Characterizations of Presburger Functions

Let $\mathcal{F}$ be the smallest class of functions on the natural numbers containing the functions $U_i^n (x_1 , \cdots ,x_n ) = x_i $, $S(x) = x + 1$, $A(x,y) = x + y$, $D(x,y) = x \dot{-} y$, $C(x,y) = (1 \dot{-} y)x$, $T_k (x) = \lfloor x/k \rfloor $ and losed under composition. It is shown that $\mathcal{F}$ is exactly the class of functions definable by Presburger formulas. Moreover, for Presburger functions with finite output range, $A(x,y)$ and $C(x,y)$ can be deleted from the list of initial functions. Characterizations of $\mathcal{F}$ and its subclasses in terms of simple programs are also given. An example is the following: A function is in $\mathcal{F}$ if and only if it is computable by a program which contains only instructions of the form $x \leftarrow x + 1$, $x \leftarrow x \dot{-} 1$, $x \leftarrow y$, and $\textbf{do } x \cdots \textbf{ end}$, where $\textbf{do}$’s cannot be nested.