1. Introduction
A cornerstone of algorithmic randomness is Schnorr’s TheoremFootnote
1
that X is Martin-Löf (ML) random iff
$K(X\!\!\upharpoonright \! n)\ge ^+ n$
for all n (Chaitin [Reference Chaitin3]). Thus X being ML-random is equivalent to having all initial segments weakly K-random. Now this result relativizes and hence X is A-random iff
$K^A(X\!\!\upharpoonright \! n)\ge ^+ n$
, so that, for example, X is 2-random, meaning X is ML-random relative to
$\emptyset '$
, iff
$K^{\emptyset '}(X\!\!\upharpoonright \! n)\ge ^+ n$
, for all n.
It is slightly surprising that
$2$
-randomness can also be characterized using K (and C) without relativization.
Theorem 1.1.
-
• (Miller [Reference Miller10], Nies, Stephan, and Terwijn [Reference Nies, Stephan and Terwijn14]) X is 2-random iff
$C(X\!\!\upharpoonright \! n)\ge ^+ n$
for infinitely many n. -
• (Miller [Reference Miller11]) X is 2-random iff
$K(X\!\!\upharpoonright \! n)\ge ^+ n +K(n)$
for infinitely many n.
Theorem 1.1 suggests the following motivating question:
Question 1.2. Is it possible to define
$(k+1)$
-randomness using unrelativized initial segment complexity K, not the relativized notion
$K^{(k)}=_{\mathrm {def}}K^{\emptyset ^{(k)}}$
? Indeed, is it possible to define
$K^{(k)}$
using K?
On the face of it, why should there be a characterization of X being
$\emptyset ^{(900)}$
-random, have a definition purely involving the behaviour
$K(X\!\!\upharpoonright \! n)$
for
$n\in \omega $
? Indeed Solovay [Reference Solovay18] (see [Reference Downey and Hirschfeldt5, Section 10.2.3]) showed that
$K^{\emptyset '}$
and K have apparently complex relationships such as
$K^{\emptyset '}(n)\le K(n)-\alpha (n) + O(\log \alpha (n)),$
where
$\alpha (n)=\min \{K(m)\mid m\ge n\}$
, a fact we use in Section 6.
Nevertheless, we’ll show that Question 1.2 has a positive answer. To prove this, we will give a new analysis of the behaviour of sets of minimal descriptions, where
$\tau $
is a minimal description of
$\sigma $
if
$U(\tau )=\sigma $
and
$|\tau |= K(\sigma )$
. Such
$\tau $
’s must be incompressible, let there be a shorter description of
$\sigma $
. For example, we will show that although every
$\Delta _2^0$
set of minimal descriptions computes
$\emptyset '$
, there are, for our purposes, sufficiently tame sets of minimal descriptions called “semilow” sets, a notion which grew from Soare’s automorphism machinery [Reference Soare16].
1.1.
$k\ge 2$
To prove that we can define
$(k+1)$
-randomness using only K, our starting point is the following attractive result.
Theorem 1.3. (Bienvenu, Muchnik, Shen, and Vereshchagin [Reference Bienvenu, Muchnik, Shen and Vereschagin2])
$$ \begin{align*} K^{\emptyset'}(\sigma)=^+\limsup_n K(\sigma \mid n). \end{align*} $$
The same result holds for C in place of K.
Here
$K(\sigma \mid n)$
is the conditional complexity of
$\sigma $
given n. The reader might think that Theorem 1.3 does the job: We could relativize the result to get
$K^{\emptyset "}(\sigma )= \limsup _n K^{\emptyset '}(\sigma \mid n)$
and then replace
$ K^{\emptyset '}(\sigma \mid n)$
by a
$\limsup _m K((\sigma \mid n)\mid m)$
.
The problem is that
$K(\sigma \mid n)$
is not really an unrelativized notion, even though no oracles appear in a standard definition such as that in Li–Vitanyi [Reference Li and Vitanyi9]
$$ \begin{align*}K(\sigma \mid n)=\min\{|y|: U(\langle y,n\rangle)=\sigma\}.\end{align*} $$
However, since n is fixed, this definition hard-codes n as an oracle. More precisely, the definition in [Reference Li and Vitanyi9] is easily seen to be equivalent to the definition of conditional complexity used in [Reference Downey and Hirschfeldt5]
$$ \begin{align*}K(\sigma \mid n)=K^{\overline{n}}(\sigma),\end{align*} $$
where
$\overline {n}$
, a self-delimited version of n, is provided as an oracle.
Perhaps we might be able to get rid of the finite oracles. You might imagine that finite oracles have little effect, but this is not true in general. Indeed, in Section 6, we will see that finite strings can have very strong compression power. There we prove the following:
Theorem 1.4. For all e there is a string
$\rho $
such that for almost all
$\tau $
,
$K(\tau \mid \rho )<K(\tau )-e.$
That is, for
$\xi =\overline {\rho }$
,
$K^\xi (\tau )<K(\tau )-e.$
So we need a new plan.
1.2. The plan
Our plan is to leverage
$K(\sigma \mid n)$
, and we will do this using Symmetry of Information (Levin and Gács [Reference Gács8], Chaitin [Reference Chaitin3]) which says that
$$ \begin{align*} K(\sigma,n)=^+ K(n)+K(\sigma \mid n^*). \end{align*} $$
Here the reader should recall that
$K(\nu ,\rho )$
is the complexity of the pair
$\langle \nu ,\rho \rangle $
, and that for any string
$\tau $
,
$\tau ^*$
is the first minimal description to occur of length
$K(\tau )$
with
$U(\tau ^*)=\tau $
. (In the case of C we will write
$\tau ^*_C$
.)
While
$\tau ^*$
is a particular minimal code for
$\tau $
(the first to appear in a fixed effective search), it may not be the only code for
$\tau $
of length
$K(\tau )$
. We will also be interested in all minimal codes, and so we adopt the following notation.
Notation 1.5. For a universal machine U (prefix-free or otherwise), let
$N_U = \{ n^* : n \in \omega \}$
, where
$n^*$
is defined based on U. Let
$M_U = \{ \rho : U(\rho )\!\!\downarrow \wedge |\rho | = K(U(\rho ))\}$
.
When U is clear from context, we will omit the subscript.
Thus
$N_U$
contains only the first minimal description for any given string, while
$M_U$
contains every minimal description. Observe that
$N_U \subseteq M_U$
.
Remark 1.6. Note that
$\tau ^*$
is interchangeable with
$(\tau , K(\tau ))$
, in that there are computable functions to uniformly pass from one to the other: in one direction,
$\tau ^* \mapsto (U(\tau ^*), |\tau ^*|)$
; in the other direction, given
$(\tau , K(\tau ))$
, we search for the first string
$\rho $
of length
$K(\tau )$
with
$U(\rho ) = \tau $
. We will use
$\tau ^*$
, but the reader who prefers may substitute
$(\tau , K(\tau ))$
anywhere this occurs in the paper.
