Finite metric bases and computability structures

Introduction

In July 2018 professor Zvonko Iljazović and myself have attended the CCA 2018 conference and presented our joint work titled "Effective compactness and uniqueness of maximal computability structures". The original presentation can be found here.

Computability structures

In the following I will mention the elements of the theory of computability structures needed to state our main result. These notions are already well known, the main contributing articles for this theory that were used for our work are listed in the references.

Let $(X,d)$ be a metric space and $(x_i)$ a sequence in $X$ . We say $(x_i)$ is an effective sequence in $(X, d)$ if the function $\mathbb{N}^2 \rightarrow \mathbb{R}$

$(i,j) \mapsto d(x_i, x_j)$

is recursive.

A finite sequence $x_0,\dots,x_n$ is an effective finite sequence if $d(x_i, x_j)$ is a recursive real number for each $i,j \in \{0, \dots, n\}$ .

If $(x_i)$ and $(y_j)$ are sequences in $X$ , we say $((x_i), (y_j))$ is an effective pair in $(X,d)$ and write $(x_i) \diamond (y_j)$ if the function $\mathbb{N}^2 \rightarrow \mathbb{R}$ ,

$(i,j) \mapsto d(x_i, y_j)$

is recursive.

Let $(X,d)$ be a metric space and $(x_i)$ a sequence in $X$ . A sequence $(y_i)$ is computable w.r.t $(x_i)$ in $(X,d)$ iff there exists a computable $F:\mathbb{N}^2\rightarrow \mathbb{N}$ such that

$d(y_i, x_{F(i, k)}) < 2^{-k}$

for all $i,k \in \mathbb{N}$ . We write $(x_i) \preceq (y_j)$ .

Let $(X, d)$ be a metric space. A set $\mathcal{S} \subseteq X^\mathbb{N}$ is a computability structure on $(X,d)$ if the following holds:

$(x_i), (y_j) \in \mathcal{S}$ , then $(x_i) \diamond (y_j)$
if $(x_i) \in \mathcal{S}$ and $(y_j) \preceq (x_i)$ , then $(y_j) \in \mathcal{S}$

We say $x$ is a computable point in $\mathcal{S}$ iff $(x,x,\dots) \in \mathcal{S}$ .

A computability structure $\mathcal{S}$ such that there exists a dense sequence $\alpha \in \mathcal{S}$ is called separable.

We say $\mathcal{S}$ is a maximal computability structure on $(X, d)$ if there exists no computability structure $\mathcal{T}$ such that $\mathcal{S} \subseteq \mathcal{T}$ and $\mathcal{S} \not = \mathcal{T}$ .

The main question

The main question we are asking and trying to answer is the following:

Question:Let $(X,d)$ be a metric space. Let $a_0,\dots,a_k \in X$ . Let $\mathcal{M}$ be a maximal computability structure in which $a_0,\dots,a_k$ are computable. Under which conditions is such $\mathcal{M}$ unique?

A known result for sub-spaces of the Euclidean space

Let $V$ be a real vector space. Let $a_0,\dots,a_k$ be vectors in $V$ . We say that $a_0,\dots, a_k$ are geometrically independent points if $a_1 - a_0, \dots, a_k - a_0$ are linearly independent vectors.

Let $X \subseteq V$ . The largest $k \in \mathbb{N}$ such that that there exist geometrically independent points $a_0,\dots, a_k \in X$ we call the affine dimension of $X$ , and write $\mathrm{dim}\ X = k$ .

The following result from [1] already answers the main question in the case where $(X,d)$ is a sub-space of the Euclidean space.

Theorem: Let $X \subseteq \mathbb{R}^n$ , $k = \mathrm{dim}\ X$ and $k \geq 1$ . If $a_0,\dots,a_{k-1}$ is a geometrically independent effective finite sequence on $X$ then there exists an unique maximal computability structure on $X$ in which $a_0,\dots,a_{k-1}$ are computable points.

Main result for more general metric spaces

We wanted to introduce for general metric spaces a notion which will be a sort of replacement to the notion of geometric independence.

