1.1 Motivations

Real numbers can be represented by Cauchy sequences, base-10 expansions, Dedekind cuts, continued fractions and a number of other representations (we will consider quite a few of them in this paper). Our goal is to analyze the computational complexity of converting one representation into another. Let us say that we have access to the Dedekind cut of the real number $\alpha $ . How hard will it be to compute a Cauchy sequence for $\alpha $ ? How hard will it be to compute the continued fraction of $\alpha $ ? Or let us say that we have access to the continued fraction of $\alpha $ , how hard will it then be to compute the base-10 expansion of $\alpha $ ? Will there be an efficient algorithm? Can it be done in polynomial time? Exponential time?

These are very natural questions to ask, but they are also naive, and the way the questions are posed above, does hardly make any sense at all. We will aim at posing such questions in a mathematically satisfactory manner, and then derive reasonably tight upper bounds on the computational complexity of a number of conversions.

At the outset we are mainly interested in classic and historically prominent representations like Cauchy sequences and Dedekind cuts, which are important in the foundation of analysis, or base-b expansions, continued fractions and representations related to the Stern–Brocot tree which have numerous applications in both pure and applied mathematics. In general, we cannot uniformly convert one of these representations into another. This is well known. For example, we cannot uniformly convert a Cauchy sequence into a base-b expansion: No algorithm with oracle access to a Cauchy sequence for a real number $\alpha $ can compute the base-b expansion of $\alpha $ because it is not possible to determine the digits in such an expansion by examining a finite part of the Cauchy sequence. The conversion is not continuous (see, e.g., [Reference Weihrauch40]), that is, we cannot determine a finite part of the output by examining a finite part of the input. For the same reason, we cannot uniformly convert Dedekind cuts to continued fractions or Cauchy sequences to Dedekind cuts.

The computational complexity of an operation is determined by imposing resource bounds on an algorithm (Turing machine) carrying out the operation. The considerations above show that, in general, it may make no sense to talk about the computational complexity of converting one representation of (the set of all) real numbers into another representation. If there does not exists a uniform conversion from one representation to another, then no algorithm can carry out the conversion, and then there is nothing to impose resource bounds on.

The absence of uniform conversions is due to the presence of the rational numbers. If we exclude the rationals and restrict our attention to the irrational numbers, then it is indeed possible to uniformly convert back and forth between any of the classic representations discussed above. We will use this insight to give a formal definition of what a representation of an irrational number is, see Definition 1.2 below. Roughly speaking, the definition states that a class of functions is a representation if it is uniformly computably equivalent to the class of Dedekind cuts of the irrational numbers. In addition to capture the classic representations discussed above, this definition is generous enough to include some representations popular in the practice of numerical computations with exact real arithmetic (Boehm et al. [Reference Boehm, Cartwright, Riggle, O’Donnell, Scherlis, Williams and Gabriel10]) and modern computable analysis (Weihrauch [Reference Weihrauch41], Ko [Reference Ko29]). We will study all sorts of representations captured by our definition, including some representations due to the authors.

We note that there exists a very general and well-developed theory of Type-2 computability, continuity, and representation of sets of continuum cardinality (some references are Weihrauch [Reference Weihrauch42], Kreitz and Weihrauch [Reference Kreitz and Weihrauch30], Brattka et al. [Reference Brattka, Hertling, Weihrauch, Cooper, Löwe and Sorbi12] and Weihrauch [Reference Weihrauch41]). This theory may serve as a foundation for modern computable analysis and allows for a wide range of representations of real and irrational numbers. In the present paper, our definition of the notion of representation (of the irrational numbers) is restrictive in the sense that it excludes some of the representations allowed in Type-2 computability. We do not regard this as a problem as our definition is instrumental rather than foundational, and its purpose is to provide a neat and transparent development of the degree structure depicted in Figure 1.

Figure 1

Subrecursive degrees (equivalence classes) of representations.

1.2 What is a representation?

Formally, a representation of the irrational numbers will be a class of functions with countable domain and codomain (typically $\mathbb {Q}$ or $\mathbb {N}$ ).Footnote ¹ Every function in the class will represent a particular irrational number, and each irrational number will be represented by some function in the class. The class of Dedekind cuts of irrational numbers will be a canonical representation to us.

