Real Recursive Functions and Baire Classes

Article type: Research Article

Authors: Mycka, Jerzy

Affiliations: Institute of Mathematics, M. Curie-Sklodowska University, pl. M. Curie-Sklodowskiej 1, 20-031 Lublin, Poland. [email protected]

Note: [] Address for correspondence: Institute of Mathematics, UMCS, pl. M. Curie-Sklodowskiej 1, 20-031 Lublin, Poland

Abstract: Recursive functions over the reals [6] have been considered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes [2]. However, one of the operators introduced in the seminal paper by Cris Moore (in 1996), the minimalization operator, creates some difficulties: (a) although differential recursion (the analog counterpart of classical recurrence) is, to some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper we use the most natural operator captured from Analysis - the operator of taking a limit - instead of the minimalization with respect to the equivalance of these operators given in [8]. In this context the natural question about coincidence between real recursive functions and Baire classes arises. To solve this problem the limit hierarchy of real recursive funcions is introduced. Relations between Baire classes, effective Baire classes and the limit hierachy are also studied.

Keywords: theory of computation, real recursive functions, hierarchy of real functions

Journal: Fundamenta Informaticae, vol. 65, no. 3, pp. 263-278, 2005