Closed Sets and Operators thereon: Representations, Computability and Complexity

TL;DR

This paper studies the complexity of representations of closed sets in Euclidean spaces within the framework of Computable Analysis, providing parameterized complexity bounds for various set operators and relating them to classical complexity classes.

Contribution

It refines the complexity analysis of representations of closed sets using second-order representations, connecting continuous and discrete complexity theory.

Findings

Parameterized complexity bounds for set operators like union and intersection

A uniform framework relating continuous representations to P/UP/NP classes

Extension of previous computability results to complexity bounds

Abstract

The TTE approach to Computable Analysis is the study of so-called representations (encodings for continuous objects such as reals, functions, and sets) with respect to the notions of computability they induce. A rich variety of such representations had been devised over the past decades, particularly regarding closed subsets of Euclidean space plus subclasses thereof (like compact subsets). In addition, they had been compared and classified with respect to both non-uniform computability of single sets and uniform computability of operators on sets. In this paper we refine these investigations from the point of view of computational complexity. Benefiting from the concept of second-order representations and complexity recently devised by Kawamura & Cook (2012), we determine parameterized complexity bounds for operators such as union, intersection, projection, and more generally function image and inversion. By indicating natural parameters in addition to the output precision, we get a uniform view on results by Ko (1991-2013), Braverman (2004/05) and Zhao & M\"uller (2008), relating these problems to the P/UP/NP question in discrete complexity theory.