Prolog in 90 minutes
Ulf Nilsson
Dept of Computer and
Information Science
Linköping University
Logic programs
A logic program describes individuals
and relations between individuals
(or properties of individuals).
The program is used to answer
queries about the world described
in the program.
Relations
Adam is a parent of Bill
Paris is the capital of France
5 is greater than 2 plus 2
X times 1 is equal to X
X is a subset of Y
5 is the maximum of 2 and 5
There is an edge from a to b
Properties
Adam is a parent
Adam is male
X plus 1 is non-zero
Paris is a capital
Grass is green
The music was loud
Queries
Who is the father of Bill?
Is there an edge from a to b?
Which town is a capital?
Who is male?
Language primitives
Constants
adam, paris, 5, 3.14, [], ´Adam’, ...
Variables
X, Y, List, _12, _, ...
Function symbols
plus/2, +/2, f/1, ...
Predicate symbols
capital/2, greater/2, non_zero/1, >/2, ...
Terms
Terms represent individuals
Constants
Variables
Compound terms
E.g. paris, X, plus(2,3), plus(2,plus(3,4))
Infix notation: 2+3
Atomic formulas
Atomic formulas describe relations:
If p is a predicate letter of arity n and
t1,...,tn are terms then p(t1,...,tn) is an
atomic formula.
E.g. capital(paris,france) greater(X,2)
Infix notation: X > 2
Logic Programs
A logic program is a set of clauses:
facts
rules
The program is used to answer queries.
Facts
A fact is an expression of the form:
A.
where A is an atomic formula.
Examples:
edge(a, X).
parent(adam, bill).
Interpretation Facts
Consider a fact A.
Declarative (logical) reading:
For all values of the variables (in A), A is true.
Procedural (operational) reading:
A is solved.
Rules
A rule is an expression of the form:
A0 :- A1, ... , An.
where each Ai is an atomic formula.
Examples:
path(X,Y) :- edge(X,Y).
father(X,Y) :- parent(X,Y), male(X).
Interpretation Rules
Consider a rule A0 :- A1, ... , An.
Declarative (logical) reading:
For all values of the variables in the rule,
A0 if A1 and...and An.
Procedural (operational) reading:
To solve A0, first solve A1, then A2 etc.
Example
gp(X,Y) :- p(X,Z), p(Z,Y).
p(X,Y) :- f(X,Y).
p(X,Y) :- m(X,Y).
f(adam,bill).
f(bill,carl).
m(anne,bill).
Queries
A query is an expression of the form:
?- A1, ..., An.
where n=0,1,2,... and A1, ..., An are
atomic formulas.
Examples:
?- father(X, bill).
?- parent(X, bill), male(X).
Interpretation Queries
Consider a query ?- A1, ... , An.
Declarative (logical) reading:
Are there values of the variables
such that A1 and...and An?
Procedural (operational) reading:
First solve A1, then A2 etc
Ground SLD-Resolution
?- A1,A2,...,An.
A1 :- B1,...,Bm.
?- B1,...,Bm,A2,...,An.
where
A1 :- B1,...,Bm is an instantiated program clause.
A Derivation
parent(X,Y) :father(X,Y).
parent(X,Y) :mother(X,Y).
father(adam,bill).
mother(anne,bill).
?- parent(adam,bill)
?- father(adam,bill)
?- true
Another Derivation
parent(X,Y) :father(X,Y).
parent(X,Y) :mother(X,Y).
father(adam,bill).
mother(anne,bill).
?- parent(anne,bill)
?- mother(anne,bill)
?- true
Full SLD-Resolution
?- A1,A2,...,An.
B0 :- B1,...,Bm.
?- (B
A1=
B0, mB,A
1,...,B
m,A
2,...,An.
1,...,B
2,...,A
n)q.
where:
• B0 :- B1,...,Bm is a renamed program clause.
• q is a solution to the equation A1 = B0.
Yet Another Derivation
?- parent(X,bill).
?- father(X,bill).
X=X1, bill=Y1, father(X1,Y1).
?- true.
X=adam, bill=bill.
parent(X1,Y1)
father(adam,bill).
Answer::-X=adam
father(X1,Y1).
And Another One...
?- gp(X,Y).
X=adam
?- p(X,Z1),
X=X1, Y=Y1,
p(Z1,Y).
p(X1,Z1), p(Z1,Y1).
Y=carl
?- f(X,Z1),
X=X2, Z1=Y2,
p(Z1,Y).
f(X2,Y2), p(Z1,Y).
?- X=adam,Z1=bill,
p(Z1,Y).
p(bill,Y).
?- f(bill,Y).
bill=X3, Y=Y3, f(X3,Y3).
?- bill=bill,
true.
Y=carl.
gp(X1,Y1)
p(X2,Y2)
p(X3,Y3)
f(adam,bill).
:f(bill,carl).
p(X1,Z1),p(Z1,Y1).
:- f(X3,Y3).
f(X2,Y2).
And a Failed One...
?- gp(X,Y).
X=bill
?- p(X,Z1),
X=X1, Y=Y1,
p(Z1,Y).
p(X1,Z1), p(Z1,Y1).
