Lecture06ThmsAsInfRules

- Lecture 6 CS 2603 Applied Logic for Hardware and Software Inference Rules Galore CS2603 Applied Logic for Hardware and Software Rex Page

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 6 —CS 2603 Applied Logic for Hardware and Software Inference Rules Galore
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 What Kind of Circuit is A B ? ± How about “Not” ? A B A B 0 0 1 0 1 1 1 0 0 1 1 1 0 B A 0 ( ¬ a) = (a False) A What goes here to make A the output ? What gate converts 0 to 1 ? What gate preserves 1 on line B ? Take this as definition of ( ¬ a) a F l s e c i r u t ± How about “True” ? ² True = ¬ False … Right? ( ¬ False) = (False False) definition of True b What goes here with 1 on B? 1 1 1 1 0 0 B A What goes here when 0 is on the A line ? A=1, B=0?
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 Modus Tollens ± Theorem (Modus Tollens) ² a b, ¬ b |– ¬ a ± Proof ² This is a homework problem 9 Proved in text, too 9 Your job – Express in natural deduction form – Convert to proof-checker notation ² We will take it as a proven theorem 9 Counting on you to succeed
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Or Introduction and Or Elimination Theorem (Or Commutes) a b |– b a Suppose ± We could prove this theorem: a |– b a ± And, we could prove this theorem: b |– b a ± Then we could apply E to conclude b a from a b remaining assumption conclusion ⎯⎯⎯ b a ⎯⎯⎯ b a a ⎯⎯⎯ { I R } b a assumptions temporarily admitted discharged by E b ⎯⎯⎯ { I L } b a a b ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { E} b a a ⎯⎯⎯ { I L } a b Or Intro Left b ⎯⎯⎯ { I R } a b Or Intro Right CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 a b a |– c b |– c ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { E} c Or Elimination
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 5 Inference Rules Galore ± Theorems become new inference rules a b, ¬ b | ¬ a {modus Tollens} a b ¬ b ⎯⎯⎯⎯⎯⎯⎯ {modTol} ¬ a premises above the line conclusion below thm cited on right (like a rule) a b ⎯⎯⎯ { Comm} b a a b ⎯⎯⎯ { Comm} b a a b b c ⎯⎯⎯⎯⎯⎯⎯ { Chain} a c
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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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- Lecture 6 CS 2603 Applied Logic for Hardware and Software Inference Rules Galore CS2603 Applied Logic for Hardware and Software Rex Page

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