Partial Derivatives of Regular Expressions over Alphabet-Invariant and User-Defined Labels
We are interested in regular expressions 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. This is the second part of a recent paper on this topic which focused on labelled graphs (transducers and automata) with alphabet-invariant and user-defined labels. In this paper we study derivatives of regular expressions over labels (atomic objects) in some set B. These labels can be any strings as long as the strings represent subsets of a certain monoid. We show that one can define partial derivative labelled graphs of type B expressions, whose transition labels can be elements of another label set X as long as X and B refer to the same monoid. We also show how to use derivatives directly to decide whether a given word pair is in the relation of a regular expression over pairing specs. Set specs and pairing specs are useful label sets allowing one to express languages and relations over large alphabets in a natural and compact way.
KeywordsAlphabet-invariant expressions Regular expressions Partial derivatives Algorithms Monoids
We are grateful to the reviewers of CIAA 2019 for their constructive suggestions for improvement. We have applied most of these suggestions, and we plan to apply the remaining ones in the journal version where more pages are allowed. The idea of using special labels on automata to denote sets is also explored in  with different objectives.
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