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Constraint Logic Programming over Integers

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# CLP(ℤ) — Constraint Logic Programming over Integers

This repository contains information about CLP(ℤ).

CLP(ℤ) requires SICStus Prolog.

As of April 2020, a version of this library ships with Scryer Prolog as library(clpz).

The present implementation builds upon a decade of experience with a precursor library which I developed for a different Prolog system. CLP(ℤ) is the more recent and conceptually more advanced implementation. To keep track of recent developments, use CLP(ℤ).

Current developments:

• increase logical purity of the implementation
• work on stronger propagation
• correct all reported issues.

CLP(ℤ) is being developed for inclusion in GUPU.

An introduction to declarative integer arithmetic is available from metalevel.at/prolog/clpz

## Using CLP(ℤ) constraints

CLP(ℤ) is an instance of the general CLP(X) scheme, extending logic programming with reasoning over specialised domains.

In the case of CLP(ℤ), the domain is the set of integers. CLP(ℤ) is a generalisation of CLP(FD) as provided by SICStus Prolog.

CLP(ℤ) constraints like `(#=)/2`, `(#\=)/2`, and `(#<)/2` are meant to be used as more general alternatives for lower-level arithmetic primitives over integers. Importantly, they can be used in all directions.

For example, consider a rather typical definition of `n_factorial/2`:

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

CLP(ℤ) constraints allow us to quite freely exchange the order of goals, obtaining for example:

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

This works in all directions, for example:

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

and also:

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

and also in the most general case:

``````?- n_factorial(N, F).
N = 0,
F = 1 ;
N = F, F = 1 ;
N = F, F = 2 ;
N = 3,
F = 6 .
``````

The advantage of using `(#=)/2` to express arithmetic equality is clear: It is a more general alternative for lower-level predicates.

In addition to providing declarative integer arithmetic, CLP(ℤ) constraints are also often used to solve combinatorial tasks with Prolog.

## Example programs

This repository contains several example programs. The main predicates are all completely pure and can be used as true relations. This means that you can use the same program to:

• find a single solution
• enumerate all solutions
• complete partially instantiated solutions
• validate fully instantiated solutions.

To get an idea of the power, usefulness and scope of CLP(ℤ) constraints, I recommend you work through the examples in the following order:

1. n_factorial.pl: Shows how to use CLP(ℤ) constraints for declarative integer arithmetic, obtaining very general programs that can be used in all directions. Declarative integer arithmetic is the simplest and most common use of CLP(ℤ) constraints. They are easy to understand and use this way, and often increase generality and logical purity of your code.

2. sendmory.pl: A simple cryptoarithmetic puzzle. The task is to assign one of the digits 0,...,9 to each of the letters S,E,N,D,M,O,R and Y in such a way that the following calculation is valid, and no leading zeroes appear:

``````     S E N D
+ M O R E
---------
= M O N E Y
``````

This example illustrates several very important concepts:

• It is the first example that shows residual constraints for the most general query. They are equivalent to the original query.

• It is good practice to separate the core relation from `labeling/2`, so that termination and determinism can be observed without an expensive search for concrete solutions.

• You can use this example to illustrate that the CLP(ℤ) system is able to propagate many things that can also be found with human reasoning. For example, due to the nature of the above calculation and the prohibition of leading zeroes, `M` is necessarily 1.

3. sudoku.pl: Uses CLP(ℤ) constraints to model and solve a simple and well-known puzzle. This example is well suited for understanding the impact of different propagation strengths: Use it to compare `all_different/1` `all_distinct/1` on different puzzles:

The small dots in each cell indicate how many elements are pruned by different consistency techniques. In many Sudoku puzzles, using `all_distinct/1` makes labeling unnecessary. Does this mean that we can forget `all_different/1` entirely?

4. magic_square.pl: CLP(ℤ) formulation of magic squares. This is a good example to learn about symmetry breaking constraints: Consider how you can eliminate solutions that are rotations, reflections etc. of other solutions, by imposing suitable further constraints. For example, the following two solutions are essentially identical, since one can be obtained from the other by reflecting elements along the main diagonal:

Can you impose additional constraints so that you get only a single solution in such cases, without losing any solutions that do not belong to the same equivalence class? How many solutions are there for N=4 that are unique up to isomorphism?

5. magic_hexagon.pl: Uses CLP(ℤ) to describe a magic hexagon of order 3. The task is to place the integers 1,...,19 in the following grid so that the sum of all numbers in a straight line (there are lines of length 3, 4 and 5) is equal to 38. One solution of this task is shown in the right picture:

This is an example of a task that looks very simple at first, yet is almost impossibly hard to solve manually. It is easy to solve with CLP(ℤ) constraints though. Use the constraint solver to show that the solution of this task is unique up to isomorphism.

