On the Hardness of Almost-Sure Termination

This paper considers the computational hardness of computing expected outcomes and deciding (universal) (positive) almost-sure termination of probabilistic programs. It is shown that computing lower and upper bounds of expected outcomes is Sigma10- and Sigma20-complete, respectively. Deciding (universal) almost-sure termination as well as deciding whether the expected outcome of a program equals a given rational value is shown to be Pi02-complete. Finally, it is shown that deciding (universal) positive almost-sure termination is Sigma20-complete (Pi30-complete).