We give a denotational model for graphical user interface (GUI) programming using the Cartesian closed category of ultrametric spaces. The ultrametric structure enforces causality restrictions on reactive systems and allows well-founded recursive definitions by a generalization of guardedness. We capture the arbitrariness of user input (e.g., a user gets to decide the stream of clicks she sends to a program) by making use of the fact that the closed subsets of an ultrametric space themselves form an ultrametric space, allowing us to interpret nondeterminism with a “powerspace ” monad. Algebras for the powerspace monad yield a model of intuitionistic linear logic, which we exploit in the definition of a mixed linear/non-linear domain-specific language for writing GUI programs. The non-linear part of the language is used for writing reactive stream-processing functions whilst the linear sublanguage naturally captures the generativity and usage constraints on the various linear objects in GUIs, such as the elements of a DOM or scene graph. We have implemented this DSL as an extension to OCaml, and give examples demonstrating that programs in this style can be short and readable.

Functional reactive programming (FRP) is an elegant and successful approach to programming reactive systems declaratively. The high levels of abstraction and expressivity that make FRP attractive as a programming model do, however, often lead to programs whose resource usage is excessive and hard to predict. In this paper, we address the problem of space leaks in discretetime functional reactive programs. We present a functional reactive programming language that statically bounds the size of the dataflow graph a reactive program creates, while still permitting use of higher-order functions and higher-type streams such as streams of streams. We achieve this with a novel linear type theory that both controls allocation and ensures that all recursive definitions are well-founded. We also give a denotational semantics for our language by combining recent work on metric spaces for the interpretation of higher-order causal functions with length-space models of spacebounded computation. The resulting category is doubly closed and hence forms a model of the logic of bunched implications.

Functional reactive programming (FRP) is an elegant approach to declaratively specify reactive systems. However, the powerful abstractions of FRP have historically made it difficult to predict and control the resource usage of programs written in this style. In this paper we give a simple type theory for higher-order functional reactive programming, as well as a natural implementation strategy for it. Our type theory simplifies and generalizes prior type systems for reactive programming. At the same time, we give a an efficient implementation strategy which eagerly deallocates old values, ruling out space and time leaks, two notorious sources of inefficiency in reactive programs. Our language neither restricts the expressive power of the FRP model, nor does it require a complex substructural type system to track the resource usage of programs. We also show that for programs well-typed under our type system, our implementation strategy of eager deallocation is safe: we show the soundness of our type system under our implementation strategy, using a novel step-indexed Kripke logical relation. 1.