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Journal articles on the topic 'Biautomatic'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 February 2022

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1

Mahdavi, Kazem, Gabe Merton, Leonidas Nguyen, Travis Schedler, and Nathan Smith. "Virtually Biautomatic Groups." Communications in Algebra 37, no. 7 (June 17, 2009): 2267–73. http://dx.doi.org/10.1080/00927870802620241.

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2

Van Wyk, Leonard. "Graph groups are biautomatic." Journal of Pure and Applied Algebra 94, no. 3 (July 1994): 341–52. http://dx.doi.org/10.1016/0022-4049(94)90015-9.

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3

Cain, Alan J., Robert D. Gray, and António Malheiro. "Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids." International Journal of Algebra and Computation 25, no. 01n02 (February 2015): 51–80. http://dx.doi.org/10.1142/s0218196715400044.

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This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids. For Chinese monoids, we first give new presentations via finite complete rewriting systems, using more lucid constructions and proofs than those given independently by Chen & Qui and Güzel Karpuz; we then construct biautomatic structures. For hypoplactic monoids, we construct finite complete rewriting systems and biautomatic structures. For sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures. Consequently, the monoid algebras corresponding to monoids of these classes are automaton algebras in the sense of Ufnarovskij.

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4

Gersten, S. M., and H. B. Short. "Rational Subgroups of Biautomatic Groups." Annals of Mathematics 134, no. 1 (July 1991): 125. http://dx.doi.org/10.2307/2944334.

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5

NEUMANN, WALTER D., and LAWRENCE REEVES. "REGULAR COCYCLES AND BIAUTOMATIC STRUCTURES." International Journal of Algebra and Computation 06, no. 03 (June 1996): 313–24. http://dx.doi.org/10.1142/s0218196796000167.

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6

Mosher, L. "Central quotients of biautomatic groups." Commentarii Mathematici Helvetici 72, no. 1 (May 1997): 16–29. http://dx.doi.org/10.1007/pl00000364.

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7

Bahls, Patrick. "Some new biautomatic Coxeter groups." Journal of Algebra 296, no. 2 (February 2006): 339–47. http://dx.doi.org/10.1016/j.jalgebra.2005.12.003.

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8

LEVITT, RENA, and JON MCCAMMOND. "TRIANGLES, SQUARES AND GEODESICS." International Journal of Algebra and Computation 22, no. 05 (July 11, 2012): 1250041. http://dx.doi.org/10.1142/s0218196712500415.

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In the early 1990s Steve Gersten and Hamish Short proved that compact nonpositively curved triangle complexes have biautomatic fundamental groups and that compact nonpositively curved square complexes have biautomatic fundamental groups. In this paper we report on the extent to which results such as these extend to nonpositively curved complexes built out a mixture of triangles and squares. Since both results by Gersten and Short have been generalized to higher dimensions, this can be viewed as a first step towards unifying Januszkiewicz and Świȧtkowski's theory of simplicial nonpositive curvature with the theory of nonpositively curved cube complexes.

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9

NEUMANN, WALTER D., and MICHAEL SHAPIRO. "AUTOMATIC STRUCTURES AND BOUNDARIES FOR GRAPHS OF GROUPS." International Journal of Algebra and Computation 04, no. 04 (December 1994): 591–616. http://dx.doi.org/10.1142/s0218196794000178.

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We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.

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10

Charney, Ruth. "Artin groups of finite type are biautomatic." Mathematische Annalen 292, no. 1 (March 1992): 671–83. http://dx.doi.org/10.1007/bf01444642.

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11

ROMAN'KOV, VITALY. "POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN." Glasgow Mathematical Journal 60, no. 1 (March 13, 2017): 209–18. http://dx.doi.org/10.1017/s0017089516000677.

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AbstractLet G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

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12

Fohry, Egbert, and Dietrich Kuske. "On Graph Products of Automatic and Biautomatic Monoids." Semigroup Forum 72, no. 3 (May 23, 2006): 337–52. http://dx.doi.org/10.1007/s00233-006-0602-9.

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13

Peifer, David. "Artin groups of extra-large type are biautomatic." Journal of Pure and Applied Algebra 110, no. 1 (July 1996): 15–56. http://dx.doi.org/10.1016/0022-4049(95)00094-1.

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14

Brady, Thomas, and Jonathan P. McCammond. "Three-generator Artin groups of large type are biautomatic." Journal of Pure and Applied Algebra 151, no. 1 (July 2000): 1–9. http://dx.doi.org/10.1016/s0022-4049(99)00094-8.

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15

Cain, Alan J., António Malheiro, and Fábio M. Silva. "The monoids of the patience sorting algorithm." International Journal of Algebra and Computation 29, no. 01 (February 2019): 85–125. http://dx.doi.org/10.1142/s0218196718500649.

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The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given. Each finite-rank [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a nontrivial identity (dependent on its rank), while the infinite rank [Formula: see text]PS monoid does not satisfy any nontrivial identity. Each [Formula: see text][Formula: see text]PS monoid of finite rank has exponential growth and does not satisfy any nontrivial identity. The complexity of the insertion algorithms is discussed. [Formula: see text]PS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is [Formula: see text] or [Formula: see text], they are also biautomatic. [Formula: see text][Formula: see text]PS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.

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16

WEISS, URI. "ON BIAUTOMATICITY OF NON-HOMOGENOUS SMALL-CANCELLATION GROUPS." International Journal of Algebra and Computation 17, no. 04 (June 2007): 797–820. http://dx.doi.org/10.1142/s0218196707003718.

