Termination of linear loops under commutative updates
We consider the following problem: given d Γ d rational matrices A_1, β¦, A_k and a polyhedral cone πββ^d, decide whether there exists a non-zero vector whose orbit under multiplication by A_1, β¦, A_k is contained in π. 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 A_1, β¦, A_k. 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 β[X_1, β¦, X_k] as well as the polynomial semiring β_β₯ 0[X_1, β¦, X_k]. The loop termination problem is then reduced to deciding whether a submodule of (β[X_1, β¦, X_k])^n contains a βpositiveβ element.
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