Categorical Semantics for Functional Reactive Programming with Temporal Recursion and Corecursion

Abstract

Functional reactive programming (FRP) makes it possible to express temporal aspects of computations in a declarative way. Recently we developed two kinds of categorical models of FRP: abstract process categories (APCs) and concrete process categories (CPCs). Furthermore we showed that APCs generalize CPCs. In this paper, we extend APCs with additional structure. This structure models recursion and corecursion operators that are related to time. We show that the resulting categorical models generalize those CPCs that impose an additional constraint on time scales. This constraint boils down to ruling out ω-supertasks, which are closely related to Zeno's paradox of Achilles and the tortoise. IntroductionFunctional reactive programming (FRP) makes it possible to express temporal aspects of computations in a declarative way. Traditional FRP is based on behaviors and events, which denote time-varying values and values attached to times, respectively. There is a Curry-Howard correspondence between traditional FRP and an intuitionistic temporal logic with "always" and "eventually" modalities [6,5]. Thereby the type constructor for behaviors corresponds to "always," and the type constructor for events corresponds to "eventually."Extending the temporal logic with "until" operators leads to an extended variant of FRP. Thereby proofs of "until" propositions correspond to a new class of FRP constructs, which we call processes. Processes in the FRP sense combine continuous and discrete aspects and generalize behaviors and events [7, Section 2]. We give an introduction to FRP with processes in Section 2.We have developed two kinds of models of FRP with processes, which can also be used to model temporal logic with "until:" Abstract process categories (APCs) [8] are defined purely axiomatically. They are an extension of temporal categories [6], which in turn build on categorical models of intuitionistic S4 [9, 2].Concrete process categories (CPCs) [7] are not defined in a purely axiomatic manner, but use concrete constructions to express time-dependence of type inhabitance and causality of FRP operations.Abstract process categories generalize concrete process categories. We describe APCs in Section 3 and CPCs in Section 5. The aim of this paper is to extend APCs and CPCs in order to model recursion and corecursion on processes. We make the following contributions:• In Section 4, we develop APCs with recursion and corecursion (R-APCs). R-APCs extend APCs with recursive comonad and completely iterative monad structures that model recursion and corecursion on processes. The extensions to APCs arise naturally as extensions of ideal comonad and ideal monad structures that APCs already contain.The FRP dialect described above corresponds to an intuitionistic temporal logic via a Curry-Howard isomorphism. Thereby time-dependent type inhabitance is related to time-dependent trueness of temporal propositions. The type constructors 0 and 1 correspond to the strong and weak "until" operators U and W from linear-time temporal logic (LTL), a...

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“…Temporal logic typing as already proposed for FRP [11][12][13] might be a starting point for such a purpose. However, we also suspect that linear types could be of some interest [2,17].…”