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. Length constraints restrict valid substitutions of variables by associating the variables of a pattern with a system (or disjunction of systems) of linear diophantine inequalities. Pattern languages with length constraints contain only words in which all variables are substituted to words with lengths that fulfill such a given set of length constraints. We consider membership, inclusion, and equivalence problems for erasing and non-erasing pattern languages with length 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 length constraints are allowed in addition to variable equality. Additionally, it is shown that the terminal-free inclusion problem-another prominent open problem in the realm of patterns-is also undecidable in this setting. It is also shown that considering regular constraints, i.e., associating variables also with regular languages as additional restrictions together with length constraints for valid substitutions, results in undecidability of the non-erasing equivalence problem. This sets a first upper bound on constraints to obtain undecidability in this case, as this problem is trivially decidable in the case of no constraints and as it has unknown decidability if only regular- or only length-constraints are considered.

翻译：模式是由终结符和变量组成的词。一个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而得到的词集合。长度约束通过将模式变量与线性丢番图不等式系统（或不等式系统的析取）相关联，从而限制变量的有效替换。带长度约束的模式语言仅包含所有变量被替换为满足给定长度约束集的词。我们研究了带长度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：在允许变量相等性之外再引入长度约束的情况下，擦除等价问题——模式研究领域最突出的未解难题之一——将变得不可判定。此外，我们还证明了在此设定下，无终结符包含问题——模式领域的另一突出未解难题——同样不可判定。研究还表明：若同时考虑正则约束（即对有效替换同时施加正则语言与长度约束的双重限制），非擦除等价问题也将不可判定。这为此类情形下获得不可判定性设定了首个约束上界，因为该问题在无约束情况下是平凡可判定的，而在仅考虑正则约束或仅考虑长度约束时其可判定性尚属未知。