Limits of the Kucera-Gacs coding method

TL;DR

This paper surveys the limitations of the Kucera-Gacs coding method for transforming arbitrary reals into Martin-Loef random reals, and introduces a new optimal coding method with logarithmic redundancy.

Contribution

It analyzes the limitations of existing coding methods and presents a new approach that achieves optimal logarithmic redundancy in coding reals into Martin-Loef random reals.

Findings

Original methods have significant redundancy limitations.

New method achieves exponential improvement with logarithmic redundancy.

Provides a comprehensive comparison of coding approaches.

Abstract

Every real is computable from a Martin-Loef random real. This well known result in algorithmic randomness was proved by Kucera and Gacs. In this survey article we discuss various approaches to the problem of coding an arbitrary real into a Martin-Loef random real,and also describe new results concerning optimal methods of coding. We start with a simple presentation of the original methods of Kucera and Gacs and then rigorously demonstrate their limitations in terms of the size of the redundancy in the codes that they produce. Armed with a deeper understanding of these methods, we then proceed to motivate and illustrate aspects of the new coding method that was recently introduced by Barmpalias and Lewis-Pye and which achieves optimal logarithmic redundancy, an exponential improvement over the original redundancy bounds.

Equations

$$\sigma$$

$$\sum _n\sum _{\sigma \in 2^{L_n}}{2}^{-L_n+m_n-{\ell }_n}=\sum _n{2}^{L_n}\cdot 2^{-L_n+m_n-{\ell }_n}=\sum _n{2}^{m_n-{\ell }_n}<\mu (P).$$

$$\sum _{i\le s}{\ell }_i\le h\left(1+\sum _{i<s}{m}_i\right)\text{and}{m}_s+\sum _{i\le s}({\ell }_i-m_i)\le h_r\left(\sum _{i<s}{m}_i\right).$$

$$L_{s+1}=M_s+m_s+\sum _{i\le s}({\ell }_i-m_i)\le n+m_s+\sum _{i\le s}({\ell }_i-m_i)\le n+h_r(M_s)\le n+h_r(n)$$

$$(U_i)$$

$$V_{t+1}\cap [\sigma ]={\left(U_{n_{t+1}}\cap [\sigma ]\right)}^{\left[\le 2^{-|\sigma |}\cdot (1-2^{-g(t)-1})\right]},$$

$$\mu (V_{t+1})\le \prod _{i=0}^t\left(1-2^{-g(i)-1}\right).$$

$${\mu }_{\sigma }(P)+{\mu }_{\sigma }(U_{n_{t+1}})-{\mu }_{\sigma }(P\cap U_{n_{t+1}})\le 1$$

$${\mathcal{T}}^{\ast }$$

Full text

**George Barmpalias

**State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand.

E-mail: barmpalias@gmail.com. Web: http://barmpalias.net **Andrew Lewis-Pye

**Department of Mathematics, Columbia House, London School of Economics, Houghton St., London, WC2A 2AE, United Kingdom.

E-mail: A.Lewis7@lse.ac.uk. Web: http://aemlewis.co.uk

1 Introduction

Information means structure and regularity, while randomness means the lack of structure and regularity. One can formalize and even quantify this intuition in the context of algorithmic randomness and Kolmogorov complexity, where the interplay between information and randomness has been a principal driving force for much of the research.

[TABLE]

This question has various answers, depending on how it is formalized, but as we are going to see in the following discussion, for sufficiently strong randomness the answer is ‘not much’.

1.1 Finite information

In the case of a finite binary sequence (string) $\sigma$, let $K(\sigma)$ denote the prefix-free complexity of $\sigma$. Then $\sigma$ is $c$-incompressible if $K(\sigma)\geq|\sigma|-c$. Here we view the underlying optimal universal prefix-free machine $U$ as a decompressor or decoder, which takes a string/program $\tau$ and may output another string $\sigma$, in which case $\tau$ is regarded as a description of $\sigma$. Then $K(\sigma)$ is the length of the shortest description of $\sigma$ and the random strings are the $c$-incompressible strings for some $c$, which is known as the randomness deficiency. It is well known that the shortest description of a string is random, i.e. there exists a constant $c$ such that each shortest description is $c$-incompressible. In other words,

[TABLE]

which may seem as a strong positive answer to Question (1.0.1), in the sense that every string $\sigma$ can be coded into a random string. The following proposition, however, points in the opposite direction:

Proposition 1.1** (Folklore).**

If $U$ is an optimal universal prefix-free machine then there exists a constant $c$ such that $U(\sigma)\uparrow$ for all strings $\sigma$ such that $K(\sigma)\geq|\sigma|+c$.111The proof of this fact is based on the idea that each string in the domain of $U$ is a prefix-free description of itself (modulo some fixed overhead). In other words, if $U(\sigma)\downarrow$ then $\sigma$ can be used to describe itself, with respect to some prefix-free machine that is then simulated by $U$, producing a $U$-description of $\sigma$ of length $|\sigma|+c$ for some constant $c$.

Viewing $U$ as a universal decompressor, Proposition (1.1) says that a sufficiently random string cannot be decoded into anything, which means that in that sense it does not effectively code any information. According to this fact, Question (1.0.1) has a strong negative answer.

1.2 Bennett’s analogy for infinite information

The notions and issues discussed in the previous section have infinitary analogues which concern coding infinite binary sequences (reals) into random reals. For sufficiently strong (yet still moderate) notions of randomness for reals (such as the randomness corresponding to statistical tests or predictions that are definable in arithmetic with two quantifiers), the answer to Question (1.0.1) is not much; such random reals cannot solve the halting problem or even compute a complete extension of Peano Arithmetic. Charles Bennett (see [Ben88]) asked if Question (1.0.1) can have a strongly positive answer, just as in the finite case, for a standard notion of algorithmic randomness such as Martin-Löf randomness. Remarkably, Kučera [Kuč85] and Gács [Gác86] gave a positive answer to Bennett’s question.

