Consider alphabet $=\{0,1\}$ and language $L_{01}=\{\begin{array}{cc}\text{w}\text{i}\text{n}& {}^{**}\text{:}w=0^n{1}^n\end{array}$ for some nonnegative $\{$:ninZ$\}.$

Prove or disprove that for each language $L$ over $,$ if $Lsube{L}_{01}$ and $L$ is regular, then $L$ is finite.

Let $L$ be a finite language over alphabet $=\{a,b\}.$ Show that for every nonnegative integer $n$ and deterministic

finite automaton $M$ that recognizes $L,$ if $M$ has exactly $n$ states, then $nlo{g}_2(|L|+1).$

Prove or disprove that for every alphabet $$ and language $L$ over $,$ if $L$ is non$-$regular, then $L^{**}$ is non$-$regular.

Consider language : $M$ is a TM that accepts at least $2023$ strings