1. Today’s Agenda  Correctness Issues

2. Why Correctness?  Programming is engineering Program is a product Program quality to be determined during production process Program Correctness is one essential aspect of quality

3.  Design Correctness Solution (design) meets requirements Verified offline (often on paper) Proof arguments  Implementation Correctness Implementation (program code) matches design Verified online (often by execution) Tests and Test Cases

4.  Output of Design step: Program Design High level solution to problem Consists of modules and module interconnections Modules are solutions to sub-problems Interconnections capture ways to combine sub-solutions

5. Module Correctness  Often reduces to algorithm correctness: Algorithm should terminate Algorithm should produce required result when it terminates.  Both arguments are fairly easy for “straight- line” programs – i.e., no loops.

6. Module Correctness  e.g., Define function to compute square (x) as return x * x  It is easy to verify the requirements: It terminates: a single step It computes the square of a given number: x 2 =x*x

7. Module Correctness  Similar argument can be applied for more than one step as well.  Problem: Given a, b, and c, solve quadratic equation – a*x 2 + b*x + c = 0 Solution X1 = (-b + √b 2 – 4ac ) / 2a X2 = (-b - √b2 – 4ac ) / 2a

8. Module Correctness  Solution: Define function quad(a, b, c, sign) disc = b*b – 4*a*c; if (sign) return (-b + sqrt(disc)) / (2 *a); else return (-b – sqrt(disc)) / (2*a);

9. Module Correctness  Termination: if sqrt terminates, quad() function terminates.  Valid results: if sqrt is correct then quad() returns correct value.  How to verify this? Contracts

10. Inter-Module Correctness  Whoever writes sqrt function, specifies input- output contract: /* Pre-condition: m > 0 Post-condition: return n such that |n * n – m| / m <.01 */ float sqrt(float m)  argument must be +ve definite

11. Inter-Module Correctness  Observation: Precondition m > 0 This is required for sqrt to be correct (or may be even to terminate). So, quad module must guarantee before invocation of sqrt: disc > 0

12. Inter-Module Correctness  The previous contract may propagate up: /* Pre-condition: b*b > 4*a*c Post-condition: return x such that | a*x*x + b*x + c | <= epsilon */ float quad(float a, float b, float c, int sign)

13. Note:  Function interfacing errors minimized due to the Pre -conditions and Post conditions.

14. Why do we need correctness ?  Testing will be easy -Helpful to the third party users  When you write a function you should write: /* Pre-condition : ……. */ /* Post-condition: …….*/

15. Module Correctness for sqrt(x) R = x/2 Err = abs(R * R – x) Err <.01? Yes return R R= (R + x/R)/2 Err = abs(R*R – x) No

16. Illustration: sqrt(9) IterationR next Err next =R 2 –x 0=x/2=9/2=4.54.5 2 -9 <0.01? N 1(4.5+9/4.5)/2=3.253.25 2 -9 <0.01N [10.5625 -9 = 1.5625] 2 (3.25+9/3.25)/2 =3.0096 3.0096 2 -9<0.01N [0.057692] 3 (3.0096 +9/ 3.0096)/2 =3.00 R next ~ sqrt(9) 3.00 2 -9 <0.01 Y Exits from while

17. Functions for quadratic eqn int main() { float a,b,c,x1,x2; scanf("%f %f %f",&a,&b,&c); x1 = quad(a,b,c,1); x2 = quad(a,b,c,-1); printf("x1 = %f x2 = %f\n",x1,x2); return(0); }

18. float quad(float a,float b,float c,int sign) { float disc,res ; disc= b*b - 4*a*c; if(disc > 0) { if(sign) res = (-b + sqrt(disc))/(2*a); else res = (-b - sqrt(disc))/(2*a); return (res); } return(-1.0); }

19. float sqrt(float x) { float r,err; r = x/2.0; err = abs(r*r - x); while((err > 0.01)||(err == 0.01)) { r = (r + x/r)/2; err = abs(r*r - x); } printf("sqrt of disc is %f\n",r); return (r); } }

20. What is a loop invariant?  Property that is maintained “invariant” by iterations in a loop.  How is it used? Verify before the loop Verify each iteration preserves it. Property on termination of loop must result in “what is expected”

21. Loop Invariants: Method to prove correctness of loops  Loop has the following appearance: [pre-condition for loop] while (Guard) [Statements in body of loop. Branching statements that lead outside the loop.] end while [post-condition for loop]

22. Loop Invariance (Example 1 ) Program to do sum of the elements in an array. Sum() { int s = 0; i = 0; // pre condn : s=0 && i=0 While(i < n) { // s is the sum of first i array elements // s = a[0] + … + a[i-1] s = s + a[i]; i = i + 1; } //post condn s = a[0] + ….+ a[i-1] Return s; //post cond : s = a[0] + ………+ a[n-1]

23. Loop Invariance (Example 2 )  Factorial (int n) fact (int n)// pre condn n>=0 { int i =1, fact = 1; // pre cond while(i<=n) { // invariance here is (fact = (i-1)! && i >0) fact = fact * i; i = i+1; } // post cond: fact = (i-1)! && i > 0 return fact; // post cond: fact = (n+1- 1)! }