Lecture12AlgebraOfSfwAndHdw

# Lecture12AlgebraOfSfwAndHdw - 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 12 — CS 2603 Applied Logic for Hardware and Software Algebra Every Which Way Boolean Algebra Predicate Algebra Software Algebra Hardware Algebra CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 Predicates and WFFs ¡ Let P be a predicate with universe of discourse U ¢ P is a collection of propositions indexed by U ¢ For each x in U, P(x) denotes a proposition in predicate P ¡ WFF grammar for predicate calculus ¢ WFF grammar for propositional calculus, plus 2 rules: ¢ Let e be a WFF in predicate calculus. Then: 9 ( ∀ x.e) and ( ∃ x.e) are also WFFs 9 ( ∀ x.e)=True means e=True for every x in U 9 ( ∃ x.e)=True means e=True for at least one x in U ¢ Free vs bound variables 9 The variable x is bound in the formulas ( ∀ x.e) and ( ∃ x.e) 9 Any variable that is not bound is free ¢ Arbitrary variables 9 A variable in proof is arbitrary if it does not occur free in any undischarged assumption of the proof 9 The term "arbitrary" is relevant only in proofs r e v i e w CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 Inference Rules of Predicate Calculus ∀ x. F(x) {y not in F(x)} ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { ∀ R} ∀ y. F(y) Renaming Variables F(x) {x arbitrary, y not in F(x)} ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ {R} F(y) ∃ x. F(x) {y not in F(x)} ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { ∃ R} ∃ y. F(y) Introducing/Eliminating Quantifiers F(x) {x arbitrary} ⎯⎯⎯⎯⎯⎯⎯⎯ { ∀ I} ∀ x. F(x) ∀ x. F(x) {universe is not empty} ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { ∀ E} F(x) ... plus the inference rules of propositional calculus Triggers a discharge? ∃ x. F(x) F(x) |– A {x not free in A} ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { ∃ E} A F(x) ⎯⎯⎯⎯ { ∃ I} ∃ x. F(x) CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 Equations of Predicate Calculus (( ∀ x.f(x)) ∧ q) = (( ∀ x.(f(x) ∧ q)) { ∧ dist over ∀ } (( ∀ x.f(x)) ∨ q) = (( ∀ x.(f(x) ∨ q)) { ∨ dist over ∀ } (( ∃ x.f(x)) ∧ q) = (( ∃ x.(f(x) ∧ q)) { ∧ dist over ∃ } (( ∃ x.f(x)) ∨ q) = (( ∃ x.(f(x) ∨ q)) { ∨ dist over ∃ } ( ∀ x. f(x)) → f(c) {7.3} f(c) → ( ∃ x. f(x)) {7.4} ( ∀ x. f(x)) = ( ∀ y. f(y)) { ∀ R} ( ∃ x. f(x)) = ( ∃ y. f(y)) { ∃ R} ( ∀ x. ¬ f(x)) = ( ¬ ( ∃ x. f(x))) {deM ∃ } ( ∃ x. ¬ f(x)) = ( ¬ ( ∀ x. f(x))) {deM ∀ } x n o t f r e e i n q ( ∀ x.(f(x) ∧ g(x))) = (( ∀ x.f(x)) ∧ ( ∀ x.g(x))) { ∀ dist over ∧ } (( ∀ x.f(x)) ∨ ( ∀ x.g(x))) → ( ∀ x.(f(x) ∨ g(x))) {7.12} (( ∃ x.f(x)) ∧ ( ∃ x.g(x))) x....
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## This note was uploaded on 04/08/2008 for the course CS 2603 taught by Professor Rexpage during the Spring '08 term at The University of Oklahoma.

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Lecture12AlgebraOfSfwAndHdw - CS2603 Applied Logic for...

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