Summary
We consider word equations where each variable occurs at most twice (quadratic systems). The satisfiability problem is NP-hard (even for a single equation), but once the lengths of a possible solution are fixed, then there is a deterministic linear time algorithm to decide 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.
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.
In the third part we show that for quadratic systems the exponent of periodicity is at most linear in the denotational length.
This work was partially supported by the French German project PROCOPE
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References
Thierry Arnoux. Untersuchungen zum Makaninschen Algorithmus. Diplomarbeit 1613, Universität Stuttgart, 1998.
Christian Choffrut and Juhani Karhumäki. Combinatorics of words. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, pages 329–438. Springer-Verlag, Berlin, Heidelberg, New York, 1997.
Volker Diekert. Makanin’s Algorithm. In M. Lothaire: Algebraic Combinatorics on Words. Cambridge University Press. A preliminary version is on the web: http://www-igm.univ-mlv.fr/ berstel/Lothaire/index.html.
Volker Diekert, Yuri Matiyasevich, and Anca Muscholl. Solving trace equations using lexicographical normal forms. In P. Degano et al., editors, Proc. 24th ICALP, Bologna (Italy) 1997, number 1256 in Lect. Notes Comp. Sci., pages 336–347. Springer-Verlag, Berlin, Heidelberg, New York, 1997.
Claudio Gutierrez. Satisfiability of word equations with constants is in exponential space. In Proc. 39th FOCS, pages 112–119, Los Alamitos, Californiax (USA), 1998. IEEE Computer Society Press.
Antoni Koscielski and Leszek Pacholski. Complexity of Makanin’s algorithm. J. Assoc. Comput. Mach., 43(4):670–684, 1996. Preliminary version in Proc. 31st FOCS, Los Alamitos, California (USA),1990.
Dexter Kozen. Lower bounds for natural proof systems. In Proc. 18th FOCS, Providence, Rhode Island (USA), pages 254–266. IEEE Computer Society Press, 1977.
Roger C. Lyndon. Equations in free groups. Transactions of the American Mathematical Society, 96, 1960.
Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1977.
Gennadií S. Makanin. The problem of solvability of equations in a free semigroup. Mat. Sb., 103(2):147–236, 1977. In Russian; English translation in:Math. USSR Sbornik, 32, 129–198, 1977.
Yuri Matiyasevich. A connection between systems of word and length equations and Hubert’s Tenth Problem. Sem. Mat. V. A. Steklov Math. Inst. Leningrad, 8:132–144, 1968. In Russian; English translation in: Seminars in Mathematics, V. A. Steklov Mathematical Institute, 8, 61–67, 1970.
Yuri Matiyasevich. Some decision problems for traces. In S. Adian et al., editors, Proc. 4th LFCS, Yaroslavl (Russia) 1997, number 1234 in Lect. Notes Comp. Sci., pages 248–257. Springer-Verlag, Berlin, Heidelberg, New York, 1997. Invited lecture.
Jakob Nielsen. Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Mathematische Annalen, 78, 1918.
Dominique Perrin. Equations in words. In H. Ait-Kaci et al., editors, Resolution of equations in algebraic structures, volume 2, pages 275–298. Academic Press, 1989.
Wojciech Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In Proc. 31st STOC, Atlanta, Georgia (USA), 1999. To appear.
John Michael Robson. Word equations with at most 2 occurrences of each variable. Preprint, LaBRI, Université de Bordeaux I, 1998.
John Michael Robson and Volker Diekert. On quadratic word equations. In Ch. Meinel et al., editors, Proc. 16th STACS, Trier (Germany), Lect. Notes Comp. Sci. Springer-Verlag, Berlin, Heidelberg, New York, 1999. To appear.
Wojciech Rytter and Wojciech Plandowski. Application of Lempel-Ziv encodings to the solution of word equations. In K. G. Larsen et al., editors, Proc. 25th ICALP, Aalborg (Denmark), number 1443 in Lect. Notes Comp. Sci., pages 731–742. Springer-Verlag, Berlin, Heidelberg, New York, 1998.
Klaus U. Schulz. Makanin’s algorithm for word equations: Two improvements and a generalization. In K.-U. Schulz, editor, Proc. IWWERT’90, Tübingen (Germany), number 572 in Lect. Notes Comp. Sci., pages 85–150. Springer-Verlag, Berlin, Heidelberg, New York, 1992.
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Diekert, V., Robson, J.M. (1999). Quadratic Word Equations. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds) Jewels are Forever. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60207-8_28
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