Inductive and Recursive Definitions

Abstract

This chapter introduces inductively defined sets and simple inductive data types such as natural numbers, words, lists and trees with emphasis on definition patterns. For these sets and data types, recursive equations in Backus-Naur form are introduced and it is shown how, e.g., propositional formulae and simple while programs can be specified in this format. Next it is explained how functions can be defined inductively on such sets and data types. Examples include the factorial function, Harmonic numbers, Fibonacci numbers, addition and multiplication on natural numbers and functions that compute the length of a word or list and the height of a tree. In contrast, recursive functions are defined in a way that does not necessarily lead to terminating evaluations. Recursive procedures are also described, with examples presented of various sorting algorithms and the Towers of Hanoi puzzle.