# Proof that the language $ww^R$ is not regular without using the pumping lemma

I am breaking my head over this. Let the alphabet $A$ be given by $A = \{a,b,c\}$ and let

$$L = \{ww^R \mid w \in A^* \}.$$

Prove that the language $L$ is not regular without using the pumping lemma, but using:

• The fact that $\{a^nb^n \in A^* \mid n \geq 0 \}$ is not regular.
• The fact that $\{w \in A^* \mid w \text{ is a palindrome} \}$ is not regular.
• Closure properties of the class of regular languages (such as if $L_1$ and $L_2$ are regular, then $L_1 \cup L_2$ is also regular, etc.).
• $L$ consists of the even length palindromes. – Hendrik Jan Dec 2 '16 at 16:31
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Let

$$L_1 = \{\ \omega\omega^r \ | \ \omega \in \Sigma^* \}$$ $$L_2 = \{\ a^nb^n \ | \ n \in \mathbb{N}\ \}$$

Because $$\Sigma = \{a, b, c\}$$ is finite, $$\Sigma^*$$ must be countable.

This means we can define a function $$f : \Sigma^* \rightarrow \mathbb{N}$$ which returns the position of a word $$\omega \in \Sigma^*$$ in an arbitrary ordering defined by $$f$$. $$f$$ is invertible, we define $$f^{-1} : \mathbb{N} \rightarrow \Sigma^*$$, which returns the word $$\omega$$ at position $$n$$ in the enumeration of $$\Sigma^*$$ as defined by $$f$$.

Now, since every word $$\omega$$ is uniquely defined by a natural number $$n$$, and every word $$\omega$$ has exactly one unique reverse $$\omega^r$$, we can define an isomorphism between $$L_1$$ and $$L_2$$:

$$g \colon\ L_1 \rightarrow L_2; \quad \omega\omega^r \mapsto a^{f(\omega)}b^{f(\omega)}$$

$$g^{-1} \colon \ L_2 \rightarrow L_1; \quad a^nb^n \mapsto f^{-1}(n) f^{-1}(n)^r$$

Since $$L_2$$ is not regular and we have a bijection between $$L_1$$ and $$L_2$$, $$L_1$$ must not be regular.

This proof works for any finite alphabet $$\Sigma$$ because the transitive closure of a finite alphabet is always countable.

• "isomorphism"? You meant bijection. It does not work; otherwise we can show $\{\ a^n \mid n \in \Bbb{N}\}$ is not regular since we can build an "isomorphism" between it and $L_2$ easily. – John L. Dec 21 '18 at 9:07
• @Apass.Jack Yes, I meant bijection. Thanks! – ThreeFx Dec 21 '18 at 9:11

Suppose that $L$ is regular. Then $L' = L \cap (a+b)^*c^2(a+b)^*$ is also regular. Another description for $L'$ is $$L' = \{ wccw^R : w \in (a+b)^* \}.$$ Consider now the regular substitution $\sigma$ given by $$\sigma(a) = \{a\}, \sigma(b) = \{b\}, \sigma(c) = \{a,b,\epsilon\}.$$ Then $L'' = \sigma(L')$ is also regular. But $L''$ is the language of palindromes.