Abstract
In this paper we introduce constructible analogs of the discrete complexity classes $\mathbf{VP}$ and $\mathbf{VNP}$ of sequences of functions. The functions in the new definitions are constructible functions on $\mathbb{R}^n$ or $\mathbb{C}^n$. We define a class of sequences of constructible functions that play a role analogous to that of $\mathbf{VP}$ in the more classical theory. The class analogous to $\mathbf{VNP}$ is defined using Euler integration. We discuss several examples, develop a theory of completeness, and pose a conjecture analogous to the $\mathbf{VP}$ vs $\mathbf{VNP}$ conjecture in the classical case. In the second part of the paper we extend the notions of complexity classes to sequences of constructible sheaves over $\mathbb{R}^n$ (or its one point compactification). We introduce a class of sequences simple constructible sheaves, that could be seen as the sheaf-theoretic analog of the Blum-Shub-Smale class $\mathbf{P}_{\mathbb{R}}$. We also define a hierarchy of complexity classes of sheaves mirroring the polynomial hierarchy, $\mathbf{PH}_{\mathbb{R}}$ in the B-S-S theory. We prove a singly exponential upper bound on the topological complexity of the sheaves in this hierarchy mirroring a similar result in the B-S-S setting. We obtain as a result an algorithm with singly exponential complexity for a sheaf-theoretic variant of the real quantifier elimination problem. Finally, we pose the natural sheaf-theoretic analogs of the classical $\mathbf{P}$ vs $\mathbf{NP}$ question, and also discuss a connection with Toda's theorem from discrete complexity theory in the context of constructible sheaves.