Theorem 1.2** (Kučera-Gács theorem).**

Every real is computable from a Martin-Löf random real.

Bennett [Ben88] commented:

[TABLE]

Here we argue that Bennett’s suggested analogy between (1.1.1) and Theorem 1.2 is not precise, in the sense that it misses the quantitative aspect of (1.1.1) – namely that the random code can be chosen short (of length the complexity of the string). It is much easier to code $\sigma$ into a random string which is much longer than $\sigma$, than code it into a random string of length at most $|\sigma|$. The analogue of ‘length of code’ for infinite codes, is the use-function in a purported Turing reduction underlying the computation of a real $X$ from a random real $Y$. The use function for the reduction is a function $f$ such that for each $n$, the first $n$ bits of $X$ can be uniformly computed from the first $f(n)$ bits of $Y$.

1.3 A quantitative version of the Kučera-Gács theorem?

The more precise version of Bennett’s suggested analogy that we have just discussed can be summarized in Table 1, where $\sigma^{\ast}$ denotes the shortest program for $\sigma$.222If there are several shortest strings $\tau$ such that $U(\tau)=\sigma$ then $\sigma^{\ast}$ denotes the one that converges the fastest. So what is the analogue of the code length in the Kučera-Gács theorem? If we code a real $X$ into a Martin-Löf random real $Y$, how many bits of $Y$ do we need in order to compute the first $n$ bits of $X$? This question has been discussed in the literature (see below) but, until recently, only very incomplete answers were known. Kučera [Kuč85] did not provide tight calculations and various textbook presentations of the theorem (e.g. Nies [Nie09, Section 3.3]) estimate the use-function in this reduction of $X$ to a Martin-Löf random $Y$ to be of the order $n^{2}$. In fact, the actual bound that can be obtained by Kučera’s method is $n\log n$. Gács used a more elaborate argument and obtained the upper bound $n+\sqrt{n}\cdot\log n$, which is $n+\mathop{\bf o}\/\left({n}\right)$, and the same bound was also obtained later by Merkle and Mihailović [MM04] who used an argument in terms of supermartingales.

1.4 Coding into random reals, since Kučera and Gács

The Kučera-Gács coding method has been combined with various arguments in order to produce Martin-Löf random reals with specific computational properties. The first application already appeared in [Kuč89], where a high incomplete Martin-Löf random real computable from the halting problem was constructed. Downey and Miller [DM06] and later Barmpalias, Downey and Ng [BDN11] presented a variety of different versions of this method, which allow some control over the degree of the random real which is coded into. Doty [Dot06] revisited the Kučera-Gács theorem from the viewpoint of constructive dimension. He characterized the asymptotics of the redundancy in computations of an infinite sequence $X$ from a random oracle in terms of the constructive dimension of $X$. We should also mention that this is not the only method for coding into members of a positive measure $\Pi^{0}_{1}$ class (or into the class of Martin-Löf random reals). Barmpalias, Lewis-Pye and Ng [BLN10] used a different method in order to show that every degree that computes a complete extension of Peano Arithmetic is the supremum of two Martin-Löf random degrees.

It is fair to say that all of these methods rely heavily on the density of reals inside a nonempty $\Pi^{0}_{1}$ class that consists entirely of Martin-Löf reals. This is also true of more recent works such as Bienvenu, Greenberg, Kučera, Nies and Turetsky [BGK*+*15], Day and Miller [DM15] and Miyabe, Nies and Zhang [MNZ15]. Khan [Kha15] explicitly studies the properties of density inside $\Pi^{0}_{1}$ classes, not necessarily consisting entirely of Martin-Löf random reals. Much of this work is concerned with lower bounds on the density that a Martin-Löf real has inside every $\Pi^{0}_{1}$ class that contains it. In our analysis of the Kučera-Gács theorem we isolate the role of density in the argument and show that, in a sense, tighter oracle-use in computations from Martin-Löf random oracles is only possible through methods that do not rely on such density requirements.

2 Coding into an effectively closed set subject to density requirements

The arguments of Kučera and Gács both provide a method for coding an arbitrary real $X$ into a member of an effectively closed set $P$ (a $\Pi^{0}_{1}$ class), and rely on certain density requirements for the set of reals $\mathcal{P}$. The connection to Theorem 1.2 is that the class of Martin-Löf random reals is the effective union of countably many $\Pi^{0}_{1}$ classes of positive measure. The only difference in the two methods is that Kučera codes $X$ one-bit-at-a-time (with each bit of $X$ coded into a specified segment of $Y$) while Gács codes $X$ block-by-block into $Y$, with respect to a specified segmentation of $X$.

2.1 Overview of the Kučera-Gács argument

In general, we can code $m_{i}$ many bits of $X$ at the $i$th coding step, using a block in $Y$ of length $\ell_{i}$, as Table 2 indicates. We leave the parameters $(m_{i}),(\ell_{i})$ unspecified for now, while in the following it will become clear what the growth of this sequence needs to be in order for the argument to work. Note that the bit-by-bit version of the coding is the special case where $m_{i}=1$ for all $i$. The basic form of the coding process (which we shall elaborate on later) can be outlined as follows.