6. n_queens.pl: Model the so-called N-queens puzzle with CLP(ℤ) constraints. This example is a good candidate to experiment with different search strategies, specified as options of `labeling/2`. For example, using the labeling strategy `ff`, you can easliy find solutions for 100 queens and more. Sample solutions for 8 and 50 queens:

Try to find solutions for larger N. Reorder the variables so that `ff` breaks ties by selecting more central variables first.

7. knight_tour.pl: Closed Knight's Tour using CLP(ℤ) constraints. This is an example of using a more complex global constraint called `circuit/1`. It shows how a problem can be transformed so that it can be expressed with a global constraint. Sample solutions, using an 8x8 and a 16x16 board:

Decide whether `circuit/1` can also be used to model tours that are not necessarily closed. If not, why not? If possible, do it.

8. tasks.pl: A task scheduling example, using the `cumulative/2` global constraint. The `min/1` labeling option is used to minimize the total duration.

## Animations

When studying Prolog and CLP(ℤ) constraints, it is often very useful to show animations of search processes. An instructional example:

N-queens animation: This visualizes the search process for the N-queens example.

You can use similar PostScript instructions to create custom animations for other examples.

## A limited alternative: Low-level integer arithmetic

Suppose for a moment that CLP(ℤ) constraints were not available in your Prolog system, or that you do not want to use them. How do we formulate `n_factorial/2` with more primitive integer arithmetic?

In our first attempt, we simply replace the declarative CLP(ℤ) constraints by lower-level arithmetic predicates and obtain:

``````n_factorial(0, 1).
n_factorial(N, F) :-
N > 0,
N1 is N - 1,
F is N * F1,
n_factorial(N1, F1).
``````

Unfortunately, this does not work at all, because lower-level arithmetic predicates are moded: This means that their arguments must be sufficiently instantiated at the time they are invoked. Therefore, we must reorder the goals and — somewhat annoyingly — change this for example to:

``````n_factorial(0, 1).
n_factorial(N, F) :-
N > 0,
N1 is N - 1,
n_factorial(N1, F1),
F is N * F1.
``````

Naive example queries inspired more by functional than by relational thinking may easily mislead us into believing that this version is working correctly:

``````?- n_factorial(6, F).
F = 720 ;
false.
``````

Another example:

``````?- n_factorial(3, F).
F = 6 ;
false.
``````

But what about more general queries? For example:

``````?- n_factorial(N, F).
N = 0,
F = 1 ;
ERROR: n_factorial/2: Arguments are not sufficiently instantiated
``````

Unfortunately, this version thus cannot be directly used to enumerate more than one solution, which is another severe drawback in comparison with the more general version.

You can make the deficiency a lot worse by arbitrarily adding a `!/0` somewhere. Using `!/0` is a quite reliable way to destroy almost all declarative properties of your code in most cases, and this example is no exception:

``````n_factorial(0, 1) :- !.
n_factorial(N, F) :-
N > 0,
N1 is N - 1,
n_factorial(N1, F1),
F is N * F1.
``````

This version appears in several places. The fact that the following interaction incorrectly tells us that there is exactly one solution of the factorial relation is apparently no cause for concern there:

``````?- n_factorial(N, F).
N = 0,
F = 1.
``````

Zero and one are the only important integers in any case, if you are mostly interested in programming at a very low level.

For more usable and general programs, I therefore recommend you stick to CLP(ℤ) constraints for integer arithmetic. You can place pure goals in any order without changing the declarative meaning of your program, just as you would expect from logical conjunction. For example:

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

Reordering pure goals can change termination properties, but it cannot incorrectly lead to failure where there is in fact a solution. Therefore, we get with the above CLP(ℤ) version for example:

``````?- n_factorial(N, 3).
<loops>
``````

And now we can reason completely declaratively about the code: Knowing that (a) CLP(ℤ) constraints are pure and can thus be reordered quite liberally and (b) that posting CLP(ℤ) constraints always terminates, we know that placing CLP(ℤ) constraints earlier can at most improve, never worsen the desirable termination properties.

Therefore, we change the definition to the version shown initially:

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

The sample query now terminates:

``````?- n_factorial(N, 3).
false.
``````

Using CLP(ℤ) constraints has allowed us to improve the termination properties of this predicate by purely declarative reasoning.

## Acknowledgments

I am extremely grateful to:

Ulrich Neumerkel, who introduced me to constraint logic programming.

Nysret Musliu, my thesis advisor, whose interest in combinatorial tasks and constraint satisfaction highly motivated me to work in this area.

Mats Carlsson, the designer and main implementor of SICStus Prolog and its superb CLP(FD) library which spawned my interest in constraints.

Constraint Logic Programming over Integers

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