Vector Reachability Problem in $\mathrm{SL}(2,\mathbb{Z})$

TL;DR

This paper proves the decidability of vector reachability in semigroups of 2x2 integer matrices with determinant 1, using language theory and geometric methods, addressing open problems in matrix semigroup theory.

Contribution

It solves two open problems on the decidability of vector and point-to-point reachability in SL(2,Z), introducing a novel approach based on formal languages and geometry.

Findings

Decidability of vector reachability in SL(2,Z) semigroups.

Decidability of point-to-point reachability over rationals.

Extension of results to matrix products on labeled graphs.

Abstract

The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are inherently difficult to solve even for problems in low dimensions and most matrix semigroup problems become undecidable in general starting from dimension three or four. This paper solves two open problems about the decidability of the vector reachability problem over a finitely generated semigroup of matrices from $\mathrm{SL}(2,\mathbb{Z})$ and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from $\mathrm{SL}(2,\mathbb{Z})$. The approach to solving reachability problems is based on the characterization of reachability paths between points which is followed by the translation of numerical problems on matrices into computational and combinatorial problems on words and formal languages. We also give a geometric interpretation of reachability paths and extend the decidability results to matrix products represented by arbitrary labelled directed graphs. Finally, we will use this technique to prove that a special case of the scalar reachability problem is decidable.