A foundational calculus for computing with streams

Computing with streams is a topic of increasing interest in modern computer science. Several proposals have suggested the use of programming languages and rewriting systems in order to formally describe programs computing over stream data. However, no stream calculus (in the sense of λ-calculus) has been exhibited for such studies yet. The challenge of providing a calculus for streams is to make possible a more general study than what is possible with stream programming languages (since no reduction strategy is fixed to start with) and more modular and compositional than stream rewriting systems (indeed a new rewriting system shall be designed for each problem). In the present work, starting from the Λμ-calculus (an extension of λ-calculus for classical logic) we design ΛS which is a calculus where streams are first-class citizens. This calculus enjoys several interesting syntactical and computational properties and, moreover, it is well adapted to the use of types in order to prove stream program properties.