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Decidability of membership problems for flat rational subsets of $\mathrm{GL}(2,\mathbb{Q})$ and singular matrices

Volker Diekert, Igor Potapov, Pavel Semukhin

TL;DR

The paper investigates membership problems for rational subsets of the semigroup of $2\times 2$ matrices over $\mathbb{Q}$, introducing flat rational sets ${\operatorname{FRat}}(M,S)$ to capture a polynomial closure over a subsemigroup $S$. It proves that ${\operatorname{FRat}}(G,H)$ forms an effective relative Boolean algebra under mild hypotheses, enabling decidability of emptiness for finite Boolean combinations of flat rational sets in ${\operatorname{FRat}}(\mathrm{GL}(2,\mathbb{Q}),\mathrm{GL}(2,\mathbb{Z}))$, and establishes a dichotomy for finitely generated subgroups of $\mathrm{GL}(2,\mathbb{Q})$ between $\mathrm{GL}(2,\mathbb{Z})$ and $\mathrm{GL}(2,\mathbb{Q})$: either $G\cong \mathrm{GL}(2,\mathbb{Z})\times \mathbb{Z}^k$ or $G$ contains an infinite-index extension of $\mathrm{BS}(1,q)$ with $q\ge2$, with decidability remaining open in the latter case. For singular targets, the authors show that membership in ${\operatorname{FRat}}(\mathbb{Q}^{2\times2},S)$ is decidable in doubly exponential time and mortality is decidable in exponential time for a related generating set. The work leverages Fatou-type transfer properties, Smith normal form decompositions, and commensurators, and introduces a flooding technique to handle mortality problems. Overall, the results substantially extend decidability boundaries for low-dimensional matrix rational-subset problems and offer a framework for further exploration of flat-rational constructions in matrix groups.

Abstract

We consider membership problems for rational subsets of the semigroup of $2\times 2$ matrices over $\mathbb{Q}$. For a semigroup $M$, the rational subsets $\mathrm{Rat}(M)$ are defined as the sets accepted by NFAs whose transitions are labeled by elements of $M$. In general, it is undecidable on inputs $m\in M$ and $R\in \mathrm{Rat}(M)$ whether $m$ belongs to $R$. Therefore, we restrict our attention to the family $\mathrm{FRat}(M,S)$ of flat rational subsets of $M$ over $S$, where $S$ is a subsemigroup of $M$. It consists of finite unions of the form $g_0L_1g_1 \cdots L_tg_t$, where $L_i\in \mathrm{Rat}(S)$ and $g_i\in M$. Assuming that the membership for $\mathrm{Rat}(S)$ is decidable, we prove various results when the membership for $\mathrm{FRat}(M,S)$ is decidable. If $H$ is a subgroup of a group $G$, then we provide a rather general condition when $\mathrm{FRat}(G,H)$ is an (effective) relative Boolean algebra. This leads to one of our main results that the emptiness problem for Boolean combinations of sets in $\mathrm{FRat}(\mathrm{GL}(2,\mathbb{Q}),\mathrm{GL}(2,\mathbb{Z}))$ is decidable. It is possible that this result cannot be pushed any further as indicated by the following dichotomy: if $G$ is a finitely generated group such that $\mathrm{GL}(2,\mathbb{Z}) < G < \mathrm{GL}(2,\mathbb{Q})$, then either $G\cong \mathrm{GL}(2,\mathbb{Z})\times \mathbb{Z}^k$ or $G$ contains an extension of the Baumslag-Solitar group $\mathrm{BS}(1,q)$ of infinite index. It is open whether the membership for rational subsets is decidable in the latter case. For singular matrices, we will show that the membership problem for $\mathrm{FRat}(\mathbb{Q}^{2\times 2},S)$ is decidable in doubly exponential time, where $S$ is the monoid generated by $\mathrm{GL}(2,\mathbb{Z})\cup \{r\in \mathbb{Q}\,\mid\,r>1\} \cup \{0,\left(\begin{smallmatrix}1 & 0\\ 0 & 0\end{smallmatrix}\right)\}$.

