Abstract
It is shown that the existential theory of \({\mathbb G}\) with rational constraints, over an HNN-extension \({\mathbb G}=\langle {\mathbb H},t; t^{-1}at=\varphi(a) (a \in A) \rangle\) is decidable, provided that the same problem is decidable in the base group \({\mathbb H}\) and that A is a finite group. The positive theory of \({\mathbb G}\) is decidable, provided that the existential positive theory of \({\mathbb G}\) is decidable and that A and ϕ(A) are proper subgroups of the base group \({\mathbb H}\) with A ∩ϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtually-free group is decidable.
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References
Dicks, W., Dunwoody, M.J.: Groups Acting on Graphs. Cambridge University Press, Cambridge (1989)
Diekert, V., Gutiérrez, C., Hagenah, C.: The existential theory of equations with rational constraints in free groups is PSPACE-complete. Inf. Comput. 202(2), 105–140 (2005)
Diekert, V., Lohrey, M.: Word Equations over Graph Products. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 156–167. Springer, Heidelberg (2003)
Diekert, V., Lohrey, M.: Existential and positive theories of equations in graph products. Theory Comput. Syst. 37(1), 133–156 (2004)
V. Diekert and A. Muscholl. Solvability of equations in free partially commutative groups is decidable. Int. J. Algebra Comput. (to appear 2006)
Durnev, V.G.: Undecidability of the positive ∀ ∃ 3-theory of a free semi-group. Sibirsky Matematicheskie Jurnal 36(5), 1067–1080 (1995)
Green, E.R.: Graph Products of Groups. PhD thesis, The University of Leeds (1990)
Khan, B., Myasnikov, A.G., Serbin, D.E.: On positive theories of groups with regular free length function. In: Manuscript (2005)
Kharlampovich, O.G., Myasnikov, A.: Tarski’s problem about the elementary theory of free groups has a positive solution. Electron. Res. Announc. AMS 4(14), 101–108 (1998)
Kościelski, A., Pacholski, L.: Makanin’s algorithm is not primitive recursive. Theor. Comput. Sci. 191(1-2), 145–156 (1998)
Lohrey, M., Sénizergues, G.: Equations in HNN-extensions. In: Manuscript (2006)
Lohrey, M., Sénizergues, G.: Positive theories of HNN-extensions and amalgamated free products. In: Manuscript (2006)
Lohrey, M., Sénizergues, G.: Rational subsets of HNN-extensions. In: Manuscript (2006)
Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Heidelberg (1977)
Makanin, G.S.: Equations in a free group. Math. USSR, Izv, 21:483–546, English translation (1983)
Makanin, G.S.: Decidability of the universal and positive theories of a free group. In: Math. USSR, Izv, pp. 75–88 (1985)
Merzlyakov, Y.I.: Positive formulas on free groups(Russian). Algebra i Logika Sem 5(4), 25–42 (1966)
Muller, D.E., Schupp, P.E.: Groups, the theory of ends, and context-free languages. J. Comput. Syst. Sci. 26, 295–310 (1983)
Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. J. ACM 51(3), 483–496 (2004)
Rips, E., Sela, Z.: Canonical representatives and equations in hyperbolic groups. Invent. Math. 120, 489–512 (1995)
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Lohrey, M., Sénizergues, G. (2006). Theories of HNN-Extensions and Amalgamated Products. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4052. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787006_43
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DOI: https://doi.org/10.1007/11787006_43
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