1. Copyright © 2006 The McGraw-Hill Companies, Inc. Programming Languages 2nd edition Tucker and Noonan Chapter 18 Program Correctness To treat programming scientifically, it must be possible to specify the required properties of programs precisely. Formality is certainly not an end in itself. The importance of formal specifications must ultimately rest in their utility - in whether or not they are used to improve the quality of software or to reduce the cost of producing and maintaining software. J. Horning

2. Copyright © 2006 The McGraw-Hill Companies, Inc. Contents 18.1 Axiomatic Semantics 18.2 Formal Methods Tools: JML 18.2.1 JML Exception Handling 18.3 Correctness of Object-Oriented Programs 18.3.1 Design by Contract 18.3.2 The Class Invariant 18.3.3 Correctness of a Queue Application 18.3.4 Final Observations 18.4 Correctness of Functional Programs

3. Copyright © 2006 The McGraw-Hill Companies, Inc. Review JML JML ExpressionMeaning requires p;p is a precondition for the call ensures p;p is a postcondition for the call signals (E e) p; when exception e is raised by the call, p is a postcondition loop_invariant p;p is a loop invariant invariant p;p is a class invariant \result == e;e is the result returned by the call \old v the value of v at entry to the call (\product int x ; p(x); e(x)) the product of e(x) for all x that satisfy p(x) (\sum int x ; p(x); e(x)) the sum of e(x) for all x that satisfy p(x) p ==> qp  q

4. Copyright © 2006 The McGraw-Hill Companies, Inc. 18.3.1 Design by Contract

5. Copyright © 2006 The McGraw-Hill Companies, Inc. The contract for a Queue class

6. Copyright © 2006 The McGraw-Hill Companies, Inc. 18.3.2 The Class Invariant A class C is formally specified if: 1. Every constructor and public method M in the classhas preconditions and postconditions, and 2. C has a special predicate called its class invariant INV which, for every object o in C, argument x and call o.M(x), must be true both before and after the call. Note: During a call, INV may temporarily become false. Why are we doing this??? i)Formal specifications provide a foundation for rigorous OO system design (e.g., “design by contract”). ii)They enable static and dynamic assertion checking of an entire OO system. iii)They enable formal correctness proof of an OO system.

7. Copyright © 2006 The McGraw-Hill Companies, Inc. 18.3.3 Correctness of a Queue Application public class Queue { private class Node { } private Node first = null; private Node last = null; private int n = 0; public void enqueue(Object v) { } public Object dequeue( ) {} public Object front() {} public boolean isEmpty() {} public int size() {} public void display() {} } state variables help define INV public methods M public constructor C = Queue() “helper” class

8. Copyright © 2006 The McGraw-Hill Companies, Inc. Adding Class-Level Specifications /*@ public model Node H, T; private represents H <- first; private represents T <- last; public invariant n == size(); @*/ private /*@ spec_public @*/ Node first = null; private /*@ spec_public @*/ Node last = null; private /*@ spec_public @*/ int n = 0; more JML JML model variables class invariant INV Notes: 1) JML model variables allow a specification to distance itself from the class’s implementation details. 2) spec_public allows JML specifications to treat a Java variable as public without forcing the code to do the same.

9. Copyright © 2006 The McGraw-Hill Companies, Inc. Adding Method Specifications /*@ ensures T.prior == \old(T) && H.prior == null && T.val.equals(v) ; @*/ public void enqueue(Object v) { last = new Node(v, last, null); if (first == null) first = last; else (last.prior).next = last; n = n+1; } Notes: 1) \old denotes the value of T at entry to enqueue. 2)The ensures clause specifies that enqueue adds v to the tail T, and that the head H is not affected.

10. Copyright © 2006 The McGraw-Hill Companies, Inc. What specifications for dequeue? /*@ @*/ public Object dequeue( ) { Object result = first.val; first = first.next; if (first != null) first.prior = null ; else last = null; n = n-1; return result; }

11. Copyright © 2006 The McGraw-Hill Companies, Inc. “Pure” methods /*@ requires n > 0; ensures \result == H.val && H == \old(H); @*/ public /*@ pure @*/ Object front() { return first.val; } Note: A method is pure if: 1) it has no non-local side effects, and 2) it is provably non-looping.

