A Finite Hierarchy of the Recursively Enumerable Real Numbers
A Finite Hierarchy of the Recursively Enumerable Real Numbers Klaus Weihrauch and Xizhong Zheng? Theoretische Informatik I, FernUniversit¨ at Hagen, 58084 Hagen, Germany
Abstract. For any set A of natural numbers, denote by xA the corresponding real number such that A is just the set of “1” positions in its binary expansion. In this paper we characterize the number xA for some classes of recursively enumerable sets A. Applying finite injury priority methods we show that there is a d-r.e. set A such that xA is not a semicomputable real number (which corresponds to the limit of computable monotonic sequence of rational numbers) and that there is an ω-r.e. set A such that xA can’t be represented as a sum of two semi-computable real numbers.
1
Introduction
A real number x is computable, if there is a computable Cauchy sequence (rn )n∈IN of rational numbers which converges effectively to x (see e.g. [3,5,6].) Where a sequence (rn )n∈IN of rational numbers is computable means that there are recursive functions a, b, c : IN → IN such that rn = (a(n) − b(n))/(c(n) + 1) for all n ∈ IN, and the sequence (rn )n∈IN converges effectively means that |rn+m − rn | < 2−n holds for all m, n ∈ IN (IN is the set of all natural numbers.) We denote the class of all computable real numbers by C0 . Here the effectivity of the convergence is crucial, because there are computable sequences of rational numbers which converge (non-effectively, of course) to non-computable real numbers (see [5,11]). P A standard example is the real number xA := n∈A 2−n for a nonrecursive r.e. set A ⊆ IN. Let a : IN → IN be an 1-1 recursive enumeration function of A, i.e., rang(a) Pn = A, then the increasing computable sequence (xn )n∈IN defined by xn := i=0 2−a(i) converges noneffectively to the noncomputable real number xA (see [4]). In fact, it is easy to see that xA is a computable real number iff A is a recursive set. Although the real number xA for a nonrecursive r.e. set A is not computable, it is still quite “effective” in the sense that we can approximate it effectively from below. We call a real number x left computable (right computable), if there is an increasing (decreasing) computable sequence of rational numbers which converges to x. A real number is called semi-computable, if it is either left computable or right computable. The class of all semi-computable real numbers ?
Contact author. Email address: [email protected]
Luboˇ s Brim et al. (Eds.): MFCS’98, LNCS 1450, pp. 798–806, 1998. c Springer-Verlag Berlin Heidelberg 1998
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is denoted by C1 . The above example shows that C0 ⊆ C1 . Obviously, xA is 6 (right) left computable, if A is (co-)r.e. On the other hand, C. G. Jockusch (see [9]) has observed that S if B is a non-recursive r.e. set, then the non-r.e. set B ⊕ B := {2n : n ∈ B} {2n + 1 : n 6∈ B} still corresponds to a left computable real number xB⊕B . Note that the set B ⊕ B above is in fact a d-r.e. set (difference of r.e. sets.) We can extend Jockusch’s result to show that there are a real (k + 1)-r.e set (which is not k-r.e.) and a real ω-r.e. set (which is not k-r.e for any k), respectively, such that their corresponding real numbers are still left computable. Because xA = xB∪C − xC if A = B \ C (set difference,) the above results show that there are many semi-computable real numbers whose differences are still semi-computable. Let C2 denote the closure of C1 under the arithmetic operations of “+” and “−”. Then it is natural to ask whether C1 = C2 holds. The answer is no. We can construct by finite injury priority method a d-r.e. set C2 A such that xA is neither left computable nor right computable, thus C1 ⊆ 6 holds. C2 is an interesting class of real numbers. It forms in fact a field, i.e. it is closed under the arithmetical operations “+”, “−”, “×” and “÷”. We will give another characterization of the elements of C2 by the limits of the “weakly effectively convergent sequences”, namely, x ∈ C P2∞iff there is a computable sequence (xn )n∈IN of rational numbers such that n=0 |xn+1 − xn | is bounded and converges to x. Compairing with the effectively convergent sequence (yn )n∈IN P which satisfies that |yn+m − yn | ≤ 2−n , for every n, m ∈ IN, hence ∞ n=0 |yn+m − yn | is bounded for every m ∈ IN, we can say that the sequence (xn )n∈IN above converges to x weakly effectively. So the real numbers in C2 can be naturally called weakly computable. Our last result shows that not every convergent computable sequence converges weakly effectively. Again by a priority injure construction we show that there is an ω-r.e. set A such that xA is not in C2 , i.e. there is no computable sequence of rational numbers which converges to xA weakly effectively. We call a real number recursively enumerable, if there is a computable sequence of rational numbers which converges to it and denote by C3 the class of all such real numbers. Our results show that (Ci : i ≤ 3) forms a noncollapsed hierarchy.
