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Logspace computations in Coxeter groups and graph groups
 In Computational and Combinatorial Group Theory and Cryptography, volume 582 of Contemporary Mathematics
, 2012
"... Abstract. Computing normal forms in groups (or monoids) is computationally harder than solving the word problem (equality testing), in general. However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal form ..."
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Abstract. Computing normal forms in groups (or monoids) is computationally harder than solving the word problem (equality testing), in general. However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal forms for important classes of groups. For Coxeter groups we show that the following algorithmic tasks can be solved by a deterministic Turing machine using logarithmic work space, only: 1. Compute the length of any geodesic normal form. 2. Compute the set of letters occurring in any geodesic normal form. 3. Compute the Parikhimage of any geodesic normal form in case that all defining relations have even length (i.e., in even Coxeter groups.) 4. For rightangled Coxeter groups we can actually compute the shortlex normal form in logspace. Next, we apply the results to rightangled Artin groups. They are also known as free partially commutative groups or as graph groups. As a consequence of our result on rightangled Coxeter groups we show that shortlex normal forms in graph groups can be computed in logspace, too. Graph groups play an important rôle in group theory, and they have a close connection to concurrency theory. As an application of our results we show that the word problem for free partially commutative inverse monoids is in logspace. This result generalizes a result of Ondrusch and the third author on free inverse monoids. Concurrent systems which are deterministic and codeterministic can be studied via inverse monoids.
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"... Communicated by [editor] Let H be a cancellative monoid and let G be an HNNextension of H with finite associated subgroups A, B ≤ H. We show that, if equations are algorithmically solvable in H, then they are also algorithmically solvable in G. The result also holds for equations with rational cons ..."
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Communicated by [editor] Let H be a cancellative monoid and let G be an HNNextension of H with finite associated subgroups A, B ≤ H. We show that, if equations are algorithmically solvable in H, then they are also algorithmically solvable in G. The result also holds for equations with rational constraints and for the existential first order theory. Analogous results are derived for amalgamated product with finite amalgamated subgroups. Keywords: Equations; groups and monoids; HNNextensions; amalgamated product. ∗State completely without abbreviations, the affiliation and mailing address, including country. Typeset in 8 pt italic. † Typeset author’s email address in 8pt italic. ‡ permanent address of G.S. 1 February 10, 2006 17:15 WSPC/INSTRUCTION FILE hnneq