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The Nielsen reduction and P-complete problems in free groups. (English) Zbl 0555.20015
This paper is written very much for the computer scientist. The authors consider various problems concerning free groups, such as: the generalized word problem; the determination of shortest coset representatives modulo a given subgroup; to decide whether two subgroups are equal or isomorphic; to decide whether a given set of elements freely generates a subgroup, to decide whether a subgroup has finite index. They show that these problems are polynomially time complete under log-space reducibility. The main tool used is a careful analysis of the Nielsen reduction process.
Reviewer: S.J.Pride

20E05 Free nonabelian groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
Full Text: DOI
[1] Avenhaus, J.; Madlener, K., String matching and algorithmic problems in free groups, Rev. coll. math., 14, 1-16, (1980) · Zbl 0514.20026
[2] Avenhaus, J.; Madlener, K., Algorithmische probleme bei einrelatorgruppen und ihre komplexität, Arch. math. logik, 19, 3-12, (1978) · Zbl 0396.03040
[3] Avenhaus, J.; Madlener, K.; Avenhaus, J.; Madlener, K., Subrekursive komplexität bei gruppen, II, Acta informatica, Acta informatica, 9, 183-193, (1978) · Zbl 0371.02020
[4] Avenhaus, J.; Madlener, K., P-complete problems in free groups, (), 42-51, Lecture Notes in Comput. Sci. · Zbl 0867.68096
[5] Avenhaus, J.; Madlener, K., The Nielsen reduction as a key problem to polynomial algorithms in free groups, (), 49-56, Lecture Notes in Comput. Sci. · Zbl 0548.68038
[6] Jones, N.D.; Laaser, W.T., Complete problems for deterministic polynomial time, Theoret. comput. sci., 3, 105-117, (1977) · Zbl 0352.68068
[7] Lipton, R.J.; Zalcstein, Y., Word problems solvable in log-space, J. ACM, 24, 522-526, (1977) · Zbl 0359.68049
[8] Lyndon, R.C.; Schupp, P.E., Combinatorial group theory, (1977), Springer Berlin · Zbl 0368.20023
[9] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory, (1976), Dover Publ New York
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