Time-bounded termination analysis for probabilistic programs with delays

This paper investigates the model of probabilistic program with delays (PPD) that consists of a few program blocks. Performing each block has an additional time-consumption—waiting to be executed—besides the running time. We interpret the operational semantics of PPD by Markov automata with a cost structure on transitions. Our goal is to measure those individual execution paths of a PPD that terminates within a given time bound, and to compute the minimum termination probability, i.e. the termination probability under a demonic scheduler that resolves the nondeterminism inherited from probabilistic programs. When running time plus waiting time is bounded, the demonic scheduler can be determined by comparison between a class of well-formed real numbers. The method is extended to parametric PPDs. When only the running time is bounded, the demonic scheduler can be determined by real root isolation over a class of well-formed real functions under Schanuel's conjecture. Finally we give the complexity upper bounds of the proposed methods.