Analysing All Polynomial Equations in ${\mathbb Z_{2^w}}$

Abstract

In this paper, we present methods for checking and inferring all valid polynomial relations in ${\mathbb Z_{2^w}}$. In contrast to the infinite field ℚ, ${\mathbb Z_{2^w}}$ is finite and hence allows for finitely many polynomial functions only. In this paper we show, that checking the validity of a polynomial invariant over ${\mathbb Z_{2^w}}$ is, though decidable, only PSPACE-complete. Apart from the impracticable algorithm for the theoretical upper bound, we present a feasible algorithm for verifying polynomial invariants over ${\mathbb Z_{2^w}}$ which runs in polynomial time if the number of program variables is bounded by a constant. In this case, we also obtain a polynomial-time algorithm for inferring all polynomial relations. In general, our approach provides us with a feasible algorithm to infer all polynomial invariants up to a low degree.