A pumping lemma for deterministic context-free languages?
There is a Pumping Lemma specifically for DCFL, under the title &quot;A Pumping Lemma for Deterministic Context-Free Languages&quot;, by Sheng Yu; Information Processing Letters 31 (1989) 47-51, doi 10.1016/0020-0190(89)90108-7. With this explicit title I must apologize that I missed it!
The online copy unfortunately has a blank spot in one of the formula, so I hope I reconstructed the result properly. Below ${}^{(1)}y$ is the first symbol of $y$ (when it exists) or $\varepsilon$ (if $y=\varepsilon$).
Lemma 1 (Pumping Lemma). Let $L$ be a DCFL. Then there exists a constant $C$ for $L$ such that for any pair of words $w,w'\in $ if
(1) $w=xy$ [?] and $w'=xz$, $|x|&gt;C$ and
(2) ${}^{(1)}y = {}^{(1)}z $, [?]
then either (3) or (4) is true:
(3) there is a factorization $x=x_1x_2x_3x_4x_5$, $|x_2x_4|\ge 1$ and $|x_2x_3x_4|\le C$, such that for all $i\ge 0$ $x_1x^i_2x_3x^i_4x_5y$ and $x_1x^i_2x_3x^i_4x_5z$  are in $L$;
(4) there exist factorizations $x=x_1x_2x_3$, $y=y_1y_2y_3$ and $z=z_1z_2z_3$,  $|x_2|\ge 1$ and $|x_2x_3|\le C$, such that for all $i\ge 0$ $x_1x^i_2x_3y_1y^i_2y_3$ and $x_1x^i_2x_3z_1z^i_2z_3$ are in $L$.
Two applications of the Lemma are given: $\{ a^ib^i \mid i\ge 0 \} \cup \{ a^ib^{2i}  \mid i\ge 0 \}$ as well as $\{ w\in\{a,b\}^* \mid w=uv, |u|=|v|, \mbox{ and } v \mbox{ contains an } a \}$ are not DCFL.
The proof uses the fact that each DCFL has an LR(1) grammar in Greibach normal form.