Logic Programming
Some "declarative" slides on logic programming and Prolog.
James Brucker
Introduction to Logic Programming
Declarative programming describes what is desired from the program, not how it should be done
Declarative language: statements of facts and propositions that must be satisfied by a solution to the program
real(x). proposition: x is a real number.
x > 0. proposition: x is greater than 0.
Declarative Languages
what is a "declarative language"?
give another example (not Prolog) of a declarative language.
SELECT * FROM COUNTRY WHERE CONTINENT = 'Asia';
Facts, Rules, ...
What is a proposition?
What are facts?
What are rules?
What is a predicate?
What is a compound term?
Facts:
fish(salmon).
likes(cat, tuna).
Predicates:
fish, likes
Compound terms:
likes(cat, X), fish(X)
Atoms:
cat, salmon, tuna
Rule:
eats(cat,X) likes(cat,X), fish(X).
A Really Simple Directed Graph
a
b c
d
(1) edge(a, b).
(2) edge(a, c).
(3) edge(c, d).
(4) path(X, X).
(5) path(X, Y)
edge(X, N), path(N, Y).
Question: What are the...
atoms
facts
rules
Clausal Form Problem: There are too many ways to express propositions.
difficult for a machine to parse or understand
Clausal form: standard form for expressing propositions
nm AAABBB 2121
Example:
path(X, Y) edge(X, N) path(N, Y).
AntecedentConsequent
Clausal Form Example
Meaning:
if there is an edge from X to N and there a path from N to Y, then there is a path from X to Y.
The above is also called a "headed Horn clause".
In Prolog this is written as a proposition or rule:
path(X, Y) edge(X, N) path(N, Y).
path(X, Y) :- edge(X, N) , path(N, Y).
Query
A query or goal is an input proposition that we want Prolog to "prove" or disprove.
A query may or may not require that Prolog give us a value that satisfies the query (instantiation).
1 ?- edge(a,b).Yes2 ?- path(c,b).No3 ?- path(c,X).X = c ;X = d ;No
Logical Operations on Propositions
What are the two operations that a logic programming language performs on propositions to establish a query?
That is, how does it satisfy a query, such as:
Unification
Unification is a process of finding values of variables (instantiation) to match terms. Uses facts.
(1-3) edge(a,b). edge(a,c). edge(c,d). (Facts)
(4) path(X,X). (Rule)
(5) path(X,Y) := edge(X,N), path(N,Y). (Rule)
?- path(a,d). This is the query (goal).
Instantiate { X=a, Y=d }, and unify path(a,d) with Rule 5.
After doing this, Prolog must satisfy:
edge(a,N). This is a subgoal.
path(N,d). This is a subgoal.
Unification in plain English
Compare two atoms and see if there is a substitution which will make them the same.
How can we unify 6 with 5?
Let X := a
Let Y := Z
1. edge(a, b). (Fact)
5. path(X, Y) :- edge(X, N) , path(N, Y).
6. path(a, Z). (Query)
Resolution
Resolution is an inference rule that allows propositions to be combined.
Idea: match the consequent (LHS) of one proposition with the antecedent (RHS term) of another.
).()( then)()( If
)()(
)()(
2112
21
21
yQyPyQyP
XQXQ
XPXP
Examples are in the textbook and tutorials.
Resolution Example
How can we unify 6 with 5?
Let X := a
Let Y := Z
Resolution:
1. edge(a, b). (Fact)
5. path(X, Y) :- edge(X, N) , path(N, Y).
6. path(a, Z). (Query)
Resolution
Resolution is an inference rule that allows propositions to be combined.
Idea: match the consequent (LHS) of one proposition with the antecedent (RHS term) of another.
).()( then)()( If
)()(
)()(
2112
21
21
yQyPyQyP
XQXQ
XPXP
Examples are in the textbook and tutorials.
How to handle failures
Prolog can work backwards towards the facts using resolution, instantiation, and unification.
