Logic Programming

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Logic Programming
Motivation. Predicate Logic Clause Form. Horn
Clauses. A backtracking Horn-clause
resolver. Sample logic programs.
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Motivation
1. Reduce the programming burden it would be
nice to have a system that could simply accept
the necessary information and the objective
(goal), and then figure out its own solution. 2.
Have a program that looks more like its own
specification. 3. Take advantage of logical
inference to automatically get many of the
consequences of the given information.
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Predicate Logic
Every Macintosh computer uses electricity. (all
x) (Macintosh(x) implies UsesElectricity(x)) vari
ables x, y, z, etc. constants a, b, c,
etc. function symbols f, g, etc. Predicate
symbols P, Q, Macintosh, UsesElectricity quan
tifiers all, exists Logical connectives not,
implies, and, or. , -gt, , V.
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Clause Form
(all x) (Macintosh(x) -gt UsesElectricity(x) UsesE
lectricy(x) V Not Macintosh(x). Any boolean
formula can be put into conjunctive normal form
(CNF). If not Y then X and not Z. Y or (X not
Z) (Y or X (Y or not Z) (X V Y) (Y V
Z) clauses (X V Y), (Y V Z) X, Y, and Z are
called literals.
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Horn Clauses
Horn clause at most one unnegated literal (X V
Y) is not a Horn clause. (Y V Z) is a Horn
clause. If X is the mother of Y and Y is a
parent of Z, then X is a grandmother of
Z. (m(X,Y) p(Y, Z)) -gt g(X,Z). grandmother(X,
Z) provided mother(X, Y) and parent (Y,
Z) g(X,Z) - m(X,Y), p(Y,Z). Edinburgh
Prolog syntax ((grandmother x z) head
(not (mother x y)) subgoal 1 (not
(parent y z))) subgoal 2
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Unification of Literals
A substitution is a set of term/variable pairs.
f(a)/x, b/y, z/w A unifier for a pair of
literals is a substitution that when applied to
both literals, makes them identical. P(x, a),
P(f(a), y) have the unifier f(a)/x, a/y
P(x), P(y) have the unifier a/x, a/y , but
they also have the unifier x/y . The latter is
more general because after unifying with x/y
we get P(x) whereas with the other it is P(a),
yet we can obtain the latter from the former with
an additional substitution a/x .
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Horn-Clause Resolution
Example of resolution with the grandmother
rules g(X, Z) - m(X, Y), p(Y, Z). m(X, Y) -
p(X, Y), f(X). g(X, Z) - p(X, Y), f(X), p(Y,
Z). more generally, from P - Q1, Q2. Q1 -
R1, R2. we obtain the resolvent P - R1, R2, Q2
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A Backtracking Horn-Clause Resolver
Maintain the rules in a database of rules. Accept
a query (goal) as a positive unit clause, e.g.,
P(a, b). Put this goal on the subgoal
list. Repeatedly try to satisfy the next clause
on the subgoal list as follows Find a rule in
the database whose head unifies with the subgoal.
Apply the same unifier to the literals in the
body of that rule and replace the subgoal by
these new literals. If the subgoal list is
empty, stop. The combined set of unifiers
includes the solution.
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Example Logic Program
(SETF DATABASE3 '( Rules ((GRANDMOTHER X Z)
(MOTHER X Y) (PARENT Y Z)) ((PARENT X Y) (MOTHER
X Y)) ((PARENT X Y) (FATHER X Y))
Facts ((MOTHER MARY JOHN)) ((FATHER JOHN
BILL)) ((FATHER PETER JAMES)) )) (QUERY
'((GRANDMOTHER X BILL)))
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Invertible List Manipulation
(SETF DATABASE4 '( (APPEND L1 L2 L3) is true
provided L3 is the result of appending L1 and
L2. ((APPEND NILL L L)) ((APPEND L NILL L))
((APPEND (CONS X L1) L2 (CONS X L3)) (APPEND L1
L2 L3)) )) (QUERY '((APPEND (CONS X (CONS Y
NILL)) (CONS Z (CONS W NILL))
L)) ) SOLUTION L(CONS X (CONS Y (CONS Z
(CONS W NILL))))
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Invertible List Manipulation (Cont.)
(QUERY '((APPEND L (CONS Z (CONS W
NILL)) (CONS X (CONS Y (CONS Z (CONS W
NILL)))))) ) SOLUTION L(CONS X (CONS Y
NILL)) Functional programming evaluation is
one-way. Logic programming evaluation can be
two-way.