Analyzing Expected Outcomes and Almost-Sure Termination of Probabilistic Programs is Hard

This paper considers the computational hardness of computing expected outcomes and deciding almost-sure termination of probabilistic programs. We show that deciding almost-sure termination and deciding whether the expected outcome of a program equals a given rational value is $\Pi^0_2$-complete. Computing lower and upper bounds on the expected outcome is shown to be recursively enumerable and $\Sigma^0_2$-complete, respectively.