Polynomial Invariants for Probabilistic Transition Systems with Unbounded Support

Abstract

We study the synthesis of polynomial invariants for probabilistic transition systems (PTS) based on martingale theory. We present tractable methods to verify that such polynomials are indeed invariants, in the sense that their expected value upon termination is the same as their value at the start of the computation. We do this by applying the Optional Stopping Theorem (OST) in the form of a specific precondition. This precondition requires the existence of an integrable dominating function for the martingale expression, which implies uniform integrability; we refer to this condition as dui. For linear PTS we simplify the dui property to proving finiteness of the expected value of an expression depending on the update matrix, the degree of the martingale expression, and the stopping time. Specifically, if all random samples have finite moments and we can verify a moment bound on the runtime of a linear loop, then we can automatically synthesise polynomial loop invariants that satisfy the OST. Notably, dui allows for the sampled distributions to have unbounded support, which is a novel contribution to the field.