Definition 1.1. The Dedekind cut of an irrational number $\alpha $ is the function $\alpha :\mathbb {Q} \longrightarrow \{0,1\}$ where

$$ \begin{align*}\alpha(q)=\begin{cases} 0 & \mbox{if } q < \alpha \\ 1 & \mbox{if } q> \alpha. \end{cases}\end{align*} $$

Each irrational number has a unique representation in this class, and we can identify an irrational number $\alpha $ with its Dedekind cut $\alpha :\mathbb {Q} \longrightarrow \{0,1\}$ .

We will take advantage of the uniqueness of the Dedekind cuts to define what a representation in general is. We refer to the functions in a representation R as R-representations. When f is an R-representation of $\alpha $ , we will require that it is possible to compute the Dedekind cut of $\alpha $ in f, that is, we will require that there exists an oracle Turing machine M such that

$$ \begin{align*}\alpha(q) = \Phi^f_M(q),\end{align*} $$

where $\Phi ^f_M$ is the function computed by M with oracle f. We will also require that at least one R-representation f of $\alpha $ can be computed in the Dedekind cut of $\alpha $ , that is, we will require that there exists an oracle Turing machine N and an R-representation f such that

$$ \begin{align*}f(x) = \Phi^\alpha_N(x),\end{align*} $$

where $\Phi ^\alpha _N$ is the function computed by N when the oracle is the Dedekind cut of $\alpha $ . We are now ready to give our formal definition.

Let us study a few examples in order to see how this definition works. We define a Cauchy sequence for $\alpha $ as a function $C:\mathbb {N}^{+} \longrightarrow \mathbb {Q}$ with the property

$$ \begin{align*}\vert C(n) - \alpha \vert < n^{-1}. \end{align*} $$

Let $\mathcal {C}$ be the class of all Cauchy sequences for all irrational numbers. We will now argue that $\mathcal {C}$ is a representation according to the definition above.

First we observe that we can compute the Dedekind cut of an irrational $\alpha $ in any Cauchy sequence for $\alpha $ . In order to compute $\alpha (q)$ , we search for the least $n>0$ such that $\vert C(n)- q\vert> n^{-1}$ . This search terminates as q is rational and $\alpha $ is irrational. If $q<C(n)$ , it will be the case that $\alpha (q)=0$ (we have $q<\alpha $ ), otherwise, we have $q>C(n)$ , and then it will be case that $\alpha (q)=1$ (we have $q>\alpha $ ). Thus there will be an oracle Turing machine M such that $\alpha = \Phi ^f_M$ whenever f is a Cauchy sequence for $\alpha $ . Now, M has the following property: For every $f\in \mathcal {C} $ there exists irrational $\alpha $ such that $\alpha = \Phi ^f_M$ . This shows that clause (1) of Definition 1.2 is satisfied.

Next we observe that we can compute a Cauchy sequence C for $\alpha $ if we have access to the Dedekind cut of $\alpha $ . We can use the Dedekind cut to find an integer a such that $a< \alpha < a+1$ . Thereafter, we can use the Dedekind cut and the equations

$$ \begin{align*}C(1)= a+ \frac{1}{2} \;\;\; \mbox{ and } \;\;\; C(i+1)= \begin{cases} C(i)- 2^{-i-1} & \mbox{if } C(i)> \alpha \\ C(i)+ 2^{-i-1} & \mbox{if } C(i)< \alpha \end{cases} \end{align*} $$

to compute $C(n)$ for arbitrary n. This is one possible way to compute a Cauchy sequence in a Dedekind cut. There are for sure other ways. Other algorithms may yield Cauchy sequences that converge faster, or slower, than the ones computed by the algorithm suggested above. Anyway, there will be an oracle Turing machine N such that we have $C = \Phi ^\alpha _N$ where C is some Cauchy sequence for $\alpha $ .

Now, in order to see that clause (2) of Definition 1.2 is satisfied, pick an arbitrary irrational $\alpha $ . Then we have $f = \Phi ^\alpha _N$ for some Cauchy sequence f, moreover, we have $\alpha = \Phi _M^f$ , and hence f is a $\mathcal {C}$ -representation of $\alpha $ . Hence, N has the following property: For every irrational $\alpha $ there exists a $\mathcal {C}$ -representation f of $\alpha $ such that $f = \Phi ^\alpha _N$ . This shows that also clause (2) is satisfied, and we conclude that $\mathcal {C}$ , that is, the class of all Cauchy sequences for all irrational numbers, is a representation.

