Ackermann's Function through Recursion and Induction
Delve into the complexity of Ackermann's Function through a detailed exploration of recursion and induction. Uncover the step-by-step reasoning behind A(5,5) and gain insights into how the function evolves with each recursive call. Follow along with the proof by mathematical induction to grasp the inner workings of this intriguing algorithmic concept.
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Presentation Transcript
Thinking about Algorithms Abstractly Recursion Jeff Edmonds York University Lecture3 COSC 3101 1
Ackermanns Function How big is A(5,5)? = Define T (n) A(k,n) k different a i.e. function for each . T k Base Case : k Proof by induction = on that , k = 0 0 0 n T T ( (n) ) T Step ( ) = 2 + k k 0 T (n) T ( T ( T ( T ( T ( T ( ) ))))) Inductive : 1 1 1 1 1 k k- k- k- k- k- k = 1 T (n) T ( T (n ) ) n applications 1 k k- k 2
Ackermanns Function k that , on induction = n applications n + = 2 Proof by 0 T (n) T ( T ( T ( T ( T ( T ( ) ))))) 1 1 1 1 1 k k- k- k- k- k- k 0 T T (n) 2 + + + + + = = 2 2 2 2 T 0 1 ( ) (n) 1 n n applications 3
Ackermanns Function k that , on induction = n applications n + = 2 Proof by 0 T (n) T ( T ( T ( T ( T ( T ( ) ))))) 1 1 1 1 1 k k- k- k- k- k- k 0 T 1 T T (n) = 2 = 2 (n) n = (n) 2 2 2 2 T 0 1 ( ) n 2 n applications 4
Ackermanns Function k that , on induction = n applications n + = 2 Proof by 0 T (n) T ( T ( T ( T ( T ( T ( ) ))))) 1 1 1 1 1 k k- k- k- k- k- k 0 T 1 T 2 T T (n) = 2 (n) n = = n 2 (n) (n) 3 = T (n) 4 5