Decidability of membership problems for flat rational subsets of $\mathrm{GL}(2,\mathbb{Q})$ and singular matrices

TL;DR

The paper investigates membership problems for rational subsets of the semigroup of matrices over , introducing flat rational sets to capture a polynomial closure over a subsemigroup . It proves that forms an effective relative Boolean algebra under mild hypotheses, enabling decidability of emptiness for finite Boolean combinations of flat rational sets in , and establishes a dichotomy for finitely generated subgroups of between and : either or contains an infinite-index extension of with , with decidability remaining open in the latter case. For singular targets, the authors show that membership in is decidable in doubly exponential time and mortality is decidable in exponential time for a related generating set. The work leverages Fatou-type transfer properties, Smith normal form decompositions, and commensurators, and introduces a flooding technique to handle mortality problems. Overall, the results substantially extend decidability boundaries for low-dimensional matrix rational-subset problems and offer a framework for further exploration of flat-rational constructions in matrix groups.

Abstract

We consider membership problems for rational subsets of the semigroup of matrices over . For a semigroup , the rational subsets are defined as the sets accepted by NFAs whose transitions are labeled by elements of . In general, it is undecidable on inputs and whether belongs to . Therefore, we restrict our attention to the family of flat rational subsets of over , where is a subsemigroup of . It consists of finite unions of the form , where and . Assuming that the membership for is decidable, we prove various results when the membership for is decidable. If is a subgroup of a group , then we provide a rather general condition when is an (effective) relative Boolean algebra. This leads to one of our main results that the emptiness problem for Boolean combinations of sets in is decidable. It is possible that this result cannot be pushed any further as indicated by the following dichotomy: if is a finitely generated group such that , then either or contains an extension of the Baumslag-Solitar group of infinite index. It is open whether the membership for rational subsets is decidable in the latter case. For singular matrices, we will show that the membership problem for is decidable in doubly exponential time, where is the monoid generated by .

Paper Structure

This paper contains 22 sections, 39 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Recognizable and rational sets in semigroups
  4. The input size of matrices and NFAs over matrices
  5. Reductions and complexity classes
  6. Boolean algebras and relative Boolean algebras
  7. The Fatou property and transfer results for rational subsets in groups
  8. Smith normal forms and commensurators
  9. Computation of the Smith normal form
  10. Commensurators
  11. Dichotomy in $\mathop\mathrm{GL}(2,\mathbb{Q})$
  12. Flat rational sets
  13. The membership problem for ${\operatorname{FRat}}(\mathop\mathrm{GL}(2,\mathbb{Q}),S)$ with $\mathop\mathrm{GL}(2,\mathbb{Z})\subseteq S$
  14. The membership problem for $\operatorname{Rat}(\mathop\mathrm{GL}(2,\mathbb{Z}))$
  15. Proof of thm:red
  16. ...and 7 more sections

Key Result

Proposition 2.3

\newlabelprop:recbool0 If $\varphi_i:M\to N_i$ recognizes subsets $L_i\subseteq M$ for $i=1,2$, then the homomorphism $M\to N_1\times N_2,\; m\mapsto (\varphi_1(m), \varphi_2(m))$ recognizes $L_1\cap L_2$ and $M\setminus L_i$ for $i=1,2$. In particular, $\operatorname{Rec}(M)$ is a Boolean algebra

Figures (4)

  • Figure 1: The upper line is a subpath of an accepting path $\pi$ in $\mathcal{A}$. The lower line is the same subpath of $\pi$ in $\mathcal{A}'$. Since $s_{-1}^2 =1$ we have $m_1\cdots m_4= m_1'\cdots m_4'\in \mathop\mathrm{SL}(2,\mathbb{Z})$.
  • Figure 1: The code of the flooding procedure.
  • Figure 2: The construction of the NFA $\mathcal{B}$ yields $h=\overline{r_p}\, a {r_q}\in {\psi}^{-1}(H)$ and $h'=\overline{r_q}\, b {r_t}\in {\psi}^{-1}(H)$.
  • Figure 2: The flooding procedure introduces a rank-$1$ transition with label $a'aa"\!\cdot\! s_0$

Theorems & Definitions (88)

  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 78 more