Membership problem in GL(2 , Z ) extended by singular matrices - PowerPoint PPT Presentation

Membership problem in GL(2 , Z ) extended by singular matrices Pavel Semukhin joint work with Igor Potapov Department of Computer Science, University of Liverpool RP, 8 September, 2017 This work was supported by EPSRC grant “Reachability problems for words, matrices and maps” (EP/M00077X/1) Pavel Semukhin Membership problem

Membership problem Let M be an n × n matrix and F = { M 1 , . . . , M k } be a finite collection of n × n matrices. Determine whether M ∈ �F� , that is, whether M = M i 1 M i 2 · · · M i t for some sequence of matrices M i 1 , M i 2 , . . . , M i t ∈ F . Pavel Semukhin Membership problem

Membership problem Let M be an n × n matrix and F = { M 1 , . . . , M k } be a finite collection of n × n matrices. Determine whether M ∈ �F� , that is, whether M = M i 1 M i 2 · · · M i t for some sequence of matrices M i 1 , M i 2 , . . . , M i t ∈ F . In case when M is the zero matrix, the above problem is called the mortality problem. Pavel Semukhin Membership problem

Known results Mortality problem (and hence the membership problem) is algorithmically undecidable for 3 × 3 matrices over integers. [Paterson, 1970] Pavel Semukhin Membership problem

Known results Mortality problem (and hence the membership problem) is algorithmically undecidable for 3 × 3 matrices over integers. [Paterson, 1970] Membership problem is decidable in PTIME for commuting matrices (over algebraic numbers) [Babai, et. al., 1996] Pavel Semukhin Membership problem

Known results Mortality problem (and hence the membership problem) is algorithmically undecidable for 3 × 3 matrices over integers. [Paterson, 1970] Membership problem is decidable in PTIME for commuting matrices (over algebraic numbers) [Babai, et. al., 1996] It is a long standing open question whether the membership problem is decidable for 2 × 2 matrices (even over integers). Pavel Semukhin Membership problem

Known results Let GL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = ± 1 } . Let SL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = 1 } . The membership problem is decidable for matrices from GL(2 , Z ) [C. Choffrut and J. Karhum¨ aki, 2005] Pavel Semukhin Membership problem

Known results Let GL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = ± 1 } . Let SL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = 1 } . The membership problem is decidable for matrices from GL(2 , Z ) [C. Choffrut and J. Karhum¨ aki, 2005] The identity problem (i.e. membership for the identity matrix) in SL(2 , Z ) is NP-complete. [B. Bell, M. Hirvensalo, I. Potapov, 2017] Pavel Semukhin Membership problem

Known results Let GL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = ± 1 } . Let SL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = 1 } . The membership problem is decidable for matrices from GL(2 , Z ) [C. Choffrut and J. Karhum¨ aki, 2005] The identity problem (i.e. membership for the identity matrix) in SL(2 , Z ) is NP-complete. [B. Bell, M. Hirvensalo, I. Potapov, 2017] The membership problem is decidable for 2 × 2 nonsingular integer matrices. [P. Semukhin and I. Potapov, 2017] Pavel Semukhin Membership problem

Known results Let GL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = ± 1 } . Let SL(2 , Z ) = { A ∈ Z 2 × 2 : det( A ) = 1 } . The membership problem is decidable for matrices from GL(2 , Z ) [C. Choffrut and J. Karhum¨ aki, 2005] The identity problem (i.e. membership for the identity matrix) in SL(2 , Z ) is NP-complete. [B. Bell, M. Hirvensalo, I. Potapov, 2017] The membership problem is decidable for 2 × 2 nonsingular integer matrices. [P. Semukhin and I. Potapov, 2017] The mortality problem is decidable for 2 × 2 integer matrices with determinant 0 , ± 1 (i.e. for matrices from GL(2 , Z ) and singular matrices) [C. Nuccio and E. Rodaro, 2008] Pavel Semukhin Membership problem

Main result Main result The membership problem is decidable for 2 × 2 integer matrices with determinant 0 , ± 1 (i.e. for matrices from GL(2 , Z ) and singular matrices) Pavel Semukhin Membership problem

Theorem (Smith Normal Form) For any matrix A ∈ Z 2 × 2 , there are matrices E, F from GL(2 , Z ) � m � 0 such that A = E F for some n, m ∈ N ∪ { 0 } . 0 nm Pavel Semukhin Membership problem

Theorem (Smith Normal Form) For any matrix A ∈ Z 2 × 2 , there are matrices E, F from GL(2 , Z ) � m � 0 such that A = E F for some n, m ∈ N ∪ { 0 } . 0 nm The numbers n and m are uniquely defined by A . The diagonal � m � 0 matrix D = is called the Smith normal form of A . 0 nm Pavel Semukhin Membership problem