2
Computable and Semi-Computable Real Numbers
This section discusses the computable and semi-computable real numbers. By definition, it is easy to see that x is left computable iff −x is right computable. Left and right computabilities are incomparable and x is computable iff it is both left and right computable. If A ⊆ IN is a r.e. (co-r.e.) set, then xA is a left (right) computable real number. We will show that the inverse is not true. Definition 1 ((cf. [10])). A r.e. set A ⊆ IN is called 1-r.e. For any k ≥ 1, set A ⊆ IN is called (k + 1)-r.e, if there are r.e. set B ⊆ IN and k-r.e. set C ⊆ IN such that A = B\C. 2-r.e. set is usually called d-r.e. P For any finite set E ⊂ IN, we define its canonical index i by i := j∈E 2j . A finite set with canonical index i is denoted by Di . A sequence (En )n∈IN of finite
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subsets of IN is called computable, iff there is a recursive function f : IN → IN such that En = Df (n) for all n ∈ IN. Proposition 1. A set A ⊆ IN is k-r.e iff there is a computable sequence (As )s∈IN of finite subsets of IN such that 1. A = lim An := n→∞
∞ ∞ T S n=0 s=n
As , and
2. |{s ∈ IN : n ∈ As+1 4 As }| ≤ k, for all n ∈ IN. S where A 4 B := (A\B) (B\A) is the symmetric difference of sets A and B, and |B| denotes the cardinality of set B. The sequence (As )s∈IN above is usually called an effective k-enumeration of A. Jockusch’s example shows that there is a d-r.e. set A such that xA is left computable. R.I. Soare [9] extended this result in several directions. He has shown, e.g., that there is a dominant d-r.e. set A and a cohensive set C such that xA and xC are left computable. Where A is dominant iff the principal function of A dominates every recursive function, T and C isTcohensive iff C is infinite and there is no r.e. set W such that W C and W C are both infinite (see [9] for exact definitions.) The next theorems give another extension of Jockusch’s observation and induce to an infinite hierarchy of left computable real numbers. Theorem 1. For any k ≥ 2, there are k-r.e. sets A, B which are not (k − 1)-r.e. such that xA and xB are left and right computable, respectively. The concept of k-r.e. set can be generalized to ω-r.e. by the Proposition 1. Definition 2. Set A ⊆ IN is called ω-r.e., iff there is a computable sequence (As )s∈IN of finite subsets of IN such that 1. A =
∞ ∞ T S n=0 s=n
As , and
2. ∀n ∈ IN∃k ∈ IN(|{s ∈ IN : n ∈ As+1 4 As }| ≤ k). The sequence (As )s∈IN is called an effective ω-enumeration of A. Theorem 2. There is an ω-r.e. sets A which is not k-r.e., for any k ∈ IN, such that xA is left computable. The same claim holds for right computability as well.
3
Weakly Computable Real Numbers
From the last section we know that there are many noncomputable semi computable real numbers whose sums and differences are still semi-computable. We will show in this section that this is not always the case, i.e., there are left computable real numbers y and z such that their difference x := y − z is neither left computable nor right computable. Hence C1 is not closed under the arithmetical operations “+” and “−”. We introduce a new notion at first.
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Definition 3. A real number x is called weakly computable, if there are two left computable real numbers y, z such that x = y − z. The set of all weakly computable real numbers is denoted by C2 . Equivalently, x is weakly computable iff there are left computable real y and right computable real z such that x = y + z. If A is a d-r.e. set, then xA is weakly computable. More generally, we can show by an easy induction on k ≥ 1, that, if A is a k-r.e set, then xA is a weakly computable real number. Now we give another characterization of weakly computable real numbers by the “weak convergence” of sequences. real numbers is weakly effectively converDefinition 4. A sequence (xn )n∈IN ofP ∞ gent (w.e. convergent for short) if , if n=0 |xn+1 − xn | is bounded. P∞ If a sequence (xn )n∈IN converges weakly effectively, i.e., n=0 |xn+1 − xn | < ∞, then “big” jumps may occur in the sequence and then may occur very late. If, however, the sequence is effectively convergent, then the big jumps must occur early. So, weakly effective convergence is a kind of weak version of effective convergence. Any effectively convergent sequence of rational numbers converges to computable real numbers. For w.e. convergent sequences, we have Theorem 3. A real number x is weakly computable, iff there is a computable sequence (xn )n∈IN of rational numbers which converges to x weakly effectively. Proof. “⇒”. Let x be a weakly computable real number. Then there are two computable increasing sequences (yn )n∈IN and (zn )n∈IN of rational numbers such that y := limn→∞ yn and z := limn→∞ zn exist and x = y − z. Let xn := yn − zn . Then (xn )n∈IN is a computable sequence of rational numbers which satisfies ∞ X
∞ X
|xn+1 − xn | ≤
n=0
(yn+1 − yn ) +
n=0
∞ X
(zn+1 − zn ) = y − y0 + z − z0
n=0
So (xn )n∈IN converges to x weakly effectively. “⇐”. Let (xn )n∈IN be a computable sequence of rational numbers which converges to x weakly effectively. Define computable sequences (yn )n∈IN and (zn )n∈IN of rational numbers by yn := x0 +
n X
· xi ) and zn := (xi+1 −
i=0
n X
· xi+1 ) (xi −
i=0
Obviously, they are both non-decreasing and bounded. Hence y := limn→∞ yn and z := limn→∞ zn exist and they are the left computable real numbers which satisfy y − z = lim (yn − zn ) = lim (x0 + n→∞
= lim (x0 + n→∞
n→∞
n X
n X
· xi ) − (xi+1 −
i=0
(xi+1 − xi )) = lim xn = x.
i=0
That is, x is weakly computable.
