Normal Form Theorem for Bar Recursive Functions of Finite Type

Abstract

Publisher Summary This chapter presents prove of Normal Form Theorem for the bar recursive functions of finite type. These functions will be represented by the bar recursive terms, which are built up from constants denoting the basic operators of explicit definition, primitive recursion and bar recursion. Rules of conversion will be introduced to express the action of these operators. The Normal Form Theorem asserts that every bar recursive term reduces, by means of a finite sequence of conversions, to a unique normal term. A normal term is one with no convertible subterms. The normal numerical terms (i.e., of type 0) are the numerals, and so, in particular, the theorem asserts that every numerical bar recursive term has a unique and computable value. The proof—that when t reduces to a normal term, the normal term is unique—is elementary. When such a normal term exists, it is called the normal form or value of t. The proof of the Normal Form Theorem will involve a modified version of the notion of convertible term, used to prove the normal form theorem for primitive recursive functions of finite type, with and without bar recursion of type 0.