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 = {M1, . . . , Mk} be a finite collection of n × n matrices. Determine whether M ∈ F, that is, whether M = Mi1Mi2 · · · Mit for some sequence of matrices Mi1, Mi2, . . . , Mit ∈ F.

Pavel Semukhin Membership problem

Membership problem Let M be an n × n matrix and F = {M1, . . . , Mk} be a finite collection of n × n matrices. Determine whether M ∈ F, that is, whether M = Mi1Mi2 · · · Mit for some sequence of matrices Mi1, Mi2, . . . , Mit ∈ 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 ∈ Z2×2 : det(A) = ±1}. Let SL(2, Z) = {A ∈ Z2×2 : 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 ∈ Z2×2 : det(A) = ±1}. Let SL(2, Z) = {A ∈ Z2×2 : 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 ∈ Z2×2 : det(A) = ±1}. Let SL(2, Z) = {A ∈ Z2×2 : 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 ∈ Z2×2 : det(A) = ±1}. Let SL(2, Z) = {A ∈ Z2×2 : 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 ∈ Z2×2, there are matrices E, F from GL(2, Z) such that A = E m nm

• F for some n, m ∈ N ∪ {0}.

Pavel Semukhin Membership problem

Theorem (Smith Normal Form) For any matrix A ∈ Z2×2, there are matrices E, F from GL(2, Z) such that A = E m nm

• F for some n, m ∈ N ∪ {0}.

The numbers n and m are uniquely defined by A. The diagonal matrix D = m nm

• is called the Smith normal form of A.

Pavel Semukhin Membership problem

Theorem (Smith Normal Form) For any matrix A ∈ Z2×2, there are matrices E, F from GL(2, Z) such that A = E m nm

• F for some n, m ∈ N ∪ {0}.

The numbers n and m are uniquely defined by A. The diagonal matrix D = m nm

• is called the Smith normal form of A.

If A ∈ Z2×2 is a singular matrix, then the Smith normal form of A is equal to t

• , where t is the gcd of the coefficients of A.

Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = {A1, . . . , An, B1, . . . , Bm}, where A1, . . . , An ∈ GL(2, Z) and B1, . . . , Bm 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 = {A1, . . . , An, B1, . . . , Bm}, where A1, . . . , An ∈ GL(2, Z) and B1, . . . , Bm are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ F. Let M = E t

• F be the Smith normal forms of M.

Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = {A1, . . . , An, B1, . . . , Bm}, where A1, . . . , An ∈ GL(2, Z) and B1, . . . , Bm are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ F. Let M = E t

• F be the Smith normal forms of M.

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 = {A1, . . . , An, B1, . . . , Bm}, where A1, . . . , An ∈ GL(2, Z) and B1, . . . , Bm are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ F. Let M = E t

• F be the Smith normal forms of M.

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 B1, . . . , Bm and two special nodes In and Fin.

Pavel Semukhin Membership problem

Theorem Given a singular 2 × 2 integer matrix M and a set F = {A1, . . . , An, B1, . . . , Bm}, where A1, . . . , An ∈ GL(2, Z) and B1, . . . , Bm are 2 × 2 singular integer matrices. Then it is decidable whether M ∈ F. Let M = E t

• F be the Smith normal forms of M.

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 B1, . . . , Bm and two special nodes In and Fin. If Bi = Ei ti

• Fi, then the weight of Bi is equal to ti.

In and Fin have weight 1.

Pavel Semukhin Membership problem

Description of G(M, F)

In Fin Bi Bj B1 Bm

Pavel Semukhin Membership problem

Description of G(M, F)

In Fin Bi Bj B1 Bm We add edges to G(M, F) according to the following rules.

Pavel Semukhin Membership problem

Description of G(M, F)

In Fin Bi Bj B1 Bm We add edges to G(M, F) according to the following rules. Recall F = {A1, . . . , An, B1, . . . , Bm} and M = E t

• F.

Let Bi = Ei ti

• Fi and Bj = Ej

tj

• Fj.

Pavel Semukhin Membership problem

Description of G(M, F)

In Fin Bi Bj B1 Bm u We add edges to G(M, F) according to the following rules. Recall F = {A1, . . . , An, B1, . . . , Bm} and M = E t

• F.

Let Bi = Ei ti

• Fi and Bj = Ej

tj

• Fj.

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

Pavel Semukhin Membership problem

Description of G(M, F)

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

Pavel Semukhin Membership problem

Description of G(M, F)

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

An edge Bi

u

− → Bj corresponds to a product BiAs1 · · · AskBj = BiCBj, where C ∈ A1, . . . , An.

Pavel Semukhin Membership problem

Description of G(M, F)

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

An edge Bi

u

− → Bj corresponds to a product BiAs1 · · · AskBj = BiCBj, where C ∈ A1, . . . , An. BiCBj = Ei ti

• FiCEj

tj

• Fj =

= Ei ti u x y z tj

• Fj = Ei

tiutj

• Fj

Pavel Semukhin Membership problem

Description of G(M, F)

In Fin Bi Bj B1 Bm v w u Recall that M = E t

• F and Bi = Ei

ti

• Fi.