n→∞
n X i=0
· xi+1 )) (xi −
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Corollary 1. A real number x is weakly computable P∞ iff there is a computable sequence (xn )n∈IN of rational numbers such that n=0 |xn+i − xn | is bounded for every i ∈ IN and limn→∞ xn = x. Proposition 2. If (xn )n∈IN is a computable sequence of computable real numbers which converges w.e. to x, then x is weakly computable. Theorem 4. The class C2 is a field generated by C1 . Now we will show that C2 extends C1 properly. Theorem 5. There is a d-r.e. set A such that xA is neither left computable, nor right computable. Proof. We construct the set A effectively in stages so that, for any s ∈ IN, the number 2s is always enumerated into A at the beginning. They can be removed from A at a later stage and then will never enter A again. The number 2s + 1 can only be enumerated into A and can not be removed from A. This makes sure that A is a d-r.e. set. We use As to denote the set constructed at the end of stage s. Our proof is a finite injury priority construction. The detail explanations about such kind of constructions can be found in [10]. To guarantee that xA is neither left computable nor right computable, we let xA diagonalize all semi-computable real numbers. Let (Mn )n∈IN be the standard effective enumeration of all Turing machines, ϕn :⊆ IN → IN the function computed by Mn . Function ϕn,s :⊆ IN → IN is defined by ϕn,s (x) := y, if Mn (x) halts and outputs y in s steps and ϕn,s (x) undefined otherwise. Let an := sup({νQ (i) ≤ 2 : i ∈ dom(ϕn )} ∪ {0}) bn := inf({νQ (i) ≥ 0 : i ∈ dom(ϕn )} ∪ {2}) an,s := max({νQ (i) ≤ 2 : i ∈ dom(ϕn,s )} ∪ {0}) bn,s := min({νQ (i) ≥ 0 : i ∈ dom(ϕn,s )} ∪ {2}) l is an effective enumeration of rational numbers. For any where νQ : IN → Q increasing computable sequence (xs )s∈IN of rational numbers from [0; 2], there is an n such that ∀s ∈ IN(xs = an,s ). And an = lims→∞ an,s for every n ∈ IN. Then L := {an : n ∈ IN} consists of all left computable real numbers of interval [0; 2]. Similarly, R := {bn : n ∈ IN} consists of all right computable real numbers of interval [0; 2]. It suffices now to make sure that the set A satisfies, for all n ∈ IN, the following requirements: R2n R2n+1
: :
xA = 6 an , xA = 6 bn .
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The strategy to satisfy a single requirement R2n is simple: we put at the beginning all even numbers into A0 . If, at some stage s + 1, there is some i such that xAs − 2−2i < an,s , then reduce the value of xAs by removing the number 2i from As . That is, define As+1 := As \ {2i} (we call R2n is attacked at this stage.) Furthermore, we define a restraint r(2n, s + 1) := µj(xAs − 2−2i + 2−j < an,s ). At any stage t + 1 > s + 1, all elements x ≤ r(2n, s + 1) are not allowed to be put into or taken out from At . Therefore, the set A := lims→∞ As satisfies that xA ≤ xAs+1 + 2−r(2n,s+1) = xAs − 2−2i + 2−r(2n,s+1) < an,s ≤ an . Hence R2n is satisfied. If no such stage s + 1 exists, then an,s ≤ xAs − 2−2i holds for all s and i, hence, say, an ≤ xA − 2−2 < xA . That is, R2n is satisfied too. The strategy for R2n+1 is similar. To accommodate all requirements Rn simultaneously, we use finite injury priority method. We give all requirements the priority in order R0 , R1 , R2 , . . . , i.e., Rm has higher priority than Rn iff m < n. At any stage s + 1, if we want to enumerate an element k into As while attacking the requirement R2n+1 , we do not need to take care for lower priority requirements Rm (with m > 2n + 1) but we have to do that for all higher priority requirements Rm (with m < 2n + 1.) In this case, the element k can be put into As at this stage only, if this does not injury any higher priority requirement, i.e., k must be bigger than all restraint r(m, s) for all m ≤ 2n+1. Note that, for any requirement Rm , one attack suffices to satisfy it, if it is not injuried any more. Then any requirement Rm can be injuried only finitely often (at most 2m − 1 times for Rm .) Eventually, every requirement can be satisfied by this strategy. The construction of (As )s∈IN is as following: Stage 0: Define A0 := {2i : i ∈ IN} and r(n, 0) := 0 for all n ∈ IN. All requirements are set to the state of unsatisfied. Stage s + 1: Given As and r(n, s) for all n. The requirement R2n requires attention, if there is an i ≤ s such that (i) r(m, s) < 2i for all m ≤ 2n; (ii) an,s > xAs − 2−2i and (iii) R2n is in the state of unsatisfied. The requirement R2n+1 requires attention, if there is an i ≤ s such that (i)’ r(m, s) < 2i + 1 for all m ≤ 2n + 1; (ii)’ bn,s < xAs + 2−(2i+1) and (iii)’ R2n+1 is in the state of unsatisfied. Choose m ≤ s minimal such that Rm requires attention. If m = 2n, then define As+1 := As \ {2i0 } ; r(e, s + 1) := r(e, s), if e 6= m and r(m, s + 1) := µi(an,s > xAs − 2−2i0 + 2−i ), where i0 is the least i which satisfies the condition (i) – (iii). S If m = 2n + 1, then define As+1 := As {2i0 + 1}; r(e, s + 1) := r(e, s), if e 6= m and r(m, s + 1) := µi(bn,s < xAs + 2−2i0 +1 − 2−i ), where i0 is the least i which satisfies the condition (i)’ – (iii)’. In both cases, we say that Rm receives attention and set it now to the state of satisfied. Furthermore, all requirements Rm0 with m0 > m are set to the state of unsatisfied. If Rm0 , m0 > m, was satisfied at stage s, then it is injured at this stage by Rm . If there is no m ≤ s such that Rm requires attention, then define simply As+1 := As and r(n, s + 1) := r(n, s) for all n ∈ IN.
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This ends the construction. It is easy to see that the construction succeeds by the following claims whose proofs are omitted here. Claim 1 A := lim An is a d-r.e. set. n→∞ Claim 2 For any n ∈ IN, Rn receives attentions at most finitely many times. Claim 3 For any n ∈ IN, Rn is eventually satisfied. It follows immediately from above theorem the following corollary. Corollary 2. There are non-semi-computable weakly computable real numbers, C2 . i.e. C1 ⊆ 6
4
Recursively Enumerable Real Numbers
We have shown that xA is weakly computable, if A is k-r.e. for any k ≥ 1 and there are also ω-r.e. set A such that xA is semi-computable (Theorem 2), hence weakly computable. It comes the question: is every xA weakly computable, if A is ω-r.e.? Or more generally, is there any convergent computable sequence of rational numbers which converges not weakly effectively? This section will answer these question positively. Theorem 6. There is a r.e. real number which is not weakly computable. Thus, C3 holds. C2 ⊆ 6 Proof. We construct a computable sequence (xn )n∈IN of rational numbers in stages such that the limit x := limn→∞ xn diagonalizes all differences of left computable real numbers. Let an and an,s be same as in the proof of Theorem 5 and define, for all n, m ∈ IN, that dhn,mi := an − am and dhn,mi,s := an,s − am,s .
