Abstract
Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies \(\varDelta ^0_2\) sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice L (e.g., the real interval \([0; 1]_\mathbb {R}\)). In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. In particular, we focus on the fuzzy n-c.e. sets which form the finite levels of this hierarchy. Intuitively, a fuzzy set is n-c.e. if its membership function can be approximated by changing monotonicity at most \(n-1\) times. We prove that the Fuzzy Ershov Hierarchy does not collapse; that, in analogy with the classical case, each fuzzy n-c.e. set can be represented as a Boolean combination of fuzzy c.e. sets; but that, contrary to the classical case, the Fuzzy Ershov Hierarchy does not exhaust the class of all \(\varDelta ^0_2\) fuzzy sets.
The work was supported by Nazarbayev University Faculty Development Competitive Research Grants 021220FD3851. Bazhenov and Ospichev were supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. San Mauro has been partially supported by the Austrian Science Fund FWF, project M 2461.
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Acknowledgements
The authors are grateful to Professor Murat Ramazanov for his hospitality during their visit to Qaraghandy, Kazakhstan, in June 2019. Part of the research contained in the paper was carried out while Bazhenov, San Mauro, and Ospichev were visiting the Department of Mathematics of Nazarbayev University, Nur-Sultan.
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Bazhenov, N., Mustafa, M., Ospichev, S., San Mauro, L. (2021). Approximating Approximate Reasoning: Fuzzy Sets and the Ershov Hierarchy. In: Ghosh, S., Icard, T. (eds) Logic, Rationality, and Interaction. LORI 2021. Lecture Notes in Computer Science(), vol 13039. Springer, Cham. https://doi.org/10.1007/978-3-030-88708-7_1
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