The Computational Complexity of Measurable Functions and Sets
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| Název: | The Computational Complexity of Measurable Functions and Sets |
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| Autoři: | null Dr. Raj Kumar |
| Zdroj: | International Journal of Scientific Research in Science, Engineering and Technology. 12:469-499 |
| Informace o vydavateli: | Technoscience Academy, 2025. |
| Rok vydání: | 2025 |
| Popis: | This paper steps beyond a standard overview of classical measure and integration theory to set a new framework for the constructive study of measurable functions and sets. It makes a trailblazing contribution by officially bringing concepts from complexity and computability theory into the analysis of continuous analysis, and doing so by developing "recursively approximable sets" and "polynomial-time approximable functions." These definitions give fundamental questions of measurability and approximation a new, computationally based perspective. The work uncovers a profound and unexpected relationship between the two disciplines, showing a set to be recursively approximable if and only if it is recursively measurable (Theorem 21). Its strongest but most unexplored result is a negative result (Theorem 24), establishing the existence of a straightforward recursive function whose level set fails to be recursively approximable. This result is as strong as a fundamental open problem of discrete complexity theory and yields new insight into the interface between continuous and discrete computation. The results of the prework form the basis of a new area of computable analysis, and it paves the way for further research areas such as an axiomatic approach to computational complexity on Banach spaces and L*-spaces and a complete characterization of recursive functions in terms of the approximability of their level sets. |
| Druh dokumentu: | Article |
| ISSN: | 2394-4099 2395-1990 |
| DOI: | 10.32628/ijsrset2512531 |
| Rights: | CC BY |
| Přístupové číslo: | edsair.doi...........bc29b73c18d81e7ebde9e20e02d71f01 |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | This paper steps beyond a standard overview of classical measure and integration theory to set a new framework for the constructive study of measurable functions and sets. It makes a trailblazing contribution by officially bringing concepts from complexity and computability theory into the analysis of continuous analysis, and doing so by developing "recursively approximable sets" and "polynomial-time approximable functions." These definitions give fundamental questions of measurability and approximation a new, computationally based perspective. The work uncovers a profound and unexpected relationship between the two disciplines, showing a set to be recursively approximable if and only if it is recursively measurable (Theorem 21). Its strongest but most unexplored result is a negative result (Theorem 24), establishing the existence of a straightforward recursive function whose level set fails to be recursively approximable. This result is as strong as a fundamental open problem of discrete complexity theory and yields new insight into the interface between continuous and discrete computation. The results of the prework form the basis of a new area of computable analysis, and it paves the way for further research areas such as an axiomatic approach to computational complexity on Banach spaces and L*-spaces and a complete characterization of recursive functions in terms of the approximability of their level sets. |
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| ISSN: | 23944099 23951990 |
| DOI: | 10.32628/ijsrset2512531 |