(1)
where h· , ·i : IN2 → IN is the Cantor’s pairing function defined by hn, mi := (n + m)(n + m + 1)/2 + m. Then D := {dn : n ∈ IN} is the set of all weakly computable real numbers by the Definition 3. It suffices now to make sure that x is a r.e. real number which satisfies all the following requirements: x 6= dn . P∞ We will construct x in such a way that x = i=0 wi 8−(i+1) and x is a limit of some computable sequence of rational numbers, where wi ∈ {0, 4} for all i ∈ IN. To satisfy the requirement Rn , we change the value of wn from 0 to 4 or from 4 to 0, if it is necessary, so that |x − dn | ≥ 8−(n+1) holds. For single Rn , the Pn−1 strategy is simple. Let rn := i=0 wi 8−(i+1) (for any given rational numbers wi with i < n) and consider the interval [rn ; rn + 8−n ]. At the beginning, let wn := 0. We redefine wn := 4, if there is an s such that dn,s ≤ rn + 2 · 8−(n+1) , set wn back to 0 later, if some t > s appears such that dn,t ≥ rn + 3 · 8−(n+1) and let wn = 4 again, if there is another s0 > t such that dn,s0 ≤ rn + 2 · 8−(n+1) , and Rn :
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so on. Every change of the value of wn is called an attack to Rn . Since (dn,s )s∈IN is not necessarily monotonic, wn may be changed many times. We denote by wns the value of wn at the end of stage s. Then wns can’t be changed infinitely often because (dn,s )s∈IN converges, hence wn := lims→∞ wns exists. We define simply x := rn + wn 8−(n+1) . By the definition, |dn,s − (rn + wns 8−(n+1) )| ≥ 8−(n+1) holds for all s ∈ IN. Therefore |dn − (rn + wn 8−(n+1) )| = |dn − x| ≥ 8−(n+1) holds. That is, x satisfies Rn . To treat all requirements simultaneously, we apply priority injury method again. Here is the construction of (xn )n∈IN : Stage s = 0. Set x0 := w00P:= 0. s Stage s + 1. Given xs := i=0 wis 8−(i+1) with wis ∈ {0, 4} for all i ≤ s. The requirement Rn , n ≤ s, requires attention, if either wns = 0 & dn,s ≤ wns = 4 & dn,s ≥
n−1 X i=0 n−1 X
wis 8−(i+1) + 2 · 8−(n+1) , wis 8−(i+1) + 3 · 8−(n+1)
or
(2)
(3)
i=0
holds. Choose the least n ≤ s such that Rn requires attention. If (2) holds, then define s if m < n; wm s+1 if m = n; := 4 (4) wm 0 if n < m ≤ s + 1. If (3) holds, then define s+1 wm
:=
s wm 0
if m < n; if n ≤ m ≤ s + 1.
(5)
In both cases we say then Rn receives attention. If no requirement Rn , n ≤ s, requires attention, then define, for all m ≤ s, simply s+1 s+1 s := wm and ws+1 := 0. wm
(6)
Ps+1 At last, set xs+1 := i=0 wis+1 8−(i+1) . This ends the construction. We can show our construction succeeds by the flowing Claims whose proofs are omitted here. Claim 1 For every n ∈ IN, the requirement Rn receives attentions at most finitely often. Claim 2 For every n ∈ IN, the limit wn := lim wns exists. s→∞ P −(i+1) w 8 . Then x = limn→∞ xn holds. Claim 3 Let x := ∞ i i=0 Claim 4 The real number x of Lemma 3 satisfies all requirements Rn for n ∈ IN.
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Corollary 3. There is an ω-r.e. set A such that xA is a r.e. real number which is not weakly computable. Proof. Let wis (s ∈ IN, i ≤ s) be same as in the proof of the Theorem 6. Note that 4 · 8−(n+1) = 2−(3n+1) . Define a computable sequence (As : s ∈ IN) of finite sunsets of IN by As := {3i + 1 ∈ IN : i ≤ s & wis = 4}. Since there are only finite many s such that wns 6= wns+1 , there are only finite many s such that As+1 4 As contains n, for every n ∈ IN. That is (As : s ∈ IN) is an effective ω-enumeration of the set A := lims→∞ As , hence A is an ω-r.e. set. On the see that A = {3n + 1 ∈ IN : wn = 4}. Then Pother hand, it is easy to P∞ xA = {2−(3n+1) : wn = 4} = n=0 wn 8−(n+1) = x. By the proof of Theorem 6, xA is a r.e. real number which is not weakly computable. Corollary 4. (Ci : i ≤ 3) forms a noncollapsed hierarchy.
