Propositional Equivalence Goal: Show how propositional equivalences are established & introduce the most important such equivalences.

Copyright © Peter Cappello Equivalence Name p  T ≡ p; p  F ≡ p Identity p  T ≡ T; p  F ≡ F Domination p  p ≡ p; p  p ≡ p Idempotent ( p) ≡ p Double negation p  q ≡ q  p; p  q ≡ q  p Commutative Copyright © Peter Cappello

Copyright © Peter Cappello Equivalence Name (p  q)  r ≡ p  (q  r ) (p  q)  r ≡ p  (q  r ) Associative p  (q  r ) ≡ (p  q)  (p  r ) p  (q  r ) ≡ (p  q)  (p  r ) Distributive (p  q) ≡  p   q (p  q) ≡  p   q DeMorgan p  (p  q ) ≡ p p  (p  q ) ≡ p Absorption p  p ≡ T p  p ≡ F Negation Copyright © Peter Cappello

Copyright © Peter Cappello Exercise A tautology is a compound proposition that always is true. Is the following a tautology? ( (p  q)  ( p  r) )  (q  r) Copyright © Peter Cappello

The Satisfiability Problem Given a function of Boolean variables, is there an assignment of values to its variables that makes it true? Is f( p, q, r ) = ( (p  q)  ( p  r) )  (q  r) satisfiable? Satisfiability is important in CS theory, algorithms, program correctness, AI, hardware design, etc. Algorithm Construct the truth table. If any assignment (row) evaluates to true, return true; Else return false. If the formula has n variables, how many rows does the truth table have? Copyright © Peter Cappello

An example satisfiability problem Let p( row, col, n ) denote the proposition “Box( row, col ) contains number n.” Using such propositions, design a compound proposition that is satisfiable if & only if n appears in some box, for 1 ≤ n ≤ 4. 1 3 4 1 Copyright © Peter Cappello

Copyright © Peter Cappello Problem Give logical expressions for a 2-bit adder, where true corresponds to 1 false corresponds to 0 For example, 01 + 11 = 100. Input: Operand 1: a1 a0 Operand 2: b1 b0 Output s2 s1 s0 That is, define 3 Boolean functions: s0( a1 , a0 , b1 , b0 ) = ? s1( a1 , a0 , b1 , b0 ) = ? s2( a1 , a0 , b1 , b0 ) = ? Copyright © Peter Cappello

Copyright © Peter Cappello Can you define a Boolean function in the C programming language? boolean[] adder( boolean a1, boolean a0, boolean b1, boolean b0 ) { . . . } Or, for an n-bit adder: boolean[] adder( boolean[] a, boolean[] b ) { . . . } For an n-bit adder, it may be useful to compute, for 0 ≤ i ≤ n, a sum bit, si and a carry bit, ci. For the sum bit, si, we may use: si = ai  bi  ci-1, where c-1 = 0 and sn = cn-1. The equation above is called a recurrence equation. What is a recurrence equation for the carry bit, ci? Copyright © Peter Cappello

Copyright © Peter Cappello Unraveling the for loop, suggests a diagram: Each box above has 3 inputs & 2 outputs, and is called a full adder. A harder problem: Compute these sum & carry bits in parallel. sn sn-1 s2 s1 s0 c0 cn-2 c2 c1 cn-1 an-1 bn-1 a2 b2 a1 b1 a0 b0 Copyright © Peter Cappello

Copyright © Peter Cappello 2011 END Copyright © Peter Cappello 2011