A Solution to Yamakami’s Problem on Non-uniform Context-free Languages

Yamakami (Theoret. Comput. Sci., 2011) studies non-uniform context-free languages. Here, the length of advice is assumed to be the same as that of an input. Let CFL and CFL/n denote the class of all context-free languages and its nonuniform version, respectively. We let CFL(2) denote the class of intersections of two context-free languages. An interesting direction of a research is asking how complex CFL(2) is, relative to CFL. Yamakami raised a problem whether there is a CFL-immune set in CFL(2) CFL/n. The best known so far is that LSPACE CFL/n has a CFL-immune set, where LSPACE denotes the class of languages recognized in logarithmic-space. We present an affirmative solution to his problem. Two key concepts of our proof are the overlapped palindrome and Yamakami’s swapping lemma. The swapping lemma is applicable to the setting where the pumping lemma (Bar-Hillel’s lemma) does not work. Our proof is an example showing how useful the swapping lemma is. In addition, by means of Kolmogorov complexity, we show the following: With respect to realtime deterministic context-free languages, the non-uniform class with parallel advice is not a subset of that with serial advice.