Hierarchy of Monotonically Computable Real Numbers Extended Abstract Robert Rettinger1 and Xizhong Zheng2 1 FernUniversit¨at Hagen 58084-Hagen, Germany 2 BTU Cottbus 03044 Cottbus, Germany [email protected] Abstract. Arealnumberx is called h-monotonically computable (h- h xs mc), for some function , if there is a computable sequence ( )s∈ Æ of rational numbers such that h(n)|x − xn|≥|x − xm| for any m ≥ n. x is called ω-monotonically computable (ω-mc) if it is h-mcfor some recursive x c h function h and, for any c ∈ Ê, is -mcif it is -mcfor the constant function h ≡ c. In this paper we discuss the properties of c-mcand ω-mc real numbers. Among others we will show a hierarchy theorem of c-mc real numbers that, for any constants c2 >c1 ≥ 1, there is a c2-mcreal number which is not c1-mcand that there is an ω-mcreal number which ω is not c-mcfor any c ∈ Ê. Furthermore, the class of all -mcreal numbers is incomparable with the class of weakly computable real numbers which is the arithmetical closure of semi-computable real numbers. 1 Introduction The effectiveness of a real number x are usually described by a computable sequence (xs)s∈N of rational numbers which converges to x with some further restrictions on the error-estimation of the approximation. The optimal situation is that we have full control over errors of the approximation. Namely, there is −e(s) an unbounded recursive function e : N → N such that |x − xs|≤2 for any s ∈ N, or equivalently, there is a computable sequence (xs)s∈N of rational −n numbers which converges to x effectively in the sense that |xn − xm|≤2 if n ≤ m. Such real numbers are called by A. Turing [16] computable.Theclassofall computable real numbers is denoted by E. There are a lot of equivalent ways to characterize this class E. In fact, corresponding to any classical definition of real numbers in mathematics there is an effectivization which induces an equivalent definition of the computable real numbers (see [9,8,17]). Typically, besides the fast converging Cauchy sequence definition mentioned above, computable real numbers can also be described by effective version of Dedekind cuts, binary or decimal expansions, and so on. Corresponding author J. Sgall, A. Pultr, and P. Kolman (Eds.): MFCS 2001, LNCS 2136, pp. 633–644, 2001. c Springer-Verlag Berlin Heidelberg 2001 634 Robert Rettinger and Xizhong Zheng The optimal error-estimation of an approximation to some real number is not always available. E. Specker [15] has found an easy example of such real numbers. Let (As)s∈N be an effective enumeration of a non-recursive r.e. set A ⊆ N.Thatis,(As)s∈N is a computable sequence of finite sets of natural ⊂ −(i+1) numbers such that As As+1 and A = s∈N As. Define xA := i∈A 2 −(i+1) s s s∈N and x := i∈As 2 .Then(x ) is a computable sequence of rational numbers which converges to xA. But an effective error-estimation for this ap- proximation is impossible, because xA is not a computable real number (it has a non-computable binary expansion). Namely, the sequence (xs)s∈N converges to x non-effectively. On the other hand, since (xs)s∈N is an increasing sequence, we get always better and better approximations xs to x with increasing index s. Such kind of real numbers have the effectiveness only weaker than that of the computable real numbers and thay are called left computable1.Theclassofall left computable real numbers is denoted by LC.TheclassLC has been widely discussed in literature (see e.g., [1,2,3,12,13,18]). Unlike the case of computable real numbers, effectivizations of classical definitions of real numbers (to the level of recursive enumerabiliy) do not induce always the same notions of left com- putability. Let’s say that x ∈ N has a r.e. binary expansion if there is a r.e. set A ⊆ N such that x = xA. Then, C.G. Jockush (cf. [12]) pointed out that the 2 real number xA⊕A¯ has a r.e. Dedekind cut but it has no r.e. binary expansion, if A ⊆ N is a non-recursive r.e. set, since A ⊕ A¯ is not r.e. In fact, as shown by Specker [15], the effectivizations of classical definitions of real numbers to the level of primitive recursiveness are also not equivalent. Left computable real numbers are not equally difficult (or easy) to be com- puted. Solovay [14] introduced a “domination relation” on the class LC to com- pare them. For any x, y ∈ LC, x dominates y if for any computable increasing sequence (ys)s∈N of rational numbers which converges to y, there is a