Decision times of infinite computations

TL;DR

This paper investigates the supremum of decision and semidecision times for sets of reals in the context of infinite computations, linking these to definable ordinals within the constructible universe.

Contribution

It determines the exact supremum of countable decision and semidecision times as specific definable ordinals, extending understanding of infinite computational complexity.

Findings

Supremum of countable decision times is σ, the supremum of Σ₁-definable ordinals.

Supremum of countable semidecision times is τ, the supremum of Σ₂-definable ordinals.

Exact values are computed for singletons as well.

Abstract

The decision time of an infinite time algorithm is the supremum of its halting times over all real inputs. The decision time of a set of reals is the least decision time of an algorithm that decides the set; semidecision times of semidecidable sets are defined similary. It is not hard to see that $\omega_1$ is the maximal decision time of sets of reals. Our main results determine the supremum of countable decision times as $\sigma$ and that of countable semidecision times as $\tau$, where $\sigma$ and $\tau$ denote the suprema of $\Sigma_1$- and $\Sigma_2$-definable ordinals, respectively, over $L_{\omega_1}$. We further compute analogous suprema for singletons.