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:
CLP(ℤ) is being developed for inclusion in
GUPU.
An introduction to declarative integer arithmetic is available from
metalevel.at/prolog/clpz
Video: https://www.metalevel.at/prolog/videos/integer_arithmetic 
For more information about pure Prolog, read The Power of Prolog.
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.
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:
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:
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.
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 fromlabeling/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.
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?
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?
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.
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 oflabeling/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 thatff breaks ties by selecting more central variables first.
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.
tasks.pl: A task scheduling example, using thecumulative/2 global constraint. The min/1 labeling option is
used to minimize the total duration.
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.
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.
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.