Similarly, any minimal code from
$M_U$
may be replaced with the corresponding
$(\tau , K(\tau ))$
. Again, given
$\rho \in M_U$
, it maps to
$(U(\rho ), |\rho |)$
. In the other direction the map is not uniform, but there is a constant bound on the number of minimal codes a string may have. Thus if
$\rho $
is a minimal code for
$\tau $
,
$\rho $
can be obtained from
$(\tau , K(\tau ))$
with only a constant amount of extra information: we begin enumerating all strings of length
$K(\tau )$
which code
$\tau $
, and we must know
$\tau $
’s position in this enumeration.
In Section 5 we will prove the following.
Theorem 1.7. For any universal prefix-free machine U,
$$ \begin{align*} \limsup_n K(\sigma \mid n^*) =^+ \limsup_{\tau \in M_U} K(\sigma\mid \tau) =^+ K^{\emptyset'}(\sigma). \end{align*} $$
Notice that by rearranging Symmetry of Information, we obtain
$K(\sigma \mid n^*) =^+ K(\sigma , n) - K(n)$
. Hence
$K^{\emptyset '}(\sigma ) =^+ \limsup _n [K(\sigma , n) - K(n)]$
, giving a definition of
$K^{\emptyset '}$
purely in terms of K without relativization.
By relativizing Theorem 1.3, we obtain
$$ \begin{align*} K^{A'}(\sigma) =^+ \limsup_n K^A(\sigma \mid n), \end{align*} $$
for all A.Footnote
2
By appropriately iterating and relativizing Theorem 1.7, we obtain a definition of
$K^{\emptyset ^{(k)}}$
for all
$k \in \omega $
, e.g.,
$$ \begin{align*} K^{\emptyset^{(2)}}(\sigma) &=^+ \limsup_n [K^{\emptyset'}(\sigma, n) - K^{\emptyset'}(n)]\\ &=^+ \limsup_n \left(\limsup_m [K(\sigma, n, m) - K(m)] - \limsup_m [K(n,m) - K(m)]\right). \end{align*} $$
From this follows a definition of k-randomness purely in terms of unrelativized K.
1.3. The complexity of the sets
$M_U$
and
$N_U$
The question arises how should we prove the
$\limsup _n K(\sigma \mid n^*)$
theorem? The answer really comes from understanding the behaviour of the set of minimal descriptions. One hint came from the unpublished work of Hirschfeldt.
Theorem 1.8 (Hirschfeldt, unpublished).
$C^{\emptyset '}(\sigma )=^+ \limsup _n C(\sigma \mid n^*_C)$
.
We will prove this result in Section 2. The method is to construct an infinite low subset of
$N_V$
, for V the machine generating C, and use some relativization tricks.
We had hoped to use this method for K, but unfortunately we were able to prove a result saying that this is impossible.
Theorem 1.9. Let U be a universal prefix-free machine, and let S be an infinite
$\Delta _2^0$
subset of
$M_U$
. Then
$\emptyset '\le _T S$
.
Note that
$M_U$
and
$N_U$
are of degree
$\mathbf {0}'$
(e.g., [Reference Downey and Hirschfeldt5]). For example,
$N_U\ge _{wtt} \Omega $
by the Coding Theorem (see [Reference Downey and Hirschfeldt5, Section 3.9]), and
$\Omega $
is wtt-complete. In view of Theorem 1.9, it would seem reasonable to suggest that they are introreducible, which in this context would mean that every infinite subset computes
$\emptyset^{\prime}$
. However, Joseph Miller proved that this is not the case. We include this also in Section 3.
We remark that the proof can also be adapted to show that an infinite
$\Delta ^0_2$
hitting set for a Solovay functionFootnote
3
must also be Turing complete. We prove these results, which are of independent interest, in Section 3.
In the end we found a way around these problems using an idea from studies in the automorphism group of the lattice of computably enumerable sets (Soare [Reference Soare16]) A set S is called semi-low iff
$$ \begin{align*} \{e\mid S\cap W_e \ne \emptyset\}\le_T \emptyset', \end{align*} $$
i.e., a pointwise version of being low. This notion was introduced as a method towards characterizing when the lattice of supersets of a c.e. set was isomorphic to the lattice of all c.e. sets. In Section 5, we will show that, although there cannot be an infinite low subset of
$M_U$
or
$N_U$
, there can be a semi-low one (which we believe to be of independent interest, especially contrasted with the previous theorem). In Section 4 we will prove that this is enough for our main result. There have been other uses of semi-lowness outside of the lattice of c.e. sets, such as Downey and Melnikov [Reference Downey and Melnikov7] in the study of abelian groups, but these seem sporadic at best.
1.4. The machine existence theorem
As we will be using it several times, we state the machine existence theorem and fix our notation surrounding it.
Definition 1.10. For a set
$A \subseteq 2^{<\omega } \times \omega $
,
$\text {wt}(A) = \sum _{(\sigma , s) \in A} 2^{-s}$
.
Theorem 1.11 (KC Theorem, Coding Theorem, or Machine Existence Theorem; see Downey and Hirschfeldt [Reference Downey and Hirschfeldt5, Section 3.6]).
If
$A \subseteq 2^{<\omega }\times \omega $
is c.e. and has
$\text {wt}(A) \le 1$
, then there is a prefix-free machine V such that for every
$(\sigma , s) \in A$
, there is a
$\rho $
with
$|\rho | = s$
and
$V(\rho ) = \sigma $
. Further, an index for V can be effectively obtained from a c.e. index for A.
Corollary 1.12. If
$A \subseteq 2^{<\omega }\times \omega $
is c.e. with
$\text {wt}(A) < \infty $
, then for all
$(\sigma , s) \in A$
,
$K(\sigma ) \le ^+ s$
.
Sets A of this form are sometimes called request sets.
The following is more of a proof technique, but we will state it as a corollary.
Corollary 1.13. As part of a uniform construction, we may effectively obtain an
$\epsilon> 0$
such that if we enumerate
$A \subseteq 2^{<\omega } \times \omega $
with
$\text {wt}(A) \le \epsilon $
, then
$K(\sigma ) \le s$
for every
$(\sigma ,s) \in A$
(observe the lack of additive constant).
Proof Fix U the universal prefix-free machine used to define K. We will generate an auxiliary c.e. set B with
$\text {wt}(B) \le 1$
. By the Recursion Theorem, we know a c.e. index for the set B we will enumerate. By the Machine Existence Theorem, this effectively gives us an index for a prefix-free machine V. From this we effectively obtain a string
$\rho $
with
$U(\rho \tau ) = V(\tau )$
for all
$\tau $
, and so
$K(\sigma ) \le t + |\rho |$
for all
$(\sigma , t) \in B$
.
