Decidability of computing all polynomial invariants for affine programs with recursive procedure calls

Determine whether there exists a decidable procedure to compute all polynomial invariants of affine programs that include recursive procedure calls. Specifically, given an affine program whose interprocedural control-flow yields non-regular path sets, ascertain if one can compute a finite basis for the vanishing ideal of all states reachable under all recursive executions, i.e., compute all polynomial invariants without imposing any a priori bound on their degree.

Background

Affine programs update integer-valued variables via linear assignments, and for non-recursive programs (where execution paths form regular languages) all polynomial invariants can be computed using procedures for the Zariski closure of sets of matrices specified by regular languages. Introducing recursive procedure calls causes the set of execution paths to become context-free rather than regular, breaking existing methods.

This open problem asks whether the general task of computing all polynomial invariants in the presence of recursion is decidable. It underpins interprocedural program analysis and connects directly to computing Zariski closures of morphic images of context-free languages.