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Finding all recursive functions that are the indicator functions of Dyadic Recursive Tree
I want to find all recursive functions $f(n)$ that are the indicator functions of Dyadic Recursive Tree of order n
I already found one solution, which is based on this argument.
Solution
Suppose $f(0) = \text{true}$.
Let $f(n) = \overline{f(f(n-1))}$
$f(n)$ is always $\text{true}$.
My question
is my argument right?
and If my argument is correct, then are there any other solutions?
A:
For every $n\in\Bbb N$ we have $f(0)=\color{red}{\color{black}{\color{red}0}}$ and $f(2^n)=\color{red}{\color{black}{\color{red}0}}$
$\begin{align}f(1)&=\overline{f(f(0))}\\
&=\overline{\overline{\color{red}{\color{black}{\color{red}0}}} }\qquad\qquad\qquad\color{red}{\color{black}{\color{red}0}}\text{by property of $\overline\quad$}\\
&=\overline{\color{red}{\color{black}{\color{red}0}}}\color{red}{\color{black}{\color{red}0}}\quad\text{by property of $\overline\quad$}\\
&=\color{red}{\color{black}{\color{red}0}}\text{since $f(1)=\color{red}{\color{black}{\color{red}0}}$}\\
f(2)&=\overline{f(f(1))}\\
&=\overline{\overline{\color{red}{\color{black}{\color{red}0}}} }\color{red}{\color{black}{\color 0b46394aab
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