A+Practical+Theory+of+Programming+09 - Independent...

Independent Composition
Dependent Composition
P . Q (sequential execution)
P and Q must have exactly the same state variables
Independent Composition
P || Q (parallel execution)
P and Q must have completely different state variables
and the state variables of the composition are those of both P and Q
Ignoring time and space variables
P || Q
=
P ∧ Q
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Independent Composition
example
in integer variables x , y , and z
x := x +1 || y := y +2 partition the variables:
put x in left part, put y and z in right part
=
x ′ = x +1 || y ′ = y +2 ∧ z ′ = z
=
x ′ = x +1 ∧ y ′ = y +2 ∧ z ′ = z
reasonable partition rule
If either x ′ or x := appears in a process specification, then x belongs to that process
(then neither x ′ nor x := can appear in the other process specification).
If neither x ′ nor x := appears at all, then x can be placed on either side of the partition.
2/21

Independent Composition
example
in variables x , y , and z
x := y || y := x partition: put x in left, y in right, z in either
=
x ′ = y ∧ y ′ = x ∧ z ′ = z
implementation of a process makes a private copy of the initial value of a variable belonging
to the other process if the other process contains an assignment to that variable
3/21

Independent Composition
example
in boolean variable b and integer variable x
b := x = x || x := x +1 replace x = x by T
=
b := T || x := x +1
example
in integer variables x and y
( x := x +1. x := x –1) || y := x
=
ok || y := x
=
y := x
4/21

Independent Composition
( x := x + y . x := x
×
y ) || ( y := x – y . y := x / y )
5/21

Independent Composition
( x := x + y . x := x
×
y ) || ( y := x – y . y := x / y )
You should have written
( x := x + y || y := x – y ). ( x := x
×
y || y := x / y )
6/21