Real number that can be computed within arbitrary precision
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel in 1912, using the intuitive notion of computability available at the time.
Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.
Informal definition.
In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1:
 ;templatestyles src="Template:Blockquote/styles.css" / ;A computable number [is] one for which there is a Turing machine which, given "n" on its initial tape, terminates with the "n"th digit of that number [encoded on its tape].
The key notions in the definition are (1) that some "n" is specified at the start, (2) for any "n" the computation only takes a finite number of steps, after which the machine produces the desired output and terminates.
An alternate form of (2) – the machine successively prints all "n" of the digits on its tape, halting after printing the "n"th – emphasizes Minsky's observation: (3) That by use of a Turing machine, a "finite" definition – in the form of the machine's state table – is being used to define what is a potentially "infinite" string of decimal digits.
This is however not the modern definition which only requires the result be accurate to within any given accuracy. The informal definition above is subject to a rounding problem called the table-maker's dilemma whereas the modern definition is not.
Formal definition.
A real number "a" is computable if it can be approximated by some computable function formula_1 in the following manner: given any positive integer "n", the function produces an integer "f"("n") such that:
formula_2
A complex number is called computable if its real and imaginary parts are computable.
Equivalent definitions.
There are two similar definitions that are equivalent:
There is another equivalent definition of computable numbers via computable Dedekind cuts. A computable Dedekind cut is a computable function formula_8 which when provided with a rational number formula_9 as input returns formula_10 or formula_11, satisfying the following conditions:
formula_12
formula_13
formula_14
An example is given by a program "D" that defines the cube root of 3. Assuming formula_15 this is defined by:
formula_16
formula_17
A real number is computable if and only if there is a computable Dedekind cut "D" corresponding to it. The function "D" is unique for each computable number (although of course two different programs may provide the same function).
Properties.
Not computably enumerable.
Assigning a Gödel number to each Turing machine definition produces a subset formula_18 of the natural numbers corresponding to the computable numbers and identifies a surjection from formula_18 to the computable numbers. There are only countably many Turing machines, showing that the computable numbers are subcountable. The set formula_18 of these Gödel numbers, however, is not computably enumerable (and consequently, neither are subsets of formula_18 that are defined in terms of it). This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals. In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the set formula_18 of machines representing computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them.
While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable. Here, for any given computable number formula_23 the well ordering principle provides that there is a minimal element in formula_18 which corresponds to formula_25, and therefore there exists a subset consisting of the minimal elements, on which the map is a bijection. The inverse of this bijection is an injection into the natural numbers of the computable numbers, proving that they are countable. But, again, this subset is not computable, even though the computable reals are themselves ordered.
Properties as a field.
The arithmetical operations on computable numbers are themselves computable in the sense that whenever real numbers "a" and "b" are computable then the following real numbers are also computable: "a + b", "a - b", "ab", and "a/b" if "b" is nonzero.
These operations are actually "uniformly computable"; for example, there is a Turing machine which on input ("A","B",formula_26) produces output "r", where "A" is the description of a Turing machine approximating "a", "B" is the description of a Turing machine approximating "b", and "r" is an formula_26 approximation of "a"+"b".
The fact that computable real numbers form a field was first proved by Henry Gordon Rice in 1954.
Computable reals however do not form a computable field, because the definition of a computable field requires effective equality.
Non-computability of the ordering.
The order relation on the computable numbers is not computable. Let "A" be the description of a Turing machine approximating the number formula_6. Then there is no Turing machine which on input "A" outputs "YES" if formula_29 and "NO" if formula_30 To see why, suppose the machine described by "A" keeps outputting 0 as formula_26 approximations. It is not clear how long to wait before deciding that the machine will "never" output an approximation which forces "a" to be positive. Thus the machine will eventually have to guess that the number will equal 0, in order to produce an output; the sequence may later become different from 0. This idea can be used to show that the machine is incorrect on some sequences if it computes a total function. A similar problem occurs when the computable reals are represented as Dedekind cuts. The same holds for the equality relation: the equality test is not computable.
While the full order relation is not computable, the restriction of it to pairs of unequal numbers is computable. That is, there is a program that takes as input two Turing machines "A" and "B" approximating numbers formula_32 and formula_33, where formula_34, and outputs whether formula_35 or formula_36 It is sufficient to use formula_26-approximations where formula_38 so by taking increasingly small formula_26 (approaching 0), one eventually can decide whether formula_35 or formula_36
Other properties.