As it works, Prolog must try each of several choices.
These choices can be stored as a tree.
?- path(a,d). The goal.
Unify: unify path(a,d)with Rule 5 by instantiate { X=a,Y=d }
Subgoal: edge(a,N).
Instantiate: N=b which is true by Fact 1.
Subgoal: path(b,d).
Unify: path(b,d)with Rule 5: path(b,d) :- edge(b,N),path(N,d)
Failure: can't instantiate edge(b,N) using any propositions.
How to handle failures (2) When a solution process fails, Prolog must undo some of the
decisions it has made.
This is called backtracking.
same as backtracking you use in recursion.
Marks a branch of the tree as failed.
How it Works (1)
There are 2 search/execution strategies that can be used by declarative languages based on a database of facts.
1. Forward Chaining
2. Backward Chaining
what are the meanings of these terms?
How it Works (2)
1. Forward Chaining
2. Backward Chaining
Which strategy does Prolog use?
Under what circumstances is one strategy more effective than the other?
Consider two cases:
large number of rules, small number of facts
small number of rules, large number of facts
PROLOG: PROgramming in LOGic
The only "logic" programming language in common use.
3 Parts of a Prolog Program
1. A database contains two kinds of information.
What information is in a database?
2. A command to read or load the database.
in Scheme you can use load("filename")
in Prolog use consult('filename')
3. A query or goal to solve.
Ancestors
ancestor(X,Y) :- parent(X,Y).
ancestor(X,Y) :- ancestor(X,Z), ancestor(Z,Y).
parent(X,Y) :- mother(X,Y).
parent(X,Y) :- father(X,Y).
father(bill, jill).
mother(jill, sam).
mother(jill, sally).
File: ancestors.pl
Query the Ancestors
?- consult('/pathname/ancestors.pl').
ancestor(bill,sam).Yes?- ancestor(bill,X).X = jill ;X = sam ;ERROR: Out of local stack?- ancestor(X,bob).ERROR: Out of local stack
Understanding the Problem
You need to understand how Prolog finds a solution.
ancestor(X,Y) :- parent(X,Y).
ancestor(X,Y) :- ancestor(X,Z), ancestor(Z,Y).
parent(X,Y) :- mother(X,Y).
parent(X,Y) :- father(X,Y).
father(bill,jill).
mother(jill,sam).
father(bob,sam).
Depth-first search causes immediate recursion
Factorial
factorial(0,1).
factorial(N,N*M) :- factorial(N-1,M). The factorial of 0 is 1.
The factorial of N is N*M if the the factorial of N-1 is M
File: factorial1.pl
?- consult('/path/factorial1.pl'). ?- factorial(0,X).X = 1Yes?- factorial(1,Y).ERROR: Out of global stack
Query Factorial
?- consult('/path/factorial1.pl'). ?- factorial(2,2).No?- factorial(1,X).ERROR: Out of global stack
?- 2*3 = 6.No?- 2*3 = 2*3.Yes
Problem: Arithmetic is not performed automatically.
?- 6 is 2*3.Yes?- 2*3 is 6.No
is(6,2*3).
l-value = r-value ?
Arithmetic via Instantiation: is
"=" simply means comparison for identity.
factorial(N, 1) :- N=0.
"is" performs instantiation if the left side doesn't have a value yet.
product(X,Y,Z) :- Z is X*Y.
this rule can answer the query:
product(3,4,N).
Answer: N = 12. but it can't answer:
product(3,Y,12).
is does not mean assignment!
This always fails: N is N - 1.
% sumto(N, Total): compute Total = 1 + 2 + ... + N.
sumto(N, 0) :- N =< 0.
sumto(N, Total) :=
Total is Subtotal + N,
N is N-1, always fails
sumto(N, Subtotal).
?- sumto(0, Sum).
Sum = 0.
Yes
?- sumto(1, Sum).
No
is : how to fix?
How would you fix this problem?