Let $a_0, a_1, a_2, \ldots $ be an infinite sequence of integers, where $a_1, a_2, a_3, \ldots $ are positive. The continued fraction $[a_0; a_1, a_2, \ldots ]$ is defined by

$$ \begin{align*}[ \ a_0; a_1, a_2,a_3, \ldots \ ] = a_0 + \frac{\displaystyle 1}{\displaystyle a_1 + \frac{\displaystyle 1}{\displaystyle a_2 + \frac{1}{a_3 + \ldots}}}. \end{align*} $$

It is well known that any irrational number can be uniquely written as an infinite continued fraction, and moreover, each infinite continued fraction equals an irrational number (rationals have finite continued fractions). There is a one-to-one correspondence between the infinite continued fractions and the irrational numbers.

Let $\mathcal {F}$ be the class of all infinite continued fractions where $[ \ a_0; a_1, a_2,a_3 \ldots \ ]$ is identified with a function f where $f(n)=a_n$ . Then $\mathcal {F}$ will be a representation according to Definition 1.2. If we have access to the continued fraction of $\alpha $ , we can compute the Dedekind cut of $\alpha $ , and if we have access to the Dedekind cut of $\alpha $ , we can compute the continued fraction of $\alpha $ . Thus there exist Turing machines M and N such that

$$ \begin{align*}\alpha = \Phi^f_M \mbox{ and } f= \Phi^\alpha_N\end{align*} $$

whenever f is the continued fraction of the irrational number $\alpha $ . Each irrational number will have one, and only one, $\mathcal {F}$ -representation, and it is easy to see that both clause (1) and clause (2) of Definition 1.2 is fulfilled.

We will not give a formal definition of a representation of all real numbers, and it is essential that Definition 1.2 is restricted to the irrational numbers. If we involve the rationals, the definition will not serve its purpose as we cannot always uniformly convert one standard representation into another, even if we are dealing with representations of a computable nature. For example, the algorithm above converting a Cauchy sequence C into a Dedekind cut, searches for a number n such that $\vert C(n)- q\vert> n^{-1}$ . This search will not terminate when C is a Cauchy sequence for the rational number q. Thus, if $\alpha $ might be rational, the algorithm does not yield a Turing machine M such that $\Phi ^C_M$ is the Dedekind cut of $\alpha $ whenever C is a Cauchy sequence for $\alpha $ , moreover, it can be proven that no such Turing machine M exists (see Mostowski [Reference Mostowski36]). We cannot uniformly convert Cauchy sequences for real numbers into Dedekind cuts, but we can uniformly convert Cauchy sequences for irrational numbers into Dedekind cuts.

The purpose of Definition 1.2 is to capture what we intuitively consider as computable representations of the real numbers, and maybe somewhat paradoxically, we enable the study of conversions among such representations by restricting the definition to the irrational numbers.

A sequence of rationals $q_0, q_1, q_2, \ldots $ containing all rational numbers less than $\alpha $ , and nothing but the rationals less than $\alpha $ , will not yield a representation according to our definitions. Such a sequence defines a unique real number $\alpha $ , but we cannot use the sequence to compute the Dedekind cut of $\alpha $ (even if $\alpha $ is irrational). By the same token, Cauchy sequences without a modulus of convergence will not yield a representation.

1.3 An ordering relation on representations

We have seen that there is an algorithm for computing the Dedekind cut of an irrational $\alpha $ in an arbitrary Cauchy sequence for $\alpha $ . The algorithm searches for the least natural number n that fulfills certain criteria. It is easy to see that such an unbounded search is necessary. We cannot convert a Cauchy sequence into a Dedekind cut if we are not allowed to carry out unbounded search. Neither can we convert a base-10 expansion into a base-3 expansion if we are not allowed to carry out unbounded search. Suppose an oracle tells us that the base-10 expansion of an irrational starts with $0.66666$ . That will not be enough for us to decide if the base-3 expansion starts with $0.1$ or $0.2$ . Thus, in order to determine which of these two options we should pick, we have to ask the oracle for the next digit of the decimal expansion, but of course, the next digit might also be 6, and so might the next after the next. The oracle may continue for an arbitrarily long time to tell us that the next digit is $6$ . Since the number is irrational the oracle will eventually yield a digit that allows to determine if the base-3 expansion starts with $0.1$ or $0.2$ , but we need to carry out an unbounded search to get that digit.

