Kolmogorov complexity and characteristic constants of formal theories of arithmetic |
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Authors: | Shingo Ibuka Makoto Kikuchi Hirotaka Kikyo |
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Institution: | 1. Department of Computer Science and Systems Engineering, Kobe University, 657‐0059, Kobe, Japan;2. Department of Information Science, Kobe University, 657‐0059, Kobe, Japan |
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Abstract: | We investigate two constants c T and r T , introduced by Chaitin and Raatikainen respectively, defined for each recursively axiomatizable consistent theory T and universal Turing machine used to determine Kolmogorov complexity. Raatikainen argued that c T does not represent the complexity of T and found that for two theories S and T , one can always find a universal Turing machine such that $c_\mathbf {S}= c_\mathbf {T}$. We prove the following are equivalent: $c_\mathbf {S}\ne c_\mathbf {T}$ for some universal Turing machine, $r_\mathbf {S}\ne r_\mathbf {T}$ for some universal Turing machine, and T proves some Π1‐sentence which S cannnot prove. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim |
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Keywords: | Characteristic constant Kolmogorov complexity MSC (2010) 03A99 03B25 |
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