On quadratic word equations

  • Authors:

  • John Michael Robson;Volker Diekert

  • Affiliations:

  • LaBRI, Université Bordeaux I, Talence Cedex, France;Institut für Informatik, Universität Stuttgart, Stuttgart, Germany

  • Venue:
  • STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
  • Year:
  • 1999

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Abstract

We investigate the satisfiability problem of word equations where each variable occurs at most twice (quadratic systems). We obtain various new results: The satisfiability problem is NP-hard (even for a single equation). The main result says that once we have fixed the lengths of a possible solution, then we can decide in linear time whether there is a corresponding solution. If the lengths of a minimal solution were at most exponential, then the satisfiability problem of quadratic systems would be NP-complete. (The inclusion in NP follows also from [21]) In the second part we address the problem with regular constraints: The uniform version is PSPACE-complete. Fixing the lengths of a possible solution doesn't make the problem much easier. The non-uniform version remains NP-hard (in contrast to the linear time result above). The uniform version remains PSPACE-complete.