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 approximation 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 computability. 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 computed. 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-Löf [6].

1 Some authors call these real numbers eﬀectively enumerable or computably enumerable. 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 computable real numbers. Interestingly, Weihrauch and Zheng [18]shownthatx is weakly computable iff there is a computable sequence (xs)s∈N of rational numbers 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 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|. Although we do not know exactly the current error |x−xn| itself, we do know that later errors are bounded by the current error some how. This requirement can be further weakened in the way that the constant c is 636 Robert Rettinger and Xizhong Zheng replaced by a function h whose value depends on the index n. Namely we can define the h-monotone convergence of the sequence (xs)s∈N to x by the condition that h(n)|x − xn|≥|x − xm| for any m ≥ n and define the h-mc real numbers accordingly. Especially we call a real number xω-mc if it is h-mc for some recursive function h.Theclassofω-mc real numbers is denoted by ω-MC.Of course, all c-mc real numbers are ω-mc. But we will show that there is an ω- mc real number which is not c-mc for any constant c. In fact we can prove that the classes ω-MC and WC are incomparable. Hence ω-MC is not covered completely by WC while all c-MC are contained in WC. Notice that, the classes E, LC, RC, WC and RA can be defined by the exis- tence of some computable sequences of rational numbers with some special properties which do not relate directly to their limits. These definitions are purely procedural. The classes c-MC and ω-MC, on the other hand, are defined in a different way. For example, the property that a sequence (xs)s∈N converges c- monoconically to x involves inevitably the limit x itself. These definitions are called abstract. Some classes, e.g. E and SC, can be defined either by procedural or abstract definitions. Classes ω-MC and WC supply a first example of incomparable classes which are defined by procedural and abstract definitions respectively. The outline of this paper is as follows. In the next section we will explain notion and notation we need and recall some related known results. The hierarchy theorem about c-mc classes is proved in Section 3. The last section discusses the relationship between the classes ω-MC and WC.

2 Preliminaries

In this section we explain some notions and notations which are needed for later sections and recall some related results. We suppose that the reader know the very basic notions and results of classic computability theory. But no knowledge from computable analysis is assumed. We denote by N, Q and R the sets of all natural, rational and real numbers, respectively. For any sets A and B, f :⊆ A → B is a partial function with dom(f) ⊆ A and range(f) ⊆ B.Iff is a total function, i.e., dom(f)=A,then it is denoted by f : A → B. The computability notions like computable (or recursive) function, recursive and r.e. (recursively enumerable) set, etc., on N are well defined and developed in classic computability theory. For example, the pairing function ·, · : N2 → N defined by m, n := (n + m)(n + m +1)/2+m is a computable function. Let π1,π2 : N → N be its inverse functions, i.e., π1 n, m = n and π2 n, m = m for any n, m ∈ N.Thenπ1 and π2 are obviously computable too. Let σ : N → Q be a coding function of Q using N defined by σ(n, m):=n/(m + 1). By this coding, the computability notions on N can be easily transferred to that of Q. For example, a function f :⊆ N → Q is computable if there is a computable function g :⊆ N → N such that f(n)= σ(g(n)) for any n ∈ dom(f), A ⊆ Q is recursive if {n ∈ N : σ(n) ∈ A} is Hierarchy of Monotonically Computable Real Numbers 637 recursive, a sequence (xn)n∈N of rational numbers is computable if there is a computable total function f : N → Q such that xn = f(n) for all n,andsoon. Frequently, we would like to diagonalize against all computable sequences of rational numbers or some subset of such sequences. In this case, an effective enumeration of all computable sequences of rational numbers would be useful. Unfortunately, the set of computable total functions are not effectively enumerable and hence there is no effective enumeration of computable sequences of rational numbers. Instead we consider simply the effective enumeration (ϕe)e∈N of all computable partial functions ϕe :⊆ N → Q. Of course, all computable sequences of rational numbers (i.e., total computable functions f : N → Q,more precisely) appear in this enumeration. Thus it suffices to carry out our diagonal- ization against this enumeration. Concretely, the enumeration (ϕe)e∈N can be defined from some effective enumeration (Me)e∈N of Turing machines. Namely, ϕe :⊆ N → Q is the function computed by the Turing machine Me.Further- more, let ϕe,s be the approximation of ϕe computed by Me until the stage s. Then (ϕe,s)e,s∈N is an uniformly effective approximation of (ϕe)e∈N such that {(e, s, n, r):ϕe,s(n) ↓= r} is a recursive set and ϕe,t(n) ↓= ϕe,s(n)=ϕe(n)for any t ≥ s,ifϕe,s(n) ↓= r,whereϕe,s(n) ↓= r means that ϕe,s(n) is defined and it has the value r. In the last section we have introduced the classes E, LC, RC, SC, WC and RA of real numbers. Some important properties about these classes are summarized again in the following theorem.

