Vector and scalar reachability problems in SL(2, Z)

Abstract This paper solves three open problems about the decidability of the vector and scalar reachability problems and the point to point reachability by fractional linear transformations over finitely generated semigroups of matrices from SL ( 2 , Z ) . Our approach to solving these problems is based on the characterization of reachability paths between vectors or points, which is then used to translate the numerical problems on matrices into computational problems on words and regular languages. We will also give geometric interpretations of these results.

[1]  Jin-Yi Cai,et al.  Multiplicative equations over commuting matrices , 1996, SODA '96.

[2]  Tero Harju,et al.  Undecidability Bounds for Integer Matrices Using Claus Instances , 2007, Int. J. Found. Comput. Sci..

[3]  Igor Potapov,et al.  The Identity Problem for Matrix Semigroups in SL2(ℤ) is NP-complete , 2017, SODA.

[4]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[5]  Igor Potapov,et al.  Reachability Problems in Quaternion Matrix and Rotation Semigroups , 2007, MFCS.

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

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

[8]  Fernando Chamizo.  Non-Euclidean visibility problems , 2006 .

[9]  A New Twist in Knot Theory , 2009 .

[10]  Tero Harju,et al.  Matrix Equations and Hilbert's Tenth Problem , 2008, Int. J. Algebra Comput..

[11]  Joël Ouaknine,et al.  On Matrix Powering in Low Dimensions , 2015, STACS.

[12]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[13]  Vincent D. Blondel,et al.  Decidable and Undecidable Problems about Quantum Automata , 2005, SIAM J. Comput..

[14]  Igor Potapov,et al.  Membership and Reachability Problems for Row-Monomial Transformations , 2004, MFCS.

[15]  Leonid Polterovich,et al.  Stable mixing for cat maps and quasi-morphisms of the modular group , 2004, Ergodic Theory and Dynamical Systems.

[16]  Juha Honkala,et al.  The freeness problem over matrix semigroups and bounded languages , 2014, Inf. Comput..

[17]  Thomas Noll.  Musical intervals and special linear transformations , 2007 .

[18]  R. Lyndon,et al.  Combinatorial Group Theory , 1977 .

[19]  Edward Witten.  SL(2;Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry , 2003 .

[20]  Igor Potapov,et al.  Matrix Semigroup Freeness Problems in SL (2, \mathbb Z) , 2016, SOFSEM.

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

[22]  Joël Ouaknine,et al.  Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences , 2013, ICALP.

[23]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[24]  Igor Potapov,et al.  Membership Problem in GL(2, Z) Extended by Singular Matrices , 2017, MFCS.

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

[26]  Between Decidability.  Skolem's Problem - On the Border , 2005 .

[27]  Igor Potapov,et al.  On undecidability bounds for matrix decision problems , 2008, Theor. Comput. Sci..

[28]  Juhani Karhumäki,et al.  On the Undecidability of Freeness of Matrix Semigroups , 1999, Int. J. Algebra Comput..

[29]  Joël Ouaknine,et al.  On the Positivity Problem for Simple Linear Recurrence Sequences, , 2013, ICALP.

[30]  W. Magnus,et al.  Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations , 1966 .

[31]  Igor Potapov.  Composition Problems for Braids , 2013, FSTTCS.

[32]  Alain Finkel,et al.  On the verification of broadcast protocols , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[33]  A. Restuccia,et al.  SL(2, Z) symmetries, supermembranes and symplectic torus bundles , 2011, 1105.3181.

[34]  Joel Ouaknine,et al.  On Termination of Integer Linear Loops , 2015, SODA.

[35]  Don Zagier,et al.  Elliptic modular forms and their applications. , 2008 .

[36]  R. Rankin.  Modular Forms and Functions , 1977 .

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

[38]  Igor Potapov,et al.  Decidability of the Membership Problem for 2 × 2 integer matrices , 2016, SODA.

[39]  Fritz Grunewald,et al.  Arithmetic Applications of the Hyperbolic Lattice Point Theorem , 1988 .