Set
$\epsilon = 2^{-|\rho |}$
, and define B by enumerating
$(\sigma , s-|\rho |)$
whenever A enumerates
$(\sigma , s)$
, provided this enumeration does not put
$\text {wt}(B)$
over 1. If
$\text {wt}(A) \le \epsilon $
, then
$(\sigma , s-|\rho |)$
is enumerated into B for every pair
$(\sigma , s) \in A$
, and so
$K(\sigma ) \le (s-|\rho |) + |\rho | = s$
, as desired.
2. Hirschfeldt’s Theorem
We prove Theorem 1.8. We will prove that
$$ \begin{align*} C^{\emptyset'}(\sigma)=\limsup_n C(\sigma \mid n^*_C). \end{align*} $$
Consider the
$\Pi _1^0$
class of sequences
$$ \begin{align*} P=\{(m_0, m_1, \dots) \mid \forall n\, [2^n \le m < 2^{n+1} \wedge C(m_n) \ge n]\}. \end{align*} $$
A simple counting argument shows that there is an appropriate
$m_n$
for every n, and so P is nonempty. Since there are only
$2^n$
options for
$m_n$
,
$P \subseteq 2^\omega $
under an appropriate effective identification. So there is a low infinite path
$L = (m_0, m_1, \dots )$
by the Low Basis Theorem.
Recall that for
$2^n \le m < 2^{n+1}$
,
$C(m) \le ^+ n$
, so fix the least d such that
$\exists ^\infty n\, C(m_n) = n+d$
, and fix an N such that
$C(m_n) \ge n+d$
for all
$n \ge N$
. Then
$$ \begin{align*} X = \{ (m_n)^*_C : n \ge N \wedge C(m_n) = n+d\} \end{align*} $$
is L-c.e. and an infinite subset of
$N_V$
, where V is the universal machine defining C. Fix
$(\rho _i)_{i \in \omega }$
an L-computable enumeration of X.
Then by relativizing Theorem 1.3, we have
$$ \begin{align*} C^{\emptyset'}(\sigma) =^+ C^{L'}(\sigma) =^+ \limsup_i C^L (\sigma \mid i). \end{align*} $$
Note that we can L-effectively pass between i and
$\rho _i$
, so
$C^L(\sigma \mid i) =^+ C^L(\sigma \mid \rho _i)$
, giving
$$ \begin{align*} C^{\emptyset'}(\sigma) =^+ \limsup_{\rho \in X} C^L(\sigma\mid \rho). \end{align*} $$
We also have
$$ \begin{align*} \limsup_{\rho \in X} C^L(\sigma\mid \rho) \le^+ \limsup_{\rho \in X}C(\sigma\mid\rho), \end{align*} $$
since oracles can only help;
$$ \begin{align*} \limsup_{\rho \in X}C(\sigma\mid\rho) \le \limsup_{n^*_C} C(\sigma\mid n^*_C) \le \limsup_{\rho \in M_V} C(\sigma\mid \rho) \le \limsup_{\rho \in 2^{<\omega}}C(\sigma\mid \rho), \end{align*} $$
since
$X \subseteq N_V \subseteq M_V \subseteq 2^{<\omega }$
, and limit supremums over larger sets are larger; and finally
$$ \begin{align*} \limsup_{\rho \in 2^{<\omega}}C(\sigma\mid \rho) =^+ C^{\emptyset'}(\sigma), \end{align*} $$
by the unrelativized version of Theorem 1.3, after an effective identification of
$2^{<\omega }$
with
$\omega $
. Picking out the relevant bits, we see that
$$ \begin{align*} C^{\emptyset'}(\sigma)=^+ \limsup_n C(\sigma\mid n^*_C) =^+ \limsup_{\rho \in M_V} C(\sigma\mid \rho). \end{align*} $$
3. No low hitting sets
We prove Theorem 1.9. Fix a universal prefix-free machine U, and suppose that
$X=\lim _s X_s$
is an infinite
$\Delta _2^0$
subset of
$M_U$
. Fix
$\epsilon $
as in Corollary 1.13; we will enumerate an appropriate set
$A \subseteq 2^{<\omega } \times \omega $
.
We describe how we code whether
$n\in \emptyset '$
. Fix
$k \in \omega $
with
$$ \begin{align*} \frac1k < 2^{-(n+2)}\epsilon. \end{align*} $$
To do this coding, we will define a k-colouring
$\chi $
on
$\text {dom}(U)$
. This colouring will be unique to n; the colourings for other values of n will have no interaction.
We declare that colour i is small if
$$ \begin{align*} \sum_{\chi(\sigma)=i}2^{-|\sigma|}\le \frac{1}{k}. \end{align*} $$
This has natural approximations: at a stage s, based on the finitely many strings we have so far coloured, a colour may still be small or may have already proven itself to be large. Note that since colours are disjoint, and we have k colours, there is always at least one small colour.
Suppose that
$\sigma $
enters
$\mbox {dom}(U)$
at stage s. Let
$$ \begin{align*} r_i=\min\{|\tau| \mid \chi(\tau)=i \land \tau\in X_s\}. \end{align*} $$
We regard this as infinite if there is no such
$\tau $
. We fix a j maximizing
$r_j$
and colour
$\chi (\sigma ) = j$
.
Suppose that n enters
$\emptyset '$
at some stage
$s+1$
. Fix a single colour j which was small at stage s; we invalidate all the strings which had colour j at stage s. That is, for every
$\sigma \in \text {dom}(U)[s]$
with
$\chi (\sigma ) = j$
, we enumerate
$(U(\sigma ), |\sigma |-1)$
into A. Provided
$\text {wt}(A) \le \epsilon $
, this will ensure that
$\sigma \not \in M_U$
for each such
$\sigma $
.
As j was small at stage s, the weight of these pairs is at most
$\frac 2k < 2^{-(n+1)}\epsilon $
. Thus, summing over the strategies for every n,
$\text {wt}(A) \le \sum _n 2^{-(n+1)}\epsilon = \epsilon $
, as required.
Claim 3.1. X has members of every colour.
Proof Fix A the set of colours which occur in X, and suppose this is not all the colours.
Fix a length n such that every colour in A occurs on a string
$\tau \in X$
with
$|\tau | \le n$
, and fix t sufficiently large such that X has converged on strings of length at most n by stage t, i.e., if
$|\tau | \le n$
, then for all
$s \ge t$
,
$X_s(\tau ) = X(\tau )$
.
Since X is infinite, it contains some
$\tau $
which enters
$\text {dom}(U)$
at some stage
$s> t$
. At stage s,
$r_i \le n$
for every
$i \in A$
, whereas
$r_i> n$
for every colour
$i \not \in A$
. So
$\chi (\tau )$
will be a colour not in A, contrary to choice of A.
We can now state our procedure for computing
$\emptyset '(n)$
from X: in the colouring for n, search for a stage t such that for every colour i, some element of X has been given colour i by stage t; then output
$\emptyset ^{\prime }_t(n)$
.
