improved. For example, the following queries now both terminate:

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

To make the predicate terminate if *any* argument is
instantiated, add the (implied) constraint `F #\= 0`

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

is the only non-terminating case of this kind.

The value of CLP(FD) constraints does *not* lie in completely
freeing us from *all* procedural phenomena. For example, the two
programs do not even have the same *termination properties* in all
cases. Instead, the primary benefit of CLP(FD) constraints is that they
allow you to try different execution orders and apply **declarative
debugging** techniques *at all*! Reordering goals (and
clauses) can significantly impact the performance of Prolog programs,
and you are free to try different variants if you use declarative
approaches. Moreover, since all CLP(FD) constraints *always terminate*,
placing them earlier can at most *improve*, never worsen, the
termination properties of your programs. An additional benefit of
CLP(FD) constraints is that they eliminate the complexity of introducing `(is)/2`

and `(=:=)/2`

to beginners, since *both* predicates are
subsumed by #=/2 when
reasoning over integers.

In the case above, the clauses are mutually exclusive *if* the
first argument is sufficiently instantiated. To make the predicate
deterministic in such cases while retaining its generality, you can use zcompare/3
to *reify* a comparison, making the different cases distinguishable
by pattern matching. For example, in this concrete case and others like
it, you can use `zcompare(Comp, 0, N)`

to obtain as `Comp`
the symbolic outcome (`<`

, `=`

, `>`

)
of 0 compared to N.

In addition to subsuming and replacing low-level arithmetic
predicates, CLP(FD) constraints are often used to solve combinatorial
problems such as planning, scheduling and allocation tasks. Among the
most frequently used **combinatorial constraints** are all_distinct/1,
global_cardinality/2
and cumulative/2.
This library also provides several other constraints like disjoint2/1
and automaton/8, which
are useful in more specialized applications.

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. CLP(FD)
constraints like #=/2, #>/2
and #\=/2 can at most
reduce, and never extend, the domains of their arguments. The
constraints in/2 and ins/2
let us explicitly state domains of CLP(FD) variables. The process of
determining and adjusting domains of variables is called constraint
**propagation**, and it is performed automatically by this library.
When the domain of a variable contains only one element, then the
variable is automatically unified to that element.

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

As another example, consider *Sudoku*: It is a popular puzzle
over integers that can be easily solved with CLP(FD) constraints.

sudoku(Rows) :- length(Rows, 9), maplist(same_length(Rows), Rows), append(Rows, Vs), Vs ins 1..9, maplist(all_distinct, Rows), transpose(Rows, Columns), maplist(all_distinct, Columns), Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is], blocks(As, Bs, Cs), blocks(Ds, Es, Fs), blocks(Gs, Hs, Is). blocks([], [], []). blocks([N1,N2,N3|Ns1], [N4,N5,N6|Ns2], [N7,N8,N9|Ns3]) :- all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]), blocks(Ns1, Ns2, Ns3). problem(1, [[_,_,_,_,_,_,_,_,_], [_,_,_,_,_,3,_,8,5], [_,_,1,_,2,_,_,_,_], [_,_,_,5,_,7,_,_,_], [_,_,4,_,_,_,1,_,_], [_,9,_,_,_,_,_,_,_], [5,_,_,_,_,_,_,7,3], [_,_,2,_,1,_,_,_,_], [_,_,_,_,4,_,_,_,9]]).

Sample query:

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

In this concrete case, the constraint solver is strong enough to find the unique solution without any search. For the general case, see search (section A.8.9).

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

?- 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.

The answers emitted by the toplevel are called *residual programs*,
and the goals that comprise each answer are called **residual goals**.
In each case above, and as for all pure programs, the residual program
is declaratively equivalent to the original query. From the residual
goals, it is clear that the constraint solver has deduced additional
domain restrictions in many cases.

To inspect residual goals, it is best to let the toplevel display them for us. Wrap the call of your predicate into call_residue_vars/2 to make sure that all constrained variables are displayed. To make the constraints a variable is involved in available as a Prolog term for further reasoning within your program, use copy_term/3. For example:

?- X #= Y + Z, X in 0..5, copy_term([X,Y,Z], [X,Y,Z], Gs). Gs = [clpfd: (X in 0..5), clpfd: (Y+Z#=X)], X in 0..5, Y+Z#=X.

