Inclusion Problems for Patterns With a Bounded Number of Variables

(Joachim Bremer, Dominik D. Freydenberger)

Abstract

We study the inclusion problems for pattern languages that are generated by patterns with a bounded number of variables. This continues the work by Freydenberger and Reidenbach by showing that restricting the inclusion problem to significantly more restricted classes of patterns preserves undecidability, at least for comparatively large bounds. For smaller bounds, we prove the existence of classes of patterns with complicated inclusion relations, and an open inclusion problem, that are related to the Collatz Conjecture. In addition to this, we give the first proof of the undecidability of the inclusion problem for NE-pattern languages that, in contrast to previous proofs, does not rely on the inclusion problem for E-pattern languages, and proves the undecidability of the inclusion problem for NE-pattern languages over binary and ternary alphabets.

Versions of the Paper

• Information and Computation. Final version, preprint.
• DLT 2010 (★ best paper award). Final version.
• more information

Additional Comments

• This paper gave me the ideas I needed to find the proofs in that paper on extended regular expressions. As you can see, the additional expressive power of extended regular expressions allows for more powerful constructions.
• The main result from this paper was used by Barcelo et al. for ECRPQs, a query language for graph databases. Building on the proof for pattern languages, Nicole Schweikardt and I managed to achieve stronger results on ECRPQs that are similar to the results on extended regular expressions.