# LARA

### Forward and backward propagation

Let denote the result of propagating backward the error conditions, that is, the negation of postcondition. We define by for all . Here for any set of states we write for the complement of , that is, the set .

If is the postcondition operator, that is, 1. Then show that where denotes the inverse of relation . In other words, computing weakest preconditions corresponds to propagating possible errors backwards.

### Galois Connection

In the lecture we talked about a special case of Galois connection, called Galois insertion. Galois connection is in general defined by two monotonic functions and between partial orders on and on , such that for all and (intuitively, the condition means that is approximated by ).

2. Show that the condition is equivalent to the conjunction of these two conditions: hold for all and .

3. Let and satisfy the condition of Galois connection. Show that the following three conditions are equivalent:

• for all • is a surjective function
• is an injective function

4. State the condition for to hold for all . When is the set of sets of concrete states and is a domain of static analysis, is it more reasonable to expect that or to be satisfied, and why?