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On groups that have normal forms computable in logspace
 AMS Sectional Meeting, Las Vegas. Paper in preparation
, 2011
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Partially commutative inverse monoids
 PROCEEDINGS OF THE 31TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS 2006), BRATISLAVE (SLOVAKIA), NUMBER 4162 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algo ..."
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Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NPcompleteness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.
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.