Download PDFOpen PDF in browserPolynomial Invariants for Probabilistic Transition Systems with Unbounded Support19 pages•Published: October 23, 2026AbstractWe 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 novel precondition. This precondition requires the existence of a dominating function for the martingale expression, which is uniformly integrable; 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 run-time of a linear loop, then we can automatically synthesize polynomial loop invariants that satisfy the OST. Notably, dui allows for the sampled distributions to be continuous and have unbounded support, which is a novel contribution to the field. We implemented and evaluated our method to symbolically compute upper and lower bounds for moments of program variables upon termination.Keyphrases: higher order moments, invariant synthesis, martingale, optional stopping, probabilistic programs, program verification In: Erika Ábrahám, Nikolaj Bjørner and Michael Rawson (editors). Proceedings of the 26th Conference on Logic for Programming, Artificial Intelligence, and Reasoning, vol 112, pages 485-503.
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