Location Based Automata for Expressions with Shuffle

Abstract

We define the notion of location for regular expressions with shuffle by extending the notion of position in standard regular expressions. Locations allow for the definition of the sets \(\mathsf {Follow}\), \(\mathsf {First}\) and \(\mathsf {Last}\) with their usual semantics. From these, we construct an automaton for regular expressions with shuffle (\(\mathcal {A}_{\mathrm {POS}}\)), which generalises the standard position/Glushkov automaton. The sets mentioned above are also the foundation for other constructions, such as the Follow automaton, and automata based on pointed expressions. As a consequence, all these constructions can now be directly generalised to regular expressions with shuffle, as well as their known relationships. We also show that the partial derivative automaton (\(\mathcal {A}_{\mathrm {PD}}\)) is a (right) quotient of the new position automaton, \(\mathcal {A}_{\mathrm {POS}}\). In previous work an automaton construction based on positions was studied (\(\mathcal {A}_{\partial pos}\)), and here we relate \(\mathcal {A}_{\mathrm {POS}}\) and \(\mathcal {A}_{\partial pos}\). Finally, we extend the construction of the prefix automaton \(\mathcal {A}_{\mathrm {Pre}}\) to the shuffle operator and show that it is not a quotient of \(\mathcal {A}_{\mathrm {POS}}\).