Algorithms (conversions, computations, etc.) that do not perform unbounded search will be referred to as subrecursive algorithms (conversions, computations, etc.), in general, the word subrecursive signifies absence of unbounded search. This terminology might not be standard, but it will be very convenient.

We will now define an ordering relation $\preceq _{S}$ over the representations. Intuitively, the relation $R_1 \preceq _{S} R_2$ will indicate that the representation $R_2$ is more informative than the representation $R_1$ . If $R_1 \preceq _{S} R_2$ holds, a Turing machine with oracle access to an $R_2$ -representation of $\alpha $ can subrecursively (yes, that means without carrying out unbounded search) compute an $R_1$ -representation of $\alpha $ . Thus, if the relation $R_1 \preceq _{S} R_2$ does not hold, it will not make much sense to talk about the computational complexity of converting an $R_2$ -representation into an $R_1$ -representation as such a conversion requires unbounded search, and thus, there will be no upper bound on the running time of a Turing machine undertaking the conversion. On the other hand, if the relation holds, it should make sense to analyze the computational complexity of the conversion. So far we have just indicated our intention with the relation; below we shall give the formal definition. First, however, we need an auxiliary definition. We are going to formulate our definition in terms of time-bounded computation, and there is need to specify what functions we admit as time bounds. We admit time-constructible functions, as defined next. This is a standard choice in complexity theory.

Clause (iii) in the definition is needed because there are functions whose computational complexity is disproportionate to the size of their values. We exclude such functions as time bounds, avoiding certain pitfalls in proofs. The class of functions we admit as time bounds includes all the functions familiar from analysis of algorithms such as polynomials (with positive coefficients), exponentials, the tower-functions, and so on.

Definition 1.4. Let t be a time-bound and let R be a representation. Then, $O(t)_R$ denotes the class of all irrational $\alpha $ in the interval $(0,1)$ such that at least one R-representation of $\alpha $ is computable by a Turing machine running in time $O(t(n))$ (where n is the length of the input).

Let $R_1$ and $R_2$ be representations. The relation $R_1 \preceq _{S} R_2$ holds if for any time-bound t there exists a time-bound s such that

$$ \begin{align*} O(t)_{R_2} \; \subseteq \; O(s)_{R_1}. \end{align*} $$

If the relation $R_1 \preceq _{S} R_2$ holds, we will say that the representation $R_1$ is subrecursive in the representation $R_2$ .

We will now study a few examples and discuss how the definition above works. Recall that ${\mathcal {C}}$ denotes the representation by Cauchy sequences (see page 4). Let ${\mathcal {D}}$ denote the representation by Dedekind cuts. It turns out that we have ${\mathcal {C}}\preceq _{S} {\mathcal {D}}$ and ${\mathcal {D}}\not \preceq _{S} {\mathcal {C}}$ . Let us see why.

Intuitively we have ${\mathcal {C}}\preceq _{S} {\mathcal {D}}$ because a Cauchy sequence for an irrational $\alpha $ in the interval $(0,1)$ can be subrecursively computed in the Dedekind cut of $\alpha $ . No unbounded search is required. We simply sets $C(1)$ equal to $1/2$ , and then we use the Dedekind cut and, e.g., the equations

$$ \begin{align*}C(i+1)= \begin{cases} C(i)- 2^{-i-1} & \mbox{if } C(i)> \alpha \\ C(i)+ 2^{-i-1} & \mbox{if } C(i)< \alpha \end{cases} \end{align*} $$

to compute $C(n)$ . Then $C:\mathbb {N}^+ \longrightarrow \mathbb {Q}$ will be a Cauchy sequence for $\alpha $ . Formally we have ${\mathcal {C}}\preceq _{S} {\mathcal {D}}$ because for every time-bound t there exists a time-bound s such that $O(t)_{{\mathcal {D}}} \; \subseteq \; O(s)_{{\mathcal {C}}}$ . A Turing machine that uses the equations above to compute $C(n)$ needs to compute the Dedekind cut n times, hence at most $2^{\Vert n\Vert }$ times where $\Vert n\Vert $ is the length of the input. Furthermore, assuming numbers are represented in binary form, the Turing machine only needs to compute the Dedekind cut for inputs of size bounded by $2^{\Vert n\Vert }$ . Hence, if the Dedekind cut of $\alpha $ is computable in time $O(t(m))$ where m is the length of the input and t is a time-bound, then a Turing machine can compute a Cauchy sequence for $\alpha $ in time $O( nt(2^{\Vert n\Vert }))$ . Thus, let s be the time-bound $s(\Vert n\Vert )= nt(2^{\Vert n\Vert })$ , and we have $O(t)_{{\mathcal {D}}} \subseteq O(s)_{{\mathcal {C}}}$ .

