Positive Almost-Sure Termination of Polynomial Random Walks

Abstract: The number of steps until termination of a probabilistic program is a random variable. Probabilistic program termination therefore requires qualitative analysis via almost-sure termination (AST), while also providing quantitative answers via positive almost-sure termination (PAST) on the expected number of steps until termination. While every program which is PAST is AST, the converse is not true. The symmetric random walk with constant step size is a prominent example of a program that is AST but not PAST. In this paper we show that a more general class of polynomial random walks is PAST. Our random walks implement a step size that is polynomially increasing in the number of loop iterations and have a constant probability $p$ of choosing either branch. We decide that such programs are PAST when the degree of the polynomial is higher than both the degree of the drift and a threshold $d_\text{min}(p)$. Our approach does not use proof rules, nor auxiliary arithmetic expressions, such as martingales or invariants. Rather, we establish an inductive bound for the cumulative distribution function of the loop guard, based on which PAST is proven. We implemented the approximation of this threshold, by combining genetic programming, algebraic reasoning and linear programming.