semantics.md

Semantics

Introduction

This document intends to describe the semantics of Hareactive. The semantics are highly inspired by Atze van der Ploegs paper Practical Principled FRP and Conal Elliotts work, in particular the papers Functional Reactive Animation and Push-Pull Functional Reactive Programming.

Future and behavior

Time is semantically equal to the reals.

type Time = ℝ

A Behavior a is denoted as a function from Time to a.

Behavior a = Time -> a

A Future a is denoted as a pair of Time and a.

Future a = (Time, a)

A Behavior is a Functor, Applicative and Monad in exactly the same way as its denotation, a function, is.

instance Functor Behavior where
  map f b = map f b

instance Applicative Behavior where
  pure = const
  ap f b = \t -> f t (b t)

instance Monad Behavior where
  m >>= f = \t -> f (m t) t

A Future is a Functor, Applicative and Monad in exactly the same way as its denotation, a pair, is.

instance Functor Future where
  map f (t, a) = (t, f a)

instance Applicative Future where
  pure a = (-Infinity, a)
  ap (ta, f) (tb, a) = (max ta tb, f a)

instance Monad Future where
  (ta, a) >>= f = let (tb, b) = f a
                  in (max ta tb, b)

Stream

A Stream is denoted as below.

Stream a = [(Time, a)]

Where the points in time are non-strictly increasing. This means that any time must be larger than or equal to any points prior to it and smaller than or equal to any points after it.