On the incomputability of computable dimension

TL;DR

This paper demonstrates that for certain simple computable subsets of the Cantor space, various notions of dimension such as Hausdorff, constructive, and computable dimensions can be incomputable, highlighting fundamental limitations in their computability.

Contribution

It introduces an iterative tree construction to show the incomputability of these dimensions in specific computable sets of the Cantor space.

Findings

Hausdorff, constructive, and computable dimensions can be incomputable for simple sets

The iterative tree construction reveals fundamental limits in dimension computability

Highlights the gap between classical and computable dimension concepts

Abstract

Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.

Equations

$$\forall w(w\in X^{\ast }\to |X{|}^{\sigma }\cdot d(w)\ge \sum _{x\in X}d(wx)).$$

$$\forall w(w\in X^{\ast }\to \frac{\mathcal{V}(w)}{|X{|}^{(1-\sigma )\cdot |w|}}=d(w))$$

$$\displaystyle\mbox{Moreover, for $0\leq t\leq p_{i}\cdot\ell_{i}$ we have\ }q_{i}\geq$$

$$q_i\le$$

$$\frac{r_i\cdot k_i+{\kappa }_i\cdot {\ell }_i}{r_i\cdot {\ell }_i+p_i\cdot {\ell }_i}=\frac{a\cdot k_i}{b\cdot {\ell }_i}$$

$$T_{i+1}:=T_{i}^{r_{i}}\cdot U_{i}\mbox{ with }U_{i}:=\left\{\begin{array}[]{ll}X^{p_{i}\cdot\ell_{i}},&\mbox{ if }q_{i+1}\geq q_{i}\mbox{ and}\\ \{u_{i}\},&\mbox{ otherwise}\end{array}\right.$$

$$|T_i|=|X{|}^{q_i\cdot {\ell }_i}.$$

Full text

Computable dimension along with constructive dimension was introduced by Lutz [siamcomp/Lutz03, ic/Lutz03] as a means for measuring the complexity of sets of infinite strings ($\omega$-words). Since then and prior to this constructive and computable dimension were investigated in connection with Hausdorff dimension (for a detailed account see [book_DowneyHirsch10, Section 13]). The results of [tocs/Hitch05, ic/St93, tcs/St07] show that the Hausdorff, constructive and computable dimensions of automaton definable sets of infinite strings (regular $\omega$-languages) are computable. In contrast to this Ko [ko98] derived examples of computable $\omega$-languages with an incomputable Hausdorff dimension.

In this paper we derive examples of computable $\omega$-languages of a simple structure which have not only incomputable Hausdorff dimension but also incomputable computable dimension. To this end we use an iteration of finite trees which resembles the tree construction of Furstenberg [Furstenberg70] (see also [ciaa/MSS18])

Lutz [siamcomp/Lutz03, ic/Lutz03] defines computable and constructive dimension via $\sigma$-(super)gales. Terwijn [terwijn04, apal/CST06] observed that this can also be done using Schnorr’s concept of combining martingales with (exponential) order functions [book_Schnorr71, Section 17]. For the computable $\omega$-languages constructed in this paper we can show that Schnorr’s concept is in some details more precise than Lutz’s approach.

1. Notation

In this section we introduce the notation used throughout the paper. By $\mathbb{N}=\{0,1,2,\ldots\}$ we denote the set of natural numbers, by $\mathbb{Q}$ the set of rational numbers, and $\mathbb{R}$ are the real numbers.

Let $X$ be an alphabet of cardinality $|X|\geq 2$. By $X^{*}$ we denote the set of finite words on $X$, including the empty word $e$, and $X^{\omega}$ is the set of infinite strings ($\omega$-words) over $X$. Subsets of $X^{*}$ will be referred to as languages and subsets of $X^{\omega}$ as $\omega$-languages.

For $w\in X^{*}$ and $\eta\in X^{*}\cup X^{\omega}$ let $w\cdot{}\eta$ be their concatenation. This concatenation product extends in an obvious way to subsets $W\subseteq X^{*}$ and $B\subseteq X^{*}\cup X^{\omega}$. We denote by $|w|$ the length of the word $w\in X^{*}$ and $\mathbf{pref}(B)$ is the set of all finite prefixes of strings in $B\subseteq X^{*}\cup X^{\omega}$.

