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 ${\mathrm{\Pi }}_2^0$-complete. Computing lower and upper bounds on the expected outcome is shown to be recursively enumerable and ${\mathrm{\Sigma }}_2^0$-complete, respectively.