# Normal form for loop-free programs

Example:

(if (x < 0) x=x+1 else x=x); (if (y < 0) y=y+x else y=y);

Without loops, after expressing conditionals using [] we obtain

c ::= x=T | assume(F) | c [] c | c ; c

Laws:

Normal form:

Each is of form for some , where each is assignment or assume. Each corresponds to one of the finitely paths from beginning to end of the acyclic control-flow graph for loop-free program.

Length of normal form with sequences of if-then-else.

We want to show:

### Verifying Each Path Separately

By normal form this is

which is equivalent to

Note: the rule also applies to infinite union of paths (e.g. generated by loops).

## Three Approaches to Generate Verification Conditions

Three equivalent formulations of Hoare triple give us three approaches:

- compute meaning of as a formula, then check Hoare triple (compositional approach for verification-condition generation)
- compute as a formula, then check entailment (forward symbolic execution)
- compute as a formula, then check (backward symbolic execution)