Computability, Logic, AI Algorithms
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It is proved that the sets of integers semi-computable this way are precisely those Σ 1 -definable over L α I and L σ 1 1 , respectively, where α I denotes the least recursively inaccessible ordinal and σ 1 1 the least Σ 1 1 -reflecting ordinal.
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Feedback computability studies oracles that provide convergence and divergence information about computability relative to that same oracle. We consider strict and loose forms of feedback computability relative to Kleene’s 2 E , the Type-2 object embodying quantification over natural numbers. We prove that the sets of integers semi-computable this way are precisely those Σ 1 -definable over L α I and L σ 1 1 , respectively, where α I denotes the least recursively inaccessible ordinal and σ 1 1 the least Σ 1 1 -reflecting ordinal. Furthermore, our construction of higher-type feedback computability through ordinal inductive fix-points serves as a natural and general framework for the study of feedback computability.
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@article{Aguilera2026Type,
title = {Type-2 feedback computability},
author = {JP Aguilera and M Soto},
journal = {Computability},
year = {2026},
doi = {10.1177/22113568251320077},
url = {https://doi.org/10.1177/22113568251320077}
}FAQ
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