5. Hoare Logic and Program Verification

Hoare Logic and Program Verification

Introduction to Hoare logic

Hoare logic is a formal system for reasoning about the correctness of computer programs. It uses assertions to describe the preconditions, postconditions, and invariants of program statements. Hoare triples, of the form {P} S {Q}, represent assertions about the behavior of programs, where P is the precondition, S is the program statement, and Q is the postcondition.

Verification of imperative programs using Hoare logic

Hoare logic allows us to reason about imperative programs by specifying preconditions and postconditions for each program statement. We can use axioms and inference rules to derive Hoare triples for complex programs, ensuring that they satisfy their specifications and behave correctly under all conditions.

Coq lab activities: Writing and verifying Hoare triples for simple imperative programs in Coq

In this lab activity, we will use Coq to write and verify Hoare triples for simple imperative programs. Coq's support for formal reasoning and proof allows us to ensure the correctness of program behavior and verify that they meet their specifications.

Lab Activities

Definition of Hoare Triples in Coq

We will define Hoare triples in Coq using the following syntax:

Definition hoare_triple (P : Assertion) (c : com) (Q : Assertion) : Prop :=
  forall st st',
    P st ->
    c / st \\ st' ->
    Q st'.

Verification of Hoare Triples for Simple Programs

We will then verify Hoare triples for simple imperative programs using Coq's proof assistant. For example, we can verify the Hoare triple for a program that increments a variable:

Example hoare_inc :
  hoare_triple (fun st => st X = 1)
               (X ::= APlus (AId X) (ANum 1))
               (fun st => st X = 2).
Proof.
  unfold hoare_triple.
  intros.
  inversion H0. subst.
  simpl in H3. inversion H3. subst.
  simpl. reflexivity.
Qed.

The above Coq code verifies the Hoare triple for a program that increments the value of variable X. The preconditions state that X is initially set to 1, and the postconditions assert that X is incremented to 2 after executing the program.