Given a regular language $L$, then it is easy to prove that there is a constant $N$ such that is $\sigma \in L$, with $\lvert \sigma \rvert \ge N$ there exist strings $\alpha$, $\beta$ and $\gamma$ such that $\lvert \alpha \beta \rvert \le N$ and $\lvert \beta \rvert \ne \epsilon$, and for all $k$ it is $\alpha \beta^k \gamma \in L$. It is widely stated that the converse isn't true, but I haven't seen any clear example. Any suggestions? Clearly the proof that the offending language isn't regular has to use stronger methods than the typical doesn't satisfy the pumping lemma. I'd be interested in simple examples, to present in introductory formal languages classes.
Languages that satisfy the pumping lemma but aren't regular?
formal languages;proof techniques
The language $\{ \$ a^nb^n \mid n \ge 1 \} \cup \{ \$^kw \mid k\neq 1, w\in \{a,b\}^* \}$ seems to be simple. The second part is regular (and can be pumped). The first part is nonregular, but can be pumped into the second part by choosing $\$$ to pump.(added) Of course, this can be generalized to $\$L \cup \{ \$^k \mid k\neq 1 \} \cdot \{a,b\}^*$ for any $L\subseteq \{a,b\}^*$. Sometimes the formulation is in the if ... then ... style: if $w$ starts with a single $\$$ then it is of the form. That I personally find less intuitive.As noted by @vonbrand the (possibly) non-regular part of the language is isolated by intersecting with $\$\{a,b\}^*$. This can be separately tested using the pumping lemma if needed.