On the hardness of analyzing probabilistic programs

We study the hardness of deciding probabilistic termination as well as the hardness of approximating expected values (e.g. of program variables) and (co)variances for probabilistic programs.Termination We distinguish two notions of probabilistic termination: Given a program P and an input sigma... 1 ...does P terminate with probability 1 on input sigma? (almost-sure termination)...is the expected time until P terminates on input sigma finite? (positive almost-sure termination) For both of these notions, we also consider their universal variant, i.e. given a program P, does P terminate on all inputs? We show that deciding almost-sure termination as well as deciding its universal variant is 20-complete in the arithmetical hierarchy. Deciding positive almost-sure termination is shown to be sigma 20-complete, whereas its universal variant is 30-complete.Expected values Given a probabilistic program P and a random variable f mapping program states to rationals, we show that computing lower and upper bounds on the expected value of f after executing P is sigma 10- and sigma 20-complete, respectively. Deciding whether the expected value equals a given rational value is shown to be 20-complete.Covariances We show that computing upper and lower bounds on the covariance of two random variables is both sigma 20-complete. Deciding whether the covariance equals a given rational value is shown to be in 30. In addition, this problem is shown to be sigma 20-hard as well as 20-hard and thus a proper 30-problem. All hardness results on covariances apply to variances as well.