% sumto(N, Total): compute Total = 1 + 2 + ... + N.
sumto(N, 0) :- N =< 0.
sumto(N, Total) :=
N1 is N-1, always fails
sumto(N1, Subtotal),
Total is Subtotal + N.
?= sumto(5, X).
Factorial revised
factorial(0,1).
factorial(N,P) :- N1 is N-1,
factorial(N1,M), P is M*N.
Meaning:
The factorial of 0 is 1.
factorial of N is P
if N1 = N-1
and factorial of N1 is M
and P is M*N.
File: factorial2.pl
Query Revised Factorial
?- consult('/path/factorial2.pl'). ?- factorial(2,2).Yes?- factorial(5,X).X = 120Yes
?- factorial(5,X).X = 120 ;ERROR: Out of local stack?- factorial(X,120).
but still has some problems...
request another solution
Factorial revised again
factorial(0,1).
factorial(N,P) :- not(N=0), N1 is N-1,
factorial(N1,M), P is M*N.
File: factorial3.pl
?- factorial(5,X).X = 120 ;No?-
Makes the rules mutually exclusive.
Readability: one clause per line
factorial(0,1).
factorial(N,P) :- not(N=0), N1 is N-1,
factorial(N1,M), P is M*N.
factorial(0,1).
factorial(N,P) :-
not(N=0),
N1 is N-1,
factorial(N1,M),
P is M*N.
Better
Finding a Path through a Graph
edge(a, b).edge(b, c). edge(b, d).edge(d, e). edge(d, f).path(X, X).path(X, Y) :- edge(X, Z), path(Z, Y).
a
b
c d
e f?- edge(a, b).Yes?- path(a, a).Yes?- path(a, e).Yes?- path(e, a).No
How To Define an Undirected Graph?
edge(a, b).edge(b, c). edge(b, d).edge(d, e). edge(d, f).edge(X, Y) := not(X=Y), edge(Y, X).path(X, X).path(X, Y) :- edge(X, Z), path(Z, Y).
a
b
c d
e f
?- edge(b, a).Yes?- path(a, b).Yes?- path(b, e).No
Queries and Answers
When you issue a query in Prolog, what are the possible responses from Prolog?
% Suppose "likes" is already in the database
:- likes(jomzaap, 219212). % Programming Languages.
Yes.
:- likes(papon, 403111). % Chemistry.
No.
:- likes(Who, 204219). % Theory of Computing?
Who = pattarin
Does this mean Papon doesn't like Chemistry?
Closed World Assumption
What is the Closed World Assumption?
How does this affect the interpretation of results from Prolog?
List Processing
[Head | Tail] works like "car" and "cdr" in Scheme. Example:
?- [H | T ] = [a,b,c,d,e].
returns:
H = a
T = [b,c,d,e]
This can be used to build lists and decompose lists. Can use [H|T] on the left side to de/construct a list:
path(X, Y, [X|P]) :-
edge(X, Node),
path(Node, Y, P).
member Predicate
Test whether something is a member of a list
?- member(a, [b,c,d]).
No.
can be used to have Prolog try all values of a list as values of a variable.
?- member(X, [a1,b2,c3,d4] ).
X = a1
X = b2
X = c3
member Predicate example Use member to try all values of a list. Useful for problems like
Queen safety enumerating possible rows and columns in a game.
% dumb function to find square root of 9
squareroot9(N) :-
member(N,[1,2,3,4,5,5,6,7,8,9]),
9 is N*N.
appending Lists
?- append([a,b],[c,d,e],L).
L = [a,b,c,d,e]
append can resolve other parameters, too:
?- append(X, [b,c,d], [a,b,c,d] ).
X = a
?- append([],[a,b,c],L).
L = [a,b,c]
Defining your own 'append'
append([], List, List).
append([Head|Tail], X, [Head|NewTail]) :-
append(Tail, X, NewTail).
