Decidability problems for meta-R-functions

Abstract

In this article we consider classes of functions defined by R-transformers, which are machines that sequentially process real numbers represented in the binary number system. The class of real functions defined by R-transformers includes all continuous and some discontinuous functions. The closure of this class under superposition produces the wider class of real meta-R-functions. Such functions are defined by finite sequences of R-transformers. Here we examine the class of meta-R-meta-R-functions. Such functions are defined by finite sequences of R-transformers. Here we examine the class of meta-R-functions and specifically the class of finite meta-R-functions. The latter are defined by finite sequences of finite R-transformers. Decidability of the equivalent problem in the class of functions defined by finite R-transformers, i.e., the class of finite R-functions, is proved elsewhere. Here we generalize this result to the class of finite meta-R-functions. We investigate not only the equivalence problem, but also the monotonicity and continuity problems. The proof is by reduction to the decidable nonemptiness problem for nondeterministic bounded-mode finite transformers with finite-turnaround counters on labeled trees. We also consider the general properties of the class of meta-R-functions.