The computable real numbers do not share all the properties of the real numbers used in analysis. For example, the least upper bound of a bounded increasing computable sequence of computable real numbers need not be a computable real number. A sequence with this property is known as a Specker sequence, as the first construction is due to Ernst Specker in 1949. Despite the existence of counterexamples such as these, parts of calculus and real analysis can be developed in the field of computable numbers, leading to the study of computable analysis.
Every computable number is arithmetically definable, but not vice versa. There are many arithmetically definable, noncomputable real numbers, including:
Both of these examples in fact define an infinite set of definable, uncomputable numbers, one for each universal Turing machine.
A real number is computable if and only if the set of natural numbers it represents (when written in binary and viewed as a characteristic function) is computable.
The set of computable real numbers (as well as every countable, densely ordered subset of computable reals without ends) is order-isomorphic to the set of rational numbers.
Digit strings and the Cantor and Baire spaces.
Turing's original paper defined computable numbers as follows:
 ;templatestyles src="Template:Blockquote/styles.css" / ;A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer formula_43 as input and produces the formula_44-th digit of the real number's decimal expansion as output.
Turing was aware that this definition is equivalent to the formula_26-approximation definition given above. The argument proceeds as follows: if a number is computable in the Turing sense, then it is also computable in the formula_26 sense: if formula_47, then the first "n" digits of the decimal expansion for "a" provide an formula_26 approximation of "a". For the converse, we pick an formula_26 computable real number "a" and generate increasingly precise approximations until the "n"th digit after the decimal point is certain. This always generates a decimal expansion equal to "a" but it may improperly end in an infinite sequence of 9's in which case it must have a finite (and thus computable) proper decimal expansion.
Unless certain topological properties of the real numbers are relevant, it is often more convenient to deal with elements of formula_50 (total 0,1 valued functions) instead of reals numbers in formula_51. The members of formula_50 can be identified with binary decimal expansions, but since the decimal expansions formula_53 and formula_54 denote the same real number, the interval formula_51 can only be bijectively (and homeomorphically under the subset topology) identified with the subset of formula_50 not ending in all 1's.
Note that this property of decimal expansions means that it is impossible to effectively identify the computable real numbers defined in terms of a decimal expansion and those defined in the formula_26 approximation sense. Hirst has shown that there is no algorithm which takes as input the description of a Turing machine which produces formula_26 approximations for the computable number "a", and produces as output a Turing machine which enumerates the digits of "a" in the sense of Turing's definition. Similarly, it means that the arithmetic operations on the computable reals are not effective on their decimal representations as when adding decimal numbers. In order to produce one digit, it may be necessary to look arbitrarily far to the right to determine if there is a carry to the current location. This lack of uniformity is one reason why the contemporary definition of computable numbers uses formula_26 approximations rather than decimal expansions.
However, from a computability theoretic or measure theoretic perspective, the two structures formula_50 and formula_51 are essentially identical. Thus, computability theorists often refer to members of formula_50 as reals. While formula_50 is totally disconnected, for questions about formula_64 classes or randomness it is easier to work in formula_50.
Elements of formula_66 are sometimes called reals as well and though containing a homeomorphic image of formula_67, formula_66 isn't even locally compact (in addition to being totally disconnected). This leads to genuine differences in the computational properties. For instance the formula_69 satisfying formula_70, with formula_71 quantifier free, must be computable while the unique formula_72 satisfying a universal formula may have an arbitrarily high position in the hyperarithmetic hierarchy.
Use in place of the reals.
The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as "e", "π", and many other transcendental numbers. Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers. The question naturally arises of whether it is possible to dispose of the full set of reals and use computable numbers for all of mathematics. This idea is appealing from a constructivist point of view, and has been pursued by the Russian school of constructive mathematics.
To actually develop analysis over computable numbers, some care must be taken. For example, if one uses the classical definition of a sequence, the set of computable numbers is not closed under the basic operation of taking the supremum of a bounded sequence (for example, consider a Specker sequence, see the section above). This difficulty is addressed by considering only sequences which have a computable modulus of convergence. The resulting mathematical theory is called computable analysis.
Implementations of exact arithmetic.
Computer packages representing real numbers as programs computing approximations have been proposed as early as 1985, under the name "exact arithmetic". Modern examples include the CoRN library (Coq), and the RealLib package (C++). A related line of work is based on taking a real RAM program and running it with rational or floating-point numbers of sufficient precision, such as the iRRAM package.
Notes.
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References.
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Further reading.
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