Download The Predicate Calculus

Chapter 2
The Predicate Calculus
Contents
• The Propositional Calculus
• The Predicate Calculus
• Using Inference Rules to Produce
Predicate Calculus
• Application: A Logic-Based Financial
Advisor
CSC411
Artificial intelligence
11
CSC411
Artificial intelligence
22
CSC411
Artificial intelligence
33
CSC411
Artificial intelligence
44
Proposition Equivalence
For propositional expressions P, Q and R:
CSC411
Artificial intelligence
55
Truth Table
Truth table for the
operator .
Truth table demonstrating
the equivalence of P Æ Q
and ÿ P ⁄ Q.
CSC411
Artificial intelligence
66
CSC411
Artificial intelligence
77
CSC411
Artificial intelligence
88
CSC411
Artificial intelligence
99
CSC411
Artificial intelligence
10
10
verify_sentence algorithm
CSC411
Artificial intelligence
11
11
CSC411
Artificial intelligence
12
12
CSC411
Artificial intelligence
13
13
Equivalence
For predicates p and q, and variables
X and Y
– Xp(X)  Xp(X)
– Xp(X)  Xp(X)
– Xp(X)  Yp(Y)
– Xp(X)  Yp(Y)
– X(p(X)q(X))  Xp(X)Yq(Y))
– X(p(X)q(X))  Xp(X)Yq(Y) )
CSC411
Artificial intelligence
14
Higher-order predicate calculi
– Allows quantified predicates
– E.g. (Likes) Likes(george, kate)
CSC411
Artificial intelligence
15
A blocks world with its predicate calculate
description.
A rule for clear block:
X(Y on(Y,X)  clear(X))
CSC411
Artificial intelligence
16
Inference
Logical inference – infer new correct
expressions from a set of true
assertions
New expressions must be consistent
with all previous expressions
CSC411
Artificial intelligence
17
CSC411
Artificial intelligence
18
CSC411
Artificial intelligence
19
CSC411
Artificial intelligence
20
CSC411
Artificial intelligence
21
Unification
Unification -- An algorithm (procedure) for
determining the substitutions needed to make two
predicate calculus expressions match
Unifier -- a set of variable substitutions that make
two expressions identical
E.g, for expression foo(X, a, goo(Y))
Expressions
Substitutions
foo(fred, a goo(Z))
{fred/X, Z/Y}
{W/X, jack/Y}
{Z/X, moo(Z)/Y}
foo(W, a, goo(jack))
foo(Z, a, goo(moo(Z)))
CSC411
Artificial intelligence
22
Unification Issues
 can be simply removed
 may be eliminated by replacing the variable with
the constant.
E.g. parent(X, tom)  parent(mary, tom)
Skolem function may be used to replace a variable
that depends on other variables
e.g. XYparent(X,Y)  parent(X,f(X))
A variable may be replaced by any terms, but not
the terms that contain it
A constant cannot be replaced by any terms
If a variable is bound to a constant, it my not be
re-bound
CSC411
Artificial intelligence
23
Composition of Unification
Substitutions
If S and S’ are two substitution sets, the the
composition of S and S’, SS’, is obtained by
applying S’ to the elements of S and adding the
result to S
E.g. Consider three sets of substitutions:
{X/Y, W/Z},
{V/X},
{a/V, f(b)/W}
Compositing the third set with the second set:
{a/X, a/V, f(b)/W}
Composing this result with the first set:
{a/Y, a/X, a/V, f(b)/Z, f(b)/W}
Composition is associative but not commutative
CSC411
Artificial intelligence
24
Most General Unifier (mgu)
The mgu for a set of expressions is unique except
for alphabetic variations
E.g. Unifying p(X) and p(Y),
– {fred/X, fred/Y} is an unifier, but not mgu
– Both {Z/X, Z/Y} and {W/X, W/Y} are mgus.
CSC411
Artificial intelligence
25
Unification Algorithm
CSC411
Artificial intelligence
26
• Unifying parents(X, father(X), mother(bill)) and
parents(bill, father(bill), Y).
• Convert to List format first
CSC411
Artificial intelligence
27
Final trace of the unification of (parents X (father
X) (mother bill)) and (parents bill (father bill) Y).
CSC411
Artificial intelligence
28
A Logic-Based Financial Advisor
The advisor helps a user to decide whether to
invest in a savings account, or the stock market
What to invest depends on their income and the
current amount they have saved
Criteria:
– Individuals with an inadequate savings account should
always make increasing the amount saved their first
priority, regardless of their income
– Individuals with adequate savings account and an
adequate income should consider a riskier but potentially
more profitable investment in the stock market
– Individuals with a lower income who already have an
adequate savings account may want to consider splitting
their surplus income between savings and stocks, to
increase the cushion in savings while attempting to
increase their income through stocks
CSC411
Artificial intelligence
29
Logic System
CSC411
Artificial intelligence
30
13
Advisor System Inference
minsavings(X)  500*X
minincome(X)  15000+(4000*X)
Consider an user with three dependents,
$220000 in savings, and steady income of
$25000
– Add facts:
9.
amount_saved(22000).
10.
earnings(25000, steady).
11.
dependents(3).
– Conjunct 10 and 11, unify with 7 under the substitution
{25000/X, 3/Y}, and use modus ponens
12.
income(inadequate).
– Conjunct 9 and 11, unify with 4 under the substitution
{22000/X, 3/Y}, and use modus ponens
13.
savings_account(adquate)
– Conjuct 12 and 13, use modus ponens with 3
14.
investment(combination)
CSC411
Artificial intelligence
31