It is sometimes convenient to regard $X^{\omega}$ as Cantor space, that is, as the product space of the (discrete space) $X$. Here open sets in $X^{\omega}$ are those of the form $W\cdot X^{\omega}$ with $W\subseteq X^{*}$. Closed are sets $F\subseteq X^{\omega}$ which satisfy the condition $F=\{\xi:\mathbf{pref}(\xi)\subseteq\mathbf{pref}(F)\}$.

For a computable domain $\mathcal{D}$, such as $\mathbb{N}$, $\mathbb{Q}$ or $X^{*}$, we refer to a function $f:\mathcal{D}\rightarrow\mathbb{R}$ as left-computable (or approximable from below) provided the set $\{(d,q):d\in\mathcal{D}\wedge q\in\mathbb{Q}\wedge q<f(d)\}$ is computably enumerable. Accordingly, a function $f:\mathcal{D}\rightarrow\mathbb{R}$ is called right-computable (or approximable from above) if the set $\{(d,q):d\in\mathcal{D}\wedge q\in\mathbb{Q}\wedge q>f(d)\}$ is computably enumerable, and $f$ is computable if $f$ is right- and left-computable. If we refer to a function $f:\mathcal{D}\rightarrow\mathbb{Q}$ as computable we usually mean that it maps the domain $\mathcal{D}$ to the domain $\mathbb{Q}$, that is, it returns the exact value $f(d)\in\mathbb{Q}$. If $\mathcal{D}=\mathbb{N}$ we write $f$ as a sequence $(q_{i})_{i\in\mathbb{N}}$.

A real number $\alpha\in\mathbb{R}$ is left-computable, right computable or computable provided the constant function $c_{\alpha}(t)=\alpha$ is left-computable, right-computable or computable, respectively. $\alpha\in\mathbb{R}$ is referred to as computably approximable if $\alpha=\lim_{i\to\infty}q_{i}$ for a computable sequence $(q_{i})_{i\in\mathbb{N}}$ of rationals. It is well-known (see e.g. [mlq/ZW01]) that there are left-computable which are not right-computable and vice versa, and that there are computably approximable reals which are neither left-computable nor right-computable.

The following approximation property is easily verified. {pty} Let $(q_{i})_{i\in\mathbb{N}}$ be a computable family of rationals converging to $\alpha$ and let $(q^{\prime}_{i})_{i\in\mathbb{N}},q^{\prime}_{i}>0,$ be a computable family of rationals converging to [math]. If $\alpha$ is not right-computable then there are infinitely many $i\in\mathbb{N}$ such that $\alpha-q_{i}>q_{i}^{\prime}$. ∎ For, otherwise, $\alpha$ as the limit of $(q_{i}+q^{\prime}_{i})_{i\in\mathbb{N}}$ would be right-computable.

2. Gales and Martingales

Hausdorff [hausdorff18] introduced a notion of dimension of a subset $Y$ of a metric space which is now known as its Hausdorff dimension, $\dim Y$; Falconer [book:Falconer03] provides an overview and introduction to this subject. In the case of the Cantor space $X^{\omega}$, Lutz [ic/Lutz03] (see also [book_DowneyHirsch10, Section 13.2]) has found an equivalent definition of Hausdorff dimension via generalisations of martingales.

Following Lutz a mapping $d:X^{*}\to[0,\infty)$ will be called an $\sigma$-supergale provided

[TABLE]

A $\sigma$-supergale $d$ is called an $\sigma$-gale if, for all $w\in X^{*}$, Eq. (1) is satisfied with equality. (Super)Martingales are $1$-(super)gales.

From Eq. (1) one easily infers that if $d,\mathcal{V}:X^{*}\to[0,\infty)$ satisfy

[TABLE]

then $d$ is a $\sigma$-(super)gale if and only if $\mathcal{V}$ is a (super)martingale. Thus (super)gales can be viewed as a combination of (super)martingales with exponential order functions in the sense of Schnorr [book_Schnorr71, Section 17] (see also [terwijn04, apal/CST06] or [book_DowneyHirsch10, Section 13.3]).

