 A famous recursive function is the Ackermann function, which, unlike the Fibonacci sequence, cannot be expressed without recursion.Applying the standard technique of proof by cases to recursively defined sets or functions, as in the preceding sections, yields structural induction — a powerful generalization of mathematical induction widely used to derive proofs in mathematical logic and computer science.Dynamic programming is an approach to optimization that restates a multiperiod or multistep optimization problem in recursive form.