Immunity and Pseudorandomness of Context-Free Languages

TL;DR

This paper explores the structural properties of context-free languages, such as immunity and pseudorandomness, demonstrating their existence, computational aspects, and implications for complexity theory.

Contribution

It introduces new classes of immune and pseudorandom context-free languages, establishing their properties and computational bounds, advancing understanding of their complexity and randomness.

Findings

Existence of REG-immune and CFL-simple languages.

Construction of REG/n-bi-immune and REG/n-pseudorandom languages.

Limits on pseudorandom generators in deterministic linear time.

Abstract

We discuss the computational complexity of context-free languages, concentrating on two well-known structural properties---immunity and pseudorandomness. An infinite language is REG-immune (resp., CFL-immune) if it contains no infinite subset that is a regular (resp., context-free) language. We prove that (i) there is a context-free REG-immune language outside REG/n and (ii) there is a REG-bi-immune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFL-simple set, where a CFL-simple language is an infinite context-free language whose complement is CFL-immune. Similar to the REG-immunity, a REG-primeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a context-free language that is REG/n-bi-primeimmune. Concerning pseudorandomness of context-free languages, we show that (v) CFL contains REG/n-pseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 1-1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write-only output tape and (vii) against REG, there is no almost 1-1 weakly pseudorandom generator computable deterministically in linear time by a single-tape Turing machine.