Following Lutz [ic/Lutz03] we define as follows. {defi} Let $F\subseteq X^{\omega}$. Then $\alpha$ is the Hausdorff dimension $\dim F$ of $F$ provided

1. (1)

for all $\sigma>\alpha$ there is a $\sigma$-supergale $d$ such that $\forall\xi(\xi\in F\to\limsup\limits_{w\to\xi}d(w)=\infty)$, and111Here $\limsup\limits_{w\to\xi}d(w)$ is an abbreviation for $\lim\limits_{n\to\infty}\sup\{d(w):w\in\mathbf{pref}(\xi)\wedge|w|\geq n\}$. 2. (2)

for all $\sigma<\alpha$ and all $\sigma$-supergales $d$ it holds $\exists\xi(\xi\in F\wedge\limsup\limits_{w\to\xi}d(w)<\infty)$.

If the $\omega$-language $F\subseteq X^{\omega}$ is closed in Cantor space and satisfies a certain balance condition Theorem 4 of [jstplann/St89] shows that the calculation of its Hausdorff dimension can be simplified. For the purposes of our investigations the following special case will suffice.

Proposition 1**.**

Let $F\subseteq X^{\omega}$ be non-empty and satisfy the conditions

1. (1)

$F=\{\xi:\mathbf{pref}(\xi)\subseteq\mathbf{pref}(F)\}$* and* 2. (2)

$|\mathbf{pref}(F)\cap w\cdot X^{k}|=|\mathbf{pref}(F)\cap v\cdot X^{k}|$* for all $k\in\mathbb{N}$ and $w,v\in\mathbf{pref}(F)$ with $|w|=|v|$.*

Then $\displaystyle\dim F=\liminf\limits_{n\to\infty}\frac{\log_{|X|}|\mathbf{pref}(F)\cap X^{n}|}{n}$ .∎

3. Iterative Tree Construction

The aim of this section is, given a sequence of rationals $(q_{i})_{i\in\mathbb{N}},0<q_{i}<1$, to construct an $\omega$-language $F\subseteq X^{\omega}$ with Hausdorff dimension $\dim F=\liminf_{i\to\infty}q_{i}$ satisfying the conditions (1) and (2) of Proposition 1.

3.1. Preliminaries

As a preparation we show how to find sequences of natural numbers $(k_{i})_{i\in\mathbb{N}}$ and $(\ell_{i})_{i\in\mathbb{N}}$ with appropriate properties such that $q_{i}=k_{i}/\ell_{i}$.

Lemma 2**.**

Let $(q_{i})_{i\in\mathbb{N}}$, $0<q_{i}<1$, $q_{i}\neq q_{i+1}$, be a family of positive rationals. Then there are families of natural numbers $(k_{i})_{i\in\mathbb{N}}$, $(\ell_{i})_{i\in\mathbb{N}}$, $(\kappa_{i})_{i\in\mathbb{N}}$, $(p_{i})_{i\in\mathbb{N}}$ and $(r_{i})_{i\in\mathbb{N}}$, such that $q_{i}=k_{i}/\ell_{i}$, $\displaystyle q_{i+1}=\frac{r_{i}\cdot k_{i}+\kappa_{i}\cdot\ell_{i}}{r_{i}\cdot\ell_{i}+p_{i}\cdot\ell_{i}}$ where \kappa_{i}=\left\{\begin{array}[]{ll}0,&\mbox{ if }q_{i}>q_{i+1}\mbox{ and}\\ p_{i},&\mbox{ if }q_{i}<q_{i+1}.\end{array}\right.

[TABLE]

Proof 3.1**.**

Let $q_{i}=k_{i}/\ell_{i}$ and $\displaystyle q_{i+1}=a/b\cdot q_{i}=\frac{a\cdot k_{i}}{b\cdot\ell_{i}}$, with $a,b\in\mathbb{N}\setminus\{0\},a\neq b$. Since $1>q_{i+1}$ we have $b\cdot\ell_{i}-a\cdot k_{i}=a\cdot\frac{q_{i}}{q_{i+1}}\cdot(1-q_{i+1})\cdot\ell_{i}>0$.

Assume $q_{i}>q_{i+1}$. Then $b>a$ and the equation

[TABLE]

has the solutions $r_{i}=a$, and $p_{i}=(b-a)=a\cdot(\frac{q_{i}}{q_{i+1}}-1)$ and $\kappa_{i}=0$.

