Bounded theories for polyspace computability

Abstract

We present theories of bounded arithmetic and weak analysis whose provably total functions (with appropriate graphs) are the polyspace computable functions. More precisely, inspired in Ferreira’s systems $\mathsf{PTCA}$, ${\mathsf{\Sigma }}_1^{\mathsf{b}}\text{-}\mathsf{NIA}$ and $\mathsf{BTFA}$ in the polytime framework, we propose analogue theories concerning polyspace computability. Since the techniques we employ in the characterization of $\mathsf{PSPACE}$ via formal systems (e.g. Herbrand’s theorem, cut-elimination theorem and the expansion of models) are similar to the ones involved in the polytime setting, we focus on what is specific of polyspace and explains the lift from $\mathsf{PTIME}$ to $\mathsf{PSPACE}$.