1. (1)

Start with a $\Pi^{0}_{1}$ class $\mathcal{P}\neq\emptyset$ which only contains (Martin-Löf) randoms. 2. (2)

Choose the length $m_{i}$ of the block coded at step $i$. 3. (3)

Choose the length $\ell_{i}=m_{i}+g(i)$ used for coding the $i$th block. 4. (4)

The oracle-use for the first $M_{n}=\sum_{i<n}m_{i}$ bits is $L_{n}=\sum_{i<n}\ell_{i}$. 5. (5)

Form a subclass $\mathcal{P}^{\ast}$ of $\mathcal{P}$ with the property that for all but finitely many $n$ and for each $X\in\mathcal{P}^{\ast}$, there are at least $2^{m_{n}}$ extensions of $X\upharpoonright_{L_{n}}$ of length $L_{n+1}$ which have infinite extensions in $\mathcal{P}^{\ast}$. 6. (6)

Argue that $\mathcal{P}^{\ast}\neq\emptyset$   (due to the growth of $(\ell_{i})$, relative to $(m_{i})$).

A crucial fact here is that if $\mathcal{P}$ is a $\Pi^{0}_{1}$ class then $\mathcal{P}^{\ast}$ is also a $\Pi^{0}_{1}$ class. In Section 2.2 we turn this outline into a modular proof, which makes the required properties of the parameters $(m_{i}),(\ell_{i})$ transparent. We will show that apart from the computability of $(m_{i}),(\ell_{i})$, the following facts characterize the necessary and sufficient constraints on the two sequences for the coding to work.

1. (i)

If $\sum_{i}2^{m_{i}-\ell_{i}}<\infty$ then there exists a $\Pi^{0}_{1}$ class of positive measure that consists entirely of Martin-Löf random reals such that $\mathcal{P}^{\ast}\neq\emptyset$; 2. (ii)

If $\sum_{i}2^{m_{i}-\ell_{i}}=\infty$ and $\mathcal{P}$ is a $\Pi^{0}_{1}$ class such that $\mathcal{P}^{\ast}\neq\emptyset$ then $\mathcal{P}$ contains a real which is not Martin-Löf random.

2.2 The general Kučera-Gács argument

We give a modular argument in terms of $\Pi^{0}_{1}$ classes, showing that every real is computable from a Martin-Löf random real, and consisting of a few simple lemmas. We use Martin-Löf’s paradigm of algorithmic randomness, much like in the original argument of Kučera and Gács.333However our presentation has been significantly assisted by Merkle and Mihailović [MM04], who phrased the argument in terms of martingales. In the next definition, recall that for finite $\sigma$, $[\sigma]$ is the set of all infinite extensions of $\sigma$.

Definition 2.1** (Extension property).**

Given a $\Pi^{0}_{1}$ class $P$ and sequences $(m_{i})$, $(\ell_{i})$ of positive integers, let $M_{n}:=\sum_{i<n}m_{i}$, $L_{n}:=\sum_{i<n}\ell_{i}$ and say that $P$ has the extension property with respect to $(m_{i})$, $(\ell_{i})$ if for each $i$, every string $\sigma$ of length $L_{i}$ with $[\sigma]\cap P\neq\emptyset$ has at least $2^{m_{i}}$ extensions $\tau$ of length $L_{i+1}$ such that $P\cap[\tau]\neq\emptyset$.

The first lemma says that subject to certain density conditions on a $\Pi^{0}_{1}$ class $P$, every real is computable from a member of $P$.

Lemma 2.2** (General block coding).**

Suppose that $P$ is a $\Pi^{0}_{1}$ class, and $(m_{i})$, $(\ell_{i})$ are computable sequences of positive integers. If $P$ has the extension property with respect to $(m_{i})$, $(\ell_{i})$, then every sequence is computable from a real in $P$ with use $L_{s+1}$ for bits in $[M_{s},M_{s+1})$.

Proof..

For any string $\sigma$ of length $L_{i}$ consider the variables $w_{0}(\sigma)[s],\dots w_{2^{m_{i}}-1}(\sigma)[s]$ for strings, which are defined dynamically according to the approximation $(P_{s})$ to $P$ as follows. At stage 0 let $w_{j}(\sigma)[0]\uparrow$ for all $j<2^{m_{i}}$. At stage $s+1$ find the least $t<2^{m_{i}}$ such that one of the following holds:

1. (a)

$w_{t}(\sigma)[s]\uparrow$; 2. (b)

$w_{t}(\sigma)[s]\downarrow$ and $[w_{t}(\sigma)[s]]\cap P_{s+1}=\emptyset$.

In case (a) look for the lexicographically least $\ell_{i}$-bit extension $\tau$ of $\sigma$ such that $[\tau]\cap P_{s+1}\neq\emptyset$ and $w_{j}(\sigma)[s]\not\simeq\tau$ for all $j<2^{m_{i}}$. If no such exists, terminate the process (hence let $w_{j}(\sigma)[n]\simeq w_{j}(\sigma)[s]$ for all $j<2^{m_{i}}$ and all $n>s$). Otherwise define $w_{t}(\sigma)[s+1]=\tau$ and go to the next stage. In case (b) let $w_{t}(\sigma)[s+1]\uparrow$ and go to the next stage.

By the hypothesis of the lemma, for every $i$ and every string $\sigma$ of length $L_{i}$ such that $[\sigma]\cap P\neq\emptyset$, the words $w_{j}(\sigma)[s]$, $j<2^{m_{i}}$ reach limits $w_{j}(\sigma)$ after finitely many stages, such that:

• •

$j\neq k\Rightarrow w_{j}(\sigma)\neq w_{k}(\sigma)$ for all $j,k<2^{m_{i}}$;

• •

$[w_{j}(\sigma)]\cap P\neq\emptyset$.

