{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies #-}
{-# LANGUAGE NoImplicitPrelude #-}
{-|
Module : Interval Algebra
Description : Implementation of Allen's interval algebra
Copyright : (c) NoviSci, Inc 2020
License : BSD3
Maintainer : bsaul@novisci.com
Stability : experimental
The @IntervalAlgebra@ module provides data types and related classes for the
interval-based temporal logic described in [Allen (1983)](https://doi.org/10.1145/182.358434)
and axiomatized in [Allen and Hayes (1987)](https://doi.org/10.1111/j.1467-8640.1989.tb00329.x).
A good primer on Allen's algebra can be [found here](https://thomasalspaugh.org/pub/fnd/allen.html).
= Design
The module is built around five typeclasses designed to separate concerns of
constructing, relating, and combining @'Interval'@s:
1. @'Intervallic'@ provides an interface to the data structure of an @'Interval'@,
defining how an @'Interval' a@ is constructed.
2. @'IntervalAlgebraic'@ provides an interface to the @'IntervalRelation's@,
the workhorse of Allen's temporal logic.
3. @'IntervalCombinable'@ provides an interface to methods of combining two
@'Interval's@.
4. @'IntervalSizeable'@ and the related @'Moment'@ provides methods for
measuring and modifying the size of an interval.
5. @'IntervalFilterable'@ provides methods for filtering 'Witherable.Filterable'
collections of intervals.
An advantage of nested typeclass design is that developers can define an
@'Interval'@ of type @a@ with just the amount of structure that they need.
== Total Ordering of @Interval@s
The modules makes the (opinionated) choice of a total ordering for @'Intervallic'@
@'Interval'@s. Namely, the ordering is based on first ordering the 'begin's
then the 'end's.
= Development
This module is under development and the API may change in the future.
-}
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE FlexibleInstances #-}
module IntervalAlgebra(
-- * Classes
Intervallic(..)
, IntervalAlgebraic(..)
, IntervalCombinable(..)
, Moment(..)
, IntervalSizeable(..)
, IntervalFilterable(..)
-- * Types
, Interval
, IntervalRelation(..)
, ComparativePredicateOf
) where
import Prelude (Eq, Ord, Show, Read, Enum(..), Bounded(..), Ordering (LT)
, Maybe(..), Either(..), String, Integer, Int, Bool(..), Num
, Foldable (maximum, minimum, foldMap, foldr)
, map, otherwise, flip, show, fst, snd, min, max, any, negate, not
, (++), (==), (&&), (<), (>), (<=), ($), (+), (-), (.))
import Data.Time as DT ( Day, addDays, diffDays, addGregorianYearsClip, calendarYear )
import Data.Semigroup ( Semigroup((<>)) )
import Data.Set(Set, fromList, difference, intersection, union, map, toList)
import Data.Ord( Ord(..), Ordering(..))
import GHC.Base (Applicative(pure))
import Witherable ( Filterable(filter) )
{- | An @'Interval' a@ is a pair of @a@s \( (x, y) \text{ where } x < y\). The
@'Intervallic'@ class provides a safe @'parseInterval'@ function that returns a
@'Left'@ error if \(y < x\) and 'unsafeInterval' as constructor for creating an
interval that may not be valid.
-}
newtype Interval a = Interval (a, a) deriving (Eq)
{- |
The 'IntervalRelation' type enumerates the thirteen possible ways that two
@'Interval' a@ objects can relate according to the interval algebra.
