How to formally prove that any regular expression can be written as a finite combination of base cases and operations?

Question

In Michael Sipser's book, "Introduction to the Theory of Computation," regular expressions are defined as follows:

Based on this definition, how can I formally prove that any regular expression can be written as a finite formula where each of the elements in the formula are either:

1. Parentheses,
2. Union operation, concatenation operation, or Kleene star operation,
3. One of the basic regular languages, i.e., a letter from the alphabet, the empty string, or the empty language.

Answer 1

What you have there is a recursive definition. Every regular expression is generated by a finite number of applications of rules 1.-6., so you can simply do mathematical induction over the number of steps required to derive your expression¹.

So start like this:

• If $R$ was derived in 1 step, then it must have the form $R = a \in \Sigma$, $R = \varepsilon$ or $R = \emptyset$. Therefore...
• Assume that all regular expressions derived in $\leq n$ steps have the desired property.
• If $R$ was derived in $n + 1$ steps, then $R$ must have the form $R = (R_1 \cup R_2), R = (R_1 \circ R_2)$ or $R = R_1^*$ where $R_1, R_2$ were derived in $\leq n$ steps. Using the induction hypothesis...

Why is every regular expression generated in a finite number of steps?

The recursive definition as written is a bit ambiguous. It only tells us which objects are regular expressions, but not which objects aren't. It is (sadly) often left implied, but when speaking about a recursively defined set, we usually mean the smallest set satisfying the requirements.

So given the definition above, we would formally define the set of regular expressions as the smallest set $\mathcal{R}$ satisfying 1. and 2. below

1. $a, \varepsilon, \emptyset \in \mathcal{R}$ for all $a \in \Sigma$
2. $(R_1 \cup R_2), (R_1 \circ R_2), (R_1^*) \in \mathcal{R}$ if $R_1, R_2 \in \mathcal{R}$

Now define the function $S: \mathbb{N} \to \wp(\mathcal{R})$ (by the recursion theorem) as follows

• $S(0) = \{\varepsilon, \emptyset\} \cup \Sigma$
• $S(n + 1) = \{(R_1 \cup R_2), (R_1 \circ R_2), (R_1^*), R_1 : R_1, R_2 \in S(n)\}$

Intuitively, think about $S(n)$ as the set of regular expressions derivable in $n$ steps or less. Define $$\mathcal{R}' := \bigcup_{n \in \mathbb{N}} S(n)$$

and note, that $\mathcal{R}' \subseteq \mathcal{R}$. Since $\mathcal{R}'$ also satisfies 1. and 2. above, it follows that $\mathcal{R} \subseteq \mathcal{R}'$ since $\mathcal{R}$ is the smallest set satisfying 1. and 2.

So $\mathcal{R} = \mathcal{R}'$.

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1 You can also use structural induction, you might have already encountered it in a similar context in a formal logic class.