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Please see CLP Z for more information. This is an instance of the general CLP X scheme section 8 , extending logic programming with reasoning over specialised domains.
Read The Power of Prolog to understand how this library is meant to be used in practice. In most cases, arithmetic constraints section A. 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.
Important concepts and principles of this library are illustrated by means of usage examples that are available in a public git repository: github. 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.
The best way to discuss applying, improving and extending CLP FD constraints is to use the dedicated clpfd tag on stackoverflow. 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. See declarative integer arithmetic section A. The arithmetic constraints section A. For example:. 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. 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:. 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. For example, the predicate:. 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.
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.
Now the point: Due to the increased flexibility and generality of CLP FD constraints, we are free to reorder the goals as follows:. In this concrete case, termination properties of the predicate are improved. For example, the following queries now both terminate:. 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.
In the case above, the clauses are mutually exclusive if the first argument is sufficiently instantiated. 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.
Each CLP FD variable has an associated set of admissible integers, which we call the variable's domain. 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. 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. 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. These predicates can be useful if you want to implement your own labeling strategies. 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. Sample query and its result actual variables replaced for readability :. 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. 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. 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. 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. Experimenting with different search strategies is easy because we have separated the core relation from the actual search. See the labeling options min Expr and max Expr , respectively.
This way, we can observe properties of the core relation in isolation, and try different labeling options without recompiling our code. 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.
Let P and Q denote reifiable constraints or Boolean variables, then:. When reasoning over Boolean variables, also consider using CLP B constraints as provided by library clpb. For example, adding constraints can yield new solutions:. This behaviour is highly problematic from a logical point of view, and it may render declarative debugging techniques inapplicable. When this flag is true , we must wrap variables that occur in arithmetic expressions with the functor?
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:.
From now on, the propagator will be invoked whenever the domains of X or Y are changed. As explained, this predicate is automatically called by the constraint solver. An example of using the new constraint:. 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! This can at most increase the generality of your programs.
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:. If you are using CLP FD to model and solve combinatorial tasks, then you typically need to specify the admissible domains of variables.
See the the section about core relations and search for more information. The variable selection strategy lets you specify which variable of Vars is labeled next and is one of:. Labeling Vars must make Expr ground.
If several such options are specified, they are interpreted from left to right, e. This generates solutions in descending order of X, and for each binding of X, solutions are generated in ascending order of Y. Labeling is always complete, always terminates, and yields no redundant solutions. See core relations and search section A. A global constraint expresses a relation that involves many variables at once.
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