This library also provides *reflection* predicates (like fd_dom/2,
fd_size/2 etc.) with
which we can inspect a variable's current domain. These predicates can
be useful if you want to implement your own labeling strategies.

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

**Modeling**. In this phase, all relevant constraints are stated.**Search**. In this phase,*enumeration predicates*are used to search for concrete solutions.

It is good practice to keep the modeling part, via a dedicated
predicate called the **core relation**, separate from the actual
search for solutions. This lets us observe termination and determinism
properties of the core relation in isolation from the search, and more
easily try different search strategies.

As an example of a constraint satisfaction problem, consider the cryptoarithmetic puzzle SEND + MORE = MONEY, where different letters denote distinct integers between 0 and 9. It can be modeled in CLP(FD) as follows:

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

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

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

From this answer, we see that this core relation *terminates*
and is in fact *deterministic*. Moreover, we see from the residual
goals that the constraint solver has deduced more stringent bounds for
all variables. Such observations are only possible if modeling and
search parts are cleanly separated.

Labeling can then be used to search for solutions in a separate predicate or goal:

?- puzzle(As+Bs=Cs), label(As). As = [9, 5, 6, 7], Bs = [1, 0, 8, 5], Cs = [1, 0, 6, 5, 2] ; false.

In this case, it suffices to label a subset of variables to find the puzzle's unique solution, since the constraint solver is strong enough to reduce the domains of remaining variables to singleton sets. In general though, it is necessary to label all variables to obtain ground solutions.

We illustrate the concepts of the preceding sections by means of the
so-called *eight queens puzzle*. The task is to place 8 queens on
an 8x8 chessboard such that none of the queens is under attack. This
means that no two queens share the same row, column or diagonal.

To express this puzzle via CLP(FD) constraints, we must first pick a
suitable representation. Since CLP(FD) constraints reason over
*integers*, we must find a way to map the positions of queens to
integers. Several such mappings are conceivable, and it is not
immediately obvious which we should use. On top of that, different
constraints can be used to express the desired relations. For such
reasons, *modeling* combinatorial problems via CLP(FD) constraints
often necessitates some creativity and has been described as more of an
art than a science.

In our concrete case, we observe that there must be exactly one queen
per column. The following representation therefore suggests itself: We
are looking for 8 integers, one for each column, where each integer
denotes the *row* of the queen that is placed in the respective
column, and which are subject to certain constraints.

In fact, let us now generalize the task to the so-called *N queens
puzzle*, which is obtained by replacing 8 by *N* everywhere it
occurs in the above description. We implement the above considerations
in the
**core relation** `n_queens/2`

, where the first argument
is the number of queens (which is identical to the number of rows and
columns of the generalized chessboard), and the second argument is a
list of *N* integers that represents a solution in the form
described above.

n_queens(N, Qs) :- length(Qs, N), Qs ins 1..N, safe_queens(Qs). safe_queens([]). safe_queens([Q|Qs]) :- safe_queens(Qs, Q, 1), safe_queens(Qs). safe_queens([], _, _). safe_queens([Q|Qs], Q0, D0) :- Q0 #\= Q, abs(Q0 - Q) #\= D0, D1 #= D0 + 1, safe_queens(Qs, Q0, D1).

Note that all these predicates can be used in *all directions*:
We can use them to *find* solutions, *test* solutions and *complete*
partially instantiated solutions.

The original task can be readily solved with the following query:

?- n_queens(8, Qs), label(Qs). Qs = [1, 5, 8, 6, 3, 7, 2, 4] .

Using suitable labeling strategies, we can easily find solutions with 80 queens and more:

?- n_queens(80, Qs), labeling([ff], Qs). Qs = [1, 3, 5, 44, 42, 4, 50, 7, 68|...] . ?- time((n_queens(90, Qs), labeling([ff], Qs))). % 5,904,401 inferences, 0.722 CPU in 0.737 seconds (98% CPU) Qs = [1, 3, 5, 50, 42, 4, 49, 7, 59|...] .

