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A Complexity Theory of Constructible Functions and Sheaves

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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 \(\mathbf {VP}\) versus \(\mathbf {VNP}\) 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 of 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. We pose the natural sheaf-theoretic analogs of the classical \(\mathbf {P}\) versus \(\mathbf {NP}\) question, and also discuss a connection with Toda’s theorem from discrete complexity theory in the context of constructible sheaves. We also discuss possible generalizations of the questions in complexity theory related to separation of complexity classes to more general categories via sequences of adjoint pairs of functors.

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Acknowledgments

The author would like to thank the anonymous referees for a very careful reading of a previous version of this paper, and for many comments and suggestions that helped to improve the paper substantially.

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA

    Saugata Basu

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Correspondence to Saugata Basu.

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Communicated by Teresa Krick and James Renegar.

Dedicated to Mike Shub on his 70-th birthday.

The author was supported in part by NSF grants CCF-0915954, CCF-1319080 and DMS-1161629 while working on this paper.

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Basu, S. A Complexity Theory of Constructible Functions and Sheaves. Found Comput Math 15, 199–279 (2015). https://doi.org/10.1007/s10208-014-9222-z

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