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Let $T:Σ^*\to Σ^*$ be an operation such that $T(L)$ is regular for all regular languages $L \in Σ^*$.

Is it possible to prove $T^∞(L)$ is regular?

$T^∞(L)=\bigcup_{i=1}^{\infty}{T^{i}\left(L\right)}$

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    $\begingroup$ "Tread carefully on infinity", said someone. The beast of infinity is beyond our finite imagination. How do you define $T^\infty(L)$? For example, $\Sigma={a}$, $L=\Sigma^*$ and $T(a)=aa$. For another example, $\Sigma=\{a,b\}$, $L=\{a\}^*$, $T(a)=b$ and $T(b)=a$. I could think of several (or maybe infinitely many?) definitions. $\endgroup$ – Apass.Jack Feb 2 at 3:50
  • $\begingroup$ can I add a constraint eg: L⊆T(L), does this make it more reasonable? $\endgroup$ – DoYouLikeMiiii Feb 2 at 3:58
  • $\begingroup$ It looks like you want $T^\infty(L) =\cap_{i=1}^\infty T^n(L)$. Can you add that to the question? However, the dust is not settled down yet. $\endgroup$ – Apass.Jack Feb 2 at 4:04
  • $\begingroup$ is this problem unsolvable yet? $\endgroup$ – DoYouLikeMiiii Feb 2 at 4:11
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    $\begingroup$ Solvable. The answer is negative. Here is a counterexample. $T(\{a^{i}\})=\{a^{2i}\}$. We can see that $T^\infty(\{a\})$ is not regular. $\endgroup$ – Apass.Jack Feb 2 at 4:12
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Colleague Apass Jack already warned about the dangers of infinity, and he also indicated a very simple example that shows that a very simple iteration leads to a non-regular language. Case closed, but I like to add an observation: iterating simple local substitutions lead to Turing power, not just non-regularity.

The single steps of a Turing machine van be encoded, and performed with a regular operation. The instruction "on reading $a$ in state $q$, write $b$, move left and change to state $p$" is coded as rule $aq \mapsto pb$ and is extended to longer strings containing a single state as $\alpha aq \beta \mapsto \alpha pb\beta $. This operation can be extended to sets of instructions, and will map a regular language into a regular language. However,iterating them will actually generated Turing machine computations!

(I have an answer somewhere with this observation, but cannot find it.)

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