Algorithmic questions for real functions

Abstract

Turing machines, in particular, finite Mill automata can be considered as a means for giving corresponding subclasses of the class of real functions. From this point of view, finite Mili automata and generalized sequences of machines were considered in [I]. Here a real function f is defined by a finite automaton A in the following way. To determine the value f(x) of the automaton A one successively transforms the input object, the expansion of the real number x in the binary (decimal) system, into the output object, which is the binary (decimal) expansion of the number y, where y = f(x). Analogously for determining real functions one can use Turing machines. The subclass of the class of real functions arising here is actually a proper subclass of the class of C-computable functions, defined below. The latter is strictly included in the class of R-functions, or real functions, and that, in its own right, is contained in the class of V-functions. Actually, each continuous real functions turns out to be a real function. V-functions are given by indeterminate R-transformers, transforming binary representations of real numbers. R-transformers with finite memory, defining real functions are said to be finite reals. The latter are richer than Mill automata, defining real functions according to [I], in two respects: they can be asynchronous and can define functions which are not strictly real. For example, the function f(x) = 3x is defined by a finitely real, but is not strictly real. For finite reals, on the basis of [2] the solvability of the equivalence problem and of some other algorithmic questions is proved. This generalizes the corresponding results found in [i] for finite Mili automata and generalized succcession machines preserving length. The solvability is also established of the equivalence problem for single-valued indeterminate finite reals with finitely turning accumulators. An algebraic characterization is given of the class of partial C-computable functions. We note that in the papers of Red'ko [3, 4] the problem of describing the algebra of computable (partially computable) functions of a real variable was formulated as open. Bui effected the construction of the algebra of computable operations on the set of rational numbers [5].