Generic complexity of the membership problem for semigroups of integer matrices

The membership problem for finitely generated subgroups (subsemigroups) of groups (semigroups) is a classical algorithmic problem, actively studied for many decades. Already for sufficiently simple groups and semigroups, this problem becomes undecidable. For example, K. A. Mikhailova in 1966 proved the undecidability of the membership problem for finitely generated subgroups (hence and for subsemigroups) of a direct product $F_2 \times F_2$ of two free groups of rank $2$. Since, by the well-known Sanov theorem, the group $F_2$ has an exact representation by integer matrices of order $2$, the group $F_2 \times F_2$ is a subgroup of the group $\text{GL}_4 (\mathbb{Z})$ of integer matrices of order $4$. It easily implies the undecidability of this problem for the group $\text{GL}_{k} (\mathbb{Z})$ for $k \geq 4$. Undecidability of the membership problem for finitely generated subsemigroups of semigroups of integer matrices of order $\geq 3$ follows from Paterson's result proved in 1970. In this paper, we propose a strongly generic algorithm deciding the membership problem for semigroups of integer matrices of arbitrary order for inputs from a subset whose sequence of frequencies exponentially fast converges to $1$ with increasing size.