1. 1. An Overview of Prolog

2. Contents An example program: defining family relations
Extending the example program by rules
A recursive rule definition
How Prolog answers questions
Declarative and procedure meaning of programs

3. An Example Program
Prolog is a programming language for symbolic, non-numeric computation.
It is specially well suited for solving problems that involve objects and relations between objects.
pam
tom
bob
liz
ann
pat
jim
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).

4. An Example Program ?- parent(bob,pat). yes ?- parent(liz,pat). no
?-parent(tom,ben).
?- parent(X,liz).
X=tom.
?- parent(bob,X).
X=ann;
X=pat;
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).

5. An Example Program Who is a parent of whom?
Find X and Y such that X is a parent of Y.
?- parent(X,Y).
X=pam
Y=bob;
X=tom
Y=liz;
...
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).

6. An Example Program Who is a grandparent of Jim?
(1) Who is a parent of Jim? Assume that this is some Y.
(2) Who is a parent of Y? Assume that this is some X.
?- parent(Y,Jim), parent(X,Y).
X=bob
Y=pat X Y
jim
parent
grandparent
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).
?- parent(X,Y),parent(Y,Jim).
will produce the same result.

7. An Example Program Who are Tom’s grandchildren?
?- parent(tom,X),parent(X,Y).
X=bob
Y=ann;
Y=pat
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).

8. An Example Program Do Ann and Pat have a common parent?
(1) Who is a parent, X, of Ann?
(2) Is (this same) X a parent of Pat?
?- parent(X,ann), parent(X,pat).
X=bob
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).

9. An Example Program
It is easy in Prolog to define a relation by stating the n-tuples of objects that satisfy the relation.
The user can query the prolog system about relations defined in the program.
The arguments of relations can be:
concrete objects, or constants, or general objects.
Questions to the system consist of one or more goals.
An answer to the question can either be positive or negative.
If several answers satisfy the question than Prolog will find as many of them as desired by the user.

10. Extending the Example Program by Rules
Adding more facts
female(pam).
male(tom).
male(bob).
female(liz).
female(pat).
female(ann).
male(jim).
sex(pam, feminine).
sex(tom, masculine).
sex(bob, masculine).
sex(liz , feminine).
sex(pat , feminine).
sex(ann , feminine).
sex(jim, masculine).

11. Extending the Example Program by Rules
We could define offspring in a similar way as the parent relation.
However, the offspring relation can be defined much more elegantly by making use of the fact that it is the inverse of parent, and that parent has already been defined.
offspring(Y,X):-parent(X,Y).
For all X and Y,
Y is an offspring of X if
X is a parent of Y.
A rule in Prolog

12. Extending the Example Program by Rules
Difference between facts and rules:
A fact is something that is always, unconditionally, true.
Rules specify things that are true if some condition is satisfied.

13. Extending the Example Program by Rules
Rules have:
a condition part (the LHS of the rule) and
a conclusion part (the RHS of the rule).
offspring(Y,X):-parent(X,Y).
head
body

14. Extending the Example Program by Rules
?- offspring(liz,tom).
Applying the rule, it becomes
offspring(liz,tom):-parent(tom,liz).
parent(tom,bob).
parent(pam,bob).
parent(tom,liz).
parent(bob,ann).
parent(bob,pat).
parent(pat,jim).
Prolog tries to find out whether
the condition part is true.

15. Extending the Example Program by Rules
The mother relation can be based on the following logical statement:
For all X and Y,
X is the mother of Y if
X is a parent of Y and X is a female.
mother(X,Y):-parent(X,Y), female(X).

16. Extending the Example Program by Rules
The grandparent relation can be immediately written in Prolog as:
grandparent(X,Z):-parent(X,Y),parent(Y,Z). X Y Z
parent
grandparent

17. Extending the Example Program by Rules
For any X and Y,
X is a sister of Y if
(1) both X and Y have the same parent, and
(2) X is a female.
sister(X,Y):-
parent(Z,X),
parent(Z,Y),
female(Z).
?- sister(ann,pat).
yes
?- sister(X,pat).
X=ann;
X=pat Z
parent
parent X Y
sister
female
Pat is a sister of herself !

18. Extending the Example Program by Rules
An improved rule for the sister relation can then be:
sister(X,Y):-
parent(Z,X),
parent(Z,Y),
female(Z),
different(X,Y).

19. A Recursive Rule DefinitionX X Z
parent
predecessor X
parent
parent Y1 Y1
parent
predecessor(X,Z):-
parent(X,Z).
predecessor
predecessor
parent Y2 Y2
parent X
parent
parent Y3 Z
predecessor
parent Y
parent
predecessor(X,Z):-
parent(X,Y1),
parent(Y1,Y2),
parent(Y2,Z). Z Z
predecessor(X,Z):-
parent(X,Y1),
parent(Y1,Y2),
parent(Y2,Y3),
parent(Y3,Z).
predecessor(X,Z):-
parent(X,Y),
parent(Y,Z).

20. A Recursive Rule Definition
There is, however, an elegant and correct formulation of the predecessor relation: it will be correct in the sense that it will work for predecessor at any depth.
The key idea is to define the predecessor relation in terms of itself.

21. A Recursive Rule Definition
For all X and Z,
X is a predecessor of Z if
there is a Y such that
(1) X is a parent of Y and
(2) Y is a predecessor of Z
predecessor(X,Z):-
parent(X,Y),
predecessor(Y,Z).

22. A Recursive Rule Definition
We have thus constructed a complete program for the predecessor relation, which consists of two rules:
one for direct predecessor and
one for indirect predecessor.
predecessor(X,Z):-
parent(X,Z).
parent(X,Y),
predecessor(Y,Z).
?- predecessor(pam,X).
X=bob;
X=ann;
X=pat;
X=jim

23. A Recursive Rule Definition
The use of predecessor itself may look surprising:
When defining something, can we use this same thing that has not yet been completely defined?
Such definitions are, in general, called recursive definitions.

24. How Prolog Answers Questions
A question to Prolog is always a sequence of one or more goals.
To answer a question, Prolog tries to satisfy all the goals.
To satisfy a goal means to demonstrate that the goal logically follows from the facts and rules in the program.
If the question contains variables, Prolog also has to find what are the particular objects for which the goals are satisfied.

25. How Prolog Answers Questions
An appropriate view of the interpretation of a Prolog program in mathematical terms is then as follows:
Prolog accepts facts and rules as a set of axioms, and the user’s question as a conjectured theorem; then it tries to prove this theorem.

26. How Prolog Answers Questions
Prolog starts with the goals and, using rules, substitutes the current goals with new goals, until new goals happen to be simple facts.
predecessor(tom,pat)
by rule pr1
by rule pr1
parent(tom,pat)
parent(tom,Y)
predecessor(Y,pat) no
Y=bob by fact parent(tom,bob)
parent(tom,Y)
predecessor(Y,pat)
by rule pr1
parent(tom,Y)
predecessor(Y,pat)

27. Declarative and Procedural Meaning
The declarative meaning is concerned only with the relation defined by the program.
The declarative meaning thus determines what will be the output of the program.
The procedural meaning determines how this output is obtained; i.e., how are the relations actually evaluated by the Prolog system.

28. Declarative and Procedural Meaning
The ability of Prolog to work out many procedural details on its own is considered to be one of its specific advantages.
It encourages the programmer to consider the declarative meaning of program relatively independently of their procedural meaning.
This is of practical importance because the declarative aspects of programs are usually easier to understand than the procedural details.