In the original presentation we used the term nice sequence. It turns out that this notion is a special case of the well-known notion of a metric base.

Let $(X, d)$ be a metric space. A subset $S \subseteq X$ is called a metric base for $(X,d)$ iff for all $x,y \in X$ the following implication holds: $d(x,s) = d(y,s)$ for all $s \in S$ implies $x=y$ .

A metric space $(X,d)$ is said to be effectively compact if there exist an effective separating sequence $\alpha$ in $(X,d)$ and a computable function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that

$X=B(\alpha _{0} ,2^{-k})\cup \dots \cup B(\alpha _{f(k)},2^{-k})$

for each $k\in \mathbb{N}$ . It is known that if $(X,d)$ is effectively compact, then for each effective separating sequence $\alpha$ in $(X,d)$ there exists such a computable function $f$ .

A metric base is exactly the notion we needed to obtain the following result, at least for effectively compact metric spaces.

Theorem: Let $(X,d)$ be an effectively compact metric space. Suppose $a_{0} ,\dots ,a_{n}$ is a metric base in $(X,d)$ and suppose that there exists a separable computability structure $\mathcal{S}$ on $(X,d)$ in which $a_{0} ,\dots ,a_{n}$ are computable points. Then $\mathcal{S}$ is a unique maximal computability structure on $(X,d)$ in which $a_{0} ,\dots ,a_{n}$ are computable points.

In fact, an even more general form of the theorem holds: the assumption of effective compactness of the space $(X,d)$ can be replaced with the assumption that $(X,d)$ has compact closed balls and there exists $\alpha$ such that the computable metric space $(X,d,\alpha)$ has the effective covering property.

Proofs will be presented in our future up-coming publication.

References

Zvonko Iljazović. Isometries and Computability Structures. Journal of Universal Computer Science, 16(18):2569--2596, 2010.
Zvonko Iljazović and Lucija Validžić. Maximal computability structures. Bulletin of Symbolic Logic, 22(4):445--468, 2016.
Alexander Melnikov. Computably isometric spaces. Journal of Symbolic Logic, 78:1055--1085, 2013.
Marian Pour-El and Ian Richards. Computability in Analysis and Physics. Springer-Verlag, Berlin-Heielberg-New York, 1989.
Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000.
M. Yasugi, T. Mori and Y. Tsujji. Effective properties of sets and functions in metric spaces with computability structure. Theoretical Computer Science}, 219:467--486, 1999.
M. Yasugi, T. Mori and Y. Tsujji. Computability structures on metric spaces. Combinatorics, Complexity and Logic Proc. DMTCS96 (D.S.~Bridges et al), Springer, Berlin}, 351--362, 1996.

Hyperbolic numbers and fractals

What happens when we change the identity defining the imaginary unit for the complex numbers to and enforcing i different from 1? We get the somewhat less known numbers called hyperbolic or split-complex numbers. The most common thing about complex numbers that comes to mind are the nice pictures of fractals and there are plenty of tools already out there to generate them. However, for the hyperbolic numbers there aren't as much fractal generators out there, so let us now make a simple tool in Python to generate these fractals as well.

First, we need to implement the basic arithmetic with hyperbolic numbers. This will enable us to write simple arithmetic expressions like z**2 + c directly instead of writing out the expressions in component form using real and imaginary parts. I made a simple Python implementation of class HyperbolicNumber in module hyperbolic doing exactly this. Using this module, we can now write a function for counting the number of iterations of f(z)=z**2+c before "escaping to infinity" in a straightforward way. In fact the formula is identical to the escape-time algorithm for fractals over complex numbers.

from hyperbolic import HyperbolicNumber

def f(z):
    w = z
    total = 0
    while mag(w) < 2.0 and total < 300:
        w = w**2 + z
        total += 1
    return total

