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 \(\varSigma _1^0\)– and \(\varSigma _2^0\)–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 \(\varPi ^0_2\)–complete. Finally, it is shown that deciding (universal) positive almost–sure termination is \(\varSigma _2^0\)–complete (\(\varPi _3^0\)–complete).

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