The Surprising Power of Restricted Programs and Gödel’s Functionals

Abstract

Consider the following imperative programming language. The programs operate on registers storing natural numbers, the input $\vec{x}$ is stored in certain registers, and a number b, called the base, is fixed to $\max(\vec{x},1)+1$ before the execution starts. The single primitive instruction $\verb/X+/$ increases the number stored in the register $\verb/X/$ by 1 modulo b. There are two control structures: the loop $\verb+while X{P}+$ executing the program $\verb+P+$ repeatedly as long as the content of the register $\verb/X/$ is different from 0; the composition $\verb+P+\texttt{;} \verb+Q+$ executing first the program $\verb/P/$, then the program $\verb/Q/$. This is the whole language. The language is natural, extremely simple, yet powerful. We will prove that it captures $\mbox{\sc linspace}$, i.e. the numerical relations decidable by such programs are exactly those decidable by Turing machines working in linear space. Variations of the language capturing other important deterministic complexity classes, like e.g. $\mbox{\sc logspace}$, $\mbox{\sc p}$ and $\mbox{\sc pspace}$ are possible (see Kristiansen and Voda [5]).