Note that the set $O(t)_{{\mathcal {D}}}$ only contains irrationals from the interval $(0,1)$ . This is important. When we compute a Cauchy sequence C for an irrational in this interval, we can simply let $C(1)=1/2$ . We cannot in general compute a Cauchy sequence for an irrational $\beta $ subrecursively in the Dedekind cut of $\beta $ . In order to set the value of $C(1)$ we will need a rational a such that $a<\beta < a+1$ , and we cannot get hold of such an a without resorting to unbounded search. Thus, if we do not restrict $O(t)_{{\mathcal {D}}}$ to irrationals in the interval $(0,1)$ , the relation ${\mathcal {C}}\preceq _{S} {\mathcal {D}}$ will not hold. It is therefore important that we are interested in representations of the fractional part of irrational numbers.

Now, let us discuss why we have ${\mathcal {D}} \not \preceq _{S} {\mathcal {C}}$ . As explained above, we have ${\mathcal {D}} \not \preceq _{S} {\mathcal {C}}$ because we cannot avoid unbounded search when we compute a Dedekind cut in a Cauchy sequence, but what does a formal proof look like? In general it is much harder to prove that the relation $\preceq _{S}$ does not hold than it is to prove that it holds. In order to prove $R_1 \preceq _{S} R_2$ , we just have to come up with a subrecursive algorithm for converting an $R_2$ -representation into an $R_1$ -representation. In order to prove $R_1 \not \preceq _{S} R_2$ , we have, according to our definitions, to prove that there is a time-bound t such that we have $O(t)_{R_2} \not \subseteq O(s)_{R_1}$ for any time-bound s. This might not be all that easy. This might call for involved diagonalization arguments. In some cases we might do with a growth argument, that is, we might be able to prove, for some time-bound t, that for any time-bound s there exists $\alpha _s\in O(t)_{R_2}$ such that any Turing machine computing an $R_1$ -representation of $\alpha _s$ will have to give very large outputs, that is, outputs whose length is not of order $O(s)$ (and thus the Turing machine cannot run in time $O(s)$ ). Growth arguments tend to be easier, or at least less tedious, than diagonalization arguments, and we will present a rather detailed proof based on a growth argument in Section 8. But we cannot prove ${\mathcal {D}} \not \preceq _{S} {\mathcal {C}}$ by a growth argument as a Turing machine computing a Dedekind cut gives outputs of length 1. Let us see how we can prove ${\mathcal {D}} \not \preceq _{S} {\mathcal {C}}$ by a diagonalization argument.

Let s be an arbitrary time-bound. We will, by standard diagonalization techniques, construct a Cauchy sequence for an irrational $\alpha $ in the interval $(0,1)$ such that $\alpha $ becomes different from each $\beta \in O(s)_{\mathcal {D}}$ , and hence we will have $\alpha \not \in O(s)_{\mathcal {D}}$ . Our construction can be carried out by a Turing machine, that is, the Cauchy sequence for $\alpha $ can be computed by a Turing machine. That Turing machine will run in time $O(t)$ for some time-bound t, and thus we have $\alpha \in O(t)_{\mathcal {C}}$ . Moreover, it will turn out that t does not depend on s. Hence, we have a time-bound t and for every time-bound s there exists $\alpha $ such that $\alpha \in O(t)_{\mathcal {C}}$ and $\alpha \not \in O(s)_{\mathcal {D}}$ . Hence, we have t such that $O(t)_{\mathcal {C}} \not \subseteq O(s)_{\mathcal {D}}$ for every s, and thus, by Definition 1.4, we can conclude that ${\mathcal {D}}\not \preceq _{S} {\mathcal {C}}$ .