As just argued, there is eventually some stage at which X intersects every colour, so this algorithm is total. Suppose first that
$n \not \in \emptyset '$
. Then certainly
$\emptyset ^{\prime }_t(n) = 0$
, as desired.
Suppose instead that
$n \in \emptyset '$
, and fix the stage
$s+1$
at which it enters. Fix the chosen colour j. Then no
$\sigma $
which received colour j at or before stage s belongs to
$M_U$
, and so cannot belong to X. Thus colour j witnesses
$t \not \le s$
, giving
$\emptyset ^{\prime }_t(n) = 1$
, as desired.
This concludes the proof of Theorem 1.9.
The same method can be used to prove the following:
Corollary 3.2. Suppose that X is an infinite
$\Delta _2^0$
set of hitting points for a Solovay function F. That is, a set S of points n where
$F(n)=^+ K(n)$
. Then
$\emptyset '\le _T S$
.
We remark that Corollary 3.2 improves a result of Bienvenu, Downey, Merkle, and Nies [Reference Bienvenu, Downey, Merkle and Nies1] who showed that the collection of all hitting points is Turing complete.
The reader should note that if
$\sigma =m^*$
, then
$\sigma $
must be weakly K-random in that
$K(\sigma )\ge ^+ |\sigma |$
. The reason is that if
$K(\sigma )<< |\sigma |$
then using the KC Theorem, we can use
$\sigma ^*$
to describe m, in a machine M we build. This would show that
$K(m)<<|\sigma |=|m^*|$
, a contradiction. This brings in to focus the question of precisely which weakly K-random strings are minimal descriptions. By the Low Basis Theorem, there are infinite low collections of weakly K-random strings. At most finitely many can be minimal descriptions. Another consequence of Theorem 1.9 is the followingFootnote
4
.
Corollary 3.3. If X is a
$\Delta ^0_2$
collection of weakly K-random strings
$($
that is,
$K(\sigma )\ge ^+ |\sigma |)$
, and
$|X \cap N_U| = \infty $
, then X computes
$\emptyset^{\prime}$
.
Proof Fix d such that for every
$\sigma \in X$
,
$K(\sigma )> |\sigma |-d$
. For every n, let
$n^*_s$
be the natural stage s approximation to
$n^*$
. This may be undefined for small s, but it will eventually converge to the true
$n^*$
. Further, if
$n^*_s\!\!\downarrow $
and
$n^*_{s+1} \neq n^*_s$
, then
$|n^*_{s+1}| < |n^*_s|$
.
Again fix
$\epsilon $
as in Corollary 1.13. Fix k with
$2^{-k} < \epsilon $
. For every n and s with
$|n_s^*|> |n^*| + k+d$
(a c.e. event), we enumerate
$(n, |n_s^*| - d)$
into A. Since for a fixed n there is at most one
$n^*_s$
of any given length, the weight of our requests is bounded by
$$ \begin{align*} \sum_{n} \sum_{i> |n^*|+k+d} 2^{-(i-d)} = 2^{-k} \sum_n 2^{-|n^*|} < 2^{-k}. \end{align*} $$
Thus
$\text {wt}(A) \le \epsilon $
, and so
$K(n^*_s) \le |n^*_s| - d$
for every such
$n^*_s$
.
It follows that if
$n^*_s \in X$
, then since
$K(n^*_s)> |n^*_s| - d$
by assumption,
$|n^*_s| \le |n^*| + k+d$
, or
$|n^*| \ge |n^*_s| - k - d$
. As in the proof of Theorem 1.8, this allows X to enumerate an infinite sequence from
$N_U$
. Since every infinite c.e. set contains an infinite computable set, and this relativizes, we get that X computes an infinite
$Y \subseteq N_U$
. As X is
$\Delta ^0_2$
, Y is as well, so
$Y \ge _T \emptyset '$
, and thus
$X \ge _T \emptyset '$
.
As we mentioned in the Introduction, Theorem 1.9 cannot be improved to show that all infinite subsets of
$N_U$
compute
$\emptyset '$
.
Theorem 3.4 (Joseph Miller, unpublished).
There is an infinite
$X \subset N_U$
which does not compute
$\emptyset '$
.
Proof Let P be a bounded
$\Pi ^0_1$
class of K-compression functionsFootnote
5
. Since we have an a priori upper bound of
$K(n) \le ^+ 2\log (n)$
, we may take
$P \subseteq 2^\omega $
. Let F be a “weakly-low for K” path. That is, there are infinitely many n with
$F(n)=^+K(n)$
. This can be shown to exist using the “low for
$\Omega $
”-Basis TheoremFootnote
6
(Downey, Hirschfeldt, Miller, and Nies [Reference Downey, Hirschfeldt, Miller and Nies6], Reimann and Slaman [Reference Reimann and Slaman15]) and the fact that low for
$\Omega $
is equivalent to “weakly low for K” (see Downey and Hirschfeldt [Reference Downey and Hirschfeldt5]).
Now fix the least c with
$K(n) = F(n)+c$
for infinitely many n, and fix an m with
$K(n) \ge F(n) + c$
for all
$n \ge m$
. F can enumerate an infinite subset of
$N_U$
:
$\{ n_s^* : n \ge m \wedge |n_s^*| = F(n)+c\}$
. Thus F computes an infinite
$X \subseteq N_U$
(again relativizing the fact that every infinite c.e. set has an infinite computable subset), Since F does not compute
$\emptyset '$
(since it is weakly low for K), X also does not compute
$\emptyset '$
.
4. Conditional complexity along semi-low sets
Semi-lowness has previously been studied for co-c.e. sets. We are interested in it for
$\Delta ^0_2$
sets, in which case it is not entirely clear that the following is the correct definitionFootnote
7
, but it is the definition relevant to our current interest.
Definition 4.1. Let
$(W_e)_{e \in \omega }$
be a standard listing of c.e. sets. A set X is semi-low if the set
$\{ e : X \cap W_e \neq \emptyset \}$
is
$\Delta ^0_2$
(i.e.,
$\le _T \emptyset '$
).
Recall Theorem 1.3:
$$ \begin{align*} K^{\emptyset'}(\sigma) =^+ \limsup_{n \in \omega} K(\sigma\mid n). \end{align*} $$
As we have seen, it can be helpful to consider
$\limsup _{n \in X} K(\sigma \mid n)$
for X infinite. It is immediate that this is
$\le ^+ K^{\emptyset '}(\sigma )$
, as we are taking a limit supremum over a smaller set. It turns out that for semi-low sets, we have equality.
Proposition 4.2. If X is semi-low and infinite, then
$K^{\emptyset '}(\sigma ) =^+ \limsup _{n \in X} K(\sigma \mid n)$
.
Proof As one direction is immediate, it remains to show that
$$ \begin{align*} K^{\emptyset'}(\sigma) \le^+ \limsup_{n \in X} K(\sigma\mid n). \end{align*} $$
We will work with request sets.
