Non-determinism in Gödel’s System T

Abstract

The calculus T − is a successor-free version of Godel’s T. It is well known that a number of important complexity classes, like e.g. the classes logspace, $\mbox{\sc p}$, $\mbox{\sc linspace}$, $\mbox{\sc etime}$ and $\mbox{\sc pspace}$, are captured by natural fragments of T − and related calculi. We introduce the calculus T ∽, which is a non-deterministic variant of T −, and compare the computational power of T ∽ and T −. First, we provide a denotational semantics for T ∽ and prove this semantics to be adequate. Furthermore, we prove that $\mbox{\sc linspace}\subseteq \mathcal {G}^{\backsim }_{0} \subseteq \mbox{\sc nlinspace}$ and $\mbox{\sc etime}\subseteq \mathcal {G}^{\backsim }_{1} \subseteq \mbox{\sc espace}$ where $\mathcal {G}^{\backsim }_{0}$ and $\mathcal {G}^{\backsim }_{1}$ are classes of problems decidable by certain fragments of T ∽. (It is proved elsewhere that the corresponding fragments of T − equal respectively $\mbox{\sc linspace}$ and $\mbox{\sc etime}$.) Finally, we show a way to interpret T ∽ in T −.