Consider the Turing functional $\Phi$ which, given oracle $Y$, works inductively as follows. Suppose that $\Phi(Y\upharpoonright_{L_{i}})\upharpoonright_{M_{i}}$ has been calculated. The functional then searches for the least pair $(j,s)$ (under a fixed effective ordering of all pairs, of order type $\omega$) such that $j<2^{m_{i}}$, $w_{j}(Y\upharpoonright_{L_{i}})[s]\downarrow$ and is a prefix of $Y$. For $\tau$ which is the $j$th string of length $m_{i}$ (under the lexicographical ordering) the functional then defines $\Phi(Y\upharpoonright_{L_{i}+\ell_{i}})=\Phi(Y\upharpoonright_{L_{i}})\ast\tau$. By construction $\Phi$ is consistent, and if $\Phi(Y\upharpoonright_{L_{i}})$ is defined it has length $M_{i}$. Finally we show that $\Phi$ is onto the Cantor space. Given $X$ we can inductively construct $Y$ such that $\Phi(Y)=X$. Suppose that we have constructed $Y\upharpoonright_{L_{i}}$ such that $\Phi(Y\upharpoonright_{L_{i}})=X\upharpoonright_{M_{i}}$ and $Y\upharpoonright_{L_{i}}$ is extendible in $P$. Let $\sigma$ be the unique string of length $m_{i}$ such that $X\upharpoonright_{M_{i}}\ast\sigma$ is a prefix of $X$. Then $w_{j}(Y\upharpoonright_{L_{i}})$ is defined for all $j<2^{m_{i}}$ and takes distinct values for different $j$. Let $t$ be the index of $\sigma$ in the lexicographical ordering of strings of length $m_{i}$. Then let $Y\upharpoonright_{L_{i+1}}=w_{t}(Y\upharpoonright_{L_{i}})$. Clearly $Y\upharpoonright_{L_{i+1}}$ is extendible in $P$ and moreover $\Phi(Y\upharpoonright_{L_{i+1}})=X\upharpoonright_{M_{i+1}}$. This completes the induction step in the construction of $Y$ and shows that $\Phi(Y)=X$. ∎

Recall that for $\sigma$ of length $n$, the $P$-density of $\sigma$ is defined to be $2^{n}\cdot\mu([\sigma]\cap P)$, where $\mu$ denotes Lebesgue measure on Cantor space.

Definition 2.3** (Density property).**

Given $P$, $(m_{i})$, $(\ell_{i})$ as in Definition 2.1 we say that $P$ has the density property with respect to $(m_{i})$, $(\ell_{i})$ if for each $n$, every string of length $L_{n}$ with $[\sigma]\cap P\neq\emptyset$ has $P$-density at least $2^{m_{n}-\ell_{n}}$.

Lemma 2.4** (Density and extensions).**

Given $P$, $(m_{i})$, $(\ell_{i})$ as in Definition 2.1, if $P$ has the density property with respect to $(m_{i})$, $(\ell_{i})$ then it also has the extension property with respect to $(m_{i})$, $(\ell_{i})$.

Proof..

This follows from the general fact that if the $P$-density of $\sigma$ is at least $2^{-t}$ for some $t$, then given any $m$, there are at least $2^{m}$ extensions $\tau$ of $\sigma$ of length $|\sigma|+t+m$ such that $[\tau]\cap P\neq\emptyset$. In order to prove the latter fact, suppose for a contradiction that it is not true. Then the $P$-density of $\sigma$ would be at most $(2^{m}-1)\cdot 2^{-m-t}=2^{-t}-2^{-m-t}<2^{-t}$ which contradicts the hypothesis. ∎

Lemma 2.5** (Lower bounds on the density).**

Let $P$ be a $\Pi^{0}_{1}$ class and let $(m_{i}),(\ell_{i})$ be computable sequences of positive integers such that $\sum_{i}2^{m_{i}-\ell_{i}}<\mu(P)$. Then there exists a $\Pi^{0}_{1}$ class $P^{\ast}\subseteq P$ which has the extension property with respect to $(m_{i}),(\ell_{i})$.

Proof..

We construct a $\Sigma^{0}_{1}$ class $Q$ in stages and let $(P_{s})$ be a $\Pi^{0}_{1}$ approximation to $P$, where each $P_{s}$ is a clopen set. A string $\sigma$ is active at stage $s+1$ if it is of length $L_{n}$ for some $n$ and $[\sigma]\cap(P_{s}-Q_{s})\neq\emptyset$. Moreover $\sigma$ of length $L_{n}$ requires attention at stage $s+1$ if it is active at this stage and the $(P_{s}-Q_{s})$-density of $\sigma$ is at most $2^{m_{n}-\ell_{n}}$. At stage $s+1$, we pick the least string of length $<s$ which requires attention (if such exists) and enumerate $[\sigma]\cap(P_{s}-Q_{s})$ into $Q$. If this enumeration occurred, we say that the construction acted on string $\sigma$ at stage $s+1$. This concludes the construction.

