As the complexity of pacemaker devices continues to grow, the importance of capturing its functional correctness requirement formally cannot be overestimated. The pacemaker system specification document by \emph{Boston Scientific} provides a widely accepted set of specifications for pacemakers. As these specifications are written in a natural language, they are not amenable for automated verification, synthesis, or reinforcement learning of pacemaker systems. This paper presents a formalization of these requirements for a dual-chamber pacemaker in \emph{duration calculus} (DC), a highly expressive real-time specification language. The proposed formalization allows us to automatically translate pacemaker requirements into executable specifications as stopwatch automata, which can be used to enable simulation, monitoring, validation, verification and automatic synthesis of pacemaker systems. The cyclic nature of the pacemaker-heart closed-loop system results in DC requirements that compile to a decidable subclass of stopwatch automata. We present shield reinforcement learning (shield RL), a shield synthesis based reinforcement learning algorithm, by automatically constructing safety envelopes from DC specifications.

Reinforcement Learning (RL) is a sampling based approach to optimization, where learning agents rely on scalar reward signals to discover optimal solutions. The specification of learning objectives as scalar rewards is tedious and error prone, and more so for real-time systems with complex time-critical requirements. This paper advocates the use of Duration Calculus (DC)—a highly expressive real-time logic with duration and length modalities—in expressing the learning objectives in model-free RL for stochastic real-time systems. On the other hand, to model stochastic real-time environments, we consider probabilistic timed automata (PTA)—Markov decision processes extended with clock variables—that provide an expressive yet computationally decidable formalism to capture real-time constraints over nondeterministic and probabilistic behaviors.The key hurdle in developing a convergent RL algorithm for DC specifications is the undecidability of the synthesis problem for PTA against general DC specifications. Inspired by the dichotomy between event-triggered and time-triggered approaches to the design of real-time systems, we present two variants of DC logic—that we dub event-triggered duration calculus (EDC) and time-triggered duration calculus (TDC)—and identify their subclasses with appealing theoretical properties. We study the decidability (and exact complexity) of the satisfiability of these calculi as well as the controller synthesis against PTA models. Based on these results, we propose a reward scheme for RL agents in such a way that guarantees that any RL algorithm maximizing rewards is guaranteed to maximize the probability of satisfaction for the given DC specification. The effectiveness of the proposed approach is demonstrated via grid-world benchmarks and a proof-of-concept case study for synthesizing control for simple cardiac pacemaker directly from a set of DC specifications.