Theorem 1 (Weihrauch and Zheng [18]). 1. The classes E, LC, RC, SC, WC and RA are all different and have the following relationships E = LC ∩ RC SC = LC ∪ RC WC RA; 2. x ∈ WC iff there is a computable sequence (xs)s∈N of rational numbers such that the sum s∈N |xs − xs+1| is finite; and 3. The classes E, WC and RA are algebraic fields. That is, they are closed under the arithmetic operations +, −, × and ÷.

Now let’s give a precise deﬁnition of c-monotonically computable real numbers.

Definition 2 (Rettinger, Zheng, Gengler and von Braunmühl [7]). Let x be any real number and c ∈ R a positive constant. 1. A sequence (xn)n∈N of real numbers converges to xc-monotonically if the sequence converges to x and satisfies the following condition

∀n, m ∈ N (m ≥ n =⇒ c ·|xn − x|≥|xm − x|). (1)

2. The real number x is called c-monotonically computable (c-mc, for short) if there is a computable sequence (xn)n∈N of rational numbers which converges to xc-monotonically. The classes of all c-mc real numbers is denoted by c-MC. Furthermore, we denote also the union c∈R+ c-MC by ∗-MC.

The next proposition follows easily from the deﬁnition. 638 Robert Rettinger and Xizhong Zheng

Proposition 3. Let x be any real number and c1,c2,c positive constants. Then 1. If c1 ≤ c2 and x is c1-mc, then it is c2-mc too, i.e., c1-MC ⊆ c2-MC; 2. For any c<1, x is c-mc iﬀ it is computable, i.e, c-MC = E;and 3. x is 1-mc iﬀ it is semi-computable. Thus, SC =1-MC.

Some further results about c-mc real numbers are summarized in the next theorem.

Theorem 4 (Rettinger, Zheng, Gengler and von Braunm¨uhl [7]). 1. A c-mc real number is also weakly computable for any constant c.Butthere is a weakly computable real number which is not c-mc for any constant c. That is, ∗-MC WC. 2. For any constant c1, there is a constant c2 >c1 such that c1-MC c2-MC. Therefore there is an inﬁnite hierarchy of the class ∗-MC.

Notice that, for any class C ⊆ R discussed in this paper, x ∈ C iff x ± n ∈ C for any x ∈ R and n ∈ N. Therefore we can assume, without loss of generality, that any real number and corresponding sequence of rational numbers discussed in this paper is usually in the interval [0; 1] except for being pointed out explicitly otherwise. Here are some further notations: For any alphabet Σ,letΣ∗ and Σω be the sets of all finite strings and infinite sequences of Σ, respectively. For u, v ∈ Σ∗, denote by uv the concatenation of v after u.Ifw ∈ Σ∗ ∪ Σω,thenw[n] denotes its n-th element. Thus, w = w[0]w[1] ···w[n − 1], if |w|, the length of w,isn, and w = w[0]w[1]w[2] ···,if|w| = ∞. The unique string of length 0 is always denoted by λ (so-called empty string). For any finite string w ∈{0; 1}∗,and number n<|w|, the restriction w n is defined by (w n)[i]:=w[i]ifi

3 A Hierarchy Theorem of c-MC Real Numbers

From Theorem 4 of last section we know that there is an inﬁnite hierarchy on the c-mc real numbers. Unfortunately, this hierarchy shown in [7] is very rough. 2 In fact, it is only shown that, for any c1 ≥ 1, there is a, say, c2 := (c1 +8) such that c1-MC = c2-MC. In this section we will prove a dense hierarchy theorem on c-mc real numbers that c1-MC = c2-MC for any real numbers c2 >c1 ≥ 1. To this end we need the following technical lemmas.