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We define the notion of V(6) presentations that naturally generalize C(4)&T(4) and C(6) small-cancellation presentations. We consider V(6) groups (i.e. groups having V(6) presentations) where every piece is of length one. No natural geometrical interpretation is known for this class of groups. However, by using combinatorial and diagrammatical arguments, we construct a biautomatic structure for these groups.

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17

Cain, Alan J., Robert D. Gray, and António Malheiro. "Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids." Journal of Algebra 423 (February 2015): 37–53. http://dx.doi.org/10.1016/j.jalgebra.2014.09.037.

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18

Holzer, Markus, and Sebastian Jakobi. "Minimal and Hyper-Minimal Biautomata." International Journal of Foundations of Computer Science 27, no. 02 (February 2016): 161–85. http://dx.doi.org/10.1142/s0129054116400050.

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We compare deterministic finite automata (DFAs) and biautomata under the following two aspects: structural similarities between minimal and hyper-minimal automata, and computational complexity of the minimization and hyper-minimization problem. Concerning classical minimality, the known results such as isomorphism between minimal DFAs, and NL-completeness of the DFA minimization problem carry over to the biautomaton case. But surprisingly this is not the case for hyper-minimization: the similarity between almost-equivalent hyper-minimal biautomata is not as strong as it is between almost-equivalent hyper-minimal DFAs. Moreover, while hyper-minimization is NL-complete for DFAs, we prove that this problem turns out to be computationally intractable, i.e., NP-complete, for biautomata.

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19

Bridson, Martin R., and Lawrence Reeves. "On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups." Science China Mathematics 54, no. 8 (August 2011): 1533–45. http://dx.doi.org/10.1007/s11425-011-4212-y.

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20

Cain, Alan J., Robert D. Gray, and António Malheiro. "Crystal monoids & crystal bases: Rewriting systems and biautomatic structures for plactic monoids of types A, B, C, D, and G2." Journal of Combinatorial Theory, Series A 162 (February 2019): 406–66. http://dx.doi.org/10.1016/j.jcta.2018.11.010.

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21

HOLZER, MARKUS, and SEBASTIAN JAKOBI. "NONDETERMINISTIC BIAUTOMATA AND THEIR DESCRIPTIONAL COMPLEXITY." International Journal of Foundations of Computer Science 25, no. 07 (November 2014): 837–55. http://dx.doi.org/10.1142/s0129054114400115.

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We investigate the descriptional complexity of nondeterministic biautomata, which are a generalization of biautomata [O. KLÍMA, L. POLÁK: On biautomata. RAIRO — Theor. Inf. Appl., 46(4), 2012]. Simply speaking, biautomata are finite automata reading the input from both sides; although the head movement is nondeterministic, additional requirements enforce biautomata to work deterministically. First we study the size blow-up when determinizing nondeterministic biautomata. Further, we give tight bounds on the number of states for nondeterministic biautomata accepting regular languages relative to the size of ordinary finite automata, regular expressions, and syntactic monoids. It turns out that as in the case of ordinary finite automata nondeterministic biautomata are superior to biautomata with respect to their relative succinctness in representing regular languages.

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22

Klíma, Ondřej, and Libor Polák. "On biautomata." RAIRO - Theoretical Informatics and Applications 46, no. 4 (June 19, 2012): 573–92. http://dx.doi.org/10.1051/ita/2012014.

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23

Holzer, Markus, and Sebastian Jakobi. "Minimization and Characterizations for Biautomata." Fundamenta Informaticae 136, no. 1-2 (2015): 113–37. http://dx.doi.org/10.3233/fi-2015-1146.

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24

Holzer, Markus, and Sebastian Jakobi. "More Structural Characterizations of Some Subregular Language Families by Biautomata." Electronic Proceedings in Theoretical Computer Science 151 (May 21, 2014): 271–85. http://dx.doi.org/10.4204/eptcs.151.19.

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25

Gilman, Robert H. "Generalized small cancellation presentations for automatic groups." Groups Complexity Cryptology, January 21, 2014. http://dx.doi.org/10.1515/gcc-2014-0007.

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AbstractBy a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length 1, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.

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26

Munro, Zachary, Damian Osajda, and Piotr Przytycki. "2-dimensional Coxeter groups are biautomatic." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, March 23, 2021, 1–20. http://dx.doi.org/10.1017/prm.2021.11.

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Let W be a 2-dimensional Coxeter group, that is, one with 1/m st + 1/m sr + 1/m tr ≤ 1 for all triples of distinct s, t, r ∈ S. We prove that W is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary W), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type W are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\widetilde {A}_3$ .

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27

Holt, Derek F., and Sarah Rees. "Biautomatic structures in systolic Artin groups." International Journal of Algebra and Computation, December 7, 2020. http://dx.doi.org/10.1142/s0218196721500193.

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28

Valiunas, Motiejus. "Isomorphism classification of Leary–Minasyan groups." Journal of Group Theory, September 21, 2021. http://dx.doi.org/10.1515/jgth-2021-0042.

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Abstract Recently, I. J. Leary and A. Minasyan [Commensurating HNN extensions: Nonpositive curvature and biautomaticity, Geom. Topol. 25 (2021), 4, 1819–1860] studied the class of groups G ⁢ ( A , L ) G(A,L) defined as commensurating HNN-extensions of Z n \mathbb{Z}^{n} . This class, containing the class of Baumslag–Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups G ⁢ ( A , L ) G(A,L) up to isomorphism.

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29

Jirásková, Galina, and Ondřej Klíma. "On linear languages recognized by deterministic biautomata." Information and Computation, July 2021, 104778. http://dx.doi.org/10.1016/j.ic.2021.104778.

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