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Hypermachines. (English) Zbl 1220.03040
Summary: The Infinite Time Turing Machine model [J. Symb. Log. 65, No. 2, 567–604 (2000; Zbl 0963.03064)] of J. D. Hamkins and A. Lewis is, in an essential sense, a “$$\Sigma _{2}$$-machine” in that it uses a $$\Sigma _{2}$$ Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate $$\Sigma _{n}$$ rule. Such machines either halt or enter an infinite loop by stage $$\zeta (n) =_{\text{df}} \mu \zeta (n) [\exists \Sigma (n) > \zeta (n) L_{\zeta (n)} \prec _{\Sigma_n} L_{\Sigma (n)}]$$, again generalising precisely the ITTM case. The collection of such machines taken together computes precisely those reals of the least model of analysis.

##### MSC:
 03D10 Turing machines and related notions 03D65 Higher-type and set recursion theory 03E45 Inner models, including constructibility, ordinal definability, and core models
##### Keywords:
transfinite time computation
Zbl 0963.03064
Full Text:
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