=== Meets, Metby
> x `meets` y
> y `metBy` x
@
x: |-----|
y: |-----|
@
=== Before, After
> x `before` y
> y `after` x
@
x: |-----|
y: |-----|
@
=== Overlaps, OverlappedBy
> x `overlaps` y
> y `overlappedBy` x
@
x: |-----|
y: |-----|
@
=== Starts, StartedBy
> x `starts` y
> y `startedBy` x
@
x: |---|
y: |-----|
@
=== Finishes, FinishedBy
> x `finishes` y
> y `finishedBy` x
@
x: |---|
y: |-----|
@
=== During, Contains
> x `during` y
> y `contains` x
@
x: |-|
y: |-----|
@
=== Equal
> x `equal` y
> y `equal` x
@
x: |-----|
y: |-----|
@
-}
data IntervalRelation a =
Meets
| MetBy
| Before
| After
| Overlaps
| OverlappedBy
| Starts
| StartedBy
| Finishes
| FinishedBy
| During
| Contains
| Equals
deriving (Eq, Show, Read)
instance Bounded (IntervalRelation a) where
minBound = Before
maxBound = After
instance Enum (IntervalRelation a) where
fromEnum r = case r of
Before -> 0
Meets -> 1
Overlaps -> 2
FinishedBy -> 3
Contains -> 4
Starts -> 5
Equals -> 6
StartedBy -> 7
During -> 8
Finishes -> 9
OverlappedBy -> 10
MetBy -> 11
After -> 12
toEnum i = case i of
0 -> Before
1 -> Meets
2 -> Overlaps
3 -> FinishedBy
4 -> Contains
5 -> Starts
6 -> Equals
7 -> StartedBy
8 -> During
9 -> Finishes
10 -> OverlappedBy
11 -> MetBy
12 -> After
instance Ord (IntervalRelation a) where
compare x y = compare (fromEnum x) (fromEnum y)
-- | The 'Set' of all 'IntervalRelation's.
intervalRelations :: Set (IntervalRelation a)
intervalRelations = fromList (Prelude.map toEnum [0..12] ::[IntervalRelation a])
-- | Find the converse of a single 'IntervalRelation'
converseRelation :: IntervalRelation a -> IntervalRelation a
converseRelation x = toEnum (12 - fromEnum x)
{-
Misc
-}
-- | Defines a predicate of two objects of type @a@.
type ComparativePredicateOf a = (a -> a -> Bool)
{- |
The @'Intervallic'@ typeclass specifies how an @'Interval' a@s is constructed.
It also includes functions for getting the @'begin'@ and @'end'@ of an @'Interval' a@.
-}
class (Ord a, Show a) => Intervallic a where
-- | Safely parse a pair of @a@s to create an @'Interval' a@.
parseInterval :: a -> a -> Either String (Interval a)
parseInterval x y
-- TODO: create more general framework for error handling
| y < x = Left $ show y ++ "<" ++ show x
| otherwise = Right $ Interval (x, y)
{- | Create a new @'Interval' a@. This function is __not__ safe as it does
not enforce that \(x < y\). Use with caution. It is meant to be helper
function in early prototyping of this package. This function may be
deprecated in future releases.
-}
unsafeInterval :: a -> a -> Interval a
unsafeInterval x y = Interval (x, y)
-- | Access the ends of an @'Interval' a@ .
begin, end :: Interval a -> a
begin (Interval x) = fst x -- \( \text{begin}(x, y) = x \)
end (Interval x) = snd x -- \( \text{end}(x, y) = y \)
{- |
The @'IntervalAlgebraic'@ typeclass specifies the functions and relational
operators for interval-based temporal logic. The typeclass defines the
relational operators for intervals, plus other useful utilities such as
@'disjoint'@, @'within'@, and @'unionPredicates'@.
-}
class (Eq a, Intervallic a) => IntervalAlgebraic a where
-- | Compare two intervals to determine their 'IntervalRelation'.
relate :: Interval a -> Interval a -> IntervalRelation a
relate x y
| x `before` y = Before
| x `after` y = After
| x `meets` y = Meets
| x `metBy` y = MetBy
| x `overlaps` y = Overlaps
| x `overlappedBy` y = OverlappedBy
| x `starts` y = Starts
| x `startedBy` y = StartedBy
| x `finishes` y = Finishes
| x `finishedBy` y = FinishedBy
| x `during` y = During
| x `contains` y = Contains
| otherwise = Equals
-- | Maps an 'IntervalRelation' to its corresponding predicate function.
predicate' :: IntervalRelation a -> ComparativePredicateOf (Interval a)
predicate' r =
case r of
Before -> before
Meets -> meets
Overlaps -> overlaps
FinishedBy -> finishedBy
Contains -> contains
Starts -> starts
Equals -> equals
StartedBy -> startedBy
During -> during
Finishes -> finishes
OverlappedBy -> overlappedBy
MetBy -> metBy
After -> after
-- | Given a set of 'IntervalRelation's return a 'List' of 'predicate' functions
-- corresponding to each relation.
