struct Team {} struct Tournament {} func runTournament(_ tournament: Tournament) -> Team { let teams = tournament.joinTeams(); let groups = tournament.emitGroups(teams); let groupResults = groups.map(tournament.runGroup); let brackets = tournament.buildBracket(groupResults); NSLog("%@", brackets); let winner = tournament.eliminate(brackets); return winner; }
Why FP Matters, John Hughes 1990: Modularity within eps (a:b:rest) | abs(a-b) <= eps = b | otherwise = within eps (b:rest) easydiff f x h = (f(x+h) − f x)/h differentiate h0 f x = map (easydiff f x) (repeat (/2) h0) within eps (differentiate h0 f x)
Graphs data Node = A | B | C | D | E | F | G | H | I | J | K | L | M | N deriving (Show, Enum, Bounded, Eq) edge from to = case (from, to) of (A, B) -> Just 4 (A, E) -> Just 6 (A, D) -> Just 7 -- ...
Build the tree of all possible paths! bruteforce :: Int -> Trace -> Node -> Node -> Tree Path bruteforce cost trace to from = Tree (Path cost trace') (map build (reachable from)) where trace' = from:trace build = bruteforce (cost + fromJust (edge from n)) trace' to
Branch and Bound data Bound a = Bound a | Unknown deriving (Show, Eq) -- | Unlike Maybe's Ord this one steers away from Unknown instance (Eq a, Ord a) => Ord (Bound a) where -- ...
minbranch :: Ord a => Tree a -> Bound a minbranch = minbranch' Unknown minbranch' :: Ord a => Bound a -> Tree a -> Bound a minbranch' bound (Tree.Node root subs) = case subs of [] -> Bound root _ -> foldr f bound subs where f [email protected](Tree.Node r _) b | Bound r <= b = branch' b sub | otherwise = b -- **prune !!!**
struct Team {} struct Tournament {} func runTournament(_ tournament: Tournament) -> Team { let teams = tournament.joinTeams(); let groups = tournament.emitGroups(teams); let groupResults = groups.map(tournament.runGroup); let brackets = tournament.buildBracket(groupResults); NSLog("%@", brackets); let winner = tournament.eliminate(brackets); return winner; }
a slight syntax conversion data Team data Tournament runTournament (tournament :: Tournament) = let teams = joinTeams tournament groups = emitGroups tournament teams groupResults = map (runGroup tournament) teams brackets = buildBracket tournament groupResults _ = nsLog brackets winner = eliminate tournament brackets in (winner :: Team)
nsLog will never be executed! data Team data Tournament runTournament (tournament :: Tournament) = let teams = joinTeams tournament groups = emitGroups tournament teams groupResults = map (runGroup tournament) teams brackets = buildBracket tournament groupResults _ = nsLog brackets winner = eliminate tournament brackets in (winner :: Team)
what type should have? sanity check: nsLog :: Show a => a -> Int nsLog x = callCFunction "NSLog" "%@" x test = let action = nsLog "hello" in x ! (\_ -> x) * not referentially transparent!
let's come up with a type for effects data Effects a = {- constructor for a computation that returns a value of type `a' -} ! :: Effects a -> (a -> Effects b) -> Effects b ! a f = {- perform computation `a', then feed its result to `f' which will give a computation `b' -} unit :: a -> Effects a unit a = {- make `a' pretend to be a computation -}
a monad, where = >>= class Functor f where fmap :: (a -> b) -> f a -> f b class Functor m => Monad m where return :: a -> m a (>>=) :: m a -> (a -> m b) -> m b also, m has kind * -> *: HKT!
map in a monad mapM :: Monad m => (a -> m b) -> [a] -> m [b] mapM f [] = return [] mapM f (x:xs) = f x >>= \x' -> mapM f xs >>= \xs' -> return (x':xs')
what about the rest of the functions? runTournament τ = do teams <- joinTeams τ groups <- emitGroups τ groupResults <- mapM (runGroup τ) groups brackets <- buildBracket τ groupResults nsLog brackets winner <- eliminate τ brackets unit winner
and a free monad for every free functor instance Functor f => Monad (Free f) where return = Return (>>=) :: Free f a -> (a -> Free f b) -> Free f b x >>= f = case x of Return a -> f a Free as -> Free (fmap (>>= f) as)
lifting to Free liftF :: Functor f => f a -> Free f a liftF = Free . fmap return act x = liftF . flip x () -- `act' for action fun0 x = liftF (x id) fun1 x = liftF . flip x id
chatbots data Query data Input data Dialog next = Ask Query (Input -> next) bot :: [Input] -> Free Dialog Input -> Int bot answers program = case program of Return x -> x Free (Ask q f) -> case answers of (x:xs) -> bot xs (f x) [] -> error "lol"
a blast from the past DSLs in objc era regularExpressionWithPattern: predicateWithFormat: constraintsWithVisualFormat: make.left.equalTo(superview.mas_left) .with.offset(padding.left)
why do this in a strict language? → OCaml's LWT: https://mirage.io/wiki/tutorial-lwt let start c = Lwt.join [ (Time.sleep_ns (Duration.of_sec 1) >>= fun () -> C.log c "Heads"); (Time.sleep_ns (Duration.of_sec 2) >>= fun () -> C.log c "Tails") ] >>= fun () -> C.log c "Finished"
data Free f a = Return a | Free (f (Free f a)) data Const c a = Const c data Free (Const c) a = Pure a | Free (Const c) data Either c a = Right a | Left c
// Staged function representing f(x, w, b) = dot(x, w) + b let f: Rep<(Float2D, Float2D, Float1D) -> Float2D> = lambda { x, w, b in x • w + b } // Staged function ’g’, type-inferred from ’f’ let g = lambda { x, w, b in let linear = f[x, w, b] // staged function application return tanh(linear) } // Gradient of ’g’ with respect to arguments ’w’ and ’b’ let dg = gradient(of: g, withRespectTo: (1, 2), keeping: 0) // ’dg’ has type: // Rep<(Float2D, Float2D, Float1D) -> (Float2D, Float2D, Float2D)> // Call staged function on input data ’x’, ’w’ and ’b’ let (dg_dw, dg_db, result) = dg[x, w, b]
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