Abstract
A pattern is a string with terminals and variables (which can be uniformly replaced by terminal words). Given a class \(\mathcal {C}\) of patterns (with variables), we say a pattern \(\alpha \) is a \(\mathcal {C}\)-(pseudo-)repetition if its skeleton – the result of removing all terminal symbols to leave only the variables – is a (pseudo-)repetition of a pattern from \(\mathcal {C}\). We introduce a large class of patterns which generalises several known classes such as the k-local and bounded scope coincidence degree patterns, and show that for this class, \( \mathcal {C}\)-(pseudo-)repetitions can be matched in polynomial time. We also show that for most classes \(\mathcal {C}\), the class of \(\mathcal {C}\)-(pseudo-)repetitions does not have bounded treewidth. Finally, we show that if the notion of repetition is relaxed, so that in each occurrence the variables may occur in a different order, the matching problem is NP-complete, even in severely restricted cases.
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Notes
- 1.
The matching problem is the same as the membership problem for pattern languages.
- 2.
The difference of whether variables can be substituted by \(\varepsilon \) may seem negligible but is crucial for some aspects of pattern languages (erasing versus non-erasing pattern languages). Since here we are concerned with the matching problem, we only point out whether results also apply to the erasing case, or how they can be extended.
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Day, J.D., Fleischmann, P., Manea, F., Nowotka, D., Schmid, M.L. (2018). On Matching Generalised Repetitive Patterns. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_22
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