Hierarchies of function classes defined by the first-value operator

Hemmerling, Armin

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 253-270.

The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov’s one within ${\Delta }_2^0$. The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff’s difference hierarchy in the Borel class ${\Delta }_2^B$. Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.

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