References 1. Ker-I Ko On the definitions of some complexity classes of real numbers, Math. Systems Theory 16(1983) 95–100. 2. A. H. Lachlan Recursive real numbers. J. Symbolic Logic 28(1963), 1–16. 3. J. Myhill Criteria of constructivity for real numbers, J. Symbolic Logic 18(1953), 7–10. 4. M. Pour-El & J. Richards Computability in Analysis and Physics. SpringerVerlag, Berlin, Heidelberg, 1989. 5. H. G. Rice Recursive real numbers, Proc. Amer. Math. Soc. 5(1954), 784–791. 6. R. M. Robinson Review of “R. Peter: ‘ Rekursive Funktionen’, Akad. Kiado. Budapest, 1951”, J. Symb. Logic 16(1951), 280. 7. H. Jr. Rogers Theory of Recursive Functions and Effective Computability McGraw-Hill, Inc. New York, 1967. 8. R. Soare Recursion theory and Dedekind cuts, Trans, Amer. Math. Soc. 140(1969), 271–294. 9. R. Soare Cohensive sets and recursive enumerable Dedekind cuts Pacific J. of Math. 31(1969), no.1, 215–231. 10. R. Soare Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin, Heidelberg, 1987. 11. E. Specter Nicht konstruktive beweisbare S¨ atze der Analysis, J. Symbolic Logic 14(1949), 145–158 12. K. Weihrauch Computability. EATCS Monographs on Theoretical Computer Science Vol. 9, Springer-Verlag, Berlin, Heidelberg, 1987.
Abstract. For any set A of natural numbers, denote by xA the corresponding real number such that A is just the set of “1” positions in its binary expansion. In this paper we characterize the number xA for some classes of recursively enumerable sets A. Applying finite injury priority methods we show that there is a d-r.e. set A such that xA is not a semicomputable real number (which corresponds to the limit of computable monotonic sequence of rational numbers) and that there is an ω-r.e. set A such that xA can’t be represented as a sum of two semi-computable real numbers.
1
Introduction
A real number x is computable, if there is a computable Cauchy sequence (rn )n∈IN of rational numbers which converges effectively to x (see e.g. [3,5,6].) Where a sequence (rn )n∈IN of rational numbers is computable means that there are recursive functions a, b, c : IN → IN such that rn = (a(n) − b(n))/(c(n) + 1) for all n ∈ IN, and the sequence (rn )n∈IN converges effectively means that |rn+m − rn | < 2−n holds for all m, n ∈ IN (IN is the set of all natural numbers.) We denote the class of all computable real numbers by C0 . Here the effectivity of the convergence is crucial, because there are computable sequences of rational numbers which converge (non-effectively, of course) to non-computable real numbers (see [5,11]). P A standard example is the real number xA := n∈A 2−n for a nonrecursive r.e. set A ⊆ IN. Let a : IN → IN be an 1-1 recursive enumeration function of A, i.e., rang(a) Pn = A, then the increasing computable sequence (xn )n∈IN defined by xn := i=0 2−a(i) converges noneffectively to the noncomputable real number xA (see [4]). In fact, it is easy to see that xA is a computable real number iff A is a recursive set. Although the real number xA for a nonrecursive r.e. set A is not computable, it is still quite “effective” in the sense that we can approximate it effectively from below. We call a real number x left computable (right computable), if there is an increasing (decreasing) computable sequence of rational numbers which converges to x. A real number is called semi-computable, if it is either left computable or right computable. The class of all semi-computable real numbers ?
Contact author. Email address: [email protected]
Luboˇ s Brim et al. (Eds.): MFCS’98, LNCS 1450, pp. 798–806, 1998. c Springer-Verlag Berlin Heidelberg 1998
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is denoted by C1 . The above example shows that C0 ⊆ C1 . Obviously, xA is 6 (right) left computable, if A is (co-)r.e. On the other hand, C. G. Jockusch (see [9]) has observed that S if B is a non-recursive r.e. set, then the non-r.e. set B ⊕ B := {2n : n ∈ B} {2n + 1 : n 6∈ B} still corresponds to a left computable real number xB⊕B . Note that the set B ⊕ B above is in fact a d-r.e. set (difference of r.e. sets.) We can extend Jockusch’s result to show that there are a real (k + 1)-r.e set (which is not k-r.e.) and a real ω-r.e. set (which is not k-r.e for any k), respectively, such that their corresponding real numbers are still left computable. Because xA = xB∪C − xC if A = B \ C (set difference,) the above results show that there are many semi-computable real numbers whose differences are still semi-computable. Let C2 denote the closure of C1 under the arithmetic operations of “+” and “−”. Then it is natural to ask whether C1 = C2 holds. The answer is no. We can construct by finite injury priority method a d-r.e. set C2 A such that xA is neither left computable nor right computable, thus C1 ⊆ 6 holds. C2 is an interesting class of real numbers. It forms in fact a field, i.e. it is closed under the arithmetical operations “+”, “−”, “×” and “÷”. We will give another characterization of the elements of C2 by the limits of the “weakly effectively convergent sequences”, namely, x ∈ C P2∞iff there is a computable sequence (xn )n∈IN of rational numbers such that n=0 |xn+1 − xn | is bounded and converges to x. Compairing with the effectively convergent sequence (yn )n∈IN P which satisfies that |yn+m − yn | ≤ 2−n , for every n, m ∈ IN, hence ∞ n=0 |yn+m − yn | is bounded for every m ∈ IN, we can say that the sequence (xn )n∈IN above converges to x weakly effectively. So the real numbers in C2 can be naturally called weakly computable. Our last result shows that not every convergent computable sequence converges weakly effectively. Again by a priority injure construction we show that there is an ω-r.e. set A such that xA is not in C2 , i.e. there is no computable sequence of rational numbers which converges to xA weakly effectively. We call a real number recursively enumerable, if there is a computable sequence of rational numbers which converges to it and denote by C3 the class of all such real numbers. Our results show that (Ci : i ≤ 3) forms a noncollapsed hierarchy.
