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. In their original definition, patterns only allow for multiple distinct occurrences of some variables to be related by the equality relation, represented by using the same variable multiple times. In an extended notion, called relational patterns and relational pattern languages, variables may be related by arbitrary other relations, achieved by using regular patterns and relating individual variables independently from the patterns structure separately. We extend the ongoing investigation of the main decision problems for patterns (namely, the equivalence problem, the inclusion problem, and the membership problem) to relational pattern languages under a wide range of relevant individual relations, providing a comprehensive foundation in all three research directions.

翻译：模式是由终结符和变量构成的字词。一个模式的语言是指通过用仅包含终结符的字词统一替换所有变量所得到的字词集合。在其最初定义中，模式仅允许某些变量的多次不同出现通过相等关系相关联，这通过多次使用相同变量来实现。在一种扩展概念中，称为关系模式和关系模式语言，变量可以通过任意其他关系相关联，这是通过使用正则模式并独立于模式结构分别关联各个变量来实现的。我们将当前对模式主要决策问题（即等价问题、包含问题和成员问题）的持续研究，扩展到涵盖广泛相关个体关系的关系模式语言，为所有三个研究方向提供了全面的基础。