Lecture11ReasoningWithPredicates

Lecture11ReasoningWithPredicates - CS2603 Applied Logic for...

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Unformatted text preview: CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 11 — CS 2603 Applied Logic for Hardware and Software What .. more rules? Reasoning with Predicates CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 What is a Predicate? ¡ Predicate ¢ Parameterized collection of propositions ¢ P(x) 9 Typically a different proposition for each x 9 Universe of discourse – Values that x may take ¡ Universe of discourse ¢ Must be specified 9 Otherwise, all bets off — muchas contradicciónes ¢ Non-empty 9 Empty universe calls for special handling 9 Default assumption: non-empty universe r e v i e w ∀ — the Universal Quantifier, Forall ¡ ∀ x.P(x) ¢ This formula is a WFF of predicate calculus whenever P(x) is a WFF of predicate calculus ¢ True if the proposition P(x) is True for every value of x in the universe of discourse ¢ False if there is some value x in the universe of discourse for which P(x) is False ¢ Equivalent to forming the Logical And of all P(x)’s ¡ Example – S predicate about sum ¢ S(n) ≡ sum[x 1 , x 2 , …, x n ] = x 1 + x 2 + … + x n ¢ ∀ n.S(n) 9 Universe of discourse: natural numbers N = {0, 1, 2, … } 9 ∀ n.S(n) means S(0) ∧ S(1) ∧ S(2) ∧ … 9 So, “ ∀ ” provides a way to write formulas that would contain an infinite number of symbols if written in propositional calculus notation (but infinitely long formulas aren’t WFFs) r e v i e w CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 ∃ —the Existential Quantifier, There Exists ¡ ∃ x.P(x) ¢ This formula is a WFF of predicate calculus whenever P(x) is a WFF of predicate calculus ¢ True if there is at least one x in the universe of discourse for which the proposition P(x) is True ¢ False if ∀ x. ¬ P(x) is True ¢ Equivalent to forming the Logical Or of all P(x)’s ¡ Example – E predicate about maximum ¢ E(n, k) ≡ maximum[s 1 , s 2 , …, s n ] = s k 9 Note: E(n, k) is an equation (a True/False proposition) ¢ ∃ k.E(23, k) 9 Universe of discourse for k in ∃ k.E(23, k): U = {1, 2, …, 23} 9 ∃ k.E(23, k) means E(23,1) ∨ E(23,2) ∨ … ∨ E(23,23) 9 Do you think ∃ k.E(23, k) is True? 9 Note: When U is finite , quantifiers not required – Clumsy to write big formulas without quantifiers, though – Without quantifiers, reasoning can be more complex, too CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 5 qsort preserves keys qsort conserves keys Another Example with ∃ 9 What about...
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Lecture11ReasoningWithPredicates - CS2603 Applied Logic for...

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