For each
$n \in \omega $
, define
$A_n = \{ (\sigma , s) : s \ge K(\sigma \mid n)\}$
. We may think of
$A_n$
as the request set generating
$K(\cdot |n)$
. Observe that
$\text {wt}(A_n) < 2$
for all n.
Note that for any finite set
$D \subset 2^{<\omega } \times \omega $
and any
$m \in \omega $
, the set
$\{ n\ge m : \text {wt}(D \cup A_n)> 2\}$
is c.e. (it is even primitive recursive with appropriate assumptions on
$K(\cdot |n)$
, but this is not necessary). Indeed this is uniform, so we may fix a total computable function e such that
$W_{e(D, m)} = \{n \ge m : \text {wt}(D\cup A_n)> 2\}$
, where D is given by a canonical index.
We will build a
$\emptyset '$
-enumerable request set B with
$\text {wt}(B) \le 2$
and such that for all
$\sigma $
, if
$s = \limsup _{n \in X} K(\sigma \mid n)$
, then
$(\sigma , s) \in B$
. By Corollary 1.12 relativized to
$\emptyset '$
, this will suffice to prove the result.
Fix an effective listing
$(\sigma _m, s_m)_{m \in \omega }$
of
$2^{<\omega }\times \omega $
such that every pair is repeated infinitely many times on the list. We define B as follows:
-
•
$B_0 = \emptyset $
. -
• Given
$B_m$
, fix
$D = B_m \cup \{(\sigma _m, s_m)\}$
. If
$X \cap W_{e(D,m)} = \emptyset $
, we let
$B_{m+1} = D$
; otherwise, we let
$B_{m+1} = B_m$
.
As X is semi-low,
$\emptyset '$
can run this construction, and so B is
$\emptyset '$
-enumerable.
Claim 4.3. For all
$n \ge m$
with
$n \in X$
,
$\text {wt}(B_m \cup A_n) \le 2$
, and thus
$\text {wt}(B) \le 2$
.
Proof Suppose not. Then as this clearly holds for
$B_0$
, we may fix
$m+1$
the least value where the claim is violated. So there is some
$n \ge m+1$
with
$n \in X$
,
$\text {wt}(B_m \cup A_n) \le 2$
and
$\text {wt}(B_{m+1} \cup A_n)> 2$
. As
$B_{m+1} \neq B_m$
, we must be in the case
$X \cap W_{e(D, m)} = \emptyset $
, with
$B_{m+1} = B_m \cup \{(\sigma _m, s_m)\} = D$
. But
$n \in X \cap W_{e(D, m)}$
, a contradiction.
That
$\text {wt}(B) \le 2$
then follows from X being infinite.
Claim 4.4. For any
$\sigma $
, if
$s = \limsup _{n \in X} K(\sigma \mid n)$
, then
$(\sigma , s) \in B$
.
Proof Fix an
$n_0$
such that for all
$n \ge n_0$
with
$n \in X$
,
$K(\sigma \mid n) \leq s$
. Then for all
$n \ge n_0$
with
$n \in X$
,
$(\sigma , s) \in A_n$
. Fix an
$m \ge n_0$
such that
$(\sigma , s) = (\sigma _m, s_m)$
. Let
$D = B_m \cup \{\sigma _m, s_m\}$
. Then for all
$n \ge m$
with
$n \in X$
,
$D \cup A_n = B_m \cup A_n$
, and
$\text {wt}(B_m \cup A_n) \le 2$
. So
$X \cap W_{e(D,m)} = \emptyset $
, and
$(\sigma , s) \in B_{m+1}$
by construction.
This completes the proof.
5. Conditional complexity along minimal codes
Fix a universal prefix-free machine U. We are interested in
$\limsup _{\tau \in M_U} K(\sigma \mid \tau )$
and
$\lim _{\tau \in N_U} K(\sigma \mid \tau )$
. First we verify that these values are machine independent.
Lemma 5.1. If U and V are universal prefix-free machines, and
$K(\cdot | \cdot )$
is defined from a third
$($
unnamed
$)$
universal prefix-free machine, then
$$ \begin{align*} \limsup_{\tau \in M_U} K(\sigma\mid \tau) =^+ \limsup_{\tau \in M_V} K(\sigma\mid \tau) =^+ \limsup_{\tau \in N_V} K(\sigma\mid \tau). \end{align*} $$
Proof By symmetry, and the fact that
$N_V \subseteq M_V$
, it suffices to show
$$ \begin{align*} \limsup_{\tau \in M_U} K(\sigma\mid \tau) \le^+ \limsup_{\tau \in N_V} K(\sigma\mid \tau). \end{align*} $$
By standard arguments, there is a constant c such that if
$\tau \in M_U$
,
$\rho \in M_V$
, and
$U(\tau ) = V(\rho )$
, then
$| |\tau | - |\rho | | \le c$
.
For each
$\tau \in 2^{<\omega }$
and
$i \in \mathbb {Z}$
with
$|i| \le c$
, let
$ \rho (\tau ,i)$
be the first
$\rho $
located with
$|\rho | = |\tau |+i$
and
$U(\tau )\!\!\downarrow = V(\rho )\!\!\downarrow $
, if such
$\rho $
exists. We define
$$ \begin{align*} B_\tau = \{ (\sigma, s) : \exists i\, \rho(\tau,i)\!\!\downarrow \wedge s \ge K(\sigma\mid \rho(\tau,i))\}. \end{align*} $$
Then
$\text {wt}(B_\tau ) \le \sum _{|i| \le c} \sum _{\sigma } 2\cdot 2^{-K(\sigma \mid \rho (\tau ,i))} \le (2c+1)\cdot 2$
, and thus these are uniformly given request sets.
It follows that
$K(\sigma \mid \tau ) \le ^+ K(\sigma \mid \rho (\tau ,i))$
for all i with
$\rho (\tau ,i)\!\!\downarrow $
. Note that if
$\tau \in M_U$
, then there is an i with
$\rho (\tau ,i)\!\!\downarrow \in N_V$
. The claim follows.
Proposition 5.2. There is a universal prefix-free machine U and an infinite, semi-low set
$X \subseteq N_U$
.
Proof Fix some universal prefix-free machine V. We define
$U(0^{3}\widehat {\phantom {\alpha }}\tau ) = V(\tau )$
for all
$\tau $
, which makes U universal while giving us the freedom to do as we like on other neighborhoods.
Let
$(N_s)_{s \in \omega }$
be the natural approximation to
$N_U$
. We will have semi-lowness requirements
$R_e$
and infiniteness requirements
$P_n$
. The strategy for each requirement will claim various strings in
$N_s$
, and each strategy will have a directive at every stage: meet or avoid. A string may only be claimed by a single strategy at a time, and a strategy will retain its claim on a string until either the string leaves
$N_s$
, or a higher priority strategy claims the string. In either case, the strategy will immediately relinquish its claim.
