Data-stream processing has continuously risen in importance as the amount of available data has been steadily increas- ing over the last decade. Besides traditional domains such as data-center monitoring and click analytics, there is an increasing number of network-enabled production machines that generate continuous streams of data. Due to their continuous nature, queries on data-streams can be more complex, and distinctly harder to understand then database queries. As users have to consider operational details, maintenance and debugging become challenging. Current approaches model data-streams as sequences, be- cause this is the way they are physically received. These models result in an implementation-focused perspective. We explore an alternate way of modeling data-streams by focusing on time-slicing semantics. This focus results in a model based on functions, which is better suited for reasoning about query semantics. By adapting the definitions of relevant concepts in stream processing to our model, we illustrate the practical useful- ness of our approach. Thereby, we link data-streams and query primitives to concepts in functional programming and mathematics. Most noteworthy, we prove that data-streams are monads, and show how to derive monad definitions for current data-stream models. We provide an abstract, yet practical perspective on data- stream related subjects based on a sound, consistent query model. Our work can serve as solid foundation for future data-stream query-languages.
Functional Reactive Programming , or FRP , is a general framework for programming hybrid systems in a high-level, declarative manner. The key ideas in FRP are its notions of behaviors and events . Behaviors are time-varying, reactive values, while events are time-ordered sequences of discrete-time event occurrences. FRP is the essence of Fran , a domain-specific language embedded in Haskell for programming reactive animations, but FRP is now also being used in vision, robotics and other control systems applications. In this paper we explore the formal semantics of FRP and how itrelates to an implementation based on streams that represent (and therefore only approximate) continuous behaviors. We show that, in the limit as the sampling interval goes to zero, the implementation is faithful to the formal, continuous semantics, but only when certain constraints on behaviors are observed. We explore the nature of these constraints, which vary amongst the FRP primitives. Our results show both the power and limitations of this approach to language design and implementation. As an example of a limitation, we show that streams are incapable of representing instantaneous predicate events over behaviors.