# Constraint Logic Programming In Prolog

**Table of Contents**

## Introduction

Prolog is a programming language based on the logic programming paradigm. Prolog uses Horn clauses to represent relations in the form of facts and rules. Together, these elements form a knowledge base that can be used to run computations or queries.

Constraint logic programming is an extension of this paradigm that introduces constraints on possible values and variables. This enables us to narrow down the search space and more quickly find solutions to optimization problems where the solution must adhere to defined constraints.

There are many subdomains to constraint logic programming, each optimized for solving a specific problem domain, including the following:

**CLP(FD)**- Integers - 1, 2, 3, ...**CLP(B)**- Booleans - true, false**CLP(R)**- Real Numbers - 1.0, 2.0, 3.14, ...**CLP(Q)**- Rational Numbers - 1/3, 2/3, 3/3, ...

Next, I'll walk you through CLP(FD) and CLP(B) to give you insight into how they can be used and for which types of problem domains.

CLP(R) and CLP(Q) are similar to CLP(FD) but contain tools optimized for solving problems specific to their respective domains.

## CLP(FD) - Integers

Using CLP(FD) we can find solutions to problems involving a set of integers within a finite domain. The domain is considered "finite" because integer variables can assume only a limited range of values.

The CLP(FD) domain includes combinatorial problems such as planning, scheduling, and allocation.

We can define different types of constraints, such as:

**Range**constraint:**X in 1..10**, indicates that variable**X**can take any integer value between 1 and 10.**Arithmetic**constraint:**X #= Y + Z**states that**X**must be the sum of**Y**and**Z**.**Global**constraints: For instance,**all_different(List)**requires all variables in the list named List to take different values.- Other constraints: E.g.,
**X #< Y**,**X #> Y**,**X #= Y**to specify less-than, greater-than, and not-equal-to constraints respectively.

For example, the Magic Square problem, in which the sums of the numbers in each row, each column, and both main diagonals are the same, can be solved this these constraings. Sudoku puzzles can also be solved using this technique.

## CLP(B) - Booleans

Using CLP(B) we can find solutions to problems involving boolean variables. This includes combinatorial problems such as verification, allocation and covering tasks

### SAT Problems

A CLP(B) problem can be seen as a general **SAT (Boolean Satisfiability Problem)**.

When solving a SAT problem, you're trying to find an assignment of truth values to boolean variables such that a given boolean formula (typically expressed in conjunctive normal form) evaluates to true.

**Problem**: Find values for A and B (either true or false) such that the following boolean formula is true: **A ∨ B** (∨ represents the OR operation).

**Solution**: The clause (A ∨ B) evaluates to true when:

- A is true and B is false.
- A is false and B is true.
- Both A and B are true.

### CNF (Conjunctive Normal Form)

Conjunctive normal form (CNF) is a way of expressing SAT problems using a combination of the following elements:

**Literals**(boolean variables): For example: The variables**A**and**B**.**Clause**(logical OR): For example: A ∨ B. This means that the statement A or B is true if at least one of the variables, A or B, is true.**Conjuction**(logical AND): For example: A ∧ B. This is true if and only if both A and B are true.**Negation**(logical NOT): For example: ¬A: This literal is true if AA is false. The "not" operation ("¬" or sometimes as "~") inverts, or negates, a variable.

### NP (Nondeterministic Polynomial Time) Problems

CLP(B) and SAT problems belong to the NP class. For problems in NP, while verifying a solution is straightforward, **finding a solution can be highly time-consuming** in the worst-case scenarios. While determining a solution for a basic expression with two boolean variables, such as A∨BA∨B, is relatively simple, the **complexity exhibits polynomial growth** when dealing with thousands of variables.

## Example: Solving the Magic Square Problem Using CLP(FD)

We'll use SWI-Prolog's CLP(FD) library and the following code to solve the Magic Square problem:

```
% Import the clpfd library which provides constraints over finite domains
:- use_module(library(clpfd)).
% Define the magic_square functor that takes a list (Square) as an argument
magic_square(Square) :-
% Assign the Square list to contain variables A through I
Square = [A,B,C,D,E,F,G,H,I],
% Constrain each variable in Square to be a value between 1 and 9
Square ins 1..9, % Each cell is in the domain from 1 to 9
% Assert that all variables in Square must have distinct values
all_different(Square), % All cells must have different values
% The following constraints ensure that the sum of numbers in each
% row, column, and diagonal is equal. The exact sum value is represented by 'Sum'.
% Row constraints: Each row should sum up to the value 'Sum'
A+B+C #= Sum,
D+E+F #= Sum,
G+H+I #= Sum,
% Column constraints: Each column should sum up to the value 'Sum'
A+D+G #= Sum,
B+E+H #= Sum,
C+F+I #= Sum,
% Diagonal constraints: Both the diagonals should sum up to the value 'Sum'
A+E+I #= Sum, % From top-left to bottom-right diagonal
C+E+G #= Sum, % From top-right to bottom-left diagonal
% Try to find a solution that satisfies all the constraints
label(Square).
```

To list all solutions we run the query in the Prolog interpreter:

```
?- magic_square(Square).
Square = [2, 7, 6, 9, 5, 1, 4, 3, 8] ;
Square = [2, 9, 4, 7, 5, 3, 6, 1, 8] ;
Square = [4, 3, 8, 9, 5, 1, 2, 7, 6] ;
Square = [4, 9, 2, 3, 5, 7, 8, 1, 6] ;
Square = [6, 1, 8, 7, 5, 3, 2, 9, 4] ;
Square = [6, 7, 2, 1, 5, 9, 8, 3, 4] ;
Square = [8, 1, 6, 3, 5, 7, 4, 9, 2] ;
Square = [8, 3, 4, 1, 5, 9, 6, 7, 2] ;
false.
```

For a more general solution for all square sizes, see Markus Triska's solution: https://github.com/triska/clpfd/blob/master/magic_square.pl

## References

- The Power of Prolog

https://www.metalevel.at/prolog

- CLP(FD)

https://github.com/triska/clpfd

https://www.metalevel.at/drt.pdf

- CLP(B)