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class Monad m where
(>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c)
return :: a -> m a
trait Monad[F[_]] {
def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C]
def unit[A](a: => A): F[A]
}
Kleisli	composition	+	unit Kleisli	composition	+	return
Kleisli Composition (fish operator) >=> compose
Bind >>= flatMap
lifts a to m a (lifts A to F[A]) return unit/pure
(>=>)::(a ->mb)->(b->mb)->(a->mc)
(>=>) = a -> (f a) >>= g
def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C]
a => flatMap(f(a))(g)
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
return :: a -> m a
-- can then implement Kleisli composition using bind
(>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c)
(>=>) = a -> (f a) >>= g
trait Monad[F[_]] {
def flatMap[A,B](ma: F[A])(f: A => F[B]): F[B]
def unit[A](a: => A): F[A]
// can then implement Kleisli composition using flatMap
def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C] =
a => flatMap(f(a))(g)
}
flatMap	+	unit bind	+	return	(Kleisli	composition	can	then	be	implemented	with	bind)
Defining	a	Monad	in	terms	of	Kleisli	composition	and	Kleisli	identity	function
Defining	Kleisli	composition	in	terms	of	flatMap	(bind)
Defining	a	Monad	in	terms	of	flatmap	(bind)	and	unit	(return)
Let’s start by reminding ourselves of a few
aspects of Monads and Kleisli composition. Philip	Schwarz				
@philip_schwarz
When we see an operation like this [function
composition: andThen]…it is interesting to look for
algebraic properties that operators like this have, and
we might ask, for instance, is this an associative
operation? So let’s find out.
Rob Norris
@tpolecat
So we have mappings between types, we have an
associative operator with an identity, at each type,
and we proved it is true by the definition of function
composition, and because there is really only one way
to define function composition, this actually follows
naturally from the type of function composition,
which I think is really interesting.
Rules	(laws)	for	function	composition
scale.bythebay.io:	Rob	Norris,	Functional	Programming	with	Effects
So what we want to
do is figure out what
this means in terms
of flatMap.
Rules	(laws)	for	Kleisli	composition
So we have two operations, pure and flatMap,
and laws telling us how they relate to each
other, and this isn’t something arbitrary, that’s
what I am trying to get across. These are things
that come naturally from the category laws,
just by analogy with pure function composition.
Rob Norris
@tpolecat
Monad	Rules	(laws)
Everything you can say about monads is on this slide, but
notice that unlike the rules for function composition, which
we proved were true and are necessarily true from the
types, this is not the case for a monad, you can satisfy this
[Monad] type and break the laws, so when we define
instances we have to verify that they meet the laws, and
Cats and Scalaz both provide some machinery to make this
very easy for you to do, so if you define [monad] instances
you have to check them [the laws].
Someone should do a conference talk on that, because it is
really important and I have never seen a talk about it.
Let’s talk about Option again. This is a monad instance for Option. I went ahead and wrote out how flatMap
works here.
Notice, this [flatMap] method could return None all the time and it would type check, but it would break
the right indentity law.
So this is why we check our laws when we implement typeclasses, it is very very important to do so.
Scala is not quite expressive enough to prove that stuff in the types, you have to do this with a second pass.
Rob Norris
@tpolecat
To	qualify	as	a	monad,	a	type	has	to	satisfy	three	
laws	that	connect	flatmap and	unit.	
Rob Norris
@tpolecat
Verifying	that	the	Option Monad satisfies	the	Monad	Laws
@odersky
@odersky
Try
NonFatal is a fairly technical thing,
essentially, an exception is fatal if it
does not make sense to export this
beyond a single thread, there are a
couple of exceptions that are, but
the vast majority of them, both
runtime exceptions and normal
exceptions are NonFatal
It	looks	like	Try might	be	a	Monad with	unit = Try
@odersky
Is	Try	a	Monad?
Is Try a Monad?
In fact it turns out that the left unit law fails.
Try in a sense trades one monad law for another
law which in this context is more useful.
I call that other law the bullet-proof principle.

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Monad Laws Must be Checked

  • 1. class Monad m where (>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c) return :: a -> m a trait Monad[F[_]] { def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C] def unit[A](a: => A): F[A] } Kleisli composition + unit Kleisli composition + return Kleisli Composition (fish operator) >=> compose Bind >>= flatMap lifts a to m a (lifts A to F[A]) return unit/pure (>=>)::(a ->mb)->(b->mb)->(a->mc) (>=>) = a -> (f a) >>= g def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C] a => flatMap(f(a))(g) class Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a -- can then implement Kleisli composition using bind (>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c) (>=>) = a -> (f a) >>= g trait Monad[F[_]] { def flatMap[A,B](ma: F[A])(f: A => F[B]): F[B] def unit[A](a: => A): F[A] // can then implement Kleisli composition using flatMap def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C] = a => flatMap(f(a))(g) } flatMap + unit bind + return (Kleisli composition can then be implemented with bind) Defining a Monad in terms of Kleisli composition and Kleisli identity function Defining Kleisli composition in terms of flatMap (bind) Defining a Monad in terms of flatmap (bind) and unit (return) Let’s start by reminding ourselves of a few aspects of Monads and Kleisli composition. Philip Schwarz @philip_schwarz
  • 2. When we see an operation like this [function composition: andThen]…it is interesting to look for algebraic properties that operators like this have, and we might ask, for instance, is this an associative operation? So let’s find out. Rob Norris @tpolecat So we have mappings between types, we have an associative operator with an identity, at each type, and we proved it is true by the definition of function composition, and because there is really only one way to define function composition, this actually follows naturally from the type of function composition, which I think is really interesting. Rules (laws) for function composition scale.bythebay.io: Rob Norris, Functional Programming with Effects
  • 3. So what we want to do is figure out what this means in terms of flatMap. Rules (laws) for Kleisli composition So we have two operations, pure and flatMap, and laws telling us how they relate to each other, and this isn’t something arbitrary, that’s what I am trying to get across. These are things that come naturally from the category laws, just by analogy with pure function composition. Rob Norris @tpolecat Monad Rules (laws)
  • 4. Everything you can say about monads is on this slide, but notice that unlike the rules for function composition, which we proved were true and are necessarily true from the types, this is not the case for a monad, you can satisfy this [Monad] type and break the laws, so when we define instances we have to verify that they meet the laws, and Cats and Scalaz both provide some machinery to make this very easy for you to do, so if you define [monad] instances you have to check them [the laws]. Someone should do a conference talk on that, because it is really important and I have never seen a talk about it. Let’s talk about Option again. This is a monad instance for Option. I went ahead and wrote out how flatMap works here. Notice, this [flatMap] method could return None all the time and it would type check, but it would break the right indentity law. So this is why we check our laws when we implement typeclasses, it is very very important to do so. Scala is not quite expressive enough to prove that stuff in the types, you have to do this with a second pass. Rob Norris @tpolecat
  • 7. @odersky Try NonFatal is a fairly technical thing, essentially, an exception is fatal if it does not make sense to export this beyond a single thread, there are a couple of exceptions that are, but the vast majority of them, both runtime exceptions and normal exceptions are NonFatal It looks like Try might be a Monad with unit = Try
  • 8. @odersky Is Try a Monad? Is Try a Monad? In fact it turns out that the left unit law fails. Try in a sense trades one monad law for another law which in this context is more useful. I call that other law the bullet-proof principle.