predicates :: Set (IntervalRelation a) -> [ComparativePredicateOf (Interval a)]
predicates x = Prelude.map predicate' (toList x)
-- | Forms a predicate function from the union of a set of 'IntervalRelation's.
predicate :: Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
predicate = unionPredicates.predicates
-- ** Algebraic operations on IntervalRelations
-- | Shortcut to creating a 'Set IntervalRelation' from a 'List'.
toSet :: [IntervalRelation a] -> Set (IntervalRelation a)
toSet = fromList
-- | Finds the complement of a 'Set IntervalRelation'.
complement :: Set (IntervalRelation a) -> Set (IntervalRelation a)
complement = difference intervalRelations
-- | Find the intersection of two 'Set's of 'IntervalRelation'
intersection :: Set (IntervalRelation a)
-> Set (IntervalRelation a)
-> Set (IntervalRelation a)
intersection = Data.Set.intersection
-- | Find the union of two 'Set's of 'IntervalRelation'
union :: Set (IntervalRelation a)
-> Set (IntervalRelation a)
-> Set (IntervalRelation a)
union = Data.Set.union
-- | Find the converse of a 'Set IntervalRelation'.
converse :: Set (IntervalRelation a)
-> Set (IntervalRelation a)
converse = Data.Set.map converseRelation
-- ** Interval algebra predicates
-- | Does x equal y?
equals :: ComparativePredicateOf (Interval a)
equals x y = x == y
-- | Does x meet y? Is y metBy x?
meets, metBy :: ComparativePredicateOf (Interval a)
meets x y = end x == begin y
metBy = flip meets
-- | Is x before y? Is x after y?
before, after :: ComparativePredicateOf (Interval a)
before x y = end x < begin y
after = flip before
-- | Does x overlap y? Is x overlapped by y?
overlaps, overlappedBy :: ComparativePredicateOf (Interval a)
overlaps x y = begin x < begin y && end x < end y && end x > begin y
overlappedBy = flip overlaps
-- | Does x start y? Is x started by y?
starts, startedBy :: ComparativePredicateOf (Interval a)
starts x y = begin x == begin y && (end x < end y)
startedBy = flip starts
-- | Synonyms for 'starts' and 'startedBy'
precedes, precededBy :: ComparativePredicateOf (Interval a)
precedes = starts
precededBy = startedBy
-- | Does x finish y? Is x finished by y?
finishes, finishedBy :: ComparativePredicateOf (Interval a)
finishes x y = begin x > begin y && end x == end y
finishedBy = flip finishes
-- | Is x during y? Does x contain y?
during, contains :: ComparativePredicateOf (Interval a)
during x y = begin x > begin y && end x < end y
contains = flip during
-- ** Interval Algebra utilities
-- | Compose a list of interval relations with _or_ to create a new
-- @'ComparativePredicateOf' 'Interval' a@. For example,
-- @unionPredicates [before, meets]@ creates a predicate function determining
-- if one interval is either before or meets another interval.
unionPredicates :: [ComparativePredicateOf (Interval a)] ->
ComparativePredicateOf (Interval a)
unionPredicates fs x y = any (\ f -> f x y) fs
-- | Operator for composing the union of two predicates
(<|>) :: ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
(<|>) f g = unionPredicates [f, g]
disjointRelations :: Set (IntervalRelation a)
disjointRelations = toSet [Before, After, Meets, MetBy]
withinRelations :: Set (IntervalRelation a)
withinRelations = toSet [During, Starts, Finishes, Equals]
-- | Are x and y disjoint ('before', 'after', 'meets', or 'metBy')?
disjoint :: ComparativePredicateOf (Interval a)
disjoint = predicate disjointRelations
-- | Are x and y not disjoint; i.e. do they share any support?
notDisjoint :: ComparativePredicateOf (Interval a)
notDisjoint = predicate (complement disjointRelations)
-- | Is x wholly within y (including endpoints) in any sense 'during',
-- 'starts', 'finishes', or 'equals'?
within :: ComparativePredicateOf (Interval a)
within = predicate withinRelations
{- |
The 'Moment' class fixes the smallest duration of an 'Intervallic a'.