First we establish an upper bound on the measure of $Q=\cup_{s}Q_{s}$. Clearly the construction can act on a string at most once. The measure that is added to $Q$ at stage $s+1$ if the construction acts on $\sigma$ of length $L_{n}$ at this stage, is at most $2^{-L_{n}+m_{n}-\ell_{n}}$. Therefore the total measure enumerated into $Q$ throughout the construction is bounded above by:

[TABLE]

It follows that $P^{\ast}:=P-Q$ is a nonempty $\Pi^{0}_{1}$ class, and by the construction we have that for every $n$ and every string $\sigma$ of length $L_{n}$, if $[\sigma]\cap P^{\ast}\neq\emptyset$ then the $P^{\ast}$-density of $\sigma$ is at least $2^{m_{n}-\ell_{n}}$. By Lemma 2.4 this means that every $P^{\ast}$-extendible string of length $L_{n}$ for some $n$ has at least $2^{m_{n}}$ many $P^{\ast}$-extendible extensions of length $L_{n}+m_{n}-(m_{n}-\ell_{n})=L_{n+1}$. Hence $P^{\ast}$ has the extension property with respect to $(m_{i})$, $(\ell_{i})$. ∎

Corollary 2.6** (General block coding).**

Suppose that $P$ is a $\Pi^{0}_{1}$ class, and $(m_{i})$, $(\ell_{i})$ are computable sequences of positive integers. If $\sum_{i}2^{m_{i}-\ell_{i}}<\mu(P)$ then every sequence is computable from a real in $P$ with use $L_{s+1}$ for bits in $[M_{s},M_{s+1})$.

Proof..

By Lemma 2.5 we can consider a $\Pi^{0}_{1}$ class $P^{\ast}\subseteq P$ which has the extension property with respect to $(m_{i})$, $(\ell_{i})$. The statement then follows by Lemma 2.2 and the fact that $P^{\ast}\subseteq P$. ∎

Note that, while Corollary 2.6 seems to require (a) $\sum_{i}2^{m_{i}-\ell_{i}}<\mu(P)$, if $P$ is of positive measure then the condition (b) $\sum_{i}2^{m_{i}-\ell_{i}}<\infty$ suffices to ensure that $\sum_{i\geq d}2^{m_{i}-\ell_{i}}<\mu(P)$ for some $d$ – meaning that (b) is sufficient to give the existence of the required functional (albeit with some added non-uniformity required in specifying the index of the reduction).

2.3 The oracle-use in the general Kučera-Gács coding argument

Recall that if $X$ can be computed from $Y$ with the use function on argument $n$ bounded by $n+g(n)$, then we say that $X$ can be computed from $Y$ with redundancy $g(n)$. Note that in the following corollary we do not need to require that $h,h_{r}$ are computable.

Corollary 2.7**.**

Suppose $(m_{i})$, $(\ell_{i})$ are computable sequences of positive integers with $\sum_{i}2^{m_{i}-\ell_{i}}<1$ and suppose $h,h_{r}$ are nondecreasing functions such that:

[TABLE]

Then if $P$ is a $\Pi^{0}_{1}$ class of positive measure, any sequence is computable from a real in $P$ with oracle-use $h$ and redundancy $h_{r}$.

Proof..

The first claim follows directly from Corollary 2.6 and for the second, recall that in the same corollary, for each $s$ and each $n\in[M_{s},M_{s+1})$, the length of the initial segment of $Y$ that is used for the computation of $X\upharpoonright_{n}$ is at most

[TABLE]

where the second inequality was obtained from the main property assumed for $h_{r}$, and the last inequality follows from the monotonicity of $h_{r}$. ∎

Without yet specifying the sequences $(m_{i})$, $(\ell_{i})$, the condition $\sum_{i}2^{m_{i}-\ell_{i}}<1$ means that a near-optimal choice for the sequence $(\ell_{i}-m_{i})$ is $\lceil 2\log(i+2)\rceil$. This means that $\sum_{i}(\ell_{i}-m_{i})$ will be of the order $\log(n!)$ or $n\log n$. We may now consider an appropriate choice for the sequence $(m_{i})$, which roughly minimizes the redundancy established in Corollary 2.7. For Kučera’s coding we have $m_{i}=1$ for all $i$ which means that the redundancy in this type of bit-by-bit coding is $n\log n$ (modulo a constant). Gács chose the sequence $m_{i}=i+1$, and the reader may verify that this growth-rate of the blocks of the coded stream gives a near-optimal redundancy in Corollary 2.7.444For example the choices $m_{i}=(i+1)^{2}$ or $m_{i}=\sqrt{i+1}$ produce redundancy considerably above Gács’ $\sqrt{n}\cdot\log n$ upper bound. In this case the function $h_{r}(n)=\sqrt{n}\cdot\log n$ satisfies the second displayed inequality of Corollary 2.7 (for almost all $n$), since $n+1+n\log n\leq\sqrt{(n+1)n/2}\cdot\log((n+1)n/2)$ for almost $n$. Hence every real is computable from a Martin-Löf random real with this redundancy, much like Gács had observed.

We can now intuitively understand how the redundancy upper bounds $n\log n$ and $\sqrt{n}\cdot\log n$, of Kučera and Gács respectively, are produced. The argument of Section 2.2 describes a coding process where in $n$ coding steps we code $M_{n}$ many bits of $X$ into $L_{n}$ many bits of $Y$. The parameter $g(i):=\ell_{i}-m_{i}$ can be seen as an overhead of the $i$th coding step, i.e. the number of additional bits we use in $Y$ in order to code the next $m_{i}$ bits of $X$. Moreover, Corollary 2.7 says that these overheads are accumulated along the coding steps and push the redundancy of the computation to become larger over time. In particular, the number $\sum_{i<n}g(i)$ is the redundancy (total overhead accumulated) corresponding to $n$ coding steps. Due to the condition $\sum_{i}2^{-g(i)}<1$ in Corollary 2.7 a representative choice for $g$ is $2\log(n+1)$, which means that $\sum_{i<n}g(i)$ needs to be of the order $\log(n!)$ or (by Stirling’s formula) $n\log n$.