Basically, the above function is all we need to start generating images of fractals. Here of course, we focus only on the famous map family { f_c(z) = z**2+c} indexed by c, but any other could be used as well. We add an additional parameter to f(z) to become f(z,c) and encapsulate it in the HyperbolicFractal class. This class will be our abstraction for a fractal calculation, which basically just calculates a 2**-k approximation of the Julia set of f_c over some given rectangular region in the hyperbolic number plane. For all the points z of a given rectangular range [centerx-rangex, centerx+rangex]x[centery-rangey, centery+rangey] we perform the call to f(z, c). To avoid recalculation every time of the same region, or part of a region, points which have already been calculated are stored in an internal cache so any subsequent calls to f(z, c) are just looked up instead of being recalculated. Here's the code.

import numpy
import matplotlib.cm as cm
import matplotlib.pylab as plt
from hyperbolic import HyperbolicNumber

class HyperbolicFractal:
    def __init__(self):
        self.hits = 0
        self.cache = {}
        
    def f(self, z, c):
        if (z, c) in self.cache:
            self.hits += 1
            return self.cache[(z, c)]
        w = z
        total = 0
        while mag(w) < 2.0 and total < 300:
            w = w**2 + c
            total += 1
        self.cache[(z, c)] = total
        return self.cache[(z, c)]

    def show(self, scale, centerx, centery, rangex, rangey, c):
        """Draw a 2**(-scale) approximation of a hyperbolic fractal 
        over a given rectangular range."""
        self.hits = 0
        totals = (centerx+rangex - (centerx-rangex))\
        * (centery+rangey - (centery-rangey))
        t = [[self.f((1.0/(2**scale)) * HyperbolicNumber(x,y), c)\
        for x in range(centerx-rangex, centerx+rangex)]\
        for y in range(centery-rangey, centery+rangey)]
        matrix = numpy.matrix(t)
        fig = plt.figure()
        plt.imshow(matrix, interpolation='bilinear', cmap=cm.RdYlGn, origin='center')
        print ("Total: {} Cache hits: {} Cache hit ratio: {}".format(totals, self.hits, (1.0 * self.hits)/totals))
        plt.show()

Let's try it out.

H = HyperbolicFractal()
H.show(7, 0, 0, 200, 200, HyperbolicNumber(-20/2**3, 0))

And there you have it! We now have a tool to look into the hyperbolic number world.

The full code for this implementation can be found on Github.

Computable metric spaces in a nutshell

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What are computable metric spaces?

We can generalize the notions of computability over the reals and in the euclidean space to metric spaces. A computable metric space is a triple $(X,d,\alpha)$ where $(X,d)$ is a metric space and $\alpha$ is a sequence in $X$ such that its image is dense in $X$ and the function $\mathbb{N}^2 \rightarrow \mathbb{R}$ defined as $(i,j) \mapsto d(\alpha_i, \alpha_j),\ \forall i,j\in\mathbb{N}$ is computable. For example, a definition of computable point is a generalization of the computable real: A point $x \in X$ is computable in $(X,d,\alpha)$ if there exists a computable function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $d(\alpha_{f(k)}, x) < 2^{-k}$ for every $k \in \mathbb{N}$ . Similarly, we can define a computable sequence of points in $X$ and functions between metric spaces that are computable,... but for subsets of $X$ it is not the same as definition of computability in $\mathbb{R}^n$ but it borrows the definitions from classical recursion theory. First we fix some computable enumeration of rational balls in $X$ for example let $(I_i)$ be a fixed such enumeration. Then for a set $S \subseteq X$ we say that $S$ is recursively enumerable $T = \{i \in \mathbb{N} | S \cap I_i \not = \emptyset\}$ is recursively enumerable i.e. there exists a computable function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(\mathbb{N}) = T$ . A subset $S$ is co-recursively enumerable if we have an algorithm that covers the complement of $S$ with rational balls. In other words, $S$ is co-recursively computable iff there exists a recursively enumerable subset $A$ of $\mathbb{N}$ such that $X \setminus S = \bigcup_{i \in A} I_i$ . A subset $S$ is computable iff $S$ is recursively computable and co-recursively computable.

We can of course generalize further and define computable topological spaces but I shall not go further into that. That is a topic for another post. Note that as we generalize more and more there are pathological spaces that do not have nice computability properties and must be more and more careful when dealing with computability.

