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We show that computing the strongest polynomial invariant for single-pathloops with polynomial assignments is at least as hard as the Skolem problem, afamous problem whose decidability has been open for almost a century. While thestrongest polynomial invariants are computable for affine loops, for polynomialloops the problem remained wide open. As an intermediate result of independentinterest, we prove that reachability for discrete polynomial dynamical systemsis Skolem-hard as well. Furthermore, we generalize the notion of invariantideals and introduce moment invariant ideals for probabilistic programs. Withthis tool, we further show that the strongest polynomial moment invariant is(i) uncomputable, for probabilistic loops with branching statements, and (ii)Skolem-hard to compute for polynomial probabilistic loops without branchingstatements. Finally, we identify a class of probabilistic loops for which thestrongest polynomial moment invariant is computable and provide an algorithmfor it.

Strong Invariants Are Hard: On the Hardness of Strongest Polynomial Invariants for (Probabilistic) Programs