In the case of Kučera’s argument, $n$ bits of $X$ are coded in $n$ coding steps, so the redundancy for the computation of $n$ bits of $X$ from $Y$ following Kučera’s argument is of the order $n\log n$. If we are free to choose $(m_{i})$, note that a fast-growing choice will make the accumulated overhead smaller (since the coding steps for any initial segment of $X$ become less) but a different type of overhead, namely the parameter $m_{s}$ in the second inequality of Corollary 2.7, pushes the redundancy higher. Gács’ choice of $m_{i}=i+1$ means that in $n$ coding steps there are $\sum_{i\leq n}m_{i}\approx n^{2}$ many bits of $X$ coded into $Y$. Hence the coding of $X\upharpoonright_{n}$ requires roughly $\sqrt{n}$ coding steps, which accumulate a total of $\sqrt{n}\cdot\log\sqrt{n}\approx\sqrt{n}\cdot\log n$ in overheads according to the previous discussion. For this reason, Gács’ redundancy is of the order $\sqrt{n}\cdot\log n$. We may observe that in Gács’ coding, the length of the next coding block $m_{n+1}$ is both:

1. (a)

the number of coding steps performed so far; 2. (b)

roughly equal to the accumulated overhead from the coding steps performed so far.

2.4 Some limits of the Kučera-Gács method

In this section we will frequently identify a set of finite strings $V$ with the $\Sigma^{0}_{1}$ class specified by $V$, i.e. the set of infinite sequences extending elements of $V$. In the following proof we use the notation $\mu_{\sigma}(C)$ for a string $\sigma$ and a set of reals $C$, which is the measure of $C$ relative to $[\sigma]$. More precisely $\mu_{\sigma}(C)=\mu(C\cup[\sigma])\cdot 2^{|\sigma|}$.

Lemma 2.8**.**

Let $P$ be a $\Pi^{0}_{1}$ class, $g$ a computable function taking positive values, such that $\sum_{i}2^{-g(i)}=\infty$. Let $(n_{i})$ be a computable sequence such that $n_{i+1}>n_{i}+g(i)$ for all $i$. If

[TABLE]

then every Martin-Löf random real $X\in\cap_{i}(P\cap U_{i})$ has a prefix in some in $U_{n_{t}}$ with $P$-density at most $2^{-g(t)}$.

Proof..

We define a uniform sequence $(V_{i})$ of $\Sigma^{0}_{1}$ classes such that $V_{t}\supseteq V_{t+1}$ for all $t$, inductively as follows. Let $V_{0}$ (as a set of finite strings) consist of all the strings of length $n_{0}$. Assuming that $V_{t}$ has been defined, for each $\sigma\in V_{t}$ define

[TABLE]

where for any real $r$ and any $\Sigma^{0}_{1}$ class $C$ with an underlying computable enumeration $C[s]$ the expression $C^{[\leq r]}$ denotes the class $C[s_{\ast}]$ where $s_{\ast}$ is the largest stage $s$ such that $\mu(C[s])\leq r$ if such a stage exists, and $s_{\ast}=\infty$ otherwise (in which case we let $C[\infty]=C$). Clearly for each $t$ the set $V_{t}$ consists of strings of length $n_{t}$. Then for each $t$ we have $\mu(V_{t+1})\leq(1-2^{-g(t)-1})\cdot\mu(V_{t})$ so

[TABLE]

By hypothesis, $\sum_{i}2^{-g(i)}=\infty$ so $\prod_{i=0}^{\infty}(1-2^{-g(i)-1})=0$. Since $g$ is computable, there exists a computable increasing sequence $(k_{i})$ such that $\prod_{i=0}^{k_{t}}(1-2^{-g(i)-1})<2^{-t}$ for all $t>0$. Hence $(V_{k_{i}})$ is a Martin-Löf test. Now let $X$ be a Martin-Löf random real with $X\in\cap_{i}(P\cap U_{i})$, as in the statement of the lemma. Since $X$ is Martin-Löf random, $X\notin\cap_{i}V_{k_{i}}=\cap_{i}V_{i}$ and there exists a maximum $t$ such $X$ has a prefix $\sigma$ in $V_{t}$. By the maximality of $t$ we have $X\not\in V_{t+1}$ and since $X\in U_{n_{t+1}}$ we must have $\mu_{\sigma}(U_{n_{t+1}})>1-2^{-g(t)-1}$, because otherwise a prefix of $X$ would enter $V_{t+1}$. Also $\mu_{\sigma}(P\cap U_{n_{t+1}})\leq 2^{|\sigma|}\cdot\mu(P\cap U_{n_{t+1}})\leq 2^{|\sigma|-n_{t+1}}$. Since $\sigma\in V_{t}$, the length of $\sigma$ is $n_{t}$. Since $n_{t+1}>n_{t}+g(t)$ we have $\mu_{\sigma}(P\cap U_{n_{t+1}})\leq 2^{-g(t)-1}$. From the fact that

[TABLE]

we can deduce that $\mu_{\sigma}(P)\leq 2^{-g(t)}$. Since $\sigma$ is a prefix of $X$ of length $n_{t}$, this concludes the proof. ∎

Corollary 2.9**.**

Suppose that $(m_{i}),(\ell_{i})$ are computable sequences of positive integers with $\sum_{i}2^{m_{i}-\ell_{i}}=\infty$. Then every $\Pi^{0}_{1}$ class consisting entirely of Martin-Löf random reals, which has the density property with respect to $(m_{i}),(\ell_{i})$, is empty.

Proof..

