Lecture12AlgebraOfSfwAndHdw

Lecture12AlgebraOfSfwAndHdw - CS2603 Applied Logic for...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 15

Lecture12AlgebraOfSfwAndHdw - CS2603 Applied Logic for...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online