Lemma 5. For any rational numbers c1,c2,a and b with 1

1. δ1 + δ2 +5" ≤ b − a; 2. c1(δ1 +2") <δ1 + δ2 + " 3. δ1 + δ2 +3" ≤ c2δ1;and 4. δ2 +2" ≤ c2δ2. Hierarchy of Monotonically Computable Real Numbers 639

Lemma 6. Let c1,c2,a,b∈ Q be any rational numbers with 1

a := aa:= a + "a:= a + "a:= a + " 0 2 1 4 3 6 5 (2) a1 := a + "a3 := a2 + δ1 a5 := a4 + δ2 a7 := b such that the following hold

x ∈ I1 & x1 ∈ I3 & x2 ∈ I5 =⇒ c1 ·|x − x1| < |x − x2|, and (3)  ∈ ∈ ∈  (x I1 & x1 I3 & x2 I5)  ∈ ∈ or (x I1 & x1,x2 I3) ⇒ ·| − |≥| − | ∈ ∈  = c2 x x1 x x2 . (4) or (x I1 & x1,x2 I5)  or (x ∈ I5 & x1,x2 ∈ I3)

Now we prove our hierarchy theorem at ﬁrst for the case of rational constants c1 and c2. The reason is, that we have to divide some interval I into subintervals according to Lemma 6. To make sure this procedure eﬀective, the interval I and numbers ai, hence also the numbers c1,c2,a,b, should be rational.

Theorem 7. For any rational numbers c1,c2 with 1

Proof. (sketch) Let c1,c2 be any rational numbers such that 1 ≤ c1

N :(xs)s∈N converges c2-monotonically to some x,and

Re :(ϕe(n))n∈N converges c1-monotonically to ye =⇒ x = ye.

The strategy to satisfy a single requirement Re is an application of Lemma 6. Let a := 0,b := 1. We fixat first three rational numbers ", δ1,δ2 which satisfy the conditions 1. – 4. of Lemma 5.Furthermore,fori<7, let ai be defined by (2)andIi := [ai; ai+1] whose length is denoted by li := ai+1 − ai.Wedenote by Io the open interval which consists of all inner points of I for any interval I ⊆ R. We take the interval (0; 1) as our base interval and will try to find out a so-called witness interval Iw ⊆ (0; 1) such that any x ∈ Iw satisfies the require- o ment Re.LetI3 be our first (default) candidate of the witness interval. If no element of the sequence (ϕe(n))n∈N appears in this interval, then it is automati- cally a correct witness interval, since the limit limn→∞ ϕe(n), if exists, will not be ∈ N ∈ o in this interval. Otherwise, if there are some s1,n1 such that ϕe,s1 (n1) I3 , o then we choose I5 as our new candidate of witness interval. In this case we denote n1 by ce(s1). Again, if there is no n2 >n1 such that ϕe(n2) comes into this 640 Robert Rettinger and Xizhong Zheng interval, then any element from this interval witnesses the requirement Re.Oth- ∈ o erwise, if ϕe,s2 (n2) I5 for some s2 >s1 and some ce(s2):=n2 >n1 = ce(s1), o then we choose the interval I1 as the new candidate of the witness interval. By the implication (3) of Lemma 6, this interval turns out to be really a correct | − | | − | ∈ o witness interval, since c1 x ϕe(n1) < x ϕe(n2) , for any x I1 , and hence (ϕe(n))n∈N does not converge to xc1-monotonically. To satisfy all requirements Re simultaneously, we implement the above strate- gies for different requirements Re on different base intervals. More precisely, we need a tree of intervals on which the above strategy can be implemented. Let I be the set of all rational subintervals of [0; 1] and Σ7 := {0, 1, ···, 6}. We define I ∗ → I ∈ ∗ atreeI on as a function I : Σ7 such that, for all w Σ7 , I(w):=[aw; bw] for aw,bw ∈ [0; 1] defined inductively by (aλ := 0,awi := aw + ai · lw)and (bλ := 1,bwi := aw + ai+1 · lw), where ai is defined by (2)andlw is the length of interval I(w) and can be defined inductively by lλ := 1 and lwi := lw · li.Notice ∈ ∈ ∗ | | that we do not distinguish ai for i Σ7 from ai for i Σ7 with i = 1, because their value are simply the same. ∈ ∗ o Now for any w Σ7 of length e,theintervalsI (w) are reserved exclusively o for the base intervals of the requirement Re. Suppose that I (w) is a current o o base interval for Re. We will try to find some subinterval I (wi) ⊂ I (w), for some i ∈{1, 3, 5},asawitnessintervalforRe such that any real number of this interval witnesses the requirement Re. More precisely, the limit limn→∞ ϕe(n), if exists, will not be in the interval I(wi), if the sequence (ϕe(n))n∈N converges c1- o monotonically. At the same time we take this witness interval I (wi)ofRe as our current base interval for the requirement Re+1 and will try to find out a o o subinterval I (wij) ⊂ I (wi)forj ∈{1, 3, 5} as a witness interval of Re+1, o o and so on. This means that, if I (w1)andI (w2) are witness intervals for Ri1 o ⊂ o and Ri2 , respectively, and i1 >i2,thenI (wi2 ) I (wi1 ). Thus, the sequence of all witness intervals for all requirements form a nested interval sequence whose common point x satisfies all requirements Re for e ∈ N. Of course, the choices of base and witness intervals have to be corrected continuously according to the behaviors of the sequences (ϕe,s(n))n∈N for different s ∈ N. The choice of the witness intervals for the requirements R0,R1, ···,Re−1 corresponds to a string w ∈{1, 3, 5}∗ of length e.Namely,foranye<|w|, o the interval I (w (e + 1)) is the base interval for Re+1 and at the same time the witness interval of requirement Re.Wedenotebyws our choice of such string at stage s, which seems correct at least for the s-th approximation sequences (ϕe,s(n))n∈N instead of (ϕe(n))n∈N for all e<|ws|.Asthe ω limit, w := lims→∞ ws ∈ Σ describes a correct sequence of witness intervals (I(w e))e∈N for all requirements (Re)e∈N, i.e., xw is not c1-mc. Let xs := aws3 for any s ∈ N. Then lims→∞ xs = xw. By 2.–3. of the Lemma 5 and the condition (4), the sequence (xs)s∈N converges in fact c2-monotonically to xw.

