1. Lecture 2: Distributed Programs and their Correctness
Anish Arora
CSE 6333 1

2.     Table of Precedence + ─ = < >   
Highest [x := E] substitution
. function application
+ ─ unary prefix operators
x / mod gcd binary arith. operators
+ ─
= < >   
 
 
Lowest  2

3. Equivalence  Associativity : (P  Q)  R  P  (Q  R)
Symmetry : P  Q  Q  P
Identity : true  Q  Q
Note:  is not ‘conjunctional’
P  Q  R ? P  Q  Q  R
Example:
Tom has 2 eyes  Tom has 1 eye  Tom has no eyes ?
(Tom has 2 eyes  Tom has 1 eye)  (Tom has 1 eye 
Tom has no eyes)

4. Negation  false   true  (P  Q)   P  Q P  Q   (P  Q)
Theorem : (P Q)  (Q P)
Proof: (P Q)  (Q P)
= { axiom : definition of }
 (P  Q)   (Q  P)
= { axiom :  (P  Q)   P  Q }
 P  Q   Q  P
= {  is symmetric, associative }
 P  P   Q  Q
=  (P  P)   (Q  Q)
{ definition of false : false   true}
= false  false
= true

5. Disjunction  , Conjunction 
Symmetry : P  Q  Q  P
Associativity : (P  Q)  R  P  (Q  R)
Idempotence : P  P  P
Distributivity of  over  : P  (Q  R)  P  Q  P  R
Excluded miracle : P  P
The Golden Rule:
P  Q  P  Q  P  Q
P  Q  P  Q  P  Q
P  Q  P  Q  P  Q
Note : P  Q  P  Q  P
P  Q  Q  P  Q

6. Quantification : Universal , Existential 
(x : R : P)  (x : true : R  P)
(x : R : true)  true
(x : R : P)   (x : R :  P)
(x : R : P)  (x : true : R  P)
(x : R : false)  false
Q: (x : false : R)  ?

7. Reasoning with Hoare Triples
{P} g  skip {P} is true
{P[x := E]} true  x := E {P} rule of textual substitution
{P} g  st {Q} equivales {P g} true  st {Q}
and
(P g)  Q
{P} g  st {P} equivales {P  g} true  st {P}

8. Reasoning with Hoare Triples (contd.)
P  Q and {Q} g  st {R} implies {P} g  st {R}
{P} g  st {Q} and Q  R implies {P} g  st {R}
{P} g  st {R} and {Q} g  st {R} implies
{P  Q} g  st {R}
{P} g  st {Q} and {P} g  st {R} implies
{P} g  st {Q  R}