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Chapter 2 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. 1. 2. 3. 4. Proposition Equivalence. For propositional expressions P , Q and R :. 5.

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Chapter 2 The Predicate Calculus

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  1. 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 Artificial intelligence 1

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  5. Proposition Equivalence For propositional expressions P, Q and R: Artificial intelligence 5

  6. Truth Table Truth table for the operator . Truth table demonstrating the equivalence of P Æ Q and ÿ P ⁄ Q. Artificial intelligence 6

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  11. verify_sentence algorithm Artificial intelligence 11

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  14. 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) ) Artificial intelligence

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

  16. A blocks world with its predicate calculate description. A rule for clear block: X(Y on(Y,X)  clear(X)) Artificial intelligence

  17. Inference • Logical inference – infer new correct expressions from a set of true assertions • New expressions must be consistent with all previous expressions Artificial intelligence

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  22. 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)) ExpressionsSubstitutions foo(fred, a goo(Z)) {fred/X, Z/Y} foo(W, a, goo(jack)) {W/X, jack/Y} foo(Z, a, goo(moo(Z))) {Z/X, moo(Z)/Y} Artificial intelligence

  23. 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 Artificial intelligence

  24. 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 Artificial intelligence

  25. 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. Artificial intelligence

  26. Unification Algorithm Artificial intelligence

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

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

  29. 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 Artificial intelligence

  30. Logic System Artificial intelligence 13

  31. 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) Artificial intelligence

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