RecursiveFunction2 2009-03-09 01:03:16 2009-03-09 01:03:16 Definition % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Intuitively, a recursive function may be defined as an integer-valued function of one or more integer variables which may be computed by a definite algorithm. In order to produce a rigorous definition, one may proceed long at least two approaches; one may define the notion of algorithm rigorously in order to complete the intuitive definition given above or one may proceed inductively, first declaring certain functions to be recursive and then specifying definite procedures by which one may construct any other recursive function starting from the initial set. In this entry, we shall concentrate on the latter approach, only making a few brief remarks regarding the former approach towards the end. \begin{enumerate} \item The constant function $c: \mathbb{Z}_+ \to \mathbb{Z}_+$ defined by $c(x) = 1$ for all $x \in \mathbb{Z}_+$ is a recursive function. \item The addition function $+: \mathbb{Z}_+^2 \to \mathbb{Z}_+$ and the multiplication function $\times: \mathbb{Z}_+^2 \to \mathbb{Z}_+$ are recursive function. \item The projection functions $I^n_m \colon \mathbb{Z}_+^n \to \mathbb{Z}_+$ with $1 \le m \le n$ defined as $I^n_m (x_1, \ldots, x_n) = x_m$ are recursive functions. \item {\it (Closure under composition)} If $f \colon \mathbb{Z}_+^n \to \mathbb{Z}_+$ is a recursive function and $g_i \colon \mathbb{Z}_+^m \to \mathbb{Z}_+$ with $i = 1, \ldots n$ are recursive functions, then $h \colon \mathbb{Z}_+^n \to \mathbb{Z}_+$, defined by $h(x_1, \ldots, x_n) = f(g_1(x_1, \ldots, x_m), \ldots, g_n(x_1, \ldots, x_m))$ is a recursive function. \item {\it (Closure under primitive recursion)}If $f \colon \mathbb{Z}_+^n \to \mathbb{Z}_+$ and $g \colon \mathbb{Z}_+^{n+2} \to \mathbb{Z}_+$ are recursive function, then $h \colon \mathbb{Z}_+^{n+1} \to \mathbb{Z}_+$, defined by the recursion $$h(n+1,x_1,\ldots,x_{k}) = g(h(n,x_1,\ldots,x_k),n,x_1,\ldots, x_k)$$ with the initial condition $$h(0,x_1,\ldots,x_k) = f(x_1,\ldots,x_k)$$ is a recursive function. \item {\it (Closure under minimization)} If $f \colon \mathbb{Z}_+^{n+1} \to \mathbb{Z}_+$ is a recursive function then $g \colon \mathbb{Z}_+^n \to \mathbb{Z}_+$ is a recursive function, where $g$ is defined to equal $y$ if there exists a $y \in \mathbb{Z}_+$ such that \begin{itemize} \item $f(0, x_1, \ldots, x_n), f(1, x_1, \ldots, x_n), \ldots, f(y, x_1, \ldots, x_n)$ are all defined, \item $f(z, x_1, \ldots, x_n) = 0$ when $1 \le z <y$, and \item $f(y, x_1, \ldots, x_n) = 0$, otherwise $g(x_1, \ldots, x_n)$ is undefined. \end{itemize} \end{enumerate}