-}
class (Intervallic a, Num b, Ord b) => Moment a b| a -> b where
moment :: b
moment = 1
{- |
The 'IntervalSizeable' typeclass provides functions to determine the size of
and to resize an 'Interval a'.
-}
class (Intervallic a, Moment a b, Num b, Ord b) => IntervalSizeable a b| a -> b where
-- | Determine the duration of an 'Interval a'.
duration :: Interval a -> b
duration x = diff (end x) (begin x)
-- | Shifts an @a@. Most often, the @b@ will be the same type as @a@.
-- But for example, if @a@ is 'Day' then @b@ could be 'Int'.
add :: b -> a -> a
-- | Takes the difference between two @a@ to return a @b@.
diff :: a -> a -> b
-- | Resize an 'Interval a' to by expanding to "left" by @l@ and to the
-- "right" by @r@. In the case that @l@ or @r@ are less than a 'moment'
-- the respective endpoints are unchanged.
expand :: b -> b -> Interval a -> Interval a
expand l r p = Interval (add s $ begin p, add e $ end p)
where s = if l < (moment @a) then 0 else negate l
e = if r < (moment @a) then 0 else r
-- | Expands an 'Interval a' to left by i.
expandl :: b -> Interval a -> Interval a
expandl i = expand i 0
-- | Expands an 'Interval a' to right by i.
expandr :: b -> Interval a -> Interval a
expandr = expand 0
-- | Safely creates an 'Interval a' using @x@ as the 'begin' and adding
-- @max moment dur@ to @x@ as the 'end'.
beginerval :: b -> a -> Interval a
beginerval dur x = Interval (x, add (max (moment @a) dur) x)
-- | Safely creates an 'Interval a' using @x@ as the 'end' and adding
-- @negate max moment dur@ to @x@ as the 'begin'.
enderval :: b -> a -> Interval a
enderval dur x = Interval (add (negate $ max (moment @a) dur) x, x)
{- |
The @'IntervalCombinable'@ typeclass provides methods for (possibly) combining
two @'Interval's@.
-}
class (IntervalAlgebraic a) => IntervalCombinable a where
-- | Maybe form a new @'Interval'@ by the union of two @'Interval'@s that 'meets'.
(.+.) :: Interval a -> Interval a -> Maybe (Interval a)
(.+.) x y
| x `meets` y = Just $ Interval (begin x, end y)
| otherwise = Nothing
-- | Creates a new @Interval@ spanning the extent x and y
extenterval :: Interval a -> Interval a -> Interval a
extenterval x y = Interval (s, e)
where s = min (begin x) (begin y)
e = max (end x) (end y)
-- | If @x@ is 'before' @y@, then form a new @Just Interval a@ from the
-- interval in the "gap" between @x@ and @y@ from the 'end' of @x@ to the
-- 'begin' of @y@. Otherwise, 'Nothing'.
(><) :: Interval a -> Interval a -> Maybe (Interval a)
(><) x y
| x `before` y = Just $ Interval ( end x, begin y )
| otherwise = Nothing
-- | If @x@ is 'before' @y@, return @f x@ appended to @f y@. Otherwise,
-- return 'extenterval' of @x@ and @y@ (wrapped in @f@). This is useful for
-- (left) folding over an *ordered* container of @Interval@s and combining
-- intervals when @x@ is *not* 'before' @y@.
(<+>):: (Semigroup (f (Interval a)), Applicative f) =>
Interval a ->
Interval a ->
f (Interval a)
(<+>) x y
| x `before` y = pure x <> pure y
| otherwise = pure ( extenterval x y )
-- | Forms a 'Just' new interval from the intersection of two intervals,
-- provided the intervals are not disjoint.
intersect :: Interval a -> Interval a -> Maybe (Interval a)
intersect x y
| disjoint x y = Nothing
| otherwise = Just $ Interval (b, e)
where b = max (begin x) (begin y)
e = min (end x) (end y)
{- |
The @'IntervalFilterable'@ class provides functions for filtering 'Filterable's of
@'Interval'@s based on @'IntervalAlgebraic'@ relations.