2
Computable and Semi-Computable Real Numbers
This section discusses the computable and semi-computable real numbers. By definition, it is easy to see that x is left computable iff −x is right computable. Left and right computabilities are incomparable and x is computable iff it is both left and right computable. If A ⊆ IN is a r.e. (co-r.e.) set, then xA is a left (right) computable real number. We will show that the inverse is not true. Definition 1 ((cf. [10])). A r.e. set A ⊆ IN is called 1-r.e. For any k ≥ 1, set A ⊆ IN is called (k + 1)-r.e, if there are r.e. set B ⊆ IN and k-r.e. set C ⊆ IN such that A = B\C. 2-r.e. set is usually called d-r.e. P For any finite set E ⊂ IN, we define its canonical index i by i := j∈E 2j . A finite set with canonical index i is denoted by Di . A sequence (En )n∈IN of finite
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subsets of IN is called computable, iff there is a recursive function f : IN → IN such that En = Df (n) for all n ∈ IN. Proposition 1. A set A ⊆ IN is k-r.e iff there is a computable sequence (As )s∈IN of finite subsets of IN such that 1. A = lim An := n→∞
∞ ∞ T S n=0 s=n
As , and
2. |{s ∈ IN : n ∈ As+1 4 As }| ≤ k, for all n ∈ IN. S where A 4 B := (A\B) (B\A) is the symmetric difference of sets A and B, and |B| denotes the cardinality of set B. The sequence (As )s∈IN above is usually called an effective k-enumeration of A. Jockusch’s example shows that there is a d-r.e. set A such that xA is left computable. R.I. Soare [9] extended this result in several directions. He has shown, e.g., that there is a dominant d-r.e. set A and a cohensive set C such that xA and xC are left computable. Where A is dominant iff the principal function of A dominates every recursive function, T and C isTcohensive iff C is infinite and there is no r.e. set W such that W C and W C are both infinite (see [9] for exact definitions.) The next theorems give another extension of Jockusch’s observation and induce to an infinite hierarchy of left computable real numbers. Theorem 1. For any k ≥ 2, there are k-r.e. sets A, B which are not (k − 1)-r.e. such that xA and xB are left and right computable, respectively. The concept of k-r.e. set can be generalized to ω-r.e. by the Proposition 1. Definition 2. Set A ⊆ IN is called ω-r.e., iff there is a computable sequence (As )s∈IN of finite subsets of IN such that 1. A =
∞ ∞ T S n=0 s=n
As , and
2. ∀n ∈ IN∃k ∈ IN(|{s ∈ IN : n ∈ As+1 4 As }| ≤ k). The sequence (As )s∈IN is called an effective ω-enumeration of A. Theorem 2. There is an ω-r.e. sets A which is not k-r.e., for any k ∈ IN, such that xA is left computable. The same claim holds for right computability as well.
3
Weakly Computable Real Numbers
From the last section we know that there are many noncomputable semi computable real numbers whose sums and differences are still semi-computable. We will show in this section that this is not always the case, i.e., there are left computable real numbers y and z such that their difference x := y − z is neither left computable nor right computable. Hence C1 is not closed under the arithmetical operations “+” and “−”. We introduce a new notion at first.
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Definition 3. A real number x is called weakly computable, if there are two left computable real numbers y, z such that x = y − z. The set of all weakly computable real numbers is denoted by C2 . Equivalently, x is weakly computable iff there are left computable real y and right computable real z such that x = y + z. If A is a d-r.e. set, then xA is weakly computable. More generally, we can show by an easy induction on k ≥ 1, that, if A is a k-r.e set, then xA is a weakly computable real number. Now we give another characterization of weakly computable real numbers by the “weak convergence” of sequences. real numbers is weakly effectively converDefinition 4. A sequence (xn )n∈IN ofP ∞ gent (w.e. convergent for short) if , if n=0 |xn+1 − xn | is bounded. P∞ If a sequence (xn )n∈IN converges weakly effectively, i.e., n=0 |xn+1 − xn | < ∞, then “big” jumps may occur in the sequence and then may occur very late. If, however, the sequence is effectively convergent, then the big jumps must occur early. So, weakly effective convergence is a kind of weak version of effective convergence. Any effectively convergent sequence of rational numbers converges to computable real numbers. For w.e. convergent sequences, we have Theorem 3. A real number x is weakly computable, iff there is a computable sequence (xn )n∈IN of rational numbers which converges to x weakly effectively. Proof. “⇒”. Let x be a weakly computable real number. Then there are two computable increasing sequences (yn )n∈IN and (zn )n∈IN of rational numbers such that y := limn→∞ yn and z := limn→∞ zn exist and x = y − z. Let xn := yn − zn . Then (xn )n∈IN is a computable sequence of rational numbers which satisfies ∞ X
∞ X
|xn+1 − xn | ≤
n=0
(yn+1 − yn ) +
n=0
∞ X
(zn+1 − zn ) = y − y0 + z − z0
n=0
So (xn )n∈IN converges to x weakly effectively. “⇐”. Let (xn )n∈IN be a computable sequence of rational numbers which converges to x weakly effectively. Define computable sequences (yn )n∈IN and (zn )n∈IN of rational numbers by yn := x0 +
n X
· xi ) and zn := (xi+1 −
i=0
n X
· xi+1 ) (xi −
i=0
Obviously, they are both non-decreasing and bounded. Hence y := limn→∞ yn and z := limn→∞ zn exist and they are the left computable real numbers which satisfy y − z = lim (yn − zn ) = lim (x0 + n→∞
= lim (x0 + n→∞
n→∞
n X
n X
· xi ) − (xi+1 −
i=0
(xi+1 − xi )) = lim xn = x.
i=0
That is, x is weakly computable.