If $q_{i}<q_{i+1}$ then $a>b$ and $r_{i}:=b\cdot\ell_{i}-a\cdot k_{i}=a\cdot(\frac{q_{i}}{q_{i+1}}\cdot\ell_{i}-k_{i})=a\cdot q_{i}\cdot(\frac{1}{q_{i+1}}-1)\cdot\ell_{i}$ and $p_{i}=\kappa_{i}:=(a-b)\cdot k_{i}=a\cdot q_{i}\cdot(1-\frac{q_{i}}{q_{i+1}})\cdot\ell_{i}$ are solutions of Eq. (5).

In view of $\kappa_{i}=0$ Eq. (3) is obvious. Eq. (4) follows inductively from $\frac{k+1}{\ell+1}\geq\frac{k}{\ell}$ whenever $0\leq k<\ell$. ∎

If the family $(q_{i})_{i\in\mathbb{N}}$ is a computable one then the families in Lemma 2 can be chosen to be computable. In addition, the values $\ell_{i}$ and $\ell_{i+1}/\ell_{i}$ can be made arbitrarily large.

3.2. Tree construction

The $\omega$-language $F$ will be the limit of the following sequence of finite trees $T_{i}$. These trees have a property similar to the one in Proposition 1 (2) which is referred to as spherical symmetry in [Furstenberg70].

We define the following auxiliary languages $T_{i}\subseteq X^{\ell_{i}}$ and $U_{i}\subseteq X^{p_{i}\cdot\ell_{i}}$.

Let $T_{0}:=X^{k_{0}}\cdot 0^{\ell_{0}-k_{0}}$ or $T_{0}:=0^{\ell_{0}-k_{0}}\cdot X^{k_{0}}$ and set

[TABLE]

where $u_{i}\in X^{p_{i}}$ is a fixed word. Then $\ell_{i+1}=(r_{i}+p_{i})\cdot\ell_{i}$. Thus $T_{i+1}$ consists of a concatenation of $r_{i}$ copies of $T_{i}$ plus an appendix $U_{i}$ of length $p_{i}\cdot\ell_{i}$. The values $r_{i}$ and $p_{i}$ are referred to as repetition or prolongation factors, respectively.

By induction one proves

[TABLE]

{pty}

The trees $T_{i}$ have the following properties. Let $\ell\leq\ell_{i}$.

1. (1)

Prefix property:* $\mathbf{pref}(T_{i+1})=\bigcup_{j=0}^{r_{i}-1}T_{i}^{j}\cdot\mathbf{pref}(T_{i})\cup T_{i}^{r_{i}}\cdot\mathbf{pref}(U_{i})$,* 2. (2)

Extension property:* $\mathbf{pref}(T_{i})\cap X^{\ell}=\mathbf{pref}(T_{i+1})\cap X^{\ell}$, and* 3. (3)

Spherical symmetry:* $\mathbf{pref}(T_{i})\cap X^{\ell}=(\mathbf{pref}(T_{i})\cap X^{\ell-1})\cdot X$ or* 4. $|\mathbf{pref}(T_{i})\cap X^{\ell}|=|\mathbf{pref}(T_{i})\cap X^{\ell-1}|.$* ∎*

3.3. The infinite tree

We define our $\omega$-language $F$ having the properties mentioned in Proposition 1 as $F:=\bigcap_{i\in\mathbb{N}}T_{i}\cdot X^{\omega}$ where the family $(T_{i})_{i\in\mathbb{N}}$ satisfies Eq. (6).

Before we proceed to further properties of $(T_{i})_{i\in\mathbb{N}}$ and $F$ we mention a general property.

Lemma 3**.**

Let $T_{i}\subseteq X^{*}$, $T_{i+1}\subseteq T_{i}\cdot X\cdot X^{*}$, $T_{i}\subseteq\mathbf{pref}(T_{i+1})$ and $F:=\bigcap_{i\in\mathbb{N}}T_{i}\cdot X^{\omega}$. Then $\mathbf{pref}(F)=\bigcup_{i\in\mathbb{N}}\mathbf{pref}(T_{i})$.

If, moreover, all $T_{i}$ are finite then $F:=\{\xi:\xi\in X^{\omega}\wedge\mathbf{pref}(\xi)\subseteq\bigcup_{i\in\mathbb{N}}\mathbf{pref}(T_{i})\}$.