Introduction to computable analysis (Part 1/2)

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What is Computable analysis anyway?

Roger Penrose in his book "The emperors new Mind" asked an interesting question regarding the Mandelbrot set: "Is the Mandelbrot set recursive?" The term "recursive" is an old name for "computable" and has nothing to do with recursion in programming.

The Mandelbrot Set (zoomed in)

Classical computability theory (also known as recursion theory) is a branch of mathematical logic that deals with studying what functions $\mathbb{N}\rightarrow \mathbb{N}$ are computable. Computable analysis is in some sense an extension of this theory. It is a branch of mathematics that applies computability theory to problems in mathematical analysis. It is concerned with questions of effectivity in analysis and its main goal is to find what parts of analysis can be described by an algorithmic procedure, one of the basic questions being what functions $f:\mathbb{R}\rightarrow\mathbb{R}$ are computable? (note the domain and codomain are now the reals).

Computable reals

Suppose we take a real number $x$ . Is $x$ computable? What do we mean by $x$ being computable? If we can find an algorithm $A$ that calculates for each input $k \in \mathbb{N}$ a rational approximation $A_k$ of $x$ ; and such that the sequence $(A_k)$ "converges fast" to $x$ then we say that $x$ is a computable real. More precisely, a real number $x \in R$ is computable iff there exists a computable function $f:\mathbb{N} \rightarrow \mathbb{Q}$ such that $|f(k) - x| < 2^{-k}$ for each $k \in \mathbb{N}$ . For example, every rational number is computable. The famous number $\pi$ is computable, and in our previous post we proved that $\sqrt{2}$ is computable. It turns out that the set of computable reals, denoted by $\mathbb{R}_C$ is a field with respect to addition and multiplication. But there exist real numbers which are not computable. This is easy to see as there is only a countable number of computable functions $\mathbb{N}\rightarrow\mathbb{N}$ , and since we know that $\mathbb{R}$ is uncountable, so we have more real numbers than we have possible algorithms, we conclude that there must be a real number that is not computable. One example of an uncomputable real is the limit of a Specker sequence. And one other interesting one is the Chaitin constant. In fact, if we take any recursively enumerable but non-recursive set $A \subseteq \mathbb{N}$ , then the number $\alpha = \sum_{i \in A} 2^{-i}$ is an example of a real number which is not computable.

Computable sequences of reals

If we have a sequence of computable reals $(x_n)$ (and hence a sequence of algorithms) we may ask if this sequence is computable i.e. is there a single algorithm that describes the whole sequence? If there exists a computable function $f : \mathbb{N}^2 \rightarrow \mathbb{Q}$ such that

$|f(n,k) - x_n| < 2^{-k}$

for each $n,k \in \mathbb{N}$ then we say that $(x_n)$ is a computable sequence. We have seen that each rational number is computable, but this is not true for sequences of rational numbers! One example of a sequence that has each term computable but it is not uniformly computable is the following:

Computable real functions

Real functions which are a main object of study in analysis can be studied in this computability setting, a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is computable iff

it maps computable sequences to computable sequences i.e. $f(x_i)$ is computable for every computable sequence $(x_i)$ ;
it is effectively uniformly continous i.e. if there exists a computable sequence thatdescribes the modulus of continuity of $f$ .

The fundamental theorem of computable analysis is that every computable real function is continuous.

There is also a notion of computability for the process of integration, derivation, solving partial differential equations, etc. and we shall look into that topic in another post.

Computable subsets of the Euclidean space

Unit circle approximated with dyadic points

Next, let's look at the subsets of the euclidean space $\mathbb{R}^n$ and ask a simple question: what subsets are computable? First, note that saying $S$ is computable iff its indicator function

$\chi_S(x) =\begin{cases} 1, & x \in S \\ 0, & x \not \in S \end{cases}$

is computable is not a useful definition since the indicator function is not continous except when $S =\mathbb{R}^n$ or $S=\emptyset$ . A subset $S$ of $\mathbb{R}^n$ is computable iff the function $\mathbb{R}^n \rightarrow \mathbb{R}$ defined as $x \mapsto d(x, S)$ is a computable function. For example, take the unit circle in $\mathbb{R}^2$ .

