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We are interested in regular expressions and transducers that represent word relations in an alphabet-invariant way—for example, the set of all word pairs u, v where v is a prefix of u independently of what the alphabet is. Current software systems of formal language objects do not have a mechanism to define such objects. We define transducers in which transition labels involve what we call set specifications, some of which are alphabet invariant. In fact, we consider automata-type objects, called labelled graphs, where each transition label can be any string, as long as that string represents a subset of a certain monoid. Then, the behaviour of the labelled graph is a subset of that monoid. We do the same for regular expressions. We obtain extensions of known algorithmic constructions on ordinary regular expressions and transducers, including partial derivative based methods, at the broad level of labelled graphs such that the computational efficiency of the extended constructions is not sacrificed. Then, for regular expressions with set specs we obtain a direct partial derivative method for membership. For transducers with set specs we obtain further algorithms that can be applied to questions about independent regular languages, in particular the witness version of the property satisfaction question.

Regular Expressions and Transducers over Alphabet-Invariant and User-Defined Labels