Hardness Results for Constant-Free Pattern Languages and Word Equations

Author Aleksi Saarela

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Aleksi Saarela
  • Department of Mathematics and Statistics, University of Turku, Finland

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Aleksi Saarela. Hardness Results for Constant-Free Pattern Languages and Word Equations. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 140:1-140:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism α such that α(xy) ≠ α(yx) for given variables x and y.

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ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Combinatorics on words
  • pattern language
  • word equation


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