Module 6

1
Module 6

• Topics
• Program behavior problems
• Input of problem is a program/algorithm
• Definition of type program
• Program correctness
• Testing versus Proving

2
Number Theory Problems

• These are problems where we investigate
  properties of numbers
• Primality
• Input Positive integer n
• Yes/No Question Is n a prime number?
• Divisor
• Input Integers m,n
• Yes/No question Is m a divisor of n?

3
Graph Theory Problems

• These are problems where we investigate
  properties of graphs
• Connected
• Input Graph G
• Yes/No Question Is G a connected graph?
• Subgraph
• Input Graphs G1 and G2
• Yes/No question Is G1 a subgraph of G2?

4
Program Behavior Problems

• These are problems where we investigate
  properties of programs and how they behave
• Give an example problem with one input program P
• Give an example problem with two input programs
  P1 and P2

5
Program Representation

• Program variables
• Abstractly, we define the type program
• graph G, program P
• More concretely, we define type program to be a
  string over the program alphabet SP a, , z,
  A, , Z, 0, , 9, punctuation, white space
• Note, many strings over SP are not legal programs
  
• We consider them to be programs that always crash
• Possible declaration of main procedure
• bool main(program P)

6
Program correctness

• How do we determine whether or not a program P we
  have written is correct?
• What are some weaknesses of this approach?
• What might be a better approach?

7
Testing versus Analyzing
Test Inputs x1 x2 x3 ...
Outputs P(x1) P(x2) P(x3) ...
Analysis of Program P
Program P
8
2 Program Behavior Problems

• Correctness
• Input
• Program P
• Yes/No Question
• Does P correctly solve the primality problem?
• Functional Equivalence
• Input
• Programs P1, P2
• Yes/No Question
• Is program P1 functionally equivalent to program
  P2