-}
class (Filterable f, IntervalAlgebraic a) => IntervalFilterable f a where
-- |Creates a function for filtering a 'Witherable.Filterable' of @Interval a@s based on a predicate
filterMaker :: ComparativePredicateOf (Interval a)
-> Interval a
-> (f (Interval a) -> f (Interval a))
filterMaker f p = Witherable.filter (`f` p)
-- | Filter a 'Witherable.Filterable' of @Interval a@s to those that 'overlaps' the @Interval a@
-- in the first argument.
filterOverlaps :: Interval a -> f (Interval a) -> f (Interval a)
filterOverlaps = filterMaker overlaps
-- | Filter a 'Witherable.Filterable' of @Interval a@s to those 'overlappedBy' the @Interval a@
-- in the first argument.
filterOverlappedBy :: Interval a -> f (Interval a) -> f (Interval a)
filterOverlappedBy = filterMaker overlappedBy
-- | Filter a 'Witherable.Filterable' of Interval as to those 'before' the @Interval a@
-- in the first argument.
filterBefore :: Interval a -> f (Interval a) -> f (Interval a)
filterBefore = filterMaker before
-- | Filter a 'Witherable.Filterable' of Interval as to those 'after' the @Interval a@
-- in the first argument.
filterAfter :: Interval a -> f (Interval a) -> f (Interval a)
filterAfter = filterMaker after
-- | Filter a 'Witherable.Filterable' of Interval as to those that 'meets' the @Interval a@
-- in the first argument.
filterMeets :: Interval a -> f (Interval a) -> f (Interval a)
filterMeets = filterMaker meets
-- | Filter a 'Witherable.Filterable' of Interval as to those 'metBy' the @Interval a@
-- in the first argument.
filterMetBy :: Interval a -> f (Interval a) -> f (Interval a)
filterMetBy = filterMaker metBy
-- | Filter a 'Witherable.Filterable' of Interval as to those 'during' the @Interval a@
-- in the first argument.
filterDuring :: Interval a -> f (Interval a) -> f (Interval a)
filterDuring = filterMaker during
-- | Filter a 'Witherable.Filterable' of Interval as to those that 'contains'
-- the @Interval a@ in the first argument.
filterContains :: Interval a -> f (Interval a) -> f (Interval a)
filterContains = filterMaker contains
-- | Filter a 'Witherable.Filterable' of Interval as to those that are 'disjoint'
-- from the @Interval a@ in the first argument.
filterDisjoint :: Interval a -> f (Interval a) -> f (Interval a)
filterDisjoint = filterMaker disjoint
-- | Filter a 'Witherable.Filterable' of Interval as to those that are 'notDisjoint'
-- from the @Interval a@ in the first argument.
filterNotDisjoint :: Interval a -> f (Interval a) -> f (Interval a)
filterNotDisjoint = filterMaker notDisjoint
-- | Filter a 'Witherable.Filterable' of Interval as to those that are 'within'
-- the @Interval a@ in the first argument.
filterWithin :: Interval a -> f (Interval a) -> f (Interval a)
filterWithin = filterMaker disjoint
{-
Instances
-}
-- | Imposes a total ordering on @'Interval' a@ based on first ordering the
-- 'begin's then the 'end's.
instance (Intervallic a) => Ord (Interval a) where
(<=) x y
| begin x < begin y = True
| begin x == begin y = end x <= end y
| otherwise = False
(<) x y
| begin x < begin y = True
| begin x == begin y = end x < end y
| otherwise = False
instance (Intervallic a, Show a) => Show (Interval a) where
show x = "(" ++ show (begin x) ++ ", " ++ show (end x) ++ ")"
instance Intervallic Int
instance IntervalAlgebraic Int
instance IntervalCombinable Int
instance Moment Int Int
instance IntervalSizeable Int Int where
add = (+)
diff = (-)
instance IntervalFilterable [] Int
instance Intervallic Integer
instance IntervalAlgebraic Integer
instance IntervalCombinable Integer
instance Moment Integer Integer
instance IntervalSizeable Integer Integer where
add = (+)
diff = (-)
instance IntervalFilterable [] Integer
instance Intervallic DT.Day
instance IntervalAlgebraic DT.Day
instance IntervalCombinable DT.Day
instance Moment DT.Day Integer
instance IntervalSizeable DT.Day Integer where
add = addDays
diff = diffDays
instance IntervalFilterable [] DT.Day