Composition in FP

Transcript

1. Composition in FP
   Vladimir Ciobanu
   Monday, August 20, 2018
   

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2. Overview
   Algebraic Recap
   Back to FP
   Monad
   Applicative
   1
   

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3. Algebraic Recap
   

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4. What Is Composition?
   Composition is essential in functional programming. We use
   mathematical abstractions (such as Semigroup and Monoid)
   to define the composition of a wide variety of data types.
   Short recap:
   • Sets, e.g., B = { , ⊥}, N = {0, 1, 2, . . .},
   T = {Void, (), Bool, Int, ((), Bool), [(Int, Bool)], . . .}
   • Operations, e.g., there are four unary operations on
   Bool (const , const ⊥, identity and not)
   • Associativity, e.g., (a ∨ b) ∨ c = a ∨ (b ∨ c)
   • Commutativity, e.g., a ∨ b = b ∨ a
   2
   

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5. Semigroup
   A semigroup is an algebraic structure consisting of a set and
   an associative binary operation.
   Given a set S and an operation +, then:
   ∀ a, b ∈ S, ∃ c ∈ S, a + b = c
   ∀ a, b, c ∈ S, (a + b) + c = a + (b + c)
   Examples of semigroups: concatenation on (non-empty) lists,
   addition and multiplication of numbers, conjunction and
   disjunction of booleans, appending parts in a path or chunks
   of a file, combining IO actions, etc.
   3
   

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6. Monoid
   A monoid is an algebraic structure consisting of a set, an
   associative binary operation, and an identity element.
   Given a set S, an operation +, and the identity element e,
   then:
   ∀ a, b ∈ S, ∃ c ∈ S, a + b = c
   ∀ a, b, c ∈ S, (a + b) + c = a + (b + c)
   ∀ a ∈ S, e + a = a + e = a
   Examples of monoids: concatenation on (possibly empty)
   lists, addition and multiplication on numbers, conjunction and
   disjunction on booleans, appending parts in a path or chunks
   of a file, combining IO actions, etc.
   4
   

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7. Semiring 1/2
   A semiring is an algebraic structure consisting of a set S, a
   commutative monoid (S, +, 0) and monoid (S, ∗, 1), such that ∗
   distributes over + and 0 annihilates ∗.
   So, given a set S, and two operations + and ∗, and the identity
   elements 0 and 1, we can say:
   5
   

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8. Semiring 2/2
   ∀ a, b ∈ S, ∃ c ∈ S, a + b = c
   ∀ a, b, c ∈ S, (a + b) + c = a + (b + c)
   ∀ a ∈ S, 0 + a = a + 0 = a
   ∀ a, b ∈ S, a + b = b + a
   ∀ a, b ∈ S, ∃ c ∈ S, a ∗ b = c
   ∀ a, b, c ∈ S, (a ∗ b) ∗ c = a ∗ (b ∗ c)
   ∀ a ∈ S, 1 ∗ a = a ∗ 1 = a
   ∀ a, b, c ∈ S, a ∗ (b + c) = (a ∗ b) + (a ∗ c)
   ∀ a, b, c ∈ S, (a + b) ∗ c = (a ∗ c) + (b ∗ c)
   ∀ a ∈ S, a ∗ 0 = 0 ∗ a = 0 6
   

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9. Semiring Example: Bool
   ∀ a, b ∈ B, ∃ c ∈ B, a ∨ b = c
   ∀ a, b, c ∈ B, (a ∨ b) ∨ c = a ∨ (b ∨ c)
   ∀ a ∈ B, ⊥ ∨ a = a ∨ ⊥ = a
   ∀ a, b ∈ B, a ∨ b = b ∨ a
   ∀ a, b ∈ B, ∃ c ∈ B, a ∧ b = c
   ∀ a, b, c ∈ B, (a ∧ b) ∧ c = a ∧ (b ∧ c)
   ∀ a ∈ B, ∧ a = a ∧ = a
   ∀ a, b, c ∈ B, a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
   ∀ a, b, c ∈ B, (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c)
   ∀ a ∈ B, a ∧ ⊥ = ⊥ ∧ a = ⊥ 7
   

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10. Semiring Example: Types
    ∀ a, b ∈ T, ∃ c ∈ T, Either a b ∼
    = c
    ∀ a, b, c ∈ T, Either (Either a b) c ∼
    = Either a (Either b c)
    ∀ a ∈ T, Either Void a ∼
    = Either a Void ∼
    = a
    ∀ a, b ∈ T, Either a b ∼
    = Either b a
    ∀ a, b ∈ T, ∃ c ∈ T, (a, b) ∼
    = c
    ∀ a, b, c ∈ T, ((a, b), c) ∼
    = (a, (b, c))
    ∀ a ∈ T, ((), a) ∼
    = (a, ()) ∼
    = a
    ∀ a, b, c ∈ T, (a, Either b c) ∼
    = Either (a, b) (a, c)
    ∀ a, b, c ∈ T, ((Either a b), c) ∼
    = Either (a, c) (b, c)
    ∀ a ∈ T, (a, Void) ∼
    = (Void, a) ∼
    = Void 8
    

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11. Back to FP
    

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12. Monoid in Haskell
    1 class Semigroup a where
    2 (<>) :: a -> a -> a
    3
    4
    5 class Semigroup a => Monoid a where
    6 mempty :: a
    7
    8
    9 instance Semigroup [a] where
    10 a <> b = a ++ b
    11
    12
    13 instance Monoid [a] where
    14 mempty = []
    9
    

