{"slug": "the-arithmetic-hierarchy-of-real-functions", "title": "The arithmetic hierarchy of real functions", "summary": "Marcus Hutter and an author published an accessible introduction to real hypercomputation in the Journal of Computer and System Sciences, focusing on applications to algorithmic information theory and building foundations for AIXI research. The work was supported by the Long-Term Future Fund (LTFF) and includes an errata note about misnumbered theorems.", "body_md": "I wrote a fairly accessible [introduction to real hypercomputation](https://www.sciencedirect.com/science/article/pii/S0890540126000738) with Marcus Hutter. The focus is on enabling applications to algorithmic information theory. This project was intended to build my technical foundations for studying AIXI, but took me a bit further afield and down some rabbit holes. In the future I will prefer to focus more tightly on AI safety.\n\nFeedback would be appreciated. In particular, I needed to introduce an extra extensionality assumption for the real domain case, which I am still not sure is necessary.\n\n**Errata:** The diagram of results currently has theorems misnumbered due to a typographical error.\n\nThanks to the LTFF for supporting my work over most of the research process.", "url": "https://wpnews.pro/news/the-arithmetic-hierarchy-of-real-functions", "canonical_source": "https://www.lesswrong.com/posts/kLiJnEfavGAxXfiNr/the-arithmetic-hierarchy-of-real-functions", "published_at": "2026-06-28 19:11:19+00:00", "updated_at": "2026-06-28 23:20:54.452325+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "ai-safety", "ai-agents", "machine-learning"], "entities": ["Marcus Hutter", "AIXI", "Long-Term Future Fund", "LTFF", "Journal of Computer and System Sciences"], "alternates": {"html": "https://wpnews.pro/news/the-arithmetic-hierarchy-of-real-functions", "markdown": "https://wpnews.pro/news/the-arithmetic-hierarchy-of-real-functions.md", "text": "https://wpnews.pro/news/the-arithmetic-hierarchy-of-real-functions.txt", "jsonld": "https://wpnews.pro/news/the-arithmetic-hierarchy-of-real-functions.jsonld"}}