Computer Science Logic and 8th Kurt Gödel Colloquium
Lars Kristiansen, Paul J. Voda: The Surprising Power of Restricted Programs and G"odel's Functionals
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We introduce a typed imperative programming language operating on natural numbers. The language has only one primitive instruction, increment modulo a base, and only two control structures, the while-loop and the composition. This extremely restricted, but still very natural language, yields an amazing computational power. We prove that the levels of a hierarchy induced by the language, exactly match the levels in the Ritchie hierarchy. (The hierarchy starts with Grzegorczyk's ${\cal E}^2$ (LINSPACE), and the union of the classes in the hierarchy equals the class of Kalmar elementary functions.)
Further, we introduce a formal system $T_\ell$ for defining and computing G\"odel functionals. The system $T_\ell$ is a restriction of G\"odel T. We have removed the successor function, and the only initial function available in $T_\ell$ is the constant function 1 (instead of T's 0). It turns out that exactly the functions in the Ritchie hierarchy can be defined in $T_\ell$. We investigate the relationship between a hierarchy induced by $T_\ell$ and the Ritchie hierarchy.