Question

A modification of the Available Expression Analysis detects when an expression is available in a particular variable: a non-trivial expression a is available in x at a label l if it has been evaluated and assigned to x on all paths leading to l and if the values of x and the variables in the expression have not changed since then. Write down the data flow equations and any auxiliary functions for this analysis

Answer 1

The data flow equations for this modification of the Available Expression Analysis can be written as x[l] = ∩ (p ∈ Pred(l)) out(p) and out(l) = a ∩ x[l]. An auxiliary function that is required for this analysis is out(p) = x[p] ∪ (in(p) - kill(p)).

Step-by-step explanation:

The data flow equations for this modification of the Available Expression Analysis can be written as:

1. x[l] = ∩ (p ∈ Pred(l)) out(p)
2. out(l) = a ∩ x[l]

Where:

• x[l] represents the set of available variables at label l
• ∩ represents the intersection operator
• Pred(l) represents the set of labels that immediately precede label l
• out(p) represents the set of available variables at label p
• a represents the expression that needs to be checked for availability

An auxiliary function that is required for this analysis is out(p) which represents the set of available variables at label p and can be defined as:

out(p) = x[p] ∪ (in(p) - kill(p))

Where:

• x[p] represents the set of available variables at label p
• ∪ represents the union operator
• in(p) represents the set of variables defined before label p
• kill(p) represents the set of variables that are redefined at label p