# Bounded time computation on metric spaces and Banach spaces

## TL;DR

This paper extends a computational complexity framework for analysis operators on metric and Banach spaces, establishing conditions for bounded-time computability and constructing efficient representations.

## Contribution

It introduces complete and regular representations ensuring bounded-time computation of metrics and explores their relationship with the metric space's properties.

## Key findings

- Bound on metric computation time translates to size bounds of compact subsets.
- Constructs admissible, complete, regular representations for Banach spaces.
- Trade-off between time bounds and encoding length for efficient computation.

## Abstract

We extend the framework by Kawamura and Cook for investigating computational complexity for operators occurring in analysis. This model is based on second-order complexity theory for functions on the Baire space, which is lifted to metric spaces by means of representations. Time is measured in terms of the length of the input encodings and the required output precision. We propose the notions of a complete representation and of a regular representation. We show that complete representations ensure that any computable function has a time bound. Regular representations generalize Kawamura and Cook's more restrictive notion of a second-order representation, while still guaranteeing fast computability of the length of the encodings. Applying these notions, we investigate the relationship between purely metric properties of a metric space and the existence of a representation such that the metric is computable within bounded time. We show that a bound on the running time of the metric can be straightforwardly translated into size bounds of compact subsets of the metric space. Conversely, for compact spaces and for Banach spaces we construct a family of admissible, complete, regular representations that allow for fast computation of the metric and provide short encodings. Here it is necessary to trade the time bound off against the length of encodings.