Propositional Equivalence
Goal: Show how propositional equivalences are
established & introduce the most important such
equivalences.
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
p  q ≡ q  p;
Double negation
pq≡qp
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Commutative
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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
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Exercise
A tautology is a compound proposition
that always is true.
Is the following a tautology?
( (p  q)  ( p  r) )  (q  r)
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The Satisfiability Problem
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
1.
Construct the truth table.
2.
If any assignment (row) evaluates to true, return true;
3.
Else return false.
If the formula has n variables, how many rows does the truth table have?
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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
1
3
n appears in some box, for 1 ≤ n ≤ 4.
4
1
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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:
• s 0 ( a1 , a0 , b1 , b 0 ) = ?
• s 1 ( a1 , a0 , b1 , b 0 ) = ?
• s 2 ( a1 , a0 , b1 , b 0 ) = ?
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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?
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Unraveling the for loop, suggests a diagram:
sn
sn-1
cn-1
s2
cn-2
an-1 bn-1
c2
s1
c0
c1
a2 b2
s0
a1 b1
0
a0 b0
Each box above has 3 inputs & 2 outputs, and is called a full adder.
A harder problem: Compute these sum & carry bits in parallel.
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END
Copyright © Peter Cappello 2011
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