computable increasing sequence (xs)s∈N of rational numbers which converges to x such that the sequence (xs)s∈N dominates (ys)s∈N in the sense that c(x − xs) ≥ y − ys for some constant c and all s ∈ N.Intuitively,ifx dominates y,thenitisnotmore difficult to approximate y than x,thusx contains at least so much information as y. If a left computable x dominates all y ∈ LC,thenx is called Ω-like by Solovay. Solovay shows that the Chaitin Ω numbers (i.e., the halting probability of a universal self-delimiting Turing machine, cf [5]) are in fact Ω-like. That is, the Chaitin Ω numbers are, in some sense, the most complicated left computable real numbers. This is also confirmed by a nice result, which is proved in stages by Chaitin [5], Solovay [14], Calude, Hertling, Khoussainov and Wang [3]and Slaman [10](seealso[2] for a good survey), that a real number x is Ω-like iff it is a Chaitin Ω number and iff it is a left computable random real number, where x is random means that its binary expansion is a random sequence of {0, 1}ω in the sense of Martin-L¨of [6]. 1 Some authors call these real numbers effectively enumerable or computably enumer- able. See e.g. [2,3,12,13] 2 A ⊕ B := {2n : n ∈ A}∪{2n +1:n ∈ B} is the join of the sets A and B and A¯ is the complement of set A. Hierarchy of Monotonically Computable Real Numbers 635 The arithmetical closure of the left computable real number class LC (under the operations +, −, ×, ÷) is denoted by WC, the class of so-called weakly com- putable real numbers. Interestingly, Weihrauch and Zheng [18]shownthatx is weakly computable iff there is a computable sequence (xs)s∈N of rational num- bers which converges to x weakly effectively in the sense that the sum of the jumps s∈N |xs − xs+1| is finite. The class WC is strictly between LC and the class RA of recursively approximable real numbers which are simply the limits of some computable sequences of rational numbers. Symmetrically, we define also the class RC of right computable real numbers by the limits of computable decreasing sequences of rational numbers. Left and right computable real numbers are all called semi-computable whose class is denoted by SC := LC ∪ RC. The semi-computability can also be characterized by the monotone convergence of the computable sequences (see [7]). A sequence (xs)s∈N converges to x monotonically means that |x − xn|≥|x − xm| holds for any m ≥ n. Then, x is semi-computable iff there is a computable sequence (xs)s∈N of rational numbers which converges to x monotonically.Noticethat, if a sequence converges monotonically, then a later element of the sequence is always a better approximation to the limit. However, since the improvement of the approximation changes during the procedure, it is still impossible to give an exact error estimation. More generally, Calude and Hertling [4] discussed c-monotone convergence for any positive constant c in the sense of c|x − xn|≥|x − xm| for any m ≥ n. They have shown that, although there are computable sequences of rational numbers which converge to computable real numbers very slowly, c-convergent computable sequences converge always fast, if their limits are computable. It is easy to see that, for different constants c, c-monotonically convergent sequences converge in different speeds. More precisely, if (xs)s∈N and (ys)s∈N converge c1- and c2-monotonically, for c1 <c2,tox and y, respectively, then, (xs)s∈N seems to be a better approximation to x than (ys)s∈N to y. Let’s call a real number x c-monotonically computable (c-mc for short) if there is a computable sequence of rational numbers which converges to xc-monotonically and denote by c-MC the class of all c-mc real numbers. It is not difficult to see that 1-MC = SC, c-MC = E if c<1andc1-MC ⊆ c2-MC if c1 ≤ c2. Furthermore, it is also shown in [7]thatc-mc real numbers are weakly computable for any constant c, but there is a weakly computable real number which is not c-mc for any constant c, i.e., c∈R+ c-MC WC. To separate the different c-mc classes, only a very rough hierarchy theorem was shown in [7] that, for any constant c1,thereisac2 >c1 such that c1-MC = c2-MC. However, it remains open in [7]whetherc1-MC = c2-MC holds for any different c1 and c2. A positive answer will be given in this paper (Theorem 7). For any n ∈ N and any sequence (xs)s∈N which converges c-monotonically to x, we know that the error of the approximation xm, for any m ≥ n,isalways bounded by c|x−xn|.