Monad Fact #2

MONAD FACT #2
@philip_schwarzslides by
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equivalence of nested flatMaps and chained flatMaps
for Kleisli arrow composition

Kleisli arrows are functions of types like A => M[B], where M is a monadic type constructor.
Consider three Kleisli arrows f, g and h:
f: A => M[B]
g: B => M[C]
h: C => M[D]
A convenient way of composing f, g and h in Scala is by using a for comprehension:
for {
b <- f(a)
c <- g(b)
d <- h(c)
} yield d
On the next slide we look at a concrete (if contrived) example.
@philip_schwarz

case class Company(name: String)
case class Driver(name: String)
case class Car(registration: String)
val ibmCompany = Company(name="IBM")
val axaCompany = Company(name="AXA")
val driverJohnSmith = Driver(name="John Smith")
val carRegisteredABC123 = Car(registration="ABC123")
val driverByCompany = Map( ibmCompany -> driverJohnSmith )
val carByDriver = Map( driverJohnSmith -> carRegisteredABC123 )
val insuranceByCar = Map( carRegisteredABC123 -> axaCompany )
val f: Company => Option[Driver] = company => driverByCompany.get(company)
val g: Driver => Option[Car] = driver => carByDriver.get(driver)
val h: Car => Option[Company] = car => insuranceByCar.get(car)
def fgh: Company => Option[Company] = company =>
for {
driver <- f(company)
car <- g(driver)
insurance <- h(car)
} yield insurance
Our domain consists of companies, drivers and cars.
Here are a multinational information technology
company, an insurance company, a driver and a car.
The CEO of IBM has a driver
whose car is insured by AXA.
Here are our three Kleisli arrows: f, g and h. They
return a monad whose type constructor is Option.
Better names for Kleisli arrows f, g and h could be
the following:
• getCEODriverOf(company)
• getCarDrivenBy(driver)
• getInsurerFor(car)
But for our purposes we can forget what the
functions are computing and just concentrate on
the fact that they are Kleisli arrows.
Here is function fgh, a
Kleisli arrow that is the
composition of f, g and h.
assert( fgh(ibmCompany) == Some(Company("AXA")) )
assert( fgh(axaCompany) == None )
Here we test that the insurer of the car driven by the driver of IBM’s CEO is AXA.
There is no insurer of the car driven by the driver of AXA’s CEO (no such car nor driver).

val fgh : Company => Option[Company] = company =>
f(company)
.flatMap { driver => g(driver)
.flatMap { car => h(car)
.map { insurance => insurance
}
}
}
for {
car <- g(driver)
insurance <- h(car)
} yield insurance
As we saw in MONAD FACT #1, the code on the left desugars to the code on the right
Mapped function insurance => insurance
is just the identity function
f(company)
.map { identity
}
}
}
The reason why a monad has a map function is that every
monad is also a functor. The map function of a functor is subject
to the following functor law:
map(x)(identity) == x.
Thanks to this law, we can simplify our code by dropping the
invocation of map.
f(company)
}
}
desugars to
simplified

At this point we can go one of two ways.
f(company)
}
}
f(company) flatMap { driver => g(driver) flatMap h }
We can simplify the above just a little by making it less verbose
Later on I’ll refer to this as
the nested flatMap function.
Or alternatively, it turns out that (see later) we can rearrange the
flatMap invocations so that rather than being nested, they are chained.
f(company)
}
}
f(company)
.flatMap { driver => g(driver) }
.flatMap { car => h(car) }
assert( fgh(ibmCompany) == Some(Company("AXA")) )
assert( fgh(axaCompany) == None )
to chained
flatMaps
from nested
flatMaps
@philip_schwarz

f(company)
.flatMap { driver => g(driver) }
.flatMap { car => h(car) }
And finally, we can simplify this a bit
val fgh: Company => Option[Company] = company =>
f(company) flatMap g flatMap h
On the next slide, I’ll be referring to the above
function as the chained flatMap function
by making it less verbose
simplified

for {
car <- g(driver)
insurance <- h(car)
} yield insurance
f(company) flatMap { driver => g(driver) flatMap h }
val fgh: Company => Option[Company] = company =>
f(company) flatMap g flatMap h
Monadic Law of Associativity
(m flatMap f) flatMap g ≡ m flatMap ( x => f(x) flatMap g )
This holds for all values m, f and g of the appropriate types (see right).
While on the left hand side of the equation the invocations of flatMap are being chained, on the
right hand side of the equation the invocations are being nested.
The two desugared versions of the for comprehension function are equivalent because
the flatMap function of a monad is subject to a monadic Law of Associativity:
an operation ✽ is associative if it doesn’t
matter whether we parenthesize it
( (x ✽ y) ✽ z) or (x ✽ (y ✽ z) )
f: A => M[B]
g: B => M[C]
h: C => M[D]
nested flatMap functionchained flatMap function
for comprehension function
e.g. in our example we have the following:
(f(a) flatMap g) flatMap h ≡ f(a) flatMap ( b => g(b) flatMap h )
m: M[A]
f: A => M[B]
g: B => M[C]
We started off with the
for comprehension
function on the right
And we refactored
it into the two
equivalent
desugared
functions below
@philip_schwarz

assert(
((company:Company) => f(company) flatMap g flatMap h)(ibmCompany) // chained flatMaps
== ((company:Company) => f(company) flatMap { driver => g(driver) flatMap h } )(ibmCompany) // nested flatMaps
)
assert(
// desugars to nested flatMaps and map
((company: Company) =>
for {
car <- g(driver)
insurance <- h(car)
} yield insurance)
(ibmCompany)
==
(Some(Company("AXA")))
)
assert(
for {
car <- g(driver)
insurance <- h(car)
} yield insurance)
(ibmCompany)
==
// chained flatMaps
((company: Company) => f(company) flatMap g flatMap h )
(ibmCompany)
)
assert(
for {
car <- g(driver)
insurance <- h(car)
} yield insurance)
(ibmCompany)
==
// nested flatMaps
((company:Company) => f(company) flatMap { driver => g(driver) flatMap h } )
(ibmCompany)
)
Monadic Law of Associativity
(m flatMap f) flatMap g ≡ m flatMap ( x => f(x) flatMap g )
The Monadic Law of Associativity in the context of this example
(f(a) flatMap g) flatMap h ≡ f(a) flatMap ( b => g(b) flatMap h )
f: A => M[B]
g: B => M[C]
h: C => M[D]
Here is a recap
m: M[A]
f: A => M[B]
g: B => M[C]

See the following for the list of all available slide decks in the MONAD FACT series
https://www.slideshare.net/pjschwarz/the-monad-fact-slide-deck-series

Monad Fact #2

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Monad Fact #2