Editing Computably enumerable set

{{Short description|Mathematical logic concept}}
{{Redirect|Enumerable set|the set-theoretic concept|Countable set}}
In [[computability theory]], a set ''S'' of [[natural numbers]] is called '''computably enumerable (c.e.)''', '''recursively enumerable (r.e.)''', '''semidecidable''', '''partially decidable''', '''listable''', '''provable''' or '''Turing-recognizable''' if:

*There is an [[algorithm]] such that the set of input numbers for which the algorithm halts is exactly ''S''. 

Or, equivalently,

*There is an [[enumeration algorithm|algorithm that enumerates]] the members of ''S''.  That means that its output is a list of all the members of ''S'':  ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ... .  If ''S'' is infinite, this algorithm will run forever, but each element of S will be returned after a finite amount of time. Note that these elements do not have to be listed in a particular way, say from smallest to largest.

The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm can run forever, and no information is returned. A set that is "completely decidable" is a [[computable set]].  The second condition suggests why ''computably enumerable'' is used. The abbreviations '''c.e.''' and '''r.e.''' are often used, even in print, instead of the full phrase.

In [[computational complexity theory]], the [[complexity class]] containing all computably enumerable sets is [[RE (complexity)|RE]]. In recursion theory, the [[Lattice (order)|lattice]] of c.e. sets under inclusion is denoted <math>\mathcal{E}</math>.

== Definition ==

A set ''S'' of natural numbers is called '''computably enumerable''' if there is a [[Computable function|partial computable function]] whose [[domain of a function|domain]] is exactly ''S'', meaning that the function is defined if and only if its input is a member of ''S''.

==Equivalent formulations==
The following are all equivalent properties of a set ''S'' of natural numbers:

;Semidecidability<nowiki>:</nowiki>
:*The set ''S'' is computably enumerable. That is, ''S'' is the domain (co-range) of a partial computable function.
:*The set ''S'' is <math>\Sigma^0_1</math> (referring to the [[arithmetical hierarchy]]).<ref>{{cite book |last1=Downey |first1=Rodney G. |last2=Hirschfeldt |first2=Denis R. |title=Algorithmic Randomness and Complexity |date=29 October 2010 |publisher=Springer Science & Business Media |isbn=978-0-387-68441-3 |page=23 |url=https://books.google.com/books?id=FwIKhn4RYzYC&pg=PA23 |language=en}}</ref>
:*There is a partial computable function ''f'' such that: <math display="block"> f(x) =
\begin{cases} 
1 &\mbox{if}\ x \in S \\
\mbox{undefined/does not halt}\ &\mbox{if}\ x \notin S
\end{cases}
</math>
;Enumerability<nowiki>:</nowiki>
:*The set ''S'' is the range of a partial computable function.
:*The set ''S'' is the range of a total computable function, or empty. If ''S'' is infinite, the function can be chosen to be [[injective]].
:*The set ''S'' is the range of a [[primitive recursive function]] or empty. Even if ''S'' is infinite, repetition of values may be necessary in this case.
;Diophantine<nowiki>:</nowiki>
:*There is a polynomial ''p'' with integer coefficients and variables ''x'', ''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'' ranging over the natural numbers such that <math display="block">x \in S \Leftrightarrow \exists a,b,c,d,e,f,g,h,i \ ( p(x,a,b,c,d,e,f,g,h,i) = 0).</math> (The number of bound variables in this definition is the best known so far; it might be that a lower number can be used to define all Diophantine sets.)
:*There is a polynomial from the integers to the integers such that the set ''S'' contains exactly the non-negative numbers in its range.

The equivalence of semidecidability and enumerability can be obtained by the technique of [[Dovetailing (computer science)|dovetailing]].

The Diophantine characterizations of a computably enumerable set, while not as straightforward or intuitive as the first definitions, were found by [[Yuri Matiyasevich]] as part of the negative solution to [[Hilbert's tenth problem|Hilbert's Tenth Problem]]. Diophantine sets predate recursion theory and are therefore historically the first way to describe these sets (although this equivalence was only remarked more than three decades after the introduction of computably enumerable sets).

