Diamonds are not forever: liveness in reactive programming with guarded recursion

Abstract

When designing languages for functional reactive programming (FRP) the main challenge is to provide the user with a simple, flexible interface for writing programs on a high level of abstraction while ensuring that all programs can be implemented efficiently in a low-level language. To meet this challenge, a new family of modal FRP languages has been proposed, in which variants of Nakano's guarded fixed point operator are used for writing recursive programs guaranteeing properties such as causality and productivity. As an apparent extension to this it has also been suggested to use Linear Temporal Logic (LTL) as a language for reactive programming through the Curry-Howard isomorphism, allowing properties such as termination, liveness and fairness to be encoded in types. However, these two ideas are in conflict with each other, since the fixed point operator introduces non-termination into the inductive types that are supposed to provide termination guarantees. In this paper we show that by regarding the modal time step operator of LTL a submodality of the one used for guarded recursion (rather than equating them), one can obtain a modal type system capable of expressing liveness properties while retaining the power of the guarded fixed point operator. We introduce the language Lively RaTT, a modal FRP language with a guarded fixed point operator and an `until' type constructor as in LTL, and show how to program with events and fair streams. Using a step-indexed Kripke logical relation we prove operational properties of Lively RaTT including productivity and causality as well as the termination and liveness properties expected of types from LTL. Finally, we prove that the type system of Lively RaTT guarantees the absence of implicit space leaks.

Modal FRP for all: Functional reactive programming without space leaks in Haskell

“…In this paper, we present Rattus, a practical modal FRP language that takes its ideas from the modal FRP calculi of Krishnaswami (2013) and Bahr et al (2019Bahr et al ( , 2021 but with a simpler and less restrictive type system that makes it attractive to use in practice. Like the Simply RaTT calculus of Bahr et al, we use a Fitch-style type system (Clouston, 2018), which extends typing contexts with tokens to avoid the syntactic overhead of the dual-context-style type system of Krishnaswami (2013).…”

Modalities and Parametric Adjoints

“…This elaboration also provides a model in the standard semantics of guarded recursion and avoids the need for more complex clock categories. Finally, we note that non-dependent variants of (single-clock) CloTT have proven useful for modelling reactive programming [5,6]; these calculi can also be encoded in Guarded FitchTT.…”

“…Other Fitch-style type theories, which were crafted for more specialized applications, have a weaker relationship with FitchTT. For example, RaTT [5,6] can be encoded in FitchTT, but this encoding would fail to capture many restrictions placed on modalities in order to ensure domainspecific theorems about RaTT (e.g. freedom from space leaks).…”

“…Instead, it adopted a 'pattern-matching' modal elimination rule, which is strictly weaker than dra/tm/unmod. 6 In addition to being weaker, this pattern-matching eliminator introduces a prohibitive overhead in certain crucial examples. For instance, there is no way to elaborate the systematic '𝜆-notation' used in Section 5 to treat the modality I ⊸ 𝐴 as a function.…”

Adjoint Reactive GUI

“…The addition of the global view or point at infinity has the benefit that it enables us to give a model that distinguishes between least and greatest fixed points [15] (i.e., inductive and coinductive types), unlike in models of guarded recursion where guarded types are bilimit-compact [6]. This will let us use the idea that temporal liveness and safety properties can be encoded in types using inductive and coinductive types [10,2].…”