Type Determination
Prolog is a weakly typed language. It provides propositions for testing the type of a variable
PREDICATE SATISFIED (TRUE) IF
var(X) X is a variable
nonvar(X) X is not a variable
atom(A) A is an atom
integer(K) K is an integer
real(R) R is a floating point number
number(N) N is an integer or real
atomic(A) A is an atom or a number or a string
functor(T,F,A) T is a term with functor F, arity A
T =..L T is a term, L is a list.
clause(H,T) H :- T is a rule in the program
Tracing the Solution
?- trace.
[trace] ?- path(a,d).
Call: (8) path(a, d) ? creep
Call: (9) edge(a, _L169) ? creep
Exit: (9) edge(a, b) ? creep
Call: (9) path(b, d) ? creep
Call: (10) edge(b, _L204) ? creep
Exit: (10) edge(b, c) ? creep
Call: (10) path(c, d) ? creep
Call: (11) edge(c, _L239) ? creep
^ Call: (12) not(c=_G471) ? creep
^ Fail: (12) not(c=_G471) ? creep
Fail: (11) edge(c, _L239) ? creep
Fail: (10) path(c, d) ? creep
Redo: (10) edge(b, _L204) ? creep
Solution Process
Stack Substitution (Instantiation)
[path(a,c), path(X,X)]
[path(a,c), path(a,a)] X = a
Undo.
[path(a,c), path(X,X)]
[path(a,c), path(c,c)] X = c
Undo.
(1) path(X,X).
(2) path(X,Y) := edge(X,Z), path(Z,Y).
?- path(a,c).
Solution Process (2)
Stack Substitution (Instantiation)
[path(a,c), path(X,Y)] (Rule 2)
[path(a,c), path(a,Y)] X = a
X = a, Y = c
edge(a,Z), path(Z,c) new subgoals
edge(a,b), path(b,c) X = a, Y = c, Z = b
path(b,c) edge(a,b) is a fact - pop it.
(1) path(X,X).
(2) path(X,Y) := edge(X,Z), path(Z,Y).
?- path(a,c).
What does this do?
% what does this do?
sub([], List).
sub([H|T], List) :-
member(H, List),
sub(T, List).
What does this do?
% what does this do?
foo([], _, []).
foo([H|T], List, [H|P]) :-
member(H, List),
foo(T, List, P).
foo([H|T], List, P) :-
not( member(H, List) ),
foo(T, List, P).
Underscore (_) means "don't care".
It accepts any value.
Max Function
Write a Prolog program to find the max of a list of numbers: max( List, X). max( [3, 5, 8, -4, 6], X).
X = 8. Strategy:
use recursion divide the list into a Head and Tail. compare X to Head and Tail. Two cases:
Head = max( Tail ). in this case answer is X is Head. X = max( Tail ) and Head < X.
what is the base case?
Max Function
% max(List, X) : X is max of List members
max([X], X). base case
max([H|Tail], H) :- 1st element is max
max(Tail, X),
H >= X.
max([H|Tail], X) :- 1st element not max
complete this
case.
Towers of Hanoi
% Move one diskmove(1,From,To,_) :- write('Move top disk from '), write(From), write(' to '), write(To), nl.
% Move more than one disk.move(N,From,To,Other) :- N>1, M is N-1, move(M,From,Other,To), move(1,From,To,_), move(M,Other,To,From).
See tutorials at:
www.csupomona.edu and
www.cse.ucsc.edu
Learning Prolog
The Textbook - good explanation of concepts
Tutorials: http://www.thefreecountry.com/documentation/
onlineprolog.shtml has annotated links to tutorials.
http://www.cse.ucsc.edu/classes/cmps112/Spring03/languages/prolog/PrologIntro.pdf last section explains how Prolog resolves queries using a stack and list of substitutions.
http://cs.wwc.edu/~cs_dept/KU/PR/Prolog.html explains Prolog syntax and semantics.
http://www.csupomona.edu/~jrfisher/www/prolog_tutorial/contents.html has many examples
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