12. Copyright © 2006 The McGraw-Hill Companies, Inc. Test driving the Queue class public class QueueTest { public static void main(String[] args) { Queue q = new Queue(); int val; int n = Integer.parseInt(args[0]); for (int i=1; i <= n; i++) q.enqueue(args[i]); System.out.print("Queue contents = "); q.display(); System.out.println("Is Queue empty? " + q.isEmpty()); System.out.println("Queue size = " + q.size()); while (! q.isEmpty()) { System.out.print(" dequeue " + q.dequeue()); System.out.print(" Queue contents = "); q.display(); } System.out.println("Is Queue empty now? " + q.isEmpty()); }

13. Copyright © 2006 The McGraw-Hill Companies, Inc. Contract test 1: normal run % jmlrac QueueTest 5 1 2 3 4 5 Queue contents = last --> 5 4 3 2 1 <-- first Is Queue empty? false Queue size = 5 dequeue 1 Queue contents = last --> 5 4 3 2 <-- first dequeue 2 Queue contents = last --> 5 4 3 <-- first dequeue 3 Queue contents = last --> 5 4 <-- first dequeue 4 Queue contents = last --> 5 <-- first dequeue 5 Queue contents = last --> <-- first Is Queue empty now? true %

14. Copyright © 2006 The McGraw-Hill Companies, Inc. Contract test 2: precondition violation Queue contents = last --> 1 2 3 4 5 <-- first … dequeue 4 Queue contents = last --> 5 <-- first dequeue 5 Queue contents = last --> <-- first Exception in thread "main" org.jmlspecs.jmlrac.runtime.JMLEntryPreconditionError: by method Queue.dequeue regarding specifications at File "../../home/allen/Desktop/workspace/myQueueTest/myqueuetest/ Queue.java", line 36, character 29 when 'this' is myqueuetest.Queue@b166b5 at myqueuetest.Queue.checkPre$dequeue$Queue(Queue.java:626) at myqueuetest.Queue.dequeue(Queue.java:771) at myqueuetest.QueueTest.main(QueueTest.java:16) Note: blame is with the caller QueueTest, since a precondition has been violated.

15. Copyright © 2006 The McGraw-Hill Companies, Inc. Contract test 3: postcondition violation Note: blame is with the supplier Queue, since a postcondition has been violated. Queue contents = last --> 5 4 3 2 1 <-- first Is Queue empty? False Exception in thread "main" org.jmlspecs.jmlrac.runtime. JMLNormalPostconditionError: by method Queue.dequeue regarding specifications at File "../../home/allen/Desktop/workspace/ myQueueTest/myqueuetest/Queue.java", line 37, character 52 when '\old(H)' is myqueuetest.Queue$Node@b166b5 '\result' is 5 'this' is myqueuetest.Queue@cdfc9cmyqueuetest.Queue@cdfc9c …

16. Copyright © 2006 The McGraw-Hill Companies, Inc. Contract test 4: invariant error % jmlrac QueueTest 5 1 2 3 4 5 Exception in thread "main" org.jmlspecs.jmlrac.runtime.JMLInvariantError: by method Queue.enqueue@post<File "../../home/allen/Desktop/workspace/ myQueueTest/myqueuetest/Queue.java", line 28, character 24> regarding specifications at File "../../home/allen/Desktop/workspace/myQueueTest/ myqueuetest/Queue.java", line 22, character 38 … Note: blame is again with the supplier Queue, since an invariant has been violated.

17. Copyright © 2006 The McGraw-Hill Companies, Inc. Class and System Correctness So far, we have only done testing; what about formal verification? 1. A class C is (formally) correct if: a. It is formally specified, and b. For every object o and every constructor and public method M in the class, the Hoare triple is valid for every argument x. 2. A system is correct if all its classes are correct.

18. Copyright © 2006 The McGraw-Hill Companies, Inc. E.g., Correctness of dequeue() /*@ requires n > 0; ensures \result.equals(\old(H).val) && H == \old(H).next; @*/ public Object dequeue( ) { Object result = first.val; first = first.next; if (first != null) first.prior = null ; To prove: else last = null; n = n-1; where: return result; } Note: We again use rules of inference and reason through the code, just like we did with Factorial. 1. 2. 3. 4. 5. 6. 7.

19. Copyright © 2006 The McGraw-Hill Companies, Inc. A “loose” correctness proof for dequeue() 1.“Loose” because a.We assume the validity of size(), and b.We omit some details. 2.The assignments in dequeue(), together with, ensure the validity of : a.Steps 1 and 7 establish \result = \old(first).val b.Steps 2-5 establish first = \old(first).next c.Step 6 and our assumption establish n = size(): I.e., n = \old(n) -1 and size() = \old(size()) -1 So, n = size(), since \old(n) = \old(size())

20. Copyright © 2006 The McGraw-Hill Companies, Inc. 18.3.4 Final Observations 1.Formal verification: a.is an enormous task for large programs. b.only proves that specifications and code agree. c.only proves partial correctness (assumes termination). 2.Tools exist for: a. Statically checking certain run-time properties of Java programs (ESC/Java2) b. formally verifying Ada programs (Spark) 3.Tools are being developed to help with formal verification of Java programs (Diacron, LOOP) 4. What is the cost/benefit of formal methods?