Abstract
The conjugacy problem is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zx z − 1 = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS 1,2 and the Baumslag(-Gersten) group G 1,2. The conjugacy problem in BS 1,2 is TC 0-complete. To the best of our knowledge BS 1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G 1,2 is an HNN-extension of BS 1,2 and its conjugacy problem is decidable G 1,2 by a result of Beese (2012). Here we show that conjugacy in G 1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G 1,2 can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G 1,2 by reducing the division problem in power circuits to the conjugacy problem in G 1,2. The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.
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References
Baumslag, G.: A non-cyclic one-relator group all of whose finite quotients are cyclic. J. Austr. Math. Soc. 10(3-4), 497–498 (1969)
Beese, J.: Das Konjugationsproblem in der Baumslag-Gersten-Gruppe. Diploma thesis, Fakultät Mathematik, Universität Stuttgart (2012) (in German)
Borovik, A.V., Myasnikov, A.G., Remeslennikov, V.N.: Generic Complexity of the Conjugacy Problem in HNN-Extensions and Algorithmic Stratification of Miller’s Groups. IJAC 17, 963–997 (2007)
Ceccherini-Silberstein, T., Grigorchuk, R.I., de la Harpe, P.: Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Tr. Mat. Inst. Steklova 224, 68–111 (1999)
Craven, M.J., Jimbo, H.C.: Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography. Groups Complexity Cryptology 4, 135–165 (2012)
Diekert, V., Laun, J., Ushakov, A.: Efficient algorithms for highly compressed data: The word problem in Higman’s group is in P. IJAC 22, 1–19 (2012)
Diekert, V., Miasnikov, A., Weiß, A.: Conjugacy in Baumslag’s group, generic case complexity, and division in power circuits. CoRR, abs/1309.5314 (2013)
Gersten, S.M.: Isodiametric and isoperimetric inequalities in group extensions (1991)
Grigoriev, D., Shpilrain, V.: Authentication from matrix conjugation. Groups Complexity Cryptology 1, 199–205 (2009)
Hesse, W.: Division is in uniform TC0. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 104–114. Springer, Heidelberg (2001)
Hesse, W., Allender, E., Barrington, D.A.M.: Uniform constant-depth threshold circuits for division and iterated multiplication. JCCS 65, 695–716 (2002)
Kapovich, I., Miasnikov, A.G., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory and random walks. J. Algebra 264, 665–694 (2003)
Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Average-case complexity and decision problems in group theory. Adv. Math. 190, 343–359 (2005)
Lyndon, R., Schupp, P.: Combinatorial Group Theory, 1st edn. (1977)
Magnus, W.: Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106, 295–307 (1932)
Miller III, C.F.: On group-theoretic decision problems and their classification. Annals of Mathematics Studies, vol. 68. Princeton University Press (1971)
Myasnikov, A., Shpilrain, V., Ushakov, A.: Group-based Cryptography. Advanced courses in mathematics. CRM Barcelona, Birkhäuser (2008)
Myasnikov, A.G., Ushakov, A., Won, D.W.: The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable. Journal of Algebra 345, 324–342 (2011)
Myasnikov, A.G., Ushakov, A., Won, D.W.: Power circuits, exponential algebra, and time complexity. IJAC 22, 51 pages (2012)
Shpilrain, V., Zapata, G.: Combinatorial group theory and public key cryptography. Appl. Algebra Eng. Comm. Comput. 17, 291–302 (2006)
Vollmer, H.: Introduction to Circuit Complexity. Springer, Berlin (1999)
Woess, W.: Random walks on infinite graphs and groups - a survey on selected topics. London Math. Soc. 26, 1–60 (1994)
Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Univ. Press (2000)
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Diekert, V., Myasnikov, A.G., Weiß, A. (2014). Conjugacy in Baumslag’s Group, Generic Case Complexity, and Division in Power Circuits. In: Pardo, A., Viola, A. (eds) LATIN 2014: Theoretical Informatics. LATIN 2014. Lecture Notes in Computer Science, vol 8392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54423-1_1
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DOI: https://doi.org/10.1007/978-3-642-54423-1_1
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