Synchronous modeling is at the heart of programming languages like
Lustre, Esterel, or Scade used routinely for implementing safety
critical control software, e.g., fly-by-wire and engine control in
planes. However, to date these languages have had limited modern
support for modeling uncertainty — probabilistic aspects of the
software's environment or behavior — even though modeling
uncertainty is a primary activity when designing a control system.

In this paper we present ProbZelus the first synchronous probabilistic
programming language. ProbZelus conservatively provides the
facilities of a synchronous language to write control software, with
probabilistic constructs to model uncertainties and perform
inference-in-the-loop.

We present the design and implementation of the language. We propose a
measure-theoretic semantics of probabilistic stream functions and a
simple type discipline to separate deterministic and probabilistic
expressions. We demonstrate a semantics-preserving compilation into a
first-order functional language that lends itself to a simple
presentation of inference algorithms for streaming models. We also
redesign the delayed sampling inference algorithm to provide efficient
streaming inference. Together with an evaluation on several reactive
applications, our results demonstrate that ProbZelus enables the
design of reactive probabilistic applications and efficient, bounded
memory inference.