We add an edge of weight v from In to Bi if there is a matrix C ∈ A1, . . . , An such that E−1CEi = v x y

• , where x, y ∈ Z.

Pavel Semukhin Membership problem

Description of G(M, F)

In Fin Bi Bj B1 Bm v w u Recall that M = E t

• F and Bi = Ei

ti

• Fi.

We add an edge of weight v from In to Bi if there is a matrix C ∈ A1, . . . , An such that E−1CEi = v x y

• , where x, y ∈ Z.

We add an edge of weight w from Bj to Fin if there is a matrix C ∈ A1, . . . , An such that FjCF −1 = w x y

• , where x, y ∈ Z.

Pavel Semukhin Membership problem

The weight of a path in G(M, F) is equal to the product of the weights of nodes and edges that occur in it.

Pavel Semukhin Membership problem

The weight of a path in G(M, F) is equal to the product of the weights of nodes and edges that occur in it. Proposition Let M = E t

• F be the Smith normal form of matrix M.

Then M ∈ F if and only if there is a path in G(M, F) from In to Fin of weight t.

Pavel Semukhin Membership problem

The weight of a path in G(M, F) is equal to the product of the weights of nodes and edges that occur in it. Proposition Let M = E t

• F be the Smith normal form of matrix M.

Then M ∈ F if and only if there is a path in G(M, F) from In to Fin of weight t. Proposition If there is a path in G(M, F) from In to Fin of weight t, then there is such path of length at most 2m log2 t + 2m + log2 t.

Pavel Semukhin Membership problem

In Fin Bi Bj B1 Bm u For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

How to decide if there is an edge from Bi to Bj of weight u?

Pavel Semukhin Membership problem

In Fin Bi Bj B1 Bm u For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

How to decide if there is an edge from Bi to Bj of weight u? The group GL(2, Z) is generated by the matrices S = −1 1

• , R =

−1 1 1

• and N =

1 −1

• .

Pavel Semukhin Membership problem

In Fin Bi Bj B1 Bm u For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

How to decide if there is an edge from Bi to Bj of weight u? The group GL(2, Z) is generated by the matrices S = −1 1

• , R =

−1 1 1

• and N =

1 −1

• .

So any matrix A ∈ GL(2, Z) is represented by a word in the alphabet Σ = {S, R, N}.

Pavel Semukhin Membership problem

A subset S of GL(2, Z) is regular if it can be described by a regular language in the alphabet Σ = {S, R, N}.

Pavel Semukhin Membership problem

A subset S of GL(2, Z) is regular if it can be described by a regular language in the alphabet Σ = {S, R, N}. A semigroup S = M1, . . . , Mk is defined by the regular expression (w1 + · · · + wk)∗, where w1, . . . , wk are words that represent the matrices M1, . . . , Mk.

Pavel Semukhin Membership problem

A subset S of GL(2, Z) is regular if it can be described by a regular language in the alphabet Σ = {S, R, N}. A semigroup S = M1, . . . , Mk is defined by the regular expression (w1 + · · · + wk)∗, where w1, . . . , wk are words that represent the matrices M1, . . . , Mk. wk w2 w1

Pavel Semukhin Membership problem

A subset S of GL(2, Z) is regular if it can be described by a regular language in the alphabet Σ = {S, R, N}. A semigroup S = M1, . . . , Mk is defined by the regular expression (w1 + · · · + wk)∗, where w1, . . . , wk are words that represent the matrices M1, . . . , Mk. wk w2 w1 Theorem (Choffrut and Karhum¨ aki, 2005) Given two regular subsets S1 and S2 of GL(2, Z), it is decidable whether the intersection S1 ∩ S2 is empty or not.

Pavel Semukhin Membership problem

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

Pavel Semukhin Membership problem

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

The set {FiCEj : C ∈ A1, . . . , An} is a regular subset of GL(2, Z).

Pavel Semukhin Membership problem

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

The set {FiCEj : C ∈ A1, . . . , An} is a regular subset of GL(2, Z). For any fixed u ∈ Z, the set u x y z

• ∈ GL(2, Z) : x, y, z ∈ Z
• is a regular subset of GL(2, Z).

Pavel Semukhin Membership problem

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

The set {FiCEj : C ∈ A1, . . . , An} is a regular subset of GL(2, Z). For any fixed u ∈ Z, the set u x y z

• ∈ GL(2, Z) : x, y, z ∈ Z
• is a regular subset of GL(2, Z).

Hence we can decide if there is an edge from Bi to Bj of weight u.

Pavel Semukhin Membership problem

For every u such that −t ≤ u ≤ t we add an edge from Bi to Bj

• f weight u if and only if there is a matrix C ∈ A1, . . . , An such

that FiCEj = u x y z

• , where x, y, z ∈ Z.

The set {FiCEj : C ∈ A1, . . . , An} is a regular subset of GL(2, Z). For any fixed u ∈ Z, the set u x y z

• ∈ GL(2, Z) : x, y, z ∈ Z
• is a regular subset of GL(2, Z).

Hence we can decide if there is an edge from Bi to Bj of weight u.

THANK YOU.

Pavel Semukhin Membership problem