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13. Monoid Usage
    1 λ> "foo" <> "bar"
    2 "foobar"
    3
    4 fold :: (Foldable t, Monoid m) => t m -> m
    5 λ> fold ["fo", "o", "bar"]
    6 "foobar"
    7
    8 foldMap :: (Monoid m, Foldable t) => (a -> m) -> t a -> m
    9 λ> foldMap Sum [1,2,3,4,5]
    10 Sum {getSum = 15}
    11
    12 λ> foldMap All [True, True, True]
    13 All {getAll = True}
    14
    15 λ> foldMap All [True, True, False]
    16 All {getAll = False}
    10
    

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14. Combining Different Types
    11
    

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15. Monad
    

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16. Monad
    When viewed from the perspective of composability, monads
    allow us to compose things that are not obtained
    independently, but rather in a more sequential manner.
    1 conn 2 someData 3 writeToDisk someData
    4
    5 -- or:
    6 getSqlConnection
    7 >>= runSomeSelectQuery
    8 >>= writeToDisk
    12
    

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17. Just a Monoid in the...
    1 -- These two functions are equivalent:
    2 (>>=) :: m a -> (a -> m b) -> m b
    3 join :: m (m a) -> m a
    4
    5 -- Forward
    6 ma >>= mab = join (mab ma)
    7 -- Backward
    8 join mma = mma >>= id
    Any Monad is a Monoid. The easiest way to look at it is by
    looking at the join operation, which takes two m’s and returns
    a single m, or combines two m’s into a single one.
    See @KenScambler’s excellent tweets about this at
    https://twitter.com/KenScambler/status/956111889519357952
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18. Applicative
    

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19. Applicative
    1 class Functor f where
    2 () :: (a -> b) -> f a -> f b
    3
    4 class Functor f => Applicative f where
    5 () :: f (a -> b) -> f a -> f b
    6 pure :: a -> f a
    7
    8 -- Applicative should really be like this.
    9 class Functor f => Apply f where
    10 () :: f (a -> b) -> f a -> f b
    11
    12 class Apply f => Applicative f where
    13 pure :: a -> f a
    14
    

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20. Applicative Example
    1 op :: A -> B -> C
    2 a :: A
    3 b :: B
    4
    5 op a b :: C
    6
    7 fa :: f A
    8 fb :: f B
    9
    10 op fa fb :: f C
    11
    12 λ> (+) Just 1 Just 2
    13 Just 3
    14
    15 λ> (+) Just 1 Nothing
    16 Nothing
    15
    

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21. Is this a semigroup / monoid?
    1 class Functor f => TupleSemigroup f where
    2 (<>) :: f a -> f b -> f (a, b)
    3
    4 class TupleSemigroup f => TupleMonoid f where
    5 mempty :: f ()
    The answer is, yes, if we alter the rules a bit to say that the
    operation is associative up to isomorphism (and that using
    the identity element results in isomorphic structures). The
    claim is these classes are equivalent to Apply and
    Applicative. How do we prove it? By implementing these in
    terms of Apply and Applicative, and then implementing
    Apply/Applicative in terms of these clases.
    16
    

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22. Equivalence
    1 -- Applicative -> Tuple*
    2 instance Apply f => TupleSemigroup f where
    3 (<>) :: f a -> f b -> f (a, b)
    4 fa <> fb = (,) fa fb
    5
    6 instance Applicative f => TupleMonoid where
    7 mempty :: f ()
    8 mempty = pure ()
    9
    10 -- Tuple* -> Applicative
    11 instance TupleSemigroup f => Apply f where
    12 () :: f (a -> b) -> f a -> f b
    13 fa2b fa = (\(a2b, a) -> a2b a) (fa2b <> fa)
    14
    15 instance TupleMonoid f => Applicative f where
    16 pure :: a -> f a
    17 pure a = const a mempty
    17
    

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23. Wait, is TupleMonoid really a Monoid?
    ∀ fa, fb ∈ T, ∃ fc ∈ T, fa <> fb ∼
    = fc ∼
    = f(a, b)
    ∀ fa, fb, fc ∈ T, (fa <> fb) <> fc ∼
    = fa <> (fb <> fc)
    f(a, b) <> fc ∼
    = fa <> f(b, c)
    f((a, b), c) ∼
    = f(a, (b, c))
    ∀ fa ∈ T, mempty <> fa ∼
    = fa <> mempty ∼
    = fa
    () <> fa ∼
    = fa <> () ∼
    = fa
    f(a, ()) ∼
    = f((), a) ∼
    = fa
    18
    

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24. Alternative
    1 -- Semigroup
    2 class Functor f => Alt f where
    3 () :: f a -> f a -> f a
    4
    5 -- Monoid
    6 class Alt f => class Plus f where
    7 empty :: f a
    8
    9 -- Near Semiring-ish
    10 class Applicative f, Plus f => Alternative f
    11
    12 -- Alternative guarantees the following laws:
    13 (f g) x == (f x) (g x)
    14 empty f == empty
    19
    

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25. Applicative Semiring Example
    1 λ> ([(+1), (+2)] [(+3), (+4)]) [1,2]
    2 [2,3,3,4,4,5,5,6]
    3
    4 λ> ([(+1), (+2)] [1,2]) ([(+3), (+4)] [1, 2])
    5 [2,3,3,4,4,5,5,6]
    6
    7 λ> [(+1), (+2)] empty
    8 []
    20
    

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26. Alternative Intuition
    To get some intuition, we can think of:
    • as: multiplication, conjunction, or carthesian product
    • pure as: 1, True, or the unit set
    • as: addition, disjunction, or concatenation
    • empty as: 0, False, or the empty set
    In a way, Applicative is a higher kinded monoid, and
    Alternative is a higher kinded semiring (except it’s not
    commutative in Plus).
    21
    

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27. Combining Different Types
    22
    

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28. Thank you for listening!
    cvlad
    vladciobanu
    cvlad.info
    23
    

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