By the density of Q in R, our hierarchy theorem for the real constants c1 and c2 follows immediately from Theorem 7 Hierarchy of Monotonically Computable Real Numbers 641

Corollary 8 (Hierarchy Theorem). For any real numbers c1,c2 with 1 ≤ c1

4 ω-Monotone Computability and Weak Computability

In this section we will extend the c-monotone computability of real numbers to a more general one, namely, ω-monotone computability. We discuss the relationships between this computability and the weak computability discussed in [1] and show that they are in fact incomparable in the sense that ω-MC ⊆ WC and WC ⊆ ω-MC.

Deﬁnition 9. Let h : N → Q be any total function. 1. A sequence (xs)s∈N of real numbers converges to xh-monotonically if the sequence converges to x and the following condition holds

∀n ∀m ≥ n (h(n)|x − xn|≥|x − xm|). (5)

2. A real number x is called h-monotonically computable (h-mc, for short) if there is a computable sequence of rational numbers which converges to xh- monotonically. The class of all h-mc real numbers is denoted by h-MC. 3. A real number x is called ω-monotonically computable (ω-mc, for short) if it is h-monotonically computable for some recursive function h.Theclassof all ω-mc real numbers is denoted by ω-MC.

By deﬁnition, a c-mc real number is h-mc for the constant function h ≡ c. Therefore c-mcrealnumbersarealsoω-mc. But we will see that not every ω-mc real number is a c-mc for some constants c. From Theorem 4, we know that not every weakly computable real number is a c-mc real number for some constant c. Next theorem shows that even the class of ω-mc real numbers does not cover the class of weakly computable real numbers.

Theorem 10. There is a weakly computable real number x which is not ω-mc.

Proof. (sketch) Let (ϕi)i∈N be an effective enumeration of computable functions ϕi :⊆ N → Q .(ϕi,s)s∈N is the uniformly effective approximation of ϕi. We will construct effectively a computable sequence (xs)s∈N of rational numbers which converges to x such that n∈N |xn − xn+1|≤c,forsomec ∈ N,andx satisfies, for all e := i, j∈N, the following requirements ϕi,ϕj are total, lim ϕi(n)=yi exists and Re : n→∞ =⇒ x = yi (6) ∀n∀m ≥ n (ϕj(n) ·|yi − ϕi(n)|≥|yi − ϕi(m)|)

The strategy for satisfying a single requirement Re is as follows. Fixa nonempty interval (a,b) as default “witness” interval and wait for t1,t2 ∈ N with t1 natural number such that lims→∞ ds = ∞.These intervals should be nested in the sense that (ae+1,s; be+1,s) ⊂ (ae,s; be,s). As witness interval for Re,(ae,s; be,s) seems “correct” at least for the approximation sequences (ϕi,s(n))n∈N (e = i, j) instead of the sequence (ϕi(n))n∈N itself. Of course we have to correct continuously our approximations according to the behaviors of (ϕi,s(n))n∈N. To this end, a priority argument is applied.

