Chapter 2 The Predicate Calculus Contents The Propositional

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 CSC 411 Artificial intelligence 11

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Proposition Equivalence For propositional expressions P, Q and R: CSC 411 Artificial intelligence 55

Truth Table Truth table for the operator . Truth table demonstrating the equivalence of P Æ Q and ÿ P ⁄ Q. CSC 411 Artificial intelligence 66

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verify_sentence algorithm CSC 411 Artificial intelligence 11 11

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Equivalence For predicates p and q, and variables X and Y – Xp(X) X p(X) – Xp(X) Yp(Y) – X(p(X) q(X)) Xp(X) Yq(Y)) – X(p(X) q(X)) Xp(X) Yq(Y) ) CSC 411 Artificial intelligence 14

Higher-order predicate calculi – Allows quantified predicates – E. g. (Likes) Likes(george, kate) CSC 411 Artificial intelligence 15

A blocks world with its predicate calculate description. A rule for clear block: X( Y on(Y, X) clear(X)) CSC 411 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 CSC 411 Artificial intelligence 17

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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))) CSC 411 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 CSC 411 Artificial intelligence 23

Composition of Unification Substitutions If S and S’ are two substitution sets, 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 CSC 411 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. CSC 411 Artificial intelligence 25

Unification Algorithm CSC 411 Artificial intelligence 26

• Unifying parents(X, father(X), mother(bill)) and parents(bill, father(bill), Y). • Convert to List format first CSC 411 Artificial intelligence 27

Final trace of the unification of (parents X (father X) (mother bill)) and (parents bill (father bill) Y). CSC 411 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 CSC 411 Artificial intelligence 29

Logic System CSC 411 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) CSC 411 Artificial intelligence 31