We will now give an algorithm for computing a Cauchy sequence C such that $\lim _n C(n) \not \in O(s)_{\mathcal {D}}$ where s is an arbitrary time-bound. We will need a standard enumeration $\{ e \}_{e\in \mathbb {N}}$ of the Turing machines, and we use $\{ e\}(x)$ to denote the execution of the e’th Turing machine on input x. Furthermore, we need a strictly increasing time-bound function S that eventually dominates any time-bound of order $O(s)$ , that is, for any $s_0$ of order $O(s)$ we have $s_0(m)< S(m)$ for all sufficiently large m. Such an S will always exist, and, moreover, it can be chosen so that $S(n)> 2n$ for all n. Note that by the definition of time-bound functions, we have a Turing machine that given n, computes $S(n)$ in at most $a_S\cdot S(n)$ steps for some constant $a_S$ . We define the sequence $d_0,d_1,d_2, \ldots $ by $d_0=2$ and $d_{i+1} = S(d_i)$ . Finally, we will need a standard computable bijection $\langle \cdot , \cdot \rangle : \mathbb {N} \times \mathbb {N} \longrightarrow \mathbb {N}$ . For any $i\in \mathbb {N}$ , our algorithm needs to compute the unique $j,e\in \mathbb {N}$ such that $\langle j , e \rangle = i$ .

The algorithm sets $C(1):= 1/2$ . If $n>1$ , the algorithm checks if there exists i such that $n = d_{3i+2}$ . If such an i does not exist, the algorithm simply set $C(n):=C(n-1)$ ; if such i exists the algorithm finds the unique $j,e$ such that $\langle j,e\rangle =i$ and sets

• • $C(n):= C(n-1)$ if $\{ e \}(C(n-1))$ does not terminate within n steps.

• • $C(n):= C(n-1) - 2^{-n}$ if $\{ e \}(C(n-1))$ terminates within n steps and outputs 0.

• • $C(n):= C(n-1) + 2^{-n}$ if $\{ e \}(C(n-1))$ terminates within n steps and outputs something other than 0.

It is clear that the algorithm indeed computes a Cauchy sequence for a real number in the interval $(0,1)$ , and it is also pretty easy to see that the length of the output $C(n)$ will be bounded by a function of order $O(n)$ if we represent numbers in binary form and code rationals in a reasonable way. We can w.l.o.g. assume that S eventually dominates any function of order $O(n)$ (we can just pick an S that increases fast enough). Thus, for all sufficiently large n, we have

(1.1) $$ \begin{align} \Vert C(n)\Vert < S(n). \end{align} $$

Now, let $\alpha = \lim _{n} C(n)$ and let $\beta \in O(s)_{\mathcal {D}}$ . We will prove that $\alpha \neq \beta $ .

The Dedekind cut of $\beta $ can be computed by a Turing machine $\{ e \}$ running in time $O(s)$ . Thus, $S(\Vert x\Vert )$ will be an upper bound on the number of steps in the computation $\{ e \}( x )$ when x is large. Pick a sufficiently large j and let $i = \langle j,e\rangle $ (we can make $d_{3i}$ as big as we want by picking a big j). Our algorithm is designed so that we have

(1.2) $$ \begin{align} C(d_{3i}) = C(d_{3i+2}-1). \end{align} $$

Hence, by (1.2) and (1.1), the number of steps in the computation $\{ e \}( C(d_{3i+2}-1) )$ will be bounded by

$$ \begin{align*}S(\Vert C(d_{3i+2}-1)\Vert) = S(\Vert C(d_{3i})\Vert) < S(S(d_{3i})) = d_{3i+2}. \end{align*} $$

Assume that the output of the computation $\{ e \}( C(d_{3i+2}-1) )$ is 0. Then we have

$$ \begin{align*}C(d_{3i+2}) = C(d_{3i+2}-1) - 2^{-d_{3i+2}} \end{align*} $$

but as $\{ e \}$ computes the Dedekind cut of $\beta $ , we have $\beta> C(d_{3i+2}-1)$ . It follows that

$$ \begin{align*}\alpha = \lim_n C(n) \le C(d_{3i+2}-1) - 2^{-d_{3i+2}} + \sum_{n> {3i+2}} 2^{-{d_n}} < C(d_{3i+2}-1) < \beta. \end{align*} $$