Experimenting with different search strategies is easy because we have separated the core relation from the actual search.

We 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, we can observe properties of the core relation in
isolation, and try different labeling options without recompiling our
code.

If necessary, we can use `once/1`

to commit to the first
optimal solution. However, it is often very valuable to see alternative
solutions that are *also* optimal, so that we can choose among
optimal solutions by other criteria. For the sake of
**purity**
and completeness, we recommend to avoid `once/1`

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

Related to optimisation with CLP(FD) constraints are
`library(simplex)`

and CLP(Q) which reason about *linear* constraints over rational
numbers.

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 CLP(B)
constraints as provided by
`library(clpb)`

.

In the default execution mode, CLP(FD) constraints still exhibit some
non-relational properties. For example, *adding* constraints can
yield new solutions:

?- X #= 2, X = 1+1. false. ?- X = 1+1, X #= 2, X = 1+1. X = 1+1.

This behaviour is highly problematic from a logical point of view, and it may render declarative debugging techniques inapplicable.

Set the Prolog flag `clpfd_monotonic`

to `true`

to make CLP(FD)
**monotonic**: This means that *adding* new constraints *cannot*
yield new solutions. When this flag is `true`

, we must wrap
variables that occur in arithmetic expressions with the functor `(?)/1`

or `(#)/1`

. For example:

?- set_prolog_flag(clpfd_monotonic, true). true. ?- #(X) #= #(Y) + #(Z). #(Y)+ #(Z)#= #(X). ?- X #= 2, X = 1+1. ERROR: Arguments are not sufficiently instantiated

The wrapper can be omitted for variables that are already constrained to integers.

We can define custom constraints. The mechanism to do this is not yet finalised, and we welcome suggestions and descriptions of use cases that are important to you.

As an example of how it can be done currently, let us define a new
custom constraint `oneground(X,Y,Z)`

, where Z shall be 1 if
at least one of X and Y is instantiated:

:- multifile 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.

CLP(FD) applications that we find particularly impressive and worth studying include:

- Michael Hendricks uses CLP(FD) constraints for flexible reasoning
about
*dates*and*times*in the`julian`

package. - Julien Cumin uses CLP(FD) constraints for integer arithmetic in
`Brachylog`

.

This library gives you a glimpse of what **SICStus
Prolog** can do. The API is intentionally mostly compatible with
that of SICStus Prolog, so that you can easily switch to a much more
feature-rich and much faster CLP(FD) system when you need it. I thank Mats
Carlsson, the designer and main implementor of SICStus Prolog, for
his elegant example. I first encountered his system as part of the
excellent
**GUPU**
teaching environment by Ulrich
Neumerkel. Ulrich was also the first and most determined tester of
the present system, filing hundreds of comments and suggestions for
improvement. Tom
Schrijvers has contributed several constraint libraries to
SWI-Prolog, and I learned a lot from his coding style and implementation
examples. Bart
Demoen was a driving force behind the implementation of attributed
variables in SWI-Prolog, and this library could not even have started
without his prior work and contributions. Thank you all!

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

*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)].

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`.

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 *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.

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.

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.

CLP(FD) constraints are one of the main reasons why logic programming approaches are picked over other paradigms for solving many tasks of high practical relevance. The usefulness of CLP(FD) constraints for scheduling, allocation and combinatorial optimization tasks is well-known both in academia and industry.

With this library, we take the applicability of CLP(FD) constraints
one step further, following the road that visionary systems like SICStus
Prolog have already clearly outlined: This library is designed to
completely subsume and *replace* low-level predicates over
integers, which were in the past repeatedly found to be a major
stumbling block when introducing logic programming to beginners.

Embrace the change and new opportunities that this paradigm allows! Use CLP(FD) constraints in your programs. The use of CLP(FD) constraints instead of low-level arithmetic is also a good indicator to judge the quality of any introductory Prolog text.

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