{-# OPTIONS -fglasgow-exts #-}

{- |

Module : $EmptyHeader$

Description : <optional short description entry>

Copyright : (c) <Authors or Affiliations>

License : GPLv2 or higher, see LICENSE.txt

Maintainer : <email>

Stability : unstable | experimental | provisional | stable | frozen

Portability : portable | non-portable (<reason>)

<optional description>

-}

module Ctl where

import Data.Set as Set

exists :: (a -> Bool) -> Set a -> Bool

exists f = Set.fold ((||) . f) False

infixr 3 `And`

infixr 2 `Or`

infixr 1 `Implies`

infixl 1 `AU`, `EU`

infix 0 |=

-- ctl formulas

data Formula a =

Top

| Bottom

| Atom a

| Not (Formula a)

| (Formula a) `And` (Formula a)

| (Formula a) `Or` (Formula a)

| (Formula a) `Implies` (Formula a)

| AX (Formula a)

| EX (Formula a)

| AF (Formula a)

| EF (Formula a)

| AG (Formula a)

| EG (Formula a)

| (Formula a) `AU` (Formula a)

| (Formula a) `EU` (Formula a)

deriving Show

-- ctl class

class CTL m a s | m -> a s where

states :: m -> Set s

next :: m -> s -> Set s

labels :: m -> s -> Set a

-- satisfaction relation

(|=) :: (CTL m a s, Ord a, Ord s) => (m, s) -> Formula a -> Bool

(m, s) |= phi = Set.member s (sat m phi)

sat :: (CTL m a s, Ord a, Ord s) => m -> Formula a -> Set s

sat m Top = states m

sat m Bottom = Set.empty

sat m (Atom a) = Set.filter (Set.member a . labels m) (states m)

sat m (Not phi) = Set.difference (states m) (sat m phi)

sat m (phi `And` psi) = Set.intersection (sat m phi) (sat m psi)

sat m (phi `Or` psi) = Set.union (sat m phi) (sat m psi)

sat m (phi `Implies` psi) = sat m (Not phi `Or` psi)

sat m (AX phi) = sat m (Not (EX (Not phi)))

sat m (EX phi) = satEX m phi

sat m (phi `AU` psi) = sat m (Not (Not psi `EU` (Not phi `And` Not psi) `Or` EG (Not psi)))

sat m (phi `EU` psi) = satEU m phi psi

sat m (EF phi) = sat m (Top `EU` phi)

sat m (EG phi) = sat m (Not (AF (Not phi)))

sat m (AF phi) = satAF m phi

sat m (AG phi) = sat m (Not (EF (Not phi)))

satEX :: (CTL m a s, Ord a, Ord s) => m -> Formula a -> Set s

satEX m phi = preE m (sat m phi)

satAF :: (CTL m a s, Ord a, Ord s) => m -> Formula a -> Set s

satAF m phi = satAF' (states m) (sat m phi)

where

satAF' x y = if x == y then y else satAF' y (Set.union y (preA m y))

satEU :: (CTL m a s, Ord a, Ord s) => m -> Formula a -> Formula a -> Set s

satEU m phi psi = satEU' (sat m phi) (states m) (sat m psi)

where

satEU' w x y = if x == y then y else satEU' w y (Set.union y (Set.intersection w (preE m y)))

preE :: (CTL m a s, Ord a, Ord s) => m -> Set s -> Set s

preE m y = Set.filter (\s-> exists (flip Set.member (next m s)) y) (states m)

preA :: (CTL m a s, Ord a, Ord s) => m -> Set s -> Set s

preA m y = Set.difference (states m) (preE m (Set.difference (states m) y))

-- test model

data My = My

data MyAtoms = P | Q | R deriving (Eq, Ord, Show)

data MyStates = S0 | S1 | S2 deriving (Eq, Ord, Show)

instance CTL My MyAtoms MyStates where

states My = Set.fromList [S0, S1, S2]

next My S0 = Set.fromList [S1, S2]

next My S1 = Set.fromList [S0, S2]

next My S2 = Set.fromList [S2]

labels My S0 = Set.fromList [P, Q]

labels My S1 = Set.fromList [Q, R]

labels My S2 = Set.fromList [R]