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I am going through some past exam paper questions on regular languages for some revision, and I am having a bit of trouble with converting general ideas into formal mathematical proofs.

The question is:

Given regular expression $S$, prove formally that $S^* = (S^*)^*$.

My problem is expressing this in a formal proof. Here is what I have worked through so far (it is a bit all over the place and just a collection of ways to express the problem mostly)

$S^* = (S^*)^*$

this implies:

$S^* \subseteq (S^*)^*$ and $S^* \supseteq (S^*)^*$

if we assume that there exists $w_k$ such that $w_k \in S^*$

then the base case for the proof is:

$k = 0$ $(w_k = \epsilon)$ (empty word, always in $S^*$ and $(S^*)^*$ by definition)

$k = 1$ $(w_k \in S^*)$

and that's kind of where my ability to reason ends.

I think the rest of it will be something like:

$w_{k+1} = w_kx$

ie. $w_k$ concatenated with $x$ where $x \in S^*$

but how can I show that $w_{k+1} \in (S^*)^*$?

Any help to push me in the right direction would be greatly appreciated.

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As you observed, to show $S^*=(S^*)^*$ it will suffice to show $S^*\subseteq(S^*)^*$ and $(S^*)^*\subseteq S^*$. Here's a sketch of a proof of both containments.

For a language $A$, define $A^*=\{x_1x_2\dots x_n\mid n\ge 0\text{ and each }x_i\in A\}$. Then (with $n=1$) it's clear that for all $x\in A$ we'll have $x\in A^*$, establishing that $S^*\subseteq(S^*)^*$.

To show containment in the opposite direction, let $x\in (S^*)^*$, then by definition we'll have $x=y_1y_2\dots y_n$ where $y_i\in S^*$ for all $1 \le i\le n$. Then, again by definition, we'll have $y_i=z_{i1}z_{i2}\dots z_{in_i}$ where $z_{ij}\in S$ for all suitable $i$. We'll then have $$ x=(z_{11}z_{12}\dots z_{1n_1})(z_{21}z_{22}\dots z_{2n_2})\dots(z_{n1}z_{n2}\dots z_{nn_n}) $$ where each $z_{ij}\in S$. Applying the definition of star one more time we see that $x\in S^*$, establishing that $(S^*)^*\subseteq S^*$ and hence the desired equality.

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Let L be the set of strings derived from S. Prove that both expressions produce exactly the strings $L_1 L_2 ... L_k$ for k >= 0 where each $L_j$ is in L, and no others.

You can also check what happens if you replace one, two, or three of the stars in the expression with “+”. With some changes the expression stays true.

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