Real Recursive Functions and Baire Classes

• Authors: Jerzy Mycka
• Venue: Fundamenta Informaticae
• Year: 2005

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.