{| style="float:right"
| [[File:Recursive enumeration of all halting Turing machines.gif|thumb|450px|A computable enumeration of the set of all Turing machines halting on a fixed input: Simulate all Turing machines (enumerated on vertical axis) step by step (horizontal axis), using the shown diagonalization scheduling. If a machine terminates, print its number. This way, the number of each terminating machine is eventually printed. In the example, the algorithm prints "9, 13, 4, 15, 12, 18, 6, 2, 8, 0, ..."]]
|}

==Examples==
* Every [[computable set]] is computably enumerable, but it is not true that every computably enumerable set is computable. For computable sets, the algorithm must also say if an input is ''not'' in the set – this is not required of computably enumerable sets.
* A [[recursively enumerable language]] is a computably enumerable subset of a [[formal language]].
* The set of all provable sentences in an effectively presented axiomatic system is a computably enumerable set.
* [[Matiyasevich's theorem]] states that every computably enumerable set is a [[Diophantine set]] (the converse is trivially true).
* The [[simple set]]s are computably enumerable but not computable.
* The [[creative set]]s are computably enumerable but not computable.
* Any [[productive set]] is '''not''' computably enumerable.
* Given a [[Gödel numbering]] <math>\phi</math> of the computable functions, the set <math>\{\langle i,x \rangle \mid \phi_i(x) \downarrow \}</math> (where <math>\langle i,x \rangle</math> is the [[Cantor pairing function]] and <math>\phi_i(x)\downarrow</math> indicates <math>\phi_i(x)</math> is defined) is computably enumerable (cf. picture for a fixed ''x''). This set encodes the [[halting problem]] as it describes  the input parameters for which each [[Turing machine]] halts.
* Given a Gödel numbering <math>\phi</math> of the computable functions, the set <math>\{ \left \langle x, y, z \right \rangle \mid \phi_x(y) = z \}</math> is computably enumerable. This set encodes the problem of deciding a function value.
* Given a partial function ''f'' from the natural numbers into the natural numbers, ''f'' is a partial computable function if and only if the graph of ''f'',  that is, the set of all pairs <math>\langle x,f(x)\rangle</math> such that ''f''(''x'') is defined, is computably enumerable.

==Properties==

If ''A'' and ''B'' are computably enumerable sets then ''A'' ∩ ''B'', ''A'' ∪ ''B'' and ''A'' × ''B'' (with the ordered pair of natural numbers mapped to a single natural number with the [[Cantor pairing function]]) are computably enumerable sets. The [[preimage]] of a computably enumerable set under a partial computable function is a computably enumerable set.

A set <math>T</math> is called '''co-computably-enumerable''' or '''co-c.e.''' if its [[complement (set theory)|complement]] <math>\mathbb{N} \setminus T</math> is computably enumerable. Equivalently, a set is co-r.e. if and only if it is at level <math>\Pi^0_1</math> of the arithmetical hierarchy. The complexity class of co-computably-enumerable sets is denoted co-RE.

A set ''A'' is [[Computable set|computable]] if and only if both ''A'' and the complement of ''A'' are computably enumerable.

Some pairs of computably enumerable sets are [[effectively separable]] and some are not.

== Remarks ==
According to the [[Church–Turing thesis]], any effectively calculable function is calculable by a [[Turing machine]], and thus a set ''S'' is computably enumerable if and only if there is some [[algorithm]] which yields an enumeration of ''S''.  This cannot be taken as a formal definition, however, because the Church–Turing thesis is an informal conjecture rather than a formal axiom.

The definition of a computably enumerable set as the ''domain'' of a partial function, rather than the ''range'' of a total computable function, is common in contemporary texts.  This choice is motivated by the fact that in generalized recursion theories, such as [[Alpha recursion theory|α-recursion theory]], the definition corresponding to domains has been found to be more natural.  Other texts use the definition in terms of enumerations, which is equivalent for computably enumerable sets.

== See also ==
* [[RE (complexity)]]
* [[Recursively enumerable language]]
* [[Arithmetical hierarchy]]

== References ==
{{reflist}}
* Rogers, H. ''The Theory of Recursive Functions and Effective Computability'', [[MIT Press]]. {{isbn|0-262-68052-1}}; {{isbn|0-07-053522-1}}.
* Soare, R. Recursively enumerable sets and degrees. ''Perspectives in Mathematical Logic.'' [[Springer-Verlag]], Berlin, 1987. {{isbn|3-540-15299-7}}.
* Soare, Robert I. Recursively enumerable sets and degrees. ''Bull. Amer. Math. Soc.'' 84 (1978), no. 6, 1149–1181.

{{Mathematical logic}}

[[Category:Computability theory]]
[[Category:Theory of computation]]