Assume that the output of the computation $\{ e \}( C(d_{3i+2}-1) )$ is different from 0. Then we have

$$ \begin{align*}C(d_{3i+2}) = C(d_{3i+2}-1) + 2^{-d_{3i+2}} \end{align*} $$

but as $\{ e \}$ computes the Dedekind cut of $\beta $ , we have $\beta < C(d_{3i+2}-1)$ . It follows that

$$ \begin{align*} \alpha = \lim_n C(n) \ge C(d_{3i+2}-1) + 2^{-d_{3i+2}} - \sum_{n> {3i+2}} 2^{d_n} > C(d_{3i+2}-1) > \beta. \end{align*} $$

This proves that $\alpha \neq \beta $ .

The same argument can be used to prove that the limit $\alpha $ is irrational: Among the Turing machines enumerated there is a machine e that computes the Dedekind cut of any given rational q; this computation can be done in linear time under a reasonable encoding of rationals, so we may assume that e has running time in $O(s)$ ; and we conclude that $\alpha \neq q$ .

We have proved that our algorithm computes a Cauchy sequence for an irrational number which cannot be in the class $O(s)_{\mathcal {D}}$ . Let us undertake a complexity analysis of a Turing machine M executing the algorithm:

• • The input to M is a natural number n (we will estimate an upper bound for M’s running time as a function of n).

• • First M will recursively compute $C(n-1)$ .

• • Then M will check if there exists i such that $n=d_{3i+2}$ . Recall that $d_0=2$ and $d_{i+1}= S(d_i)$ where S is a time-bound function. It is possible to check if such an i exists in time $O(n)$ . Briefly, M computes $d_0,d_1,d_2\dots $ until it either hits j such that $n=d_j$ , or j such that $n<d_j$ , or j such that the computation of $d_j$ exceeds $a_S n$ time (the upper bound that we have on this computation time is $a_S S(d_{j-1})$ ). In all cases, the computation of the last $d_j$ is either completed or stopped after $O(n)$ steps. The computation of each of the previous elements of the sequence, needed for computing $C(n)$ , also takes $O(n)$ steps, and there are at most $\log n$ such elements, because the sequence $d_i$ grows at least exponentially. We conclude that this computation runs in $O(n\log n)$ time.

• • If M finds i such that $n=d_{3i+2}$ , then M will compute $j,e$ such that $\langle j,e\rangle = i$ , check if the computation $\{ e \}(C(n-1))$ terminates within n steps, and finally, compute the output. All this can be done in time $O(n^2)$ on a multi-tape Turing machine.

Since the computation of $C(n)$ takes time $O(n^2)$ , provided we have $C(n-1)$ , we conclude that M runs in time $O(\sum _{i=1}^n i^2) = O(n^{3})$ , where $n\in \mathbb {N}$ is the input, and thus in time $O(2^{3\Vert n\Vert })$ where $\Vert n\Vert $ is the length of the input (time complexity is always stated as a function of the input bit-length). This allows us to conclude that M computes a Cauchy sequence for an irrational in the class $O(2^{3\Vert n\Vert })_{\mathcal {C}}$ . We also know that this irrational is not in the class $O(s)_{\mathcal {D}}$ , and recall that s was an arbitrary chosen time-bound. Hence, we have $O(2^{3\Vert n\Vert })_{\mathcal {C}} \not \subseteq O(s)_{\mathcal {D}}$ for any time-bound s (note that the witness for the non-containement depends on s, and further its computation time depends on the constant $a_S$ , but it is always in $O(2^{3\Vert n\Vert })$ ). This proves that ${\mathcal {D}} \not \preceq _{S} {\mathcal {C}}$ .

Our proof that ${\mathcal {D}} \not \preceq _{S} {\mathcal {C}}$ is meant to illustrate how our definitions work. In the current paper we will in general not formally prove results of the form $R_1\not \preceq _{S} R_2$ , but for the benefit of the reader we will to a certain extent provide informal explanations and intuitive arguments of why it is impossible to subrecursively convert an $R_2$ -representation into an $R_1$ -representation (full proofs may be long and involved, and they are available elsewhere, see Section 1.4).