We will build a c.e. set
$A \subseteq 2^{<\omega } \times \omega $
. As we will argue, the sum
$\sum 2^{-|\sigma |}$
over all strings
$\sigma $
which are ever claimed in the construction will be bounded by 1/2. The first time a string
$\sigma $
is claimed by a strategy (i.e., it was unclaimed at all previous stages), we will immediately enumerate
$(k, |\sigma |-1)$
into A, where k is larger than any value yet seen in the construction. As the previous sum is bounded by 1/2,
$\text {wt}(A) \le 1$
. By the Machine Existence Theorem, we uniformly obtain the index of a corresponding prefix-free machine Q such that for every such pair
$(k, |\sigma |-1) \in A$
, there is a
$\tau $
with
$|\tau | = |\sigma |-1$
and
$Q(\tau ) = k$
.
We define
$U(1\widehat {\phantom {\alpha }}\tau ) = Q(\tau )$
for all
$\tau $
. Suppose
$\sigma $
is first claimed at stage s, and so we enumerate
$(k, |\sigma |-1)$
into A for some large k. Then for the appropriate
$\tau $
,
$|1\widehat {\phantom {\alpha }}\tau | = |\sigma |$
and
$U(1\widehat {\phantom {\alpha }}\tau ) = k$
. By the largeness of k,
$N_s$
contains no codes for k, so
$\tau $
will belong to
$N_U$
unless V enumerates a sufficiently shorter code at some subsequent stage. The idea is that whenever a potential element of
$N_U$
is claimed, we ensure it is replaced with a new element of the same length or shorter.
This completes the description of U, apart from describing how strategies claim strings. We order our requirements
$R_0, P_0, R_1, P_1, \dots $
. At stage s, we consider the first s requirements in order, implementing the following strategies.
Strategy for
$P_n$
:
$P_n$
will always have the meet directive, and will claim at most one string at a time. At stage s, if it retains a claimed string from the previous stage (i.e.,
$s>0$
,
$P_n$
had a claimed string at stage
$s-1$
, that string remains in
$N_s$
, and that string has not been claimed by a higher priority strategy earlier in stage s), then we take no further action. Otherwise, if there is a string in
$N_s$
of length at least
$2n+4$
and unclaimed by any higher priority strategy,
$P_n$
claims the least one (in some effective ordering). If
$P_n$
did not retain a claimed string, and there is no appropriate string to claim, we simply do nothing.
Strategy for
$R_e$
: The behaviour of
$R_e$
is a yo–yo: it continues to claim strings until the weight of the claimed strings surpasses
$2^{-(2e+4)}$
, at which point it stops and lets those strings bleed away. Once it has lost all of its claims, the strategy begins claiming new strings again. The details follow.
At stage s, let C be the set of strings which
$R_e$
retains the claim on from the previous stage: those strings which it claimed at stage
$s-1$
, which remain in
$N_s$
, and which were not claimed by a higher priority strategy earlier in stage s. If
$R_e$
was not considered at stage
$s-1$
(possibly because
$s = 0$
), then
$C=\emptyset $
.
Let
$w = \sum _{\sigma \in C} 2^{-|\sigma |}$
. Our directive for
$R_e$
at stage s will depend on w and on
$R_e$
’s directive at stage
$s-1$
:
-
• If
$w = 0$
(i.e.,
$C = \emptyset $
), then
$R_e$
has the directive avoid at stage s. -
• If
$0 < w \le 2^{-(2e+4)}$
, then
$R_e$
retains the same directive as it had at the previous stage (
$w> 0$
entails that
$R_e$
was considered at the previous stage). -
• If
$w> 2^{-(2e+4)}$
, then
$R_e$
has the directive meet at stage s.
If our directive for
$R_e$
at stage s is meet, we take no further action at this stage.
If our directive for
$R_e$
at stage s is avoid, and there is a string in
$N_s \cap W_{e,s}$
of length at least
$2e+4$
and unclaimed by any higher priority strategy, then
$R_e$
claims the least such string (in some effective ordering). Our action for
$R_e$
at this stage is then complete. Note that we claim no more than one string for
$R_e$
at each stage.
This completes the construction.
Definition of X:
Let
$X_s$
be the set of strings
$\sigma $
claimed by a strategy with the meet directive at stage s.
Observe that if
$\sigma \in N_U$
is claimed by a strategy, then there are only two possibilities: that strategy may retain its claim on
$\sigma $
forever, or the claim on
$\sigma $
may pass to a higher priority strategy. As we will argue, each
$R_e$
strategy changes its directive only finitely many times. It follows that
$X = \lim _s X_s$
is defined.
Verification:
First we must keep our promises.
Claim 5.3. At any stage s, for the strategy
$R_e$
, the value
$w = w(e,s)$
is at most
$2^{-(2e+3)}$
.
Proof At each stage,
$R_e$
claims at most one string, and that string will always have length at least
$2e+4$
. So
$w(e,s+1) - w(e,s) \le 2^{-(2e+4)}$
. Further,
$R_e$
only claims a string if
$w(e,s) \le 2^{-(2e+4)}$
, so
$w(e,s+1)$
is at most
$2^{-(2e+4)} + 2^{-(2e+4)} = 2^{-(2e+3)}$
.
Claim 5.4. The sum
$\sum 2^{-|\sigma |}$
over all strings
$\sigma $
which are ever claimed in the course of the construction is at most
$1/2$
.
Proof If a string
$\sigma $
is claimed during the construction, there are three possible fates for
$\sigma $
: 1) there is some n such that
$P_n$
claims
$\sigma $
at almost every stage; 2) there is some e such that
$R_e$
claims
$\sigma $
at almost every stage; 3)
$\sigma $
leaves
$N_U$
. We consider each case in turn.
Fix n. By construction, there is at most one string which is ultimately claimed by
$P_n$
, and such a string has length at least
$2n+4$
. So the sum
$\sum 2^{-|\sigma |}$
over all strings
$\sigma $
ultimately claimed by
$P_n$
is bounded by
$2^{-(2n+4)}$
.
Fix e. Let
$\hat {C}$
be the set of strings
$\sigma $
such that
$R_e$
claims
$\sigma $
at almost every stage. Then each
$\sigma \in \hat {C}$
contributes to almost every
$w(e,s)$
, so
$\sum _{\sigma \in \hat {C}}2^{-|\sigma |} \le \sup _s w(e,s) \le 2^{-(2e+3)}$
.
Finally, consider strings which leave
$N_U$
. We will split this case into two subcases, based on whether the given string extends
$000$
, and so was introduced by our copying of V, or it extends 1, and so was introduced by our other actions. Since the extensions of
$000$
in the domain of U form an antichain, the sum
$\sum 2^{-|\sigma |}$
over all strings in the first subcase is bounded by
$1/8$
.
