Lower bound technique for length-reducing automata

The class of growing context-sensitive languages (GCSL) was proposed as a naturally defined subclass of context-sensitive languages whose membership problem is solvable in polynomial time. Growing context-sensitive languages and their deterministic counterpart called Church–Rosser languages (CRL) complement the Chomsky hierarchy in a natural way, as the classes filling the gap between context-free languages and context-sensitive languages. They possess characterizations by a natural machine model, length-reducing two-pushdown automata (lrTPDA). We introduce a lower bound technique for lrTPDAs. Using this technique, we prove the conjecture of McNaughton, Narendran and Otto that the set of palindromes is not in CRL. As a consequence we obtain that CFL∩coCFL as well as UCFL∩coUCFL are not included in CRL, where UCFL denotes the class of unambiguous context-free languages; this solves an open problem posed by Beaudry, Holzer, Niemann and Otto. Another corollary is that CRL is a strict subset of GCSL∩coGCSL.