The equivalence relation $\equiv _S$ induces a degree structure on the representations. The directed graph in Figure 1 gives an overview of the relationship between some natural degrees (equivalence classes). The nodes depict degrees of representations, and each degree is labeled with one of the most well-known representations in the degree. For two representations $R_1$ and $R_2$ , there is a directed path from a node labeled $R_1$ to a node labeled $R_2$ if and only if $R_2 \prec _S R_1$ . Thus, if there is a directed path from $R_1$ to $R_2$ , we can subrecursively convert an $R_1$ -representation into an $R_2$ -representation, and if there is no directed path from $R_1$ to $R_2$ , we cannot subrecursively convert an $R_1$ -representation into an $R_2$ -representation. Unfortunately we are not able to accurately depict the complex relationship between the degrees of the base-b expansions and the degrees of the base-b sum approximations from below and above (for $b=2,3,4,\ldots )$ , but our graph gives a rough idea of what this world looks like. See Section 5 for more on how these degrees relate to each other.

1.4 Our goals and some references

We present a (degree) theory of representations (of irrational numbers) which is based on Turing machines and standard complexity theory. This theory should be considered as a recast and an improvement of the theory developed in Kristiansen [Reference Kristiansen31] [Reference Kristiansen32] and Georgiev et al. [Reference Georgiev, Kristiansen and Stephan20] which is based on honest functions and subrecursive classes. The two approaches studying representations of reals and conversions between them are essentially the same, even if, e.g., the reducibility relation $\preceq _{S}$ is never formally defined in any other paper. It follows more or less straightforwardly from results proved in [Reference Kristiansen31] [Reference Kristiansen32] [Reference Georgiev, Kristiansen and Stephan20] that the picture drawn in Figure 1 is correct.

Recently Georgiev [Reference Georgiev, Vedova, Dundua, Lempp and Manea18] has identified some interesting degrees which are not depicted in Figure 1, moreover, Ben-Amram & Kristiansen [Reference Ben-Amram and Kristiansen7] have proved that the structure is a distributive lattice with a minimal and a maximal degree. Two other recent papers which shed light on the degree structure, are Georgiev [Reference Georgiev19] and Hiroshima & Kawamura [Reference Hiroshima, Kawamura, Vedova, Dundua, Lempp and Manea22].

In papers like [Reference Kristiansen31] [Reference Kristiansen32] [Reference Georgiev, Kristiansen and Stephan20], and furthermore Georgiev [Reference Georgiev, Vedova, Dundua, Lempp and Manea18] [Reference Georgiev19] and Kristiansen [Reference Kristiansen, De Mol, Weiermann, Manea and Fernández-Duque33], the authors are just concerned with the existence or inexistence of subrecursive conversions. In cases where the relation $R_1\preceq _{S} R_2$ holds, they never analyze the computational complexity of converting an $R_2$ -representation into an $R_1$ -representation, and they do not make any effort to find efficient conversions. In this paper we will care about such matters. Indeed, from now on such matters will be our primary concern: We will prove that the relation $R_1\preceq _{S} R_2$ holds by imposing tight upper bounds on the running time of an oracle Turing machine which converts an $R_2$ -representation into an $R_1$ -representation. We will also give upper bounds on the number of oracle calls required and the size of those calls.

We will impose resource bounds on oracle Turing machines that convert irrational numbers in the interval $(0,1)$ . The reader should note that many of our results can be easily generalized to oracle Turing machines converting arbitrary real numbers (including rationals). Moreover, many of these generalizations are trivial. We leave the details to the interested reader.

1.5 Overview of the paper

In Section 2 we provide the necessary preliminaries on complexity theory, oracle Turing machines and the Stern-Brocot tree. In Section 3, we introduce the representation by Weihrauch intersections. The equivalence class of this representation will be the minimal degree in the degree structure depicted in Figure 1 (a detailed proof can be found in [Reference Ben-Amram and Kristiansen7]).

In Section 4 we study conversions between representations subrecursively equivalent to the representation by Cauchy sequences. In Section 5 we study conversions between representations subrecursively equivalent to representations by base-b expansions and base-b sum approximations. In Section 6 we treat representations subrecursively equivalent to the representation by Dedekind cuts, and thereafter, in Section 7, representations subrecursively equivalent to the representation by left/right best approximations.

The degree structure will be discussed further in Section 8. Finally, in Section 9, we study conversions between representations subrecursively equivalent to the representation by continued fractions. It is proved in [Reference Ben-Amram and Kristiansen7] that equivalence class of the continued fractions is the maximal degree in our degree structure.