Consider now the second subcase. Note that strings never leave
$N_U$
because of our action; instead, if
$\sigma $
leaves
$N_U$
, then there must be some
$\tau \in N_V$
with
$V(\tau ) = U(\sigma )$
and
$|\tau | < |\sigma |-3$
. As we always choose our values large, if
$\sigma _0$
and
$\sigma _1$
are distinct strings from this subcase,
$U(\sigma _0) \neq U(\sigma _1)$
, so the corresponding
$\tau $
s are distinct. So summing over the
$\sigma $
of this subcase, we have
$\sum 2^{-|\sigma |} < \frac 18 \sum _{\tau \in N_V} 2^{-|\tau |} < \frac 18$
. Putting these all together, our desired sum is bounded by
$$ \begin{align*}6 \sum_{n \in \omega} 2^{-(2n+4)} + \sum_{e \in \omega} 2^{-(2e+3)} + \frac18 + \frac18 \le \frac12, \end{align*} $$
as desired.
Claim 5.5. Each
$R_e$
changes its directive only finitely many times.
Proof Suppose not. Then there is a sequence of stages
$s_0 < s_1 < \dots $
such that
$R_e$
has directive avoid at stage
$s_i$
, and has directive meet at stage
$s_i+1$
, for every i. In order to switch from meet at stage
$s_i+1$
back to avoid at stage
$s_{i+1}$
, every string claimed by
$R_e$
at stage
$s_i+1$
must either be stolen by a higher priority strategy or leave
$N_U$
, both of which are irreversible. Thus the strings which contribute to
$w(e,s_i+1)$
must be entirely different from those which contribute to
$w(e, s_j+1)$
for
$j \neq i$
. But
$w(e,s_i+1)> 2^{-(2e+4)}$
for every i, and the strings which contribute to
$w(e,s_i+1)$
all belong to
$\text {dom}(U)$
, a contradiction.
Thus we may speak of an
$R_j$
strategy’s ultimate directive.
Our promises being met, the construction of X is as described. Now we verify that X has the desired properties.
Claim 5.6. For each n, the strategy
$P_n$
ultimately claims a string which it never renounces its claim upon, and thus X is infinite.
Proof Fix n. It suffices to argue that there is some string in
$N_U$
of length at least
$2n+4$
which is never claimed by any strategy.
Fix
$s_0$
such that
$N_{s_0}$
has converged on all strings of length less than
$2n+4$
. We build a sequence
$\sigma _0, \sigma _1, \ldots \in N_U$
:
-
• Fix some
$\sigma _0 \in N_U \setminus \text {dom}(U_{s_0})$
. Such a
$\sigma _0$
must exist, as U is universal (and in particular, surjective). -
• If
$\sigma _i$
is eventually claimed by some strategy, then the construction responds by enumerating a
$\tau $
into
$\text {dom}(U)$
with
$|\tau | = |\sigma _i|$
and
$U(\tau )$
not any previously seen value. It may be that
$\tau $
is not a minimal code, but there is some
$\sigma _{i+1} \in N_U$
with
$U(\sigma _{i+1}) = U(\tau )$
, and
$\sigma _{i+1}$
enters
$\text {dom}(U)$
after the stage at which
$\sigma _i$
is claimed (by the largeness of
$U(\tau )$
).
Inductively, we see that
$U(\sigma _i) \neq U(\sigma _j)$
for any
$j < i$
, and so the sequence
$\sigma _0, \sigma _1, \dots $
is injective. Further,
$|\sigma _{i+1}| \le |\sigma _i|$
. As
$\sum _{\sigma \in N_U} 2^{-|\sigma |} < 1$
, this sequence must be finite, so there is some
$\sigma _i$
which is never claimed by any strategy. Since
$\sigma _i$
enters
$\text {dom}(U)$
after stage
$s_0$
,
$|\sigma _i| \ge 2n+4$
by choice of
$s_0$
.
Claim 5.7. X is semi-low.
Proof We give an algorithm for determining whether
$X \cap W_e = \emptyset $
, using oracle
$\emptyset '$
. First,
$\emptyset '$
can determine a stage
$s_0$
such that every
$R_j$
strategy with
$j \le e$
has settled on its ultimate directive, and such that each
$P_n$
strategy with
$n < e$
has made its ultimate claim.
We may ignore those
$R_j$
with
$j \le e$
which have avoid as their ultimate directive. For the remaining, they have claimed some finitely many strings by stage
$s_0$
, and none will ever claim another string. With oracle
$\emptyset '$
, we can examine the entire finite collection to determine if there is a string
$\sigma $
among them which remains claimed by its current strategy forever, and with
$\sigma \in W_e$
.
We claim that there is such a
$\sigma $
if and only if
$X \cap W_e \neq \emptyset $
. In the one direction, if there is such a
$\sigma $
, then
$\sigma \in X$
by construction, so
$\sigma \in X \cap W_e$
.
In the other direction, if there is no such
$\sigma $
, note that this implies that
$R_e$
’s ultimate directive is avoid—if it were meet, then
$R_e$
’s strings are amongst those examined, so it must eventually renounce its claim to all of them, resulting in
$R_e$
changing directive to avoid, contrary to choice of
$s_0$
. Now for any
$\tau \in W_e \cap N_U$
, we consider two cases:
$|\tau | < 2e+4$
and
$|\tau | \ge 2e+4$
.
If
$|\tau | < 2e+4$
, then
$\tau $
is too short to be claimed by any strategy of lower priority than
$R_e$
, and by assumption
$\tau $
cannot be ultimately claimed by any strategy of higher priority with ultimate directive meet. So
$\tau \not \in X$
.
If
$|\tau | \ge 2e+4$
, then
$\tau $
will eventually be claimed by
$R_e$
, by construction, and so
$\tau \not \in X$
.
This completes the proof.
Corollary 5.8. For some, and hence any, universal prefix-free machine U,
$$ \begin{align*} \limsup_{\tau\in N_U} K(\sigma\mid \tau) = \limsup_{\tau \in M_U} K(\sigma\mid \tau) =^+ K^{\emptyset'}(\sigma). \end{align*} $$
Proof As this is independent of choice of machine, let U and X be as in Proposition 5.2. Then
$$ \begin{align*} K^{\emptyset'}(\sigma) =^+ \limsup_{\tau \in X} K(\sigma\mid \tau) \le \limsup_{\tau \in N_U} K(\sigma\mid \tau) \le \limsup_{\tau \in M_U} K(\sigma\mid \tau) \le \limsup_{\tau \in 2^{<\omega}} K(\sigma\mid \tau) =^+ K^{\emptyset'}(\sigma), \end{align*} $$
where the first equality is by Proposition 4.2, the last is by Theorem 1.3, and the inequalities are by subset.
6. Where the limsup’s live, and finite strings as oracles
Here we collect some miscellaneous results about finite strings as oracles. The first is motivated by our (numerous!) failed attempts to prove Theorem 1.7 before we finally discovered the method of Sections 4 and 5.