We apply Lemma 2.8 with $n_{k}=L_{k}=\sum_{i<k}\ell_{i}$ and $g(i)=\ell_{i}-m_{i}$. First note that $n_{k+1}=n_{k}+\ell_{k}>n_{k}+g(k)$ because $g(k)<\ell_{k}$, so the hypothesis of Lemma 2.8 for $(n_{i})$ holds. Second, for each $i$ let $\sigma_{i}^{\ast}$ be the leftmost $P$-extendible string of length $i$ and let $U_{i}$ consist of $\sigma_{i}^{\ast}$ as well as the strings of length $i$ which are lexicographically to the left of $\sigma_{i}^{\ast}$. Then $(U_{i})$ is uniformly c.e. and $\mu(P\cap U_{i})=\mu(P\cap[\sigma_{i}^{\ast}])\leq 2^{-i}$ for all $i$. Now suppose that $P$ is non-empty and consider the leftmost path $X$ through $P$. By our assumptions regarding $P$, the real $X$ is Martin-Löf random, so by Lemma 2.8 there exists some $k$ such that the $P$-density of $X\upharpoonright_{L_{k}}$ is less than $2^{m_{k}-\ell_{k}}$. This means that there is a $P$-extendible string of length $L_{k}$ with $P$-density below $2^{m_{k}-\ell_{k}}$, so $P$ does not have the density property with respect to $(m_{i}),(\ell_{i})$. ∎

Corollary 2.10** (Lower bounds on the density inside a $\Pi^{0}_{1}$ class of random reals).**

Let $P$ be a nonempty $\Pi^{0}_{1}$ class consisting entirely of Martin-Löf random reals, let $g$ be a computable function, and let $(L_{i})$ be an increasing sequence of positive integers such that $L_{t+1}>L_{t}+g(t)$ for all $t$. Then the the following are equivalent:

1. (a)

For every $i$ the $P$-density of any $P$-extendible string of length $L_{i}$ is $\Omega(2^{-g(i)})$ 2. (b)

$\sum_{i}2^{-g(i)}<\infty$**

where the asymptotic notation $\Omega(2^{-g(i)})$ means $\geq 2^{-g(i)-c}$ for some constant $c$.

3 Coding into randoms without density assumptions

In [BLP16] a new coding method was introduced which allows for coding every real into a Martin-Löf random real with optimal, logarithmic redundancy. We call this method density-free coding as it does not rely on density assumptions inside $\Pi^{0}_{1}$ classes, which is also the reason why it gives an exponentially better redundancy upper bound.

Lemma 3.1** (Density-free coding, from [BLP16]).**

Let $(u_{i})$ be a nondecreasing computable sequence and let $\mathcal{P}$ be a $\Pi^{0}_{1}$ class. If $\sum_{i}2^{i-u_{i}}<\mu(\mathcal{P})$ then every binary stream is uniformly computable from some member of $\mathcal{P}$ with oracle-use $(u_{i})$.

Note that by letting $P$ be a $\Pi^{0}_{1}$ class of Martin-Löf random reals of sufficiently large measure, Lemma 3.1 shows that every real is computable from a Martin-Löf random real with use $n+2\log n$, i.e. with logarithmic redundancy. On the other hand in [BLPT16] it was shown that this is optimal, in the sense that if $\sum_{i}2^{i-u_{i}}=\infty$ then there is a real which is not computable from any Martin-Löf random real with use $n\mapsto u_{n}$. In particular, given a real $\epsilon$, redundancy $\epsilon\cdot\log n$ in a computation from a random oracle is possible for every real if and only if $\epsilon>1$.

We shall not give a proof of Lemma 3.1. Instead, we will discuss some aspects of this more general coding method, which contrasts the more restricted Kučera-Gács coding whose limitations we have already explored.

3.1 Coding as a labelling task

Coding every real into a path through a tree $\mathcal{T}$ in the Cantor space involves constructing a Turing functional $\Phi$ which is onto the Cantor space, even when it is restricted to $\mathcal{T}$. In fact, this is normally done in such a way that there is a subtree $\mathcal{T}^{\ast}$ of $\mathcal{T}$ such that $\Phi$ is a bijection between $[\mathcal{T}^{\ast}]$ and $2^{\omega}$. In this case we refer to $\mathcal{T}^{\ast}$ as the code-tree. Suppose we fix $\mathcal{T}$ and consider constructing a functional for which the use $u_{n}$ on argument $n$ does not depend upon the oracle. Then the functional $\Phi$ can be constructed as a partial computable ‘labelling’ of the finite branches of $\mathcal{T}$. If the label $x_{\sigma}$ is placed on $\tau$, this means that $\Phi$ outputs $\sigma$ when $\tau$ is the oracle. If we also suppose that the use function is strictly increasing, then the labelling might be assumed to satisfy the following conditions:

1. (1)

only strings of lengths $u_{i},i\in\mathbb{N}$ of $\mathcal{T}$ can have a label; 2. (2)

the labels placed on strings of length $u_{i}$ of $\mathcal{T}$ are of the type $x_{\sigma}$ where $|\sigma|=i$; 3. (3)

if label $x_{\sigma}$ exists in $\mathcal{T}$ then all labels $x_{\rho}$, $\rho\in 2^{\leq|\sigma|}$ exist in $\mathcal{T}$; 4. (4)

each string in $\mathcal{T}$ can have at most one label; 5. (5)

if $\rho$ of length $u_{k}$ in $\mathcal{T}$ has label $x_{\sigma}$ then for each $i<k$, $\rho\upharpoonright_{u_{i}}$ has label $x_{\sigma\upharpoonright_{i}}$.

The reader may take a minute to view the Kuc̃era coding as detailed in Section 2.2 as a labelling satisfying the properties (1)-(5) above. It is clear that:

[TABLE]

The new coding behind Lemma 3.1 is also a labelling process, but in this case the code-tree can be much more complex.

