A chain of inclusion relations in computable analysis

Abstract

Introduction. Within the subfield of the reals, the computable numbers, it is possible to develop an analysis with constructive definitions for all the usual concepts: sequence, convergent sequence, Cauchy sequence, function, uniformly continuous function, integral, etc. The Russian mathematicians Markov, Ceitin, Zaslavskil, Sanin, and others, have developed such an analysis in an extensive series of publications. In a recent article [1], we gave an exposition of an analysis which is equivalent to theirs, though differing in terminology (we use the same expressions as in real analysis), and in the means of construction. Our constructions are exclusively in terms of programs, a formally defined concept which seems especially suitable for the treatment of the computable numbers. To summarize briefly, if vl, v2, * * , vu are rational valued variables, and we permit a certain number of operations on these variables, called computation steps, including one which terminates activities, the halt step, then a program is given by a list of computation steps. If P is a program, P(a) denotes the value of the variable v1 after the program halts, when initially vi = a, V2=V3=. . . *=vu=O. If P does not halt P(a) is undefined. To every program P is assigned a unique positive integer Np, its descriptive integer. A rational valued function of rational variables is called programmable if it can be realized as a program. A computable number is defined by a certain type of programmable function, called a computable process. A key role in proving negative results is played by the programmable function U(n, m, a), which equals 1 if n=Np and P(a) is defined after the execution of no more than m computation steps; otherwise U(n, m, a) equals zero. For a description of the logic employed in computable analysis, we refer to an article by Sanin [5], which deals with this subject in detail. In [1] definitions were given of functions f of a computable number variable x, and certain types of these functions: pointwise continuous functions fp, uniformly continuous functions fu, and integrable functions fr. In addition, bounded functions fB and functions of bounded variation fBv may be defined in an obvious manner, and below we define functions of limit variation fLVy If a, b are any two fixed (computable) numbers with a<b, and we use brackets