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Extending our own and others' earlier approaches to reasoning abouttermination of probabilistic programs, we propose and prove a new rule fortermination with probability one, also known as "almost-certain termination".The rule uses both (non-strict) super martingales and guarantees of progress,together, and it seems to cover significant cases that earlier methods do not.In particular, it suffices for termination of the unbounded symmetric randomwalk in both one- and two dimensions: for the first, we give a proof; for thesecond, we use a theorem of Foster to argue that a proof exists.Non-determinism (i.e. demonic choice) is supported; but we do currentlyrestrict to discrete distributions.

A new rule for almost-certain termination of probabilistic- and demonic programs