Program Correctness

Program Correctness William Groesbeck, Sam Delaney

Program Correctness • What is program correctness?

Program Correctness • What is program correctness? • A program is said to be correct if it produces the correct output for every possible input

Program Correctness • Two types of correctness: • Total – algorithm always returns at least one correct solution • Partial – if a solution is returned, it will be correct

Program Correctness • Tony Hoare – • Introduced the idea of partial correctness • Also the creator of the quick sort – one of the most commonly used and studied sorting algorithms

Program Correctness • Partial correctness with respect to assertions: • A program can be partially correct with respect to the initial assertion (represented with p) and the final assertion (represented by q). • Notation: p{S}q – Hoare triple

Program Correctness • Example 1, Section 4.5: Show that the program segment y := 2, z := x+y is correct with respect to the initial assertion p: x=1 and the final assertion q: z=3

Program Correctness • Solution: • Suppose that p is true, so x=1. According to the program segment, y is then assigned 2. Then z is assigned x+y, or 1+2 making z=3. Thus, S is correct with respect to both the initial and the final assertions. p{S}q is true.

Program Correctness • Example 2: int identity(int x){ x=x*1; return x; }

Program Correctness • Curry-Howard Correspondence • Deep result in proof theory that states that a proof of functional correctness in constructive logic corresponds to a certain program in the lambda calculus • Converting a proof in this way is called program extraction.

Program Correctness • Lambda Calculus: • Formal system designed to investigate function definition • Greatly influenced functional programming language including Lisp, ML, and Haskell • Consists of a single variable substitution and a single function definition scheme. • Undecidability

Program Correctness • Lambda Calculus Expressions • f(x) = x + 2 • λ x. x+2 • f(3) = (λ x. x+2) 3 • (λ f. f 3)(λ x. x+2) • = (λ x. x+2)3 = 3+2 =5

Program Correctness • Example 2 (Revised): int identity(int x){ x=x*2; return x; } = (λ x. x*2)

Program Correctness • Example 3 : int identity(int x){ x=x*2; return x; } = (λ x. x*2) If x=3? (λ f. f 3) (λ x. x*2)

Program Correctness • Conclusion: • Partial / Total Correctness • Tony Hoare • Partial correctness with respect to… • Curry-Howard Correspondence • Lambda Calculus • Bonus! • Name two things that Tony Hoare was best known for

Program Correctness • Questions?

Program Correctness