We consider three important and well-studied algorithmic problems in grouptheory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspa...
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We consider three important and well-studied algorithmic problems in grouptheory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conjugacy problem. We have a general result for graph products, but even in the special case of a graph group the result is new. Graph groups are closely linked to the theory of Mazurkiewicz traces which form an algebraic model for concurrent processes. Our proofs are combinatorial and based on well-known concepts in trace theory. We also use rewriting techniques over traces. For the group-theoretical part we apply Bass-Serre theory. But as we need explicit formulae and as we design concrete algorithms all our group-theoretical calculations are completely explicit and accessible to non-specialists. (C) 2015 Elsevier Ltd. All rights reserved.
We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) eq...
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ISBN:
(纸本)9781450386883
We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been intensively studied for various classes of groups in recent years. In many cases, it turns out that the set of all solutions on an exponent equation is a semilinear set that can be constructed effectively. Such groups are called knapsack semilinear. The class of knapsack semilinear groups is quite rich and it is closed under many group theoretic constructions, e.g., finite extensions, graph products, wreath products, amalgamated free products with finite amalgamated subgroups, and HNN-extensions with finite associated subgroups. On the other hand, arbitrary HNN-extensions do not preserve knapsack semilinearity. In this paper, we consider the knapsack semilinearity of HNN-extensions, where the stable letter t acts trivially by conjugation on the associated subgroup A of the base group G. We show that under some additional technical conditions, knapsack semilinearity transfers from the base group G to the HNN-extension. These additional technical conditions are satisfied in many cases, e.g., when A is a centralizer in G or A is a quasiconvex subgroup of the hyperbolic group G.
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the...
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ISBN:
(纸本)9783939897354
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is is decidable in polynomial time. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The Word Problem of Higman's group is decidable in polynomial time. It is due to the last result that we were forced to advance the theory of power circuits.
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 zxz(-1) = y in G. The conjug...
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ISBN:
(纸本)9783642544231
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 zxz(-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 BS1,2 and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS1,2 is TC0-complete. To the best of our knowledge BS1,2 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 BS1,2 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.
The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result use...
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ISBN:
(纸本)9783031206238;9783031206245
The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed data structure of power circuits allowing for a non-elementary compression of integers. Later this was extended in two directions: Laun showed that the same applies to generalized Baumslag groups G(1, q) for q >= 2 and we established that the word problem of the Baumslag group G(1,2) can be solved in TC2. In this work we further improve upon both previous results by showing that the word problems of all the generalized Baumslag groups G(1, q) can be solved in TC1 - even for negative q. Our result is based on using refined operations on reduced power circuits. Moreover, we prove that the conjugacy problem in G(1, q) is strongly generically in TC1 (meaning that for "most" inputs it is in TC1). Finally, for every fixed g is an element of G(1, q) conjugacy to g can be decided in TC1 for all inputs.
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the...
详细信息
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: (1) We define a modified reduction procedure on power circuits which runs in quadratic time, thereby improving the known cubic time complexity. The improvement is crucial for our other results. (2) We improve the complexity of the Word Problem for the Baumslag group to cubic time, thereby providing the first practical algorithm for that problem. (The algorithm has been implemented and is available in the CRAG library.) (3) The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.
Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to fin...
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Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pre-groups and in particular to free products with amalgamation, HNN- extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.
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