We say that Re has a higher priority than Re1 if ee, i.e., the requirement Re is injured at this stage. The witness intervals for Re has to be redefined at a later stage again. On the other hand, if some correct witness interval I for Re is defined by the above strategy, then I witnesses the requirement Re and we don’t need to do anything more for Re unless it is destroyed again. To avoid any unnecessary multiple action, we will set the requirement Re into the state of “satisfied” if it receives attention. If it is injuried, then set it back to the state “unsatisfied”. Only the requirement of state “unsatisfied” can receive attention. In this way we guarantee that any e requirement Re can be injured at most 2 − 1 times and can receive attention e at most 2 times. This means also that the limit (ae; be) := lims→∞(ae,s; be,s) exists and it is a correct witness interval for Re. At any stage s, we define xs to be the middle point of the smallest witness interval defined at stage s. This guarantees that xs locates in all current defined witness intervals. On the other hand, we choose the interval (ae,s; be,s)small −2e enough that its length not longer as, say, 2 .Thenwehavealsothatbe − ae ≤ −2e 2 , hence the nested interval sequence ((ae; be))e∈N has a unique common Hierarchy of Monotonically Computable Real Numbers 643 point x which is in fact the limit lims→∞ xs. The limit x satisfies all requirement Re, hence it is not ω-mc. Furthermore, by the construction, we have | − |≤ −2e ∈ xs xs+1 2 if Re receives attention at stage s + 1, because xs,xs+1 −2e e (ae,s; be,s). This implies that s∈N |xs − xs+1|≤ e∈N 2 · 2 ≤ 2sinceRe receives attention at most 2e times. This means that x is weakly computable.

By Theorem 4,anyc-monotonically computable real number is also weakly computable. Next theorem shows that this is not the case any more for ω-mc real number.

Theorem 11. There is an ω-mc real number which is not weakly computable.

Proof. (sketch) Let (ϕi)i∈N be an effective enumeration of computable partial functions ϕi :⊆ N → Q and (ϕi,s)s∈N its uniform approximation. We will define a computable sequence (xs)s∈N of rational numbers and a recursive function h : N → N such that the limit x := lims→∞ xs exists and satisfies, for all i ∈ N, the following requirements

N :(xs)s∈N converges to xh-monotonuically, and Ri : |ϕi(s) − ϕi(s +1)|≤i & yi := lim ϕi(s)=⇒ yi = x. s∈N s∈N

Since every sequence (ϕi(s))s∈N have infinitely many different indices i by the Padding Lemma, the real number x can’t be weakly computable, if all requirements Ri are satisfied. The strategy to satisfy a single requirement Ri is quite straightforward. We take the interval [0; 1] as the base interval and define at ≤ first x0 := 1/7. If there is some m1 and s1 such that ϕi,s1 (m1) 3/7, then define xs1 := 5/7. Again, at stage s2 >s1,ifthereisanm2 >m1 such that ≥ ϕi,s2 (m2) 4/7, then define xs2 := 1/7, and so on. Otherwise, xs+1 takes simply the same value as xs.Now,if s∈N |ϕi(s) − ϕi(s +1)|≤i,thenxs can be redefined at most 7 · i times (suppose that ϕi(s) ∈ [0; 1]) and the last value x := lims→∞ xs has at least the distant 1/7fromyi := lims→∞ ϕi(s), if this limit exists. Of coures the sequence (xs)s∈N defined in this way does not satisfy the requirement N, because some elements xs of the sequence are simply equal to the limit x which destroies the monotone convergence. Fourtunately, this problem disappears if we implement a stragegy to satisfy all requirement Ri simultan- ∗ → I uously. In this case, we need an interval tree I : Σ7 whichisdefinedby −(i+1) −|w| I(w):=[aw; bw], where aw := i<|w| w[i] · 7 and bw := aw +7 ,for ∈ ∗ ∈ ∗ all w Σ7 .TheintervalI(w)forw Σ7 of length i is reserved for the requirement Ri. We will define xs := aw1 or xs := aw5 in order to guarantee that −(i+1) the limit yi := lims→∞ ϕi(s), if exists, has at least the distant 7 from xs, hence the limit lims→∞ xs is also different from yi. On the other hand, this construction guarantees also that the element xs defined at stage s for satisfying Ri −(i+2) has at least a distant 7 from the limit lims→∞ xs. Therefore the sequence (i+2) (xs)s∈N converges h-monotonically for h(s):=7 . 644 Robert Rettinger and Xizhong Zheng

Since every c-mc real number is weakly computable, the Theorem 11 implies that there is an ω-mc real number which is not c-mc for any constant c, i.e., ∗-MC ω-MC. Besides, Theorem 11 and Theorem 10 implies directly the following corollary. Corollary 12. The classes of weakly computable and ω-monotonically computable real numbers are incomparable, i.e., WC ⊆ ω-MC & ω-MC ⊆ WC.

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