LTLFuture.hs

{-# LANGUAGE GADTs #-}
-- |
-- Copyright  : (c) Ivan Perez, 2017-2022
-- License    : BSD-style (see the LICENSE file in the distribution)
-- Maintainer : ivan.perez@keera.co.uk
--
-- Linear Temporal Logics based on SFs.
--
-- This module contains a definition of LTL with Next on top of Signal
-- Functions.
--
-- LTL predicates are parameterized over an input. A basic proposition is a
-- Signal Function that produces a boolean function.
module FRP.Yampa.LTLFuture
    ( TPred(..)
    , evalT
    )
  where

-- External imports
import FRP.Yampa (DTime, SF, evalFuture)

-- Internal imports
import FRP.Yampa.Stream (SignalSampleStream, evalSF, firstSample)

-- | Type representing future-time linear temporal logic predicates with until
-- and next.
data TPred a where
  SP         :: SF a Bool -> TPred a
  And        :: TPred a -> TPred a -> TPred a
  Or         :: TPred a -> TPred a -> TPred a
  Not        :: TPred a -> TPred a
  Implies    :: TPred a -> TPred a -> TPred a
  Always     :: TPred a -> TPred a
  Eventually :: TPred a -> TPred a
  Next       :: TPred a -> TPred a
  Until      :: TPred a -> TPred a -> TPred a

-- | Evaluates a temporal predicate at time t=0 with a concrete sample stream.
--
-- Returns 'True' if the temporal proposition is currently true.
evalT :: TPred a -> SignalSampleStream a -> Bool
evalT (SP sf)         = \stream -> firstSample $ fst $ evalSF sf stream
evalT (And t1 t2)     = \stream -> evalT t1 stream && evalT t2 stream
evalT (Or  t1 t2)     = \stream -> evalT t1 stream || evalT t2 stream
evalT (Not t1)        = \stream -> not (evalT t1 stream)
evalT (Implies t1 t2) = \stream -> not (evalT t1 stream) || evalT t2 stream
evalT (Always  t1)    = \stream ->
  evalT t1 stream && evalT (Next (Always t1)) stream

evalT (Eventually t1) = \stream ->
  case stream of
    (a, [])             -> evalT t1 stream
    (a1, (dt, a2) : as) -> evalT t1 stream
                             || evalT (tauApp (Eventually t1) a1 dt) (a2, as)

evalT (Until t1 t2) = \stream ->
  (evalT t1 stream && evalT (Next (Until t1 t2)) stream)
    || evalT t2 stream

evalT (Next t1) = \stream ->
  case stream of
    (a, [])             -> True -- This is important. It determines how
                                -- always and next behave at the end of the
                                -- stream, which affects that is and isn't a
                                -- tautology. It should be reviewed very
                                -- carefully.
    (a1, (dt, a2) : as) -> evalT (tauApp t1 a1 dt) (a2, as)

-- | Tau-application (transportation to the future)
tauApp :: TPred a -> a -> DTime -> TPred a
tauApp pred sample dtime =
  tPredMap (\sf -> snd (evalFuture sf sample dtime)) pred

-- | Apply a transformation to the leaves (to the SFs)
tPredMap :: (SF a Bool -> SF a Bool) -> TPred a -> TPred a
tPredMap f (SP sf)         = SP (f sf)
tPredMap f (And t1 t2)     = And (tPredMap f t1) (tPredMap f t2)
tPredMap f (Or t1 t2)      = Or (tPredMap f t1) (tPredMap f t2)
tPredMap f (Not t1)        = Not (tPredMap f t1)
tPredMap f (Implies t1 t2) = Implies (tPredMap f t1) (tPredMap f t2)
tPredMap f (Always t1)     = Always (tPredMap f t1)
tPredMap f (Eventually t1) = Eventually (tPredMap f t1)
tPredMap f (Next t1)       = Next (tPredMap f t1)
tPredMap f (Until t1 t2)   = Until (tPredMap f t1) (tPredMap f t2)