One avenue we pursued was attempting to determine for which m does
$K(\sigma \mid m)$
achieve
$\limsup _n K(\sigma \mid n)$
. A natural candidate is the nondeficiency stages: fix a computable enumeration
$(a_m)_{m \in \omega }$
of
$\emptyset '$
, and define
$$ \begin{align*} E = \{ m : (\forall n> m)\, [a_m < a_n]\}. \end{align*} $$
This is the basis for the method of true stages (see Montalbán [Reference Montalbán12] for a modern interpretation for higher level priority arguments, but the idea going back to Dekker [Reference Dekker4], as per Soare [Reference Soare17, Chapter V 2.5]), where the elements of E are employed because they make correct guesses about
$\emptyset '$
(as we shall see in a moment).
However, this turns out to be approaching from the wrong direction. Since
$K^{\emptyset '}(\sigma ) = \limsup _n K(\sigma \mid n)$
, to find places where the limit supremum is achieved, we are not concerned with doing as well as
$\emptyset '$
—we are concerned with doing no better than
$\emptyset '$
. Thus we are looking not for n which are powerful, but for those which are weak.
The following result says that for
$m \in E$
,
$K(\sigma \mid m)$
does much better than
$\limsup _n K(\sigma \mid n)$
.
Proposition 6.1.
$$ \begin{align*} \limsup_{m \in E} K(\sigma \mid m) =^+ K^{\emptyset^{(2)}}(\sigma). \end{align*} $$
Proof For any m, define
$\tau _m \in 2^{<\omega }$
to be the string of length
$a_m$
such that
$\tau _m(x) = 1$
iff
$x = a_n$
for some
$n < m$
. Note that
$m \mapsto \tau _m$
is effective, so
$K(\sigma \mid m) \le ^+ K(\sigma \mid \tau _m)$
. Also,
$\sigma _m \prec \emptyset '$
iff
$m \in E$
, so
$$ \begin{align*} \limsup_{m \in E} K(\sigma \mid m) &\le^+ \limsup_{m \in E} K(\sigma \mid \emptyset'\!\!\upharpoonright\! a_m)\\ &\le \limsup_n K(\sigma \mid \emptyset'\!\!\upharpoonright\! n)\\ &=^+ \limsup_n K^{\emptyset'}(\sigma\mid n). \end{align*} $$
Conversely,
$\emptyset '$
can compute the increasing enumeration of E,
$E = \{ b_0 < b_1 < \dots \}$
, so
$K^{\emptyset '}(\sigma \mid n) \le ^+ K(\sigma \mid b_n)$
, giving
$$ \begin{align*} \limsup_{m \in E} K(\sigma \mid m) =^+ \limsup_n K^{\emptyset'}(\sigma\mid n). \end{align*} $$
By Theorem 1.3 relativized to
$\emptyset '$
, this is (up to an additive constant)
$K^{\emptyset ^{(2)}}(\sigma )$
.
The reader might note the following somewhat paradoxical situation. The natural proof to show that
$K^{\emptyset '}(\sigma )\le ^+ \limsup _n K(\sigma \mid n)$
is to fully approximate
$K^{\emptyset '}(\sigma )[n]$
at each stage n, where both the computations and oracles are approximated for n stages. We would do this as part of the computation of
$K^{\overline {n}}(\sigma )$
for some machine
$M^{\overline {n}}(\sigma )$
via the Machine Existence Theorem as mentioned above. Therefore, for all stages
$t>n$
it can only be that
$K(\sigma \mid n)[t]\le ^+ K(\sigma \mid n)[n] \le ^+ K^{\emptyset '}(\sigma )[n].$
The true value of
$K^{\emptyset '}(\sigma )$
must have been achieved at a true stage, but we see above, it does not happen at almost all true stages. Thus it must be achieved at infinitely many non-true stages s, but where
$K^{\emptyset '}[s]=^+ K^{\emptyset '}(\sigma ).$
We don’t really understand the characteristics of such “almost true” stages s. We also point out that the limsups appear to be achieved for different s’s for different
$\sigma $
’s.
Theorem 1.3 says that for almost any string
$\sigma $
, almost any finite oracle can aid in the compression of
$\sigma $
. A priori, however, there is no reason to expect there to be a single finite oracle which aids in the compression of almost every string. Nevertheless, this is the case.
Theorem 6.2. For all e there is a string
$\rho $
such that for almost all
$\tau $
,
$K(\tau \mid \rho )<K(\tau )-e.$
That is for
$\xi =\overline {\rho }$
,
$K^\xi (\tau )<K(\tau )-e.$
Our proof is based on the conditional complexity variant of the Machine Existence Theorem, which we first state.
Proposition 6.3. Suppose
$A \subseteq 2^{<\omega } \times \omega \times 2^{<\omega }$
is a c.e. set such that for every
$\tau \in 2^{<\omega }$
,
$\sum _{(\sigma , s, \tau ) \in A} 2^{-s} \le 1$
. Then for all
$(\sigma , s, \tau ) \in A$
,
$K(\sigma \mid \tau ) \le ^+ s$
.
Proof of Theorem 6.2
We will enumerate a c.e. set A. Fix an effective bijection
$\tau = \langle D, k \rangle $
between
$\tau \in 2^{<\omega }$
and pairs
$\langle D, k \rangle $
with
$D \subset 2^{<\omega }$
finite and
$k \in \omega $
. For
$\tau = \langle D, k \rangle $
, for every
$\sigma \in 2^{<\omega } \setminus D$
, we enumerate
$(\sigma , t-k, \tau )$
into A for every
$t \ge K(\sigma )$
, provided doing so does not cause
$\sum _{(\sigma , s, \tau ) \in A} 2^{-s}$
to exceed 1.
Fix the constant c such that
$K(\sigma \mid \tau ) \le s+c$
for every
$(\sigma ,s, \tau ) \in A$
. As
$\sum _\sigma 2^{-K(\sigma )} < 1$
, there is some finite D such that
$\sum _{\sigma \not \in D} 2^{-K(\sigma )} < 2^{-(e+c+1)}$
. Fix
$\tau = \langle D, e+c \rangle $
. Then
$\sum _{\sigma \not \in D} \sum _{t \ge K(\sigma } 2^{-(t - e - c)} < 1$
, so
$(\sigma , t-e-c, \tau )$
is successfully enumerated into A for all such
$\sigma $
and t. Thus
$K(\sigma \mid \tau ) \le K(\sigma ) - e - c+c = K(\sigma )-e$
for all
$\sigma \not \in D$
.
Define
$\rho $
to be e-compressing if for almost all
$\tau $
,
$K(\tau \mid \rho )<K(\tau )-e$
.
Question 6.4. What can be said about the set
$C_e=\{\rho \mid \rho $
is e-compressing
$\}$
?
Question 6.5. Is there a definition of k-randomness purely involving C?
Acknowledgements
We thank Denis Hirschfeldt, Joseph Miller, and André Nies for helpful discussions.
Funding
All authors were supported by the Marsden Fund of New Zealand, with Lu on a Postdoctoral Fellowship.
Open access