Theorem (Smith Normal Form) For any matrix A ∈ Z 2 × 2 , there are matrices E, F from GL(2 , Z ) � m � 0 such that A = E F for some n, m ∈ N ∪ { 0 } . 0 nm The numbers n and m are uniquely defined by A . The diagonal � m � 0 matrix D = is called the Smith normal form of A . 0 nm If A ∈ Z 2 × 2 is a singular matrix, then the Smith normal form of A � t � 0 is equal to , where t is the gcd of the coefficients of A . 0 0 Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = { A 1 , . . . , A n , B 1 , . . . , B m } , where A 1 , . . . , A n ∈ GL(2 , Z ) and B 1 , . . . , B m are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ �F� . Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = { A 1 , . . . , A n , B 1 , . . . , B m } , where A 1 , . . . , A n ∈ GL(2 , Z ) and B 1 , . . . , B m are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ �F� . � t � 0 Let M = E F be the Smith normal forms of M . 0 0 Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = { A 1 , . . . , A n , B 1 , . . . , B m } , where A 1 , . . . , A n ∈ GL(2 , Z ) and B 1 , . . . , B m are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ �F� . � t � 0 Let M = E F be the Smith normal forms of M . 0 0 We will construct a graph G ( M, F ) with the property: M ∈ �F� if and only if there is a path in G ( M, F ) from an initial to a final node of weight t . Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = { A 1 , . . . , A n , B 1 , . . . , B m } , where A 1 , . . . , A n ∈ GL(2 , Z ) and B 1 , . . . , B m are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ �F� . � t � 0 Let M = E F be the Smith normal forms of M . 0 0 We will construct a graph G ( M, F ) with the property: M ∈ �F� if and only if there is a path in G ( M, F ) from an initial to a final node of weight t . Graph G ( M, F ) has m nodes labelled by singular matrices B 1 , . . . , B m and two special nodes In and Fin . Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = { A 1 , . . . , A n , B 1 , . . . , B m } , where A 1 , . . . , A n ∈ GL(2 , Z ) and B 1 , . . . , B m are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ �F� . � t � 0 Let M = E F be the Smith normal forms of M . 0 0 We will construct a graph G ( M, F ) with the property: M ∈ �F� if and only if there is a path in G ( M, F ) from an initial to a final node of weight t . Graph G ( M, F ) has m nodes labelled by singular matrices B 1 , . . . , B m and two special nodes In and Fin . � t i � 0 If B i = E i F i , then the weight of B i is equal to t i . 0 0 In and Fin have weight 1 . Pavel Semukhin Membership problem

Description of G ( M, F ) B j B i In Fin B 1 B m Pavel Semukhin Membership problem

Description of G ( M, F ) B j B i In Fin B 1 B m We add edges to G ( M, F ) according to the following rules. Pavel Semukhin Membership problem

Description of G ( M, F ) B j B i In Fin B 1 B m We add edges to G ( M, F ) according to the following rules. � t � 0 Recall F = { A 1 , . . . , A n , B 1 , . . . , B m } and M = E F . 0 0 � t i � � t j � 0 0 Let B i = E i F i and B j = E j F j . 0 0 0 0 Pavel Semukhin Membership problem

Description of G ( M, F ) u B j B i In Fin B 1 B m We add edges to G ( M, F ) according to the following rules. � t � 0 Recall F = { A 1 , . . . , A n , B 1 , . . . , B m } and M = E F . 0 0 � t i 0 � � t j 0 � Let B i = E i F i and B j = E j F j . 0 0 0 0 For every u such that − t ≤ u ≤ t we add an edge from B i to B j of weight u if and only if there is a matrix C ∈ � A 1 , . . . , A n � such � u � x that F i CE j = , where x, y, z ∈ Z . y z Pavel Semukhin Membership problem

Description of G ( M, F ) For every u such that − t ≤ u ≤ t we add an edge from B i to B j of weight u if and only if there is a matrix C ∈ � A 1 , . . . , A n � such � u � x that F i CE j = , where x, y, z ∈ Z . y z Pavel Semukhin Membership problem

Description of G ( M, F ) For every u such that − t ≤ u ≤ t we add an edge from B i to B j of weight u if and only if there is a matrix C ∈ � A 1 , . . . , A n � such � u � x that F i CE j = , where x, y, z ∈ Z . y z u An edge B i − → B j corresponds to a product B i A s 1 · · · A s k B j = B i CB j , where C ∈ � A 1 , . . . , A n � . Pavel Semukhin Membership problem