?- f(X,Z1),
X=X2, Z1=Y2,
p(Z1,Y).
f(X2,Y2), p(Z1,Y).
?- X=bill,Z1=carl,
p(Z1,Y).
p(carl,Y).
?- f(carl,Y).
carl=X3, Y=Y3, f(X3,Y3).
?- fail.
gp(X1,Y1)
p(X2,Y2)
p(X3,Y3)
FAILURE!!!
:f(bill,carl).
p(X1,Z1),p(Z1,Y1).
:- f(X3,Y3).
f(X2,Y2).
SLD-Tree
?- gp(X,Y).
?- p(X,Z),p(Z,Y).
X=adam
?- f(X,Z),p(Z,Y).
?- p(bill,Y).
X=anne
?- m(X,Z),p(Z,Y).
?- p(carl,Y).
?- p(bill,Y).
?- f(carl,Y). ?- m(carl,Y).
?- f(bill,Y).
?- f(bill,Y). ?- m(bill,Y).
?- true.
Y=carl
?- fail.
?- fail.
?-?-fail.
true.
Y=carl
?- m(bill,Y).
?- fail.
Example
/* inv(In, Out) */
inv(0, 1).
inv(1, 0).
/* and(In1, In2, Out) */
and(0, 0, 0).
and(0, 1, 0).
and(1, 0, 0).
and(1, 1, 1).
/* or(In1, In2, Out) */
or(0, 0, 0).
or(0, 1, 1).
or(1, 0, 1).
or(1, 1, 1).
/* nand(In1, In2, Out) */
nand(X, Y, Z) :and(X, Y, Tmp),
inv(Tmp, Z).
Database
lecturer(Lecturer,Course) :course(Course,_,Lecturer,_).
teaches(Lect,Day) :course(_, time(Day,_,_), Lect, _).
duration(Course,Length) :course(Course,time(_,S,F),_,_),
plus(S,Length,F).
occupied(Room,Day,Time) :course(_,time(Day,S,F),_,Room),
S =< Time,
Time =< F.
% Database
course(logic, time(monday, 8, 10), dave, a12).
...
Recursion
b
d
f
c
e
g
a
edge(a,b).
edge(a,c).
edge(b,d).
edge(c,d).
edge(d,e).
edge(f,g).
path(Node,Node).
path(Node1,Node3) :edge(Node1,Node2),
path(Node2,Node3).
List Notation
.
a
a
.
.
b
b
c
c
.(a, .(b, .(c, [])))
[]
More On List Notation
The empty list: []
A non-empty list: .(X,Y) or [X|Y]
Syntactic Sugar:
[b] instead of [b|[]] and .(b, [])
[a,b] instead of [a|[b]] and [a|[b|[]]]
[a,b|X] instead of [a|[b|X]]
List manipulation
list([ ]).
list([X|Xs]) :- list(Xs).
member(X,[X|Xs]).
member(X,[Y|Ys]) :- member(X,Ys).
append([ ],Ys,Ys).
append([X|Xs],Ys,[X|Zs]) :- append(Xs,Ys,Zs).
List Manipulation
% reverse(A, B)
% B is A in reverse order
reverse([ ],[ ]).
reverse([X|Xs],Zs) :- reverse(Xs,Ys), append(Ys,[X],Zs).
% Alternative version
reverse(Xs,Ys) :- reverse(Xs,[ ],Ys).
reverse([ ],Ys,Ys).
reverse([X|Xs],Acc,Ys) :- reverse(Xs,[X|Acc],Ys).
Insertion Sort
% sort(A,B)
% B is a sorted version of A
sort([X|Xs],Ys) :- sort(Xs,Zs), insert(X,Zs,Ys).
sort([ ],[ ]).
% insert(A,B,C)
% if B is a sorted list, then C is sorted
% and contains all elements in B plus A
insert(X,[ ],[X]).
insert(X,[Y|Ys],[Y|Zs]) :- X > Y, insert(X,Ys,Zs).
insert(X,[Y|Ys],[X,Y|Ys]) :- X =< Y.
Binary Trees
% binary_tree(A)
% A is a binary tree
binary_tree(void).
binary_tree(tree(Element,Left,Right)) :binary_tree(Left), binary_tree(Right).
% tree_member(A,B)
% A is a node in the tree B
tree_member(X,tree(X,_,_)).
tree_member(X,tree(_,Left,_)) :- tree_member(X,Left).
tree_member(X,tree(_,_,Right)) :- tree_member(X,Right).
Built In Predicates
setof(X, p(X), S)
~ S is the set of all X such that p(X)
bagof(X, p(X), B)
~ B is the sequence of all X such that p(X)
findall(X, p(X), B)
B is the sequence of all X such that p(X)
Negation
Prolog contains a weak form of negation
called “negation as failure”.
Written: \+ p(a)
A query ?- \+ p(a) succeeds if the query
?- p(a) fails finitely.
Robust only when the goal contains no
variables. (Use only as a test!)
Example Negation
a
b
on_top(X) :- \+ blocked(X).
blocked(X) :- on(Y, X).
on(a, b).
%---------------------------?- on_top(a).
yes