Composition Problems for Braids

In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with 3 strands, B_3, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-hard for braids with only 3 strands. The membership problem is decidable for B_3, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least 5 strands, but decidability of these problems for B_4 remains open. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms.

[1]  Igor Potapov,et al.  On the Computational Complexity of Matrix Semigroup Problems , 2012, Fundam. Informaticae.

[2]  Stepan Yu Orevkov,et al.  Quasipositivity Problem for 3-Braids , 2003 .

[3]  Christian Choffrut,et al.  Some decision problems on integer matrices , 2005, RAIRO Theor. Informatics Appl..

[4]  Igor Potapov,et al.  Mortality for 2×2 Matrices Is NP-Hard , 2012, MFCS.

[5]  Louis H. Kauffman,et al.  Quantizing braids and other mathematical structures: the general quantization procedure , 2011, Defense + Commercial Sensing.

[6]  Patrick Dehornoy,et al.  Ordering Braids , 2008 .

[7]  Igor Potapov,et al.  On the Undecidability of the Identity Correspondence Problem and its Applications for Word and Matrix Semigroups , 2010, Int. J. Found. Comput. Sci..

[8]  F. A. Garside,et al.  THE BRAID GROUP AND OTHER GROUPS , 1969 .

[9]  Jean-Camille Birget,et al.  Two-letter group codes that preserve aperiodicity of inverse finite automata , 2007 .

[10]  Alexander A. Razborov,et al.  The Set of Minimal Braids is co-NP-Complete , 1991, J. Algorithms.

[11]  François Nicolas,et al.  On the decidability of semigroup freeness , 2008, RAIRO Theor. Informatics Appl..

[12]  Evelyne Contejean,et al.  Avoiding Slack Variables in the Solving of Linear Diophantine Equations and Inequations , 1997, Theor. Comput. Sci..

[13]  Paul E. Schupp,et al.  Membership Problem for the Modular Group , 2007, SIAM J. Comput..

[14]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[15]  Jeffrey Shallit,et al.  Automata and Reduced Words in the Free Group , 2009, ArXiv.

[16]  Владимир Николаевич Безверхний,et al.  О неразрешимости проблемы сопряженности подгрупп в группе крашеных кос $R_5$@@@Undecidability of the conjugacy problem for subgroups in the colored braid group $R_5$ , 1999 .