Enumerator (computer science)

An enumerator ${\displaystyle E}$ is usually represented as a 2-tape Turing machine. One working tape, and one print tape. It can be defined by a 7-tuple, following the notation of a Turing machine: ${\displaystyle E=\langle Q,\Sigma ,\Gamma ,\delta ,q_{0},q_{print},q_{reject}\rangle }$ where

• ${\displaystyle Q}$ is a finite, non-empty set of states.
• ${\displaystyle \Sigma }$ is a finite, non-empty set of the output / print alphabet
• ${\displaystyle \Gamma }$ is a finite, non-empty set of the working tape alphabet
• ${\displaystyle \delta }$ is the transition function. ${\displaystyle \delta :Q\times \Gamma \rightarrow Q\times \Gamma \times \{L,R\}\times \Sigma _{\epsilon }}$
• ${\displaystyle q_{0}\in Q}$ is the start state
• ${\displaystyle q_{print}\in Q}$ is the print state.
• ${\displaystyle q_{reject}}$ is the reject state with ${\displaystyle q_{print}\neq q_{reject}}$

A language over a finite alphabet is Turing Recognizable if and only if it can be enumerated by an enumerator. This shows Turing recognizable languages are also recursively enumerable.

Proof

A Turing Recognizable language can be Enumerated by an Enumerator

Consider a Turing Machine ${\displaystyle M}$ and the language accepted by it be ${\displaystyle L(M)}$. Since the set of all possible strings over the input alphabet ${\displaystyle \Sigma }$ i.e. the Kleene Closure ${\displaystyle \Sigma ^{*}}$ is a countable set, we can enumerate the strings in it as ${\displaystyle s_{1},s_{2},\dots ,s_{i},}$ etc. Then the Enumerator enumerating the language ${\displaystyle L(M)}$ will follow the steps:

1 for i = 1,2,3,...
2 Run ${\displaystyle M}$ with input strings ${\displaystyle s_{1},s_{2},\dots ,s_{i}}$ for ${\displaystyle i}$-steps
3 If any string is accepted, then print it. 

Now the question comes whether every string in the language ${\displaystyle L(M)}$ will be printed by the Enumerator we constructed. For any string ${\displaystyle w}$ in the language ${\displaystyle L(M)}$ the TM ${\displaystyle M}$ will run finite number of steps(let it be ${\displaystyle k}$ for ${\displaystyle w}$) to accept it. Then in the ${\displaystyle k}$-th step of the Enumerator ${\displaystyle w}$ will be printed. Thus the Enumerator will print every string ${\displaystyle M}$ recognizes but a single string may be printed several times.

An Enumerable Language is Turing Recognizable

It's very easy to construct a Turing Machine ${\displaystyle M}$ that recognizes the enumerable language ${\displaystyle L}$. We can have two tapes. On one tape we take the input string and on the other tape, we run the enumerator to enumerate the strings in the language one after another. Once a string is printed in the second tape we compare it with the input in the first tape. If its a match, then we accept the input, else reject.

Sipser, Michael (2012). Introduction to the Theory of Computation - International Edition. ISBN 978-1-133-18781-3.