Parametrised second-order complexity theory with applications to the study of interval computation

TL;DR

This paper extends second-order complexity theory with parameters to analyze a broader class of representations, enabling a rigorous complexity framework for interval computation and real function evaluation.

Contribution

It introduces a parameterized complexity framework for represented spaces, generalizing size functions and applying it to develop a complexity theory for interval computation.

Findings

Interval representation is polytime equivalent to Cauchy representation.

Interval function representation is strictly smaller than the modulus-based representation.

Interval-based function representation is optimal for polytime evaluation.

Abstract

We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the complexity of a given name. This parameter generalises the size function which is usually used in second-order complexity theory and therefore also central to the framework of Kawamura and Cook. The complexity of an algorithm is measured in terms of its running time as a second-order function in the parameter, as well as in terms of how much it increases the complexity of a given name, as measured by the parameters on the input and output side. As an application we develop a rigorous computational complexity theory for interval computation. In the framework of Kawamura and Cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory. In our new framework this representation is polytime equivalent to the usual Cauchy representation based on dyadic rational approximation. By contrast, the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation, which encodes a modulus of continuity. Furthermore, the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable.