On the Hierarchy of ∆02-Real Numbers
(A Survey)
Xizhong Zheng
Theoretische Informatik
Brandenburgische Technische Universität
Cottbus, Germany
Dagstuhl, Nov. 15 – 17, 2004
Contents
1. A Finite Hierarchy
•
•
•
•
Computable Reals
C.e. (left computable) Reals
D-c.e. (weakly computable) Reals
DBC Reals
2. Ershov’s Hierarchy
• Binary Computability
• Dedekind Computability
• Cauchy Computability
3. Hierarchy based on Divergence Bounding
4. Monotone Computability Hierarchy
• c-Monotone Computability
• Semi-Computability and c-Computability
• ω-Monotone Computability
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I. A Finite Hierarchy
• Computable Reals
• Semi-computable Reals
• Weakly computable Reals
• Divergence bounded computable Reals
• Computably approximable reals
2
Computable Reals (EC)
A real number x is computable if there is a computable sequence (xs) of rational numbers
which converges to x effectively in one of the following senses:
• (∀n ∈ N)(|x − xn| ≤ 2−n);
• (∀n, s ∈ N)(s ≥ e(n) =⇒ |x − xs| ≤ 2−n) for a computable function e;
• (∀n ∈ N)(|xn − xn+1| ≤ 2−n)
• (∀n, m ∈ N)(m ≥ n =⇒ |xn − xm| ≤ 2−n)
The following are equivalent (Robinson 1951):
• (Cauchy) x is computable;
• (Dedekind) Lx := {r ∈ Q : r < x} is computable; and
• (Binary) x = xA :=
−(i+1)
i∈A 2
P
for a computable set A ⊆ N.
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Computably Enumerable Reals (CE)
A real x is computably enumerable (c.e., or left computable) if there is an increasing
computable sequence (xs) of rational numbers which converges to x.
The following are equivalent:
• x is c.e.;
• Lx := {r ∈ Q : r < x} is a c.e. set;
• (Calude et al, 1998) The binary expansion of x is strongly ω-c.e.
A set A ⊆ N is strongly ω-c.e. if there exists a computable sequence (As) of finite sets
converging to A such that
(∀n, s)(n ∈ As − As+1 =⇒ (∃m < n)(m ∈ As+1 − As)).
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Binary c.e. 6= Dedekind c.e.
(Jockusch 1969): Not every c.e. real has a c.e. binary expansion.
Let A := {a0, a1, a2, . . . , } be c.e. but not computable and B := A ⊕ A = 2A ∪ (2A + 1).
Then:
• B is not c.e.,
• For As := {a0, a1, . . . , as}, Bs := As ⊕ As and xs := xAs . Then
xs+1 := xAs+1⊕As+1 = xAs⊕As + 2−2as+1 − 2−(2as+1+1) > xs
• lim xs = xB , and hence xB is a c.e. real.
xA is called strongly c.e., if A is a c.e. set (Downey 2001).
In general, a c.e. real xA is called n-strongly c.e. if A is an n-c.e. set. This forms a proper
hierarchy of c.e. reals.
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Semi-Computable Reals (SC)
Left computable (c.e.) and right computable (co-c.e.) are called semi-computable.
• (Weihrauch and Z. 1998) x is semi-computable iff there is a computable sequence (xs) of
rational numbers which converges to x monotonically in the sense that
(∀s, t)(s < t =⇒ |x − xs| ≤ x − xt|).
• (Soare, 1969) If xA is semi-computable, then A is λn.2n-c.e.;
• (Ambos-Spies, Weihrauch and Z. 2000) If xA⊕B is semi-computable and the sets A and B
are c.e., then either A ≤T B or B ≤T A.
• The class SC of semi-computable reals is NOT closed under arithmetical operators.
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CE
co−CE
EC
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D-c.e. Reals (DCE)
A real x is called d-c.e. (difference of c.e.) if there are c.e. reals y, z such that x = y − z.
D-c.e. reals are also called weakly computable because of the following results:
• (Weihrauch, Z. 1998) x is d-c.e. iff there is a computable sequence (xs) of rational numbers
which converges to x weakly effectively in the sense that
X
|xs − xs+1| ≤ c
for a constant c;
• (WZ 1998) The class DCE of all d-c.e. reals is a field;
• (Z. 1999) There are c.e. reals y, z such that x := y − z does not have an ω-c.e. Turing
degree. (where deg(xA) := deg(A))
• (Downey, Wu and Z. 2003) Any ω-c.e. Turing degree contains a d-c.e. real, but not every
∆02-degree contain a d-c.e. real.
• (AWZ 2000) If x2A is d-c.e., the A is λn.23n-c.e.
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DCE
CE
co−CE
EC
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Divergence Bounded Computable Reals (DBC)
A real x is called divergence bounded computable (DBC) if there is a d-c.e. real y and a
total computable real function f such that f (y) = x.
The name DBC comes from the following results:
• (Rettinger, Romain and Z, 2001) x is dbc iff there is a computable sequence (xs) of rational
numbers which converges to x and a computable function h such that, for any n, (xs) has
at most h(n) pairs non-overlapping indices (i, j) with |xi − xj | ≥ 2−n.
• (RRZ 2001) The class DBC is a field.
• (Rettinger and Z 2004) The classes of Turing degrees of DCE, DBC are all different from
the class of ∆02-degrees.
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DBC
DCE
CE
co−CE
EC
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Computably Approximable Reals (CA)
A real x is called computably approximable (c.a.) if it is the limit of a computable sequence
(xs) of rational numbers.
• x is c.a. iff it has a ∆02 binary expansion and iff it has a ∆02-Turing degree.
(c.a. reals are also called ∆02-reals)
• (Ho 1999) x is c.a. iff there is a ∅0-computable sequence (xs) of rational numbers which
converges to x effectively. (∅0 is the halting problem.)
• The class CA is a field.
• The class CA is closed under computable real functions.
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CA
DBC
DCE
CE
co−CE
EC
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II. Ershov’s Hierarchy
• Binary Computability
• Dedekind Computability
• Cauchy Computability
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Original Ershov’s Hierarchy (1968)
A set A ⊆ N is h-c.e. if A has a computable h-enumeration (As), i.e., lim As = A and
A0 = ∅ & (∀n)(|{s ∈ N : n ∈ As∆As+1}| ≤ h(n)).
• A is c.e., if A is h-c.e. for the constant function h(n) = 1;
• A is k-c.e., if A is h-c.e. for the constant function h(n) = k; and
• A is ω-c.e., if A is h-c.e. for a computable function h.
Theorem.
(Hierarchy Theorem, Ershov 1968, 1970)
• There is a (k + 1)-c.e. set which is not k-c.e. for any k ∈ N;
• There is an ω-c.e. set which is not k-c.e. for any k; and
• There is an f -c.e. set which is not g-c.e. if (∃∞n)(f (n) > g(n)).
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h-Binary Computable Reals (h-b EC)
A real xA is called h-binary computable if its binary expansion A is an h-c.e. set.
• The hierarchy theorem holds for binary computable reals;
• 1-b EC ( CE and k-b EC * SC for all k ≥ 2;
• CE * ∗-b EC ( DCE; and
• ω-b EC is incomparable with DCE
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CA
DBC
DCE
ω− bEC
*−bEC
k−bEC
co−CE
CE
EC
1−bEC
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h-Dedekind Computable Reals (h-dEC)
A real x is called h-Dedekind computable if the cut Lx := {r ∈ Q : r < x} is an h-c.e.
set.
• The hierarchy theorem does not hold. Actually we have k-dEC = 2-dEC = SC for all
k ≥ 2;
• ω-dEC = ω-b EC.
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h-Cauchy Computable Reals (h-cEC)
A real x is called h-Cauchy computable if there is a computable sequence (xs) of rational
numbers which converges to x h-effectively in the sense that: for any n, there are at most h(n)
pairs of non-overlapping indices i, j ≥ n such that
2n < |xi − xj | ≤ 2−n+1.
Example: 0-cEC = EC and ω-EC = DBC
Rettinger and Z. (2003) show that:
• f -cEC ( g-cEC for computable functions f, g with (∀∞n)(f (n) < g(n));
• k-cEC ⊆ DCE for all k;
• k-cEC (k > 0) is incomparable with SC;
• All classes k-cEC (k > 0) and ∗-cEC are not closed under addition; and
• ω-b EC = ω-dEC ( ω-cEC.
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CA
DBC
ω− cEC
DCE
*−cEC
k−cEC
co−CE
CE
EC
0−cEC
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III. h-Bounded Computable Reals (h-BC)
A real x is called h-bounded computable if there is a computable sequence (xs) of rational
numbers which converges to x h-bounded effectively in the sense that, for any n, there are at
most h(n) non-overlapping indices i, j such that
|xi − xj | > 2−n.
• EC ⊆ id-BC and k-BC = Q for any constant k (there is no Ershov Hierarchy);
• For computable functions f, g we have g-BC 6= f -BC if
(∀c)(∃∞m)(|f (m) − g(m)| > c);
• Let C be a class of functions. If for any f, g ∈ C and any c, there is an h ∈ C with
(∀n)(f (n + c) + g(n + c) ≤ h(n)), the C-BC is a field; and
• Let oe(2n) := {f : f is computable and f ∈ o(2n)}. Then
SC * oe(2−n) and DCE ( o(2n).
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VI. Monotonically Computable Reals (MC)
A real x is called h-monotonically computable if there is a computable sequence (xs) of
rational numbers which converges to x h-monotonically in the sense that:
(∀n, m ∈ N)(n < m =⇒ h(n)|x − xn| ≤ |x − xm|).
The classes h-MC, c-MC (c ∈ R+) and ω-MC are defined accordingly. MC := ∪c∈Rc-MC.
Barmpalias, Retting and Z., (2003) show that
• 1-MC = SC;
• SC ( MC ( DCE;
• (Dense Hierarchy) c1-MC ( c2-MC for any c1 > c1 ≥ 1;
• If h is an increasing and unbounded computable function, then h-MC = ω-MC; and
• ω-MC is incomparable with both DCE and DBC.
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CA
DBC
ω− MC
DCE
MC
CE
co−CE
1−MC
EC
c−MC
(c<1)
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Around 1-Monotone Computability
Facts: For a computable function h : N → (0, 1] we have
• EC ⊆ h-MC ⊆ SC; and
• h-MC = EC if h(n) ≤ c < 1
Generally, Barmpalias, Rettinger and Z. (2003) show that
P
• hMC = EC if the sum (1 − h(n)) = ∞;
• h-MC = SC if the sum
P
(1 − h(n)) is finite and computable; and
P
• EC ( h-MC ( SC if the sum (1 − h(n)) is finite and is not computable.
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