上下文无关语言的免疫性和伪随机性
Immunity and Pseudorandomness of Context-Free Languages
摘要：
我们讨论了上下文无关语言的计算复杂性，集中在两个著名的结构特性-免疫性和伪随机性。无限语言是REG免疫的（分别是，CFL免疫），如果它不包含是正则的无限子集（分别为，上下文无关）语言。我们证明了（i）在REG/n之外存在一个上下文无关的REG-免疫语言;（ii）存在一个REG-双免疫语言，它可以用对数空间确定性地计算.我们还表明，（iii）有一个CFL简单的集合，其中CFL简单的语言是一个无限的上下文无关的语言，其补充是CFL免疫。类似于REG免疫，REG素免疫语言没有多项式稠密的子集也是正则的。我们进一步证明了（iv）存在REG/n-bi-primeimmune的上下文无关语言。关于上下文无关语言的伪随机性，我们证明了（v）CFL包含REG/n-伪随机语言。最后证明了：（vi）在REG/n下，存在一个几乎1-1伪随机发生器，它可在配有只写输出带的非确定下推自动机中计算;（vii）在REG下，不存在几乎1-1弱伪随机发生器，它可在线性时间内由单带图灵机确定计算.
英文摘要：
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.
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