n→∞
n X i=0
· xi+1 )) (xi −
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Corollary 1. A real number x is weakly computable P∞ iff there is a computable sequence (xn )n∈IN of rational numbers such that n=0 |xn+i − xn | is bounded for every i ∈ IN and limn→∞ xn = x. Proposition 2. If (xn )n∈IN is a computable sequence of computable real numbers which converges w.e. to x, then x is weakly computable. Theorem 4. The class C2 is a field generated by C1 . Now we will show that C2 extends C1 properly. Theorem 5. There is a d-r.e. set A such that xA is neither left computable, nor right computable. Proof. We construct the set A effectively in stages so that, for any s ∈ IN, the number 2s is always enumerated into A at the beginning. They can be removed from A at a later stage and then will never enter A again. The number 2s + 1 can only be enumerated into A and can not be removed from A. This makes sure that A is a d-r.e. set. We use As to denote the set constructed at the end of stage s. Our proof is a finite injury priority construction. The detail explanations about such kind of constructions can be found in [10]. To guarantee that xA is neither left computable nor right computable, we let xA diagonalize all semi-computable real numbers. Let (Mn )n∈IN be the standard effective enumeration of all Turing machines, ϕn :⊆ IN → IN the function computed by Mn . Function ϕn,s :⊆ IN → IN is defined by ϕn,s (x) := y, if Mn (x) halts and outputs y in s steps and ϕn,s (x) undefined otherwise. Let an := sup({νQ (i) ≤ 2 : i ∈ dom(ϕn )} ∪ {0}) bn := inf({νQ (i) ≥ 0 : i ∈ dom(ϕn )} ∪ {2}) an,s := max({νQ (i) ≤ 2 : i ∈ dom(ϕn,s )} ∪ {0}) bn,s := min({νQ (i) ≥ 0 : i ∈ dom(ϕn,s )} ∪ {2}) l is an effective enumeration of rational numbers. For any where νQ : IN → Q increasing computable sequence (xs )s∈IN of rational numbers from [0; 2], there is an n such that ∀s ∈ IN(xs = an,s ). And an = lims→∞ an,s for every n ∈ IN. Then L := {an : n ∈ IN} consists of all left computable real numbers of interval [0; 2]. Similarly, R := {bn : n ∈ IN} consists of all right computable real numbers of interval [0; 2]. It suffices now to make sure that the set A satisfies, for all n ∈ IN, the following requirements: R2n R2n+1
: :
xA = 6 an , xA = 6 bn .
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803
The strategy to satisfy a single requirement R2n is simple: we put at the beginning all even numbers into A0 . If, at some stage s + 1, there is some i such that xAs − 2−2i < an,s , then reduce the value of xAs by removing the number 2i from As . That is, define As+1 := As \ {2i} (we call R2n is attacked at this stage.) Furthermore, we define a restraint r(2n, s + 1) := µj(xAs − 2−2i + 2−j < an,s ). At any stage t + 1 > s + 1, all elements x ≤ r(2n, s + 1) are not allowed to be put into or taken out from At . Therefore, the set A := lims→∞ As satisfies that xA ≤ xAs+1 + 2−r(2n,s+1) = xAs − 2−2i + 2−r(2n,s+1) < an,s ≤ an . Hence R2n is satisfied. If no such stage s + 1 exists, then an,s ≤ xAs − 2−2i holds for all s and i, hence, say, an ≤ xA − 2−2 < xA . That is, R2n is satisfied too. The strategy for R2n+1 is similar. To accommodate all requirements Rn simultaneously, we use finite injury priority method. We give all requirements the priority in order R0 , R1 , R2 , . . . , i.e., Rm has higher priority than Rn iff m < n. At any stage s + 1, if we want to enumerate an element k into As while attacking the requirement R2n+1 , we do not need to take care for lower priority requirements Rm (with m > 2n + 1) but we have to do that for all higher priority requirements Rm (with m < 2n + 1.) In this case, the element k can be put into As at this stage only, if this does not injury any higher priority requirement, i.e., k must be bigger than all restraint r(m, s) for all m ≤ 2n+1. Note that, for any requirement Rm , one attack suffices to satisfy it, if it is not injuried any more. Then any requirement Rm can be injuried only finitely often (at most 2m − 1 times for Rm .) Eventually, every requirement can be satisfied by this strategy. The construction of (As )s∈IN is as following: Stage 0: Define A0 := {2i : i ∈ IN} and r(n, 0) := 0 for all n ∈ IN. All requirements are set to the state of unsatisfied. Stage s + 1: Given As and r(n, s) for all n. The requirement R2n requires attention, if there is an i ≤ s such that (i) r(m, s) < 2i for all m ≤ 2n; (ii) an,s > xAs − 2−2i and (iii) R2n is in the state of unsatisfied. The requirement R2n+1 requires attention, if there is an i ≤ s such that (i)’ r(m, s) < 2i + 1 for all m ≤ 2n + 1; (ii)’ bn,s < xAs + 2−(2i+1) and (iii)’ R2n+1 is in the state of unsatisfied. Choose m ≤ s minimal such that Rm requires attention. If m = 2n, then define As+1 := As \ {2i0 } ; r(e, s + 1) := r(e, s), if e 6= m and r(m, s + 1) := µi(an,s > xAs − 2−2i0 + 2−i ), where i0 is the least i which satisfies the condition (i) – (iii). S If m = 2n + 1, then define As+1 := As {2i0 + 1}; r(e, s + 1) := r(e, s), if e 6= m and r(m, s + 1) := µi(bn,s < xAs + 2−2i0 +1 − 2−i ), where i0 is the least i which satisfies the condition (i)’ – (iii)’. In both cases, we say that Rm receives attention and set it now to the state of satisfied. Furthermore, all requirements Rm0 with m0 > m are set to the state of unsatisfied. If Rm0 , m0 > m, was satisfied at stage s, then it is injured at this stage by Rm . If there is no m ≤ s such that Rm requires attention, then define simply As+1 := As and r(n, s + 1) := r(n, s) for all n ∈ IN.
