Recursion and Induction: Reasoning About Recursive Programs Greg - - PowerPoint PPT Presentation

Recursion and Induction: Reasoning About Recursive Programs

Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin

Proving Properties of power2

• Recall the recursive program power2 presented last time

power2 0 = 1 power2 (n+1) = 2 * (power2 n)

• Claim: For all nonnegative integers n, the value of the power2 function

applied to n is 2n

• While this claim may seem relatively obvious, how can we formally

prove it?

Theory in Programming Practice, Plaxton, Spring 2005

Proving Properties of count

• Recall the recursive program count we presented last time

count 0 = 0 count n | even n = count (n ‘div‘ 2) | odd n = count (n ‘div‘ 2) + 1

• For any nonnegative integer n, let C(n) denote the value returned by

the count function applied to argument n

• Claim 1: For all nonnegative integers n, C(n) equals the number of 1s

in the binary representation of n

Theory in Programming Practice, Plaxton, Spring 2005

Proving Properties of count

• Claim 2: For all nonnegative integers m and n,

C(m + n) ≤ C(m) + C(n)

• Hint: Use induction over pairs (m, n) (ordered lexicographically, say),

and consider cases depending on the parity of m and n

Theory in Programming Practice, Plaxton, Spring 2005

Proving Properties of fib

• Recall the recursive program we presented last time for computing the

Fibonacci numbers fib 0 = 0 fib 1 = 1 fib (n + 2) = (fib n) + (fib (n+1))

• For any nonnegative integer n, let F(n) denote the value returned by

the fib function applied to argument n

• Claim: For all integers m and n such that m > 0 and n ≥ 0

F(m + n) = F(m − 1) · F(n) + F(m) · F(n + 1) – Hint: Fix m arbitrarily and use induction on n alone

Theory in Programming Practice, Plaxton, Spring 2005

Proving Properties of fib and gcd

• Recall Euclid’s gcd algorithm

gcd m n | m == n = m | m > n = gcd (m - n) n | n > m = gcd m (n - m)

• For any nonnegative integers m and n, let g(m, n) denote the value

returned by the gcd function applied to arguments m and n

• Claim: For all positive integers n

g(F(n), F(n + 1)) = 1

Theory in Programming Practice, Plaxton, Spring 2005