Patterns are words with terminals and variables. The language of a pattern is the set of words obtained by uniformly substituting all variables with words that contain only terminals. Regular constraints restrict valid substitutions of variables by associating with each variable a regular language representable by, e.g., finite automata. Pattern languages with regular constraints contain only words in which each variable is substituted according to a set of regular constraints. We consider the membership, inclusion, and equivalence problems for erasing and non-erasing pattern languages with regular constraints. Our main result shows that the erasing equivalence problem, one of the most prominent open problems in the realm of patterns, becomes undecidable if regular constraints are allowed in addition to variable equality.

翻译：模式是由终结符和变量构成的词。模式的语言是通过将所有变量一致替换为仅包含终结符的词所得到的词集合。正则约束通过为每个变量关联一个可由有限自动机表示的正则语言来限制变量的有效替换。带正则约束的模式语言仅包含其中每个变量均按照一组正则约束进行替换的词。我们研究了带正则约束的擦除与非擦除模式语言的成员判定、包含判定及等价判定问题。我们的主要结果表明，擦除等价问题——模式领域最突出的未解决问题之一——在允许变量相等性之外还允许正则约束时，将变得不可判定。