We consider the following problem: given dΓd rational matrices A1β,β¦,Akβ and a polyhedral cone CβRd, decide
whether there exists a non-zero vector whose orbit under multiplication by
A1β,β¦,Akβ is contained in C. This problem can be
interpreted as verifying the termination of multi-path while loops with linear
updates and linear guard conditions. We show that this problem is decidable for
commuting invertible matrices A1β,β¦,Akβ. The key to our decision
procedure is to reinterpret this problem in a purely algebraic manner. Namely,
we discover its connection with modules over the polynomial ring
R[X1β,β¦,Xkβ] as well as the polynomial semiring
Rβ₯0β[X1β,β¦,Xkβ]. The loop termination problem is then
reduced to deciding whether a submodule of (R[X1β,β¦,Xkβ])n contains a ``positive'' element.Comment: 6 page