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Metric Temporal Logic can express temporally evolving properties withtime-critical constraints or time-triggered constraints for real-time systems.This paper extends the Metric Interval Temporal Logic with a distributioneventuality operator to express time-sensitive missions for a systeminteracting with a dynamic, probabilistic environment. This formalism enablesus to describe the probabilistic occurrences of random external events as partof the task specification and event-triggered temporal constraints for theintended system's behavior. The main contributions of this paper are two folds:First, we propose a procedure to translate a specification into a stochastictimed automaton. Second, we develop an approximate-optimal probabilisticplanning problem for synthesizing the control policy that maximizes theprobability for the planning agent to achieve the task, provided that theexternal events satisfy the specification. The planning algorithm employs atruncation in the clocks for the timed automaton to reduce the planning in acountably infinite state space to a finite state space with a bounded errorguarantee. We illustrate the method with a robot motion planning example.

Policy Synthesis for Metric Interval Temporal Logic with Probabilistic Distributions