Decision times of infinite computations

Merlin Carl; Philipp Schlicht; Philip Welch

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arXiv · 2011.04942

Decision times of infinite computations

Merlin Carl·Philipp Schlicht·Philip Welch

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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.

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BibTeX RIS

Merlin Carl, Philipp Schlicht, Philip Welch. 2020-11-10. Decision times of infinite computations. https://arxiv.org/abs/2011.04942

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