1. Chapter 5: Sequences, Mathematical Induction, and Recursion 5.5 Application: Correctness of Algorithms 1 [P]rogramming reliability – must be an activity of an undeniably mathematical nature…You see, mathematics is about thinking, and doing mathematics is always trying to think as well as possible. – Edsger W. Dijkstra, 1930 – 2002

2. Syntax errors – Failure to declare a variable – Using a restricted keyword as a variable name – Writing ik instead of if Logical errors 5.5 Application: Correctness of Algorithms2

3. In this section, we will – give an overview of the general format of correctness proofs, and – go into details of one crucial technique called the loop invariant procedure We now need to switch from using the term program which refers to a particular programming language, and use the more general term algorithm. 5.5 Application: Correctness of Algorithms3

4. Both the initial and final states can be expressed as predicates involving the input and output variables. Pre-condition for the algorithm – the predicate describing the initial state. Post-condition for the algorithm – the predicate describing the final state. 5.5 Application: Correctness of Algorithms4

5. Algorithm to compute a product of nonnegative integers. – Pre-condition: The input variables m and n are nonnegative integers. – Post-condition: The output variable p equals mn. 5.5 Application: Correctness of Algorithms5

6. The annotated loop has the following appearance. [Pre-condition for the loop] while (Guard) [Statements in the body of the loop. None contain branching statements that lead outside the loop.] end while [Post-condition for the loop] 5.5 Application: Correctness of Algorithms6

7. Correct w.r.t pre- and post-conditions A loop is defined as correct w.r.t pre- and post-conditions iff whenever the algorithm variables satisfy the pre-condition for the loop and the loop terminates after a finite number of steps, the algorithm variables satisfy the post- condition for the loop. 5.5 Application: Correctness of Algorithms7

8. Let a while loop with guard G be given, together with pre- and post-conditions that are predicates in the algorithm variables. Also let a predicate I(n), called the loop invariant, be given. If the following four properties are true, then the loop is correct w.r.t. its pre- and post- conditions. 5.5 Application: Correctness of Algorithms8

9. I.Basis Property The pre-condition for the loop implies that I(0) is true before the first iteration of the loop. II.Inductive Property For all integers k  0, if the guard G and the loop invariant I(k) are both true before an iteration of the loop, then I(k + 1) is true after an iteration of the loop. 5.5 Application: Correctness of Algorithms9

10. III. Eventual Falsity of Gaurd After a finite number of iterations of the loop, the guard G becomes false. IV. Correctness of the Post-Conditions If N is the least number of iterations after which G is false and I(N) is true, then the values of the algorithm variables will be as specified in the post-condition of the loop. 5.5 Application: Correctness of Algorithms10

11. Show that if the predicate is true before entry to the loop, then it is also true after the loop. Loop:while (m  0 and m  100) m := m + 4 n := n – 2 end while Predicate:m + n is odd 5.5 Application: Correctness of Algorithms11

12. Use the loop invariant theorem to prove the correctness of the loop w.r.t. the pre- and post-condtions. [Pre-condition: largest = A[1] and i =1] while (i  m) 1. i := i + 1 2. if A[i] > largest then largest := A[i] end while [Post-condition: largest = max value of A[1], A[2], …, A[m]] 5.5 Application: Correctness of Algorithms12