A pumping lemma for a class of languages \(\varvec{\mathcal {C}}\) is often used to show particular languages are not in \(\varvec{\mathcal {C}}\). In contrast, we show that a pumping lemma for a class of languages \(\varvec{\mathcal {C}}\) can be used to study the computational complexity of the predicate “\(\in \varvec{\mathcal {C}}\)” via highly efficient many-one reductions. In this paper, we use extended regular expressions (EXREGs, introduced in Câmpeanu et al. (Int. J. Foundations Comput. Sci. 14(6), 1007–1018, 2003)) as an example to illustrate the proof technique and establish the complexity of the predicate “is an EXREG language” for several classes of languages. Due to the efficiency of the reductions, both productiveness (a stronger form of non-recursive enumerability) and complexity results can be obtained simultaneously. For example, we show that the predicate “is an EXREG language” is productive (hence, not recursively enumerable) for context-free grammars, and is Co-NEXPTIME-hard for context-free grammars generating bounded languages. The proof technique is easy to use and requires only a few conditions. This suggests that for any class of languages \(\varvec{\mathcal {C}}\) having a pumping lemma, the language class comparison problems (e.g., does a given context-free grammar generate a language in \(\varvec{\mathcal {C}}\)?) are almost guaranteed to be hard. So, pumping lemmas sometimes could be “harmful” when studying computational complexity results.

一类语言的泵引理\(\varvec{\mathcal {C}}\)通常用于表明特定语言不在\(\varvec{\mathcal {C}}\)中。相比之下，我们证明一类语言的泵引理\(\varvec{\mathcal {C}}\)可用于研究谓词“ \(\in \varvec{\mathcal {C” }}\) ”通过高效的多一归约。在本文中，我们使用扩展正则表达式（EXREGs，Câmpeanu et al. (Int. J. Foundations Comput. Sci. 14 (6), 1007–1018, 2003) 中介绍的）作为示例来说明证明技术并建立几类语言的谓词“is an EXREG language”的复杂性。由于约简的效率，可以同时获得生产力（非递归可枚举性的更强形式）和复杂性结果。例如，我们证明谓词“是一种 EXREG 语言”对于上下文无关语法是有效的（因此，不可递归枚举），并且对于生成有界语言的上下文无关语法来说是 Co-NEXPTIME-hard。该证明技术易于使用，并且只需要几个条件。这表明对于任何具有泵引理的语言\(\varvec{\mathcal {C}}\)，语言类比较问题（例如，给定的上下文无关语法是否生成\(\varvec{ \mathcal {C}}\) ?) 几乎肯定是困难的。因此，在研究计算复杂性结果时，抽引理有时可能是“有害的”。