On the computational and descriptional complexity of multi-pattern languages

Abstract

We study the complexity of a restricted version of the universality problem: testing equivalence to ${\{0,1\}}^{⁎}$ for languages that are either ${\{0,1\}}^{⁎}$ or ${\{0,1\}}^{⁎}-\left\{w\right\}$ where $w\in {\{0,1\}}^{⁎}$. We show that this restricted problem for multi-pattern languages (MPL) is NP-hard, and for multi-pattern languages with regular substitutions (MPL_REG) is not recursively enumerable. Moreover, this undecidability result holds for any class of languages that effectively contains $\left\{w\#w\right|w\in {\{0,1\}}^{⁎}\}$, and is effectively closed under union, and concatenation with regular sets. Consequently, it has broad applicability and extends to many generalizations of regular languages. Sufficient conditions are then presented for a language predicate to be as hard as the restricted universality problem. By applying these conditions, we develop a uniform method for showing undecidability and complexity results simultaneously via highly efficient many-one reductions. In addition, the properties of this restricted universality problem can be used to investigate the descriptional complexity of multi-patterns. We show that the trade-off between multi-patterns and regular expressions is not exponential-size bounded. Non-recursive trade-offs between multi-patterns (or multi-patterns with regular substitutions) and numerous classes of language descriptors are also established.