$\mathbb{S}^1 = \{(x,y): x^2 + y^2 =1\}$

. The distance function is

$d((x,y), \mathbb{S}^1) = |x^2 + y^2 - 1|, \forall x,y \in \mathbb{R}^2.$

These are just some basic definitions and facts, but also interesting questions are studied in computable analysis like for example what mappings preserve computability of objects? Is for example the Mandelbrot set computable? What about Julia Sets? For them, there are nice results presented in the book "Computability of Julia Sets" by Braverman and Yampolsky. Although Julia Sets have been thoroughly researched in the book, for the Mandelbrot set we still don't know the answer.

Basically for anything we do in analysis (finding derivatives, integration, finding roots, solving PDEs ...) we may ask: "Is it computable?" and that is a topic for another post.

(to be continued)

Another look at the square root of two

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The other day I was thinking about approximating $\sqrt{2}$ in terms of segments of size $1/2^k$ for $k=0,1,2,\dots$ , The question I was interested was "What is the least number of segments of length $1/2^k$ that we need to cross over $\sqrt{2}$ starting from zero?" and one particular sequence of integers I came up with was this:

$2, 3, 6, 12, 23, 46, 91, 182, 363, 725,\dots.$

Sadly, at the time of writing this post the Online Encyclopedia of Integer Sequences did not return anything for this sequence.

The sequence represents the numerators for dyadic rationals which approximate $\sqrt{2}$ within precision $2^{-k}$ (within k-bits of precision), but is this sequence computable? In what follows I assume that the reader is familiar with the basic definitions of mu-recursive functions.

It turns out that our sequence can be described by a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$

$f(k) = \mu.n [n^2 > 2^{(2k+1)}]$

for all $k\geq0$ .

The idea of the function $f$ is that for each $k$ it outputs the number of $1/2^k$ segments that fit into $\sqrt{2}$ plus one.

Since for each $k$ , the value $f(k)$ is the smallest integer such that $f(k)/2^k > \sqrt{2}$ we have

$(f(k) - 1)/2^k < \sqrt{2} < f(k)/2^k$

for each $k\geq 0$ .

From this we obtain the error bound $|f(k)/2^k - \sqrt{2}| < 2^{-k}$ .

Approximation of $\sqrt{2}$ within precision $2^{-k}$ can be "represented" by a natural number $f(k)$ . To get the actual approximation by a dyadic rational just divide by $2^k$ but what we want is to avoid division. We want to calculate only with natural numbers!

Implementation

The function f has a straightforward (although inefficient) implementation in Python.

def calcSqrt2Segments(k):
    n = 0
    while(n*n <= 2**(2*k+1)):
        n = n + 1
    return n

This is in fact terribly inefficient, but in computability theory efficiency is not the goal, the goal is usually to prove that something is uncomputable even if you have all the resources of time and space at your disposal. Nevertheless, here is a slightly better version we get if we notice that the next term $f(k+1)$ is about twice as large as $f(k)$ with a small correction.

def calcSqrt2segmentsRec(k):
    if k == 0:
        return 2
    else:
        res = 2*calcSqrt2segmentsRec(k-1) - 1
        if res*res <= 2**(2*k+1):
           res = res + 1
        return res

This of course suffers from stack overflow problems, so memoization is a natural remedy.

def calcSqrt2segmentsMemo(k):
    m = dict()
    m[0] = 2
    for j in range(1, k+1):
        m[j] = 2*m[j-1] - 1
        if m[j]*m[j] <= 2**(2*j+1):
            m[j] = m[j] + 1
    return m[k]

Memoization can use-up memory and we can do better. We don't need to memorize the whole sequence up to k to calculate $f(k+1)$ we only need the value $f(k)$ .

def calcSqrt2segmentsBest(k):
    m = 2
    for j in range(1, k+1):
        m = 2*m - 1
        if m*m <= 2**(2*j+1):
            m = m + 1
    return m

