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Keywords = analog of Rice’s theorem

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12 pages, 271 KiB  
Article
On Productiveness and Complexity in Computable Analysis Through Rice-Style Theorems for Real Functions
by Jingnan Xie, Harry B. Hunt and Richard E. Stearns
Mathematics 2024, 12(20), 3248; https://doi.org/10.3390/math12203248 - 17 Oct 2024
Cited by 1 | Viewed by 975
Abstract
This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep [...] Read more.
This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep reexamination of Hilbert’s tenth problem and the equivalence to the identically 0 function problem, extending the undecidability results of these problems into the realm of productiveness. Additionally, we study the complexity of the equivalence to the identically 0 function problem over different domains. We then construct highly efficient many-one reductions to establish Rice-style theorems for the study of real functions. Specifically, we show that many predicates, including those related to continuity, differentiability, uniform continuity, right and left differentiability, semi-differentiability, and continuous differentiability, are as hard as the equivalence to the identically 0 function problem. Due to their high efficiency, these reductions preserve nearly any level of complexity, allowing us to address both complexity and productiveness results simultaneously. By demonstrating these results, which highlight a more nuanced and potentially more intriguing aspect of real function theory, we provide new insights into how various properties of real functions can be analyzed. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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