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DOI: 10.6084/m9.figshare.13686439
Shagrir & Pitowsky (2003) present a device named HM, and they defend that “HM is a physical system that computes the function h”, where “h characterizes the self-halting states of Turing machines.”
If that were true, then HM would falsify Church's thesis and, as I agree completely with Post (1936) in that Church’s thesis is a natural law stating the calculating limitations of the members of our own species Homo sapiens, then I suspected that they had overlooked something so I determined to investigate it. The last paragraph summarizes my conclusion.
Contrary to Shagrir & Pitowsky’s “contention that HM is a physical system that computes the function h” (page 88), our analysis shows that, either HM is not a physical system, but an idealization that ignores its errors, or, if we take into account its errors, then HM does not compute the function h.
Shagrir & Pitowsky's idealization abstracts away errors, and thus they do not discuss how badly errors affect infinite computations. Andréka et al. (2018) address this issue, which they resolve by using an unbounded fleet of robots, that is, by using an idealization, too, though now one that abstracts away energy and other resources. In both cases, my argument is that an idealization cannot falsify a law of nature, because only observations and measurements can falsify a law of nature, see Subjectivist Propaganda, in this case the law of Post (1936) also known as Church's thesis, see Proof of Church's Thesis.
When the error rate is not absolutely zero, infinite computations are necessarily erroneous, rendering them uninformative. This restricts the practical value of those hypercomputers that perform infinite computations.
Link to the page of my paper on “Errors in Infinite Computations” in figshare, and direct link to the pdf file.
These are the external references used in his page:
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Última actualización: 2023-03-01.
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