After defining a simple timing function and testing our memoization and "best" version we see that the best version is not so good in terms of running time when compared with memoization, they are both approximately about the same in terms of speed (the times are in seconds):

>>> timing(calcSqrt2segmentsMemo, 10000)
0.6640379428863525
>>> timing(calcSqrt2segmentsBest, 10000)
0.6430368423461914
>>> timing(calcSqrt2segmentsMemo, 20000)
3.3311898708343506
>>> timing(calcSqrt2segmentsBest, 20000)
3.3061890602111816
>>> timing(calcSqrt2segmentsMemo, 30000)
8.899508953094482
>>> timing(calcSqrt2segmentsBest, 30000)
8.848505973815918
>>> timing(calcSqrt2segmentsMemo, 50000)
30.04171895980835
>>> timing(calcSqrt2segmentsBest, 50000)
29.96371293067932
>>> timing(calcSqrt2segmentsMemo, 100000)
164.99343705177307
>>> timing(calcSqrt2segmentsBest, 100000)
168.6026430130005

Neverteless, we can use our "best" program to calculate $\sqrt{2}$ to arbritrary number of decimal places (in base 10). First, we need to determine how many bits are sufficient to calculate $\sqrt{2}$ to $n$ decimal places in base 10. Simple calculation yields:

$k > \lceil n log_2 10 \rceil.$

In Python we need a helper function that calculates this bound that avoids the nasty log and ceiling functions, so here it is:

def calcNumDigits(n):
    res = 1
    for j in range(2, n+1): 
        if( j%3 == 0 or j%3 == 1 ):
          res = res + 3
        else:
          res = res + 4
    return res

At last, calculating $\sqrt{2}$ to arbritrary precision is done with this simple code below, returning a String with the digits of $\sqrt{2}$ in base 10. Note once more, that all this calculation is done only with the natural numbers and here we are using Python's powerful implementation of arithmetic with arbritrary long integers (which is not as nicely supported for decimals at least at the time of writing this post).

Note also, that instead of dividing $f(k)$ by $2^k$ we are multiplying it with $5^k$ to get a natural number with all of it's digits being an approximation of $\sqrt{2}$ with k bits of precision. This natural number we are then converting to a string, inserting a decimal point '.' and returning this as our result. So, here is the code:

def calcSqrt2(n):
    k = calcNumDigits(n)
    res = str(5**k * calcSqrt2segmentsBest(k))
    return res[0:1] + '.' + res[1:n]

Running it gives us nice results:

>>> calcSqrt2(30)
'1.41421356237309504880168872421'
>>> calcSqrt2(300)
'1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140799'
>>> calcSqrt2(3000)
'1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008344491148185876555542864551233142199263113325179706084365597043528564100879185007603610091594656706768836055717400767569050961367194013249356052401859991050621081635977264313806054670102935699710424251057817495310572559349844511269227803449135066375687477602831628296055324224269575345290288387684464291732827708883180870253398523381227499908123718925407264753678503048215918018861671089728692292011975998807038185433325364602110822992792930728717807998880991767417741089830608003263118164279882311715436386966170299993416161487868601804550555398691311518601038637532500455818604480407502411951843056745336836136745973744239885532851793089603738989151731958741344288178421250219169518755934443873961893145499999061075870490902608835176362247497578588583680374579311573398020999866221869499225959132764236194105921003280261498745665996888740679561673918595728886424734635858868644968223860069833526427990562831656139139425576490620651860216472630333629750756978706066068564981600927187092921531323682'

Note:
This can of course be generalized to calculating the square root of any number and of any order. This is an easy exercise for the reader.

The Hitchhiker's Guide to Computable Mathematics

Konrad Burnik

Finite metric bases and computability structures

Introduction

Computability structures

The main question

A known result for sub-spaces of the Euclidean space

Main result for more general metric spaces

References

Hyperbolic numbers and fractals

Computable metric spaces in a nutshell

Introduction to computable analysis (Part 1/2)

Another look at the square root of two