A Complexity Theory of Constructible Functions and Sheaves

In this paper we introduce constructible analogs of the discrete complexity classes VP and VNP of sequences of functions. The functions in the new definitions are constructible functions on R n or C n . We define a class of sequences of constructible functions that play a role analogous to that of V...

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Vydáno v:Foundations of computational mathematics Ročník 15; číslo 1; s. 199 - 279
Hlavní autor: Basu, Saugata
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.02.2015
Springer Nature B.V
Témata:
Analogs
Applications of Mathematics
Complexity
Complexity theory
Computational mathematics
Computer Science
Construction
Economics
Geometry
Hierarchies
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Sequences
Sheaves
Primary 14P10
Adjoint functors
Secondary 68W30
Polynomial hierarchy
Constructible functions
Constructible sheaves
14P25
Complexity classes
ISSN:1615-3375, 1615-3383
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Shrnutí:In this paper we introduce constructible analogs of the discrete complexity classes VP and VNP of sequences of functions. The functions in the new definitions are constructible functions on R n or C n . We define a class of sequences of constructible functions that play a role analogous to that of VP in the more classical theory. The class analogous to VNP is defined using Euler integration. We discuss several examples, develop a theory of completeness, and pose a conjecture analogous to the VP versus 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 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 P R . We also define a hierarchy of complexity classes of sheaves mirroring the polynomial hierarchy, PH 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 P versus 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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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-014-9222-z