Textbook examples of recursion

We discuss properties of recursive schemas related to McCarthy's " 91 function " and to Takeuchi's triple recursion. Several theorems are proposed as interesting candidates for machine verification, and some intriguing open questions are raised. John McCarthy and Ikuo Takeuchi introduced interesting recurrence equations as they were exploring the properties of recursive programs. McCarthy's function [7] f (x) = if x > 100 then x − 10 else f f (x + 11) has become known as the " 91 function, " since it turns out that f (x) = 91 for all x ≤ 101. Takeuchi's function [13] is a triple recursion t(x, y, z) = if x ≤ y then y else t t(x − 1, y, z), t(y − 1, z, x), t(z − 1, x, y) , which has proved useful for benchmark testing of Lisp systems because the recursion terminates only after the definition has been expanded a large number of times (assuming that previously computed values are not remembered). Neither of these functions is of practical importance, because no reasonable programmer would ever want to carry out such recursive computations on a realistic problem. Yet both functions are quite instructive because they illustrate important problems and techniques that arise when we consider the task of verifying computer programs formally. Therefore they make excellent examples for textbooks that discuss recursion. The purpose of this paper is to obtain new information about f (x) and t(x, y, z) and about several closely related functions. Several of the theorems proved below should provide good test material for automated verification systems. A few open problems are stated, illustrating the fact that extremely simple recursions can lead to quite difficult questions. 1. The 91 function. It is appropriate to begin by studying the 91 function, because 1991 is the year of John McCarthy's 64th birthday (and because a computer scientist's most significant birthday is the 64th). McCarthy originally wrote down the definition of f (x), as shown above, because he wanted to study a simple recursion whose properties could not be deduced by ordinary mathematical induction. After studying the definition, he was pleasantly surprised to discover that it had the totally unexpected " 91 property. " The 91 function certainly belongs to the set of significant textbook examples, because it is mentioned on at least 14 pages of Zohar Manna's well known text, Mathematical Theory of Computation [6]. The …