The emptiness problem for indexed language is exponential‐time complete

TLDR: This paper reduces the problem to the emptiness problem for the class of indexed languages in the logarithmic space, indicating the exponential-time difficulty of the emptinessproblem for the indexed language.

Abstract: The class of indexed languages properly includes the class of context-free languages and is properly included in the class of context-dependent languages [1]. The emptiness problem (the problem of determining whether or not the given language is empty) is polynomial-time complete for the class of context-free languages and is undecidable for the class of context-dependent languages. The recognition problem (the problem, given a language L and word w, of determining whether or not w belongs to L) is polynomial-time complete for the class of context-free languages and is polynomialspace complete for the class of contextdependent languages. This paper shows that both the emptiness and recognition problems are exponential-time complete for the class of indexed languages. It is known in the pebble game [2] that the problem of determining whether or not the first player has the winning strategy is exponential-time complete. This paper reduces the problem to the emptiness problem for the class of indexed languages in the logarithmic space, indicating the exponential-time difficulty of the emptiness problem for the indexed language. Since Aho has shown that the problem can be answered in exponential time, the exponential-time completeness is shown. The exponential-time difficulty is also directly indicated from the fact that the emptiness problem is exponential-time complete. Consequently, the recognition problem is also exponential-time complete.