The structure of this lecture is similar to the previous one:
Today, we extend the approach from last lecture (both generating and proving formulas) to support data structures such as references and arrays.
We use weakest preconditions, although you could also use strongest postconditions or any other variants of the conversion from programs to formulas.
Can we derive wp(Q,havoc(x))?
By wp semantics of havoc and assume
wp(Q, havoc(x); assume(x=t)) =
while [inv I] (F) c1
⇒
assert(I); havoc(x1,...,xn); (those that can be modified in the loop) assume(I); assume(~F);
Provided that I is inductive. We can check that I is inductive by proving:
assume(I); assume(F); c1; assert(I);
in isolation. Or proving
havoc(x1,...,xn) assume(I); assume(F); c1; assert(I);
at any point. If we combine these two, we get:
⇒
assert(I); havoc(x1,...,xn); assume(I); (assume(~F) [] assume(F); c1; assert I; assume(false));
Benefit: if there is x_{n+1} that is not changed, we do not need to write its properties in the loop invariant. This can make loop invariant shorter.
If then else expressions.
Desugaring if-then-else.
Avoiding exponential explosion using flattening.
Array bounds checking.
Objects as references, null as an object.
Program with class declaration
class Node { Node left, right; }
How can we represent fields?
Possible mathematical model: fields as functions from objects to objects.
left : Node => Node right : Node => Node
What is the meaning of assignment?
x.f = y
left, right - uninterpreted functions (can have any value, depending on the program, unlike arithmetic functions such as +,-,* that have fixed interpretation).
Null checks.
All objects exist.
The allocated object set.
Why we need so many arguments.
Again, the second part! More technical. But, often you can use these things as a black box.
Properties of relations
Equality is a congruence with respect to all function symbols.
More information on congruence closure algorithm: