Discrete Structures

1. Discrete Structures Chapter 5: Sequences, Mathematical Induction, and Recursion 5.5 Application: Correctness of Algorithms [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 5.5 Application: Correctness of Algorithms

2. Programming Errors • 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 Algorithms

3. Programming Errors • 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 Algorithms

4. Assertions • 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 Algorithms

5. Example of Assertions • 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 Algorithms

6. Loop Invariants • 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 Algorithms

7. Definition • Correct w.r.t pre- and post-conditions A loop is defined as correct w.r.t pre- and post-conditionsiff 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 Algorithms

8. Theorem 5.5.1 – Loop Invariant Theorem • 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 Algorithms

9. Theorem 5.5.1 – Loop Invariant Theorem • Basis Property The pre-condition for the loop implies that I(0) is true before the first iteration of the loop. • 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 Algorithms

10. Theorem 5.5.1 – Loop Invariant Theorem 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 Algorithms

11. Example – pg. 288 #2 • 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 Algorithms

12. Example – pg. 288 #7 • 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 Algorithms