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This ends the construction. It is easy to see that the construction succeeds by the following claims whose proofs are omitted here. Claim 1 A := lim An is a d-r.e. set. n→∞ Claim 2 For any n ∈ IN, Rn receives attentions at most finitely many times. Claim 3 For any n ∈ IN, Rn is eventually satisfied. It follows immediately from above theorem the following corollary. Corollary 2. There are non-semi-computable weakly computable real numbers, C2 . i.e. C1 ⊆ 6
4
Recursively Enumerable Real Numbers
We have shown that xA is weakly computable, if A is k-r.e. for any k ≥ 1 and there are also ω-r.e. set A such that xA is semi-computable (Theorem 2), hence weakly computable. It comes the question: is every xA weakly computable, if A is ω-r.e.? Or more generally, is there any convergent computable sequence of rational numbers which converges not weakly effectively? This section will answer these question positively. Theorem 6. There is a r.e. real number which is not weakly computable. Thus, C3 holds. C2 ⊆ 6 Proof. We construct a computable sequence (xn )n∈IN of rational numbers in stages such that the limit x := limn→∞ xn diagonalizes all differences of left computable real numbers. Let an and an,s be same as in the proof of Theorem 5 and define, for all n, m ∈ IN, that dhn,mi := an − am and dhn,mi,s := an,s − am,s .
(1)
where h· , ·i : IN2 → IN is the Cantor’s pairing function defined by hn, mi := (n + m)(n + m + 1)/2 + m. Then D := {dn : n ∈ IN} is the set of all weakly computable real numbers by the Definition 3. It suffices now to make sure that x is a r.e. real number which satisfies all the following requirements: x 6= dn . P∞ We will construct x in such a way that x = i=0 wi 8−(i+1) and x is a limit of some computable sequence of rational numbers, where wi ∈ {0, 4} for all i ∈ IN. To satisfy the requirement Rn , we change the value of wn from 0 to 4 or from 4 to 0, if it is necessary, so that |x − dn | ≥ 8−(n+1) holds. For single Rn , the Pn−1 strategy is simple. Let rn := i=0 wi 8−(i+1) (for any given rational numbers wi with i < n) and consider the interval [rn ; rn + 8−n ]. At the beginning, let wn := 0. We redefine wn := 4, if there is an s such that dn,s ≤ rn + 2 · 8−(n+1) , set wn back to 0 later, if some t > s appears such that dn,t ≥ rn + 3 · 8−(n+1) and let wn = 4 again, if there is another s0 > t such that dn,s0 ≤ rn + 2 · 8−(n+1) , and Rn :
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so on. Every change of the value of wn is called an attack to Rn . Since (dn,s )s∈IN is not necessarily monotonic, wn may be changed many times. We denote by wns the value of wn at the end of stage s. Then wns can’t be changed infinitely often because (dn,s )s∈IN converges, hence wn := lims→∞ wns exists. We define simply x := rn + wn 8−(n+1) . By the definition, |dn,s − (rn + wns 8−(n+1) )| ≥ 8−(n+1) holds for all s ∈ IN. Therefore |dn − (rn + wn 8−(n+1) )| = |dn − x| ≥ 8−(n+1) holds. That is, x satisfies Rn . To treat all requirements simultaneously, we apply priority injury method again. Here is the construction of (xn )n∈IN : Stage s = 0. Set x0 := w00P:= 0. s Stage s + 1. Given xs := i=0 wis 8−(i+1) with wis ∈ {0, 4} for all i ≤ s. The requirement Rn , n ≤ s, requires attention, if either wns = 0 & dn,s ≤ wns = 4 & dn,s ≥
n−1 X i=0 n−1 X
wis 8−(i+1) + 2 · 8−(n+1) , wis 8−(i+1) + 3 · 8−(n+1)
or
(2)
(3)
i=0
holds. Choose the least n ≤ s such that Rn requires attention. If (2) holds, then define s if m < n; wm s+1 if m = n; := 4 (4) wm 0 if n < m ≤ s + 1. If (3) holds, then define s+1 wm
:=
s wm 0
if m < n; if n ≤ m ≤ s + 1.
(5)
In both cases we say then Rn receives attention. If no requirement Rn , n ≤ s, requires attention, then define, for all m ≤ s, simply s+1 s+1 s := wm and ws+1 := 0. wm
(6)
Ps+1 At last, set xs+1 := i=0 wis+1 8−(i+1) . This ends the construction. We can show our construction succeeds by the flowing Claims whose proofs are omitted here. Claim 1 For every n ∈ IN, the requirement Rn receives attentions at most finitely often. Claim 2 For every n ∈ IN, the limit wn := lim wns exists. s→∞ P −(i+1) w 8 . Then x = limn→∞ xn holds. Claim 3 Let x := ∞ i i=0 Claim 4 The real number x of Lemma 3 satisfies all requirements Rn for n ∈ IN.
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Corollary 3. There is an ω-r.e. set A such that xA is a r.e. real number which is not weakly computable. Proof. Let wis (s ∈ IN, i ≤ s) be same as in the proof of the Theorem 6. Note that 4 · 8−(n+1) = 2−(3n+1) . Define a computable sequence (As : s ∈ IN) of finite sunsets of IN by As := {3i + 1 ∈ IN : i ≤ s & wis = 4}. Since there are only finite many s such that wns 6= wns+1 , there are only finite many s such that As+1 4 As contains n, for every n ∈ IN. That is (As : s ∈ IN) is an effective ω-enumeration of the set A := lims→∞ As , hence A is an ω-r.e. set. On the see that A = {3n + 1 ∈ IN : wn = 4}. Then Pother hand, it is easy to P∞ xA = {2−(3n+1) : wn = 4} = n=0 wn 8−(n+1) = x. By the proof of Theorem 6, xA is a r.e. real number which is not weakly computable. Corollary 4. (Ci : i ≤ 3) forms a noncollapsed hierarchy.
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