3.2 Fully labelable trees

If $(u_{i})$ is an increasing sequence of positive integers, a $(u_{i})$-tree $T$ is a subset of $\{\lambda\}\cup(\cup_{i}2^{u_{i}})$ which contains the empty string and is downward closed, in the sense that for each $\sigma\in 2^{u_{i+1}}\cap T$, the string $\sigma\upharpoonright_{u_{i}}$ belongs to $T$. The elements of a $(u_{i})$-tree $T$ are called nodes and the $t$-level of $T$ consists of the nodes of $T$ of length $u_{t}$. The full binary tree of height $k$ is $2^{\leq k}$ ordered by the prefix relation. Note that a $(u_{i})$-tree is a tree, in the sense that it is a partially ordered set (with respect to the prefix relation) in which the predecessors of each member are linearly ordered. Hence given any $k\in\mathbb{N}$, we may talk about a $(u_{i})$-tree being isomorphic to the full binary tree of height $k$. When we talk about two trees being isomorphic, it is in this sense that we shall mean it – as partially ordered sets. A labelling of a $(u_{i})$-tree is a partial map from the nodes of the tree to the set of labels $\{x_{\sigma}\ |\ \sigma\in 2^{<\omega}\}$ which satisfies properties (1)-(5) of the previous section. A full labelling of a $(u_{i})$-tree is a labelling $\{x_{\sigma}\ |\ \sigma\in 2^{<\omega}\}$ with the property that for every $\sigma$ there exists a node on the $(u_{i})$-tree which has label $x_{\sigma}$.

A $(u_{i})$-tree is called fully labelable if there exists a full labelling of it. Figure 5 illustrates some examples of fully labelable trees of height 3. Note that here the nodes are binary strings (hence nodes of the full binary tree) but since they are nodes of a $(u_{i})$-tree, they can have more than two branches. Clearly, if $T_{0}\subseteq T_{1}$ are $(u_{i})$-trees and $T_{0}$ is fully labelable, then $T_{1}$ is also fully labelable. These definitions can be easily adapted to finite $(u_{i})$-trees (where the height is the length of its longest leaf). Figure 5 shows some examples of fully labelable finite $(u_{i})$-trees, while Figure 6 shows some examples of finite $(u_{i})$-trees which are not fully labelable.

Clearly any $(u_{i})$-tree which is isomorphic to the full binary tree, is fully labelable. The success of the Kučera coding was based on this fact, along with the fact that a $\Pi^{0}_{1}$ class of sufficient measure contains such a canonical tree (subject to the growth of $(u_{i})$). A similar remark can be made about the slightly more general Gács coding. We have already demonstrated that the density property that guarantees the extension property cannot be expected to hold if the growth of $(u_{i})$ is significantly less than $n+\sqrt{n}\cdot\log n$. Hence more efficient coding methods, such as the one behind Lemma 3.1, need to rely on a wider class of labelable trees.

Given two trees $T_{0},T_{1}$ (thought of as partially ordered sets), we say that $T_{0}$ is splice-reducible to $T_{1}$ if we can obtain $T_{1}$ from $T_{0}$ via a series of operations on the nodes of $T_{0}$, each consisting of splicing two sibling nodes into one – i.e. the two sibling nodes $u_{0}$ and $u_{1}$ are replaced by a single node $u$, for which the set of elements $>u$ is isomorphic to the set of nodes strictly greater than $u_{0}$ union the set of nodes strictly greater than $u_{1}$. The following result points to a concrete difference between Kučera coding and the general coding from [BLP16]: in Kučera coding the code-tree is an isomorphic copy of the full binary tree, while in [BLP16] the code-tree is only splice-reducible to an isomorphic copy of the full binary tree.555A similar remark can be made with respect to the Gács coding, only that instead of binary trees we need to consider a homogeneous trees, in the sense that for each level, every node of that level has the same number of successors.

Theorem 3.2**.**

Given a $(u_{i})$-tree $T$, the following are equivalent:

1. (a)

$T$* is a fully labelable $(u_{i})$-tree;* 2. (b)

$T$* is splice-reducible to an isomorphic copy of the full binary tree.*

Proof..

Suppose that $T$ is fully labelable. We describe how to produce the full binary tree by a repeated application of the splice operation between siblings of $T$. Fix a full labelling of $T$ and obtain the minimal fully labeled tree $T^{\prime}$ from $T$ by splicing the unlabelled nodes of $T$ onto labelled ones. Now all nodes of $T^{\prime}$ are labelled. Then gradually, starting from the first level and moving toward the last level of $T^{\prime}$, splice siblings with identical labels. Inductively, by the properties of the assumed labelling, the resulting $(u_{i})$-tree is isomorphic to the full binary tree.

Conversely, assume that $T$ is splice-reducible to a $(u_{i})$-tree which is isomorphic to the full binary tree. Then reversing the splice operations behind this reduction, we get a sequence of node splitting operations that transform an isomorphic $(u_{i})$-copy of the full binary tree into $T$. Since this $(u_{i})$-copy of the full binary tree has a full labeling, by making these labels persistent during the series of splitting operations that lead to $T$, we get a full labelling of $T$. ∎

The work in [BLP16] shows that if $(u_{i})$ is an increasing computable sequence, then any tree of measure more than $\sum_{i}2^{i-u_{i}}<\mu(\mathcal{P})$ has a full labelling. Moreover, such a labelling has a $\Pi^{0}_{1}$ approximation, given any $\Pi^{0}_{1}$ approximation of $\mathcal{P}$.