PyMonad 1.2

Installation

Using pip:

pip install PyMonad

Or download the package and run:

python setup.py install

from the project directory.

Imports

Import the entire package:

from pymonad import *

Or just a single monad type:

from pymonad.Maybe import *

If you’re not importing everything but want to use curried functions:

from pymonad.Reader import curry

Curried Functions and Partial Application

To define a curried function use the @curry decorator:

@curry
def add(x, y):
        return x + y

@curry
def func(x, y, z):
        # Do something with x, y and z.
        ...

The above fuctions can be partially applied by passing them less than their full set of arguments:

add(7, 8)                       # Calling 'add' normally returns 15 as expected.
add7 = add(7)           # Partial application: 'add7' is a function taking one argument.
add7(8)                         # Applying the final argument retruns 15...
add7(400)                       # ... or 407, or whatever.

# 'func' can be applied in any of the following ways.
func(1, 2, 3)           # Call func normally.
func(1, 2)(3)           # Partially applying two, then applying the last argument.
func(1)(2, 3)           # Partially applying one, then applying the last two arguments.
func(1)(2)(3)           # Partially applying one, partially applying again, then applying the last argument.

Function Composition

Curried functions can be composed with the * operator. Functions are applied from right to left:

# Returns the first element of a list.
@curry
def head(aList):
        return aList[0]

# Returns everything except the first element of the list.
@curry
def tail(aList):
        return aList[1:]

second = head * tail            # 'tail' will be applied first, then its result passed to 'head'
second([1, 2, 3, 4])            # returns 2

You can also compose partially applied functions:

@curry
def add(x, y):
        return x + y

@curry
def mul(x, y):
        return x * y

comp = add(7) * mul(2)          # 'mul(2)' is evaluated first, and it's result passed to 'add(7)'
comp(4)                                         # returns 15

# Composition order matters!
comp = mul(2) * add(7)
comp(4)                                         # returns 22

Functors, Applicative Functors, and Monads

All Monads are also Applicative Functors, and all Applicative Functors are also Functors, though the same is not necessarily true in reverse. All the types included with PyMonad are defined as all three but you can define new types however you want.

All of these types ultimately derive from Container which simply holds a value and provides some basic value equality checking. The method getValue() will allow you to “extract” the value of monadic computations if/when necessary.

Functors

All functors define the fmap method which can be invoked via the fmap operator *. fmap takes functions which operate on simple types – integers, strings, etc. – and allows them to operate of functor types:

from pymonad.Maybe import *
from pymonad.List import *

# 'neg' knows nothing about functor types...
def neg(x):
        return -x

# ... but that doesn't stop us from using it anyway.
neg * Just(9)                           # returns Just(-9)
neg * Nothing                           # returns Nothing
neg * List(1, 2, 3, 4)          # returns List(-1, -2, -3, -4)

Notice that the function is on the left and the functor type is on the right. If you think of * as a sort of fancy opening paren, then normal calls and fmap calls have basically the same structure:

------------------------------------------------------------------
                                function                open            argument                close
Normal call               neg                    (                         9                      )
fmap call                 neg                    *                       Just(9)
------------------------------------------------------------------

Notice that * is also the function composition operator. In fact, curried functions are instances of the Reader monad, and fmap -ing a function over another function is the same thing as function composition.

Applicative Functors

Functors allow you to use normal functions of a single argument – like neg above – with functor types. Applicative Functors extend that capability – via amap and its operator & – allowing you to use normal functions of multiple arguments with functor types:

# 'add' operates on simple types, not functors or applicatives...
def add(x, y):
        return x + y

# ... but we're going to use it on those types anyway.
# Note that we're still using '*' but now in conjunction with '&'
add * Just(7) & Just(8)                                 # returns Just(15)
add * Nothing & Just(8)                                 # returns Nothing
add * Just(7) & Nothing                                 # returns Nothing
add * List(1, 2, 3) & List(4, 5, 6)             # returns List(5, 6, 7, 6, 7, 8, 7, 8, 9)

If * is a fancy paren, & is the fancy comma used to separate arguments.

Monads

Monads allow you to sequence a series of calculations within than monad using the bind operator >>.

The first argument to >> is a monad type. The second argument is a function which takes a single, non-monad argument and returns an instance of the same monad:

from pymonad.List import *
from pymonad.Reader import curry

# Takes a simple number type and returns a 'List' containing that value and it's negative.
def positive_and_negative(x):
        return List(x, -x)

# You can call 'positive_and_negative' normally.
positive_and_negative(9)                # returns List(9, -9)

# Or you can create a List...
x = List(9)

# ... and then use '>>' to apply positive_and_negative'
x >> positive_and_negative              # also returns List(9, -9)

# But 'x' could also have more than one value...
x = List(1, 2)
x >> positive_and_negative              # returns List(1, -1, 2, -2)

# And of course you can sequence partially applied functions.
@curry
def add_and_sub(x, y):
        return List(y + x, y - x)

List(2) >> positive_and_negative >> add_and_sub(3)              # creates List(2)
                                                                                                                # applies positive_and_negative: List(2, -2)
                                                                                                                # then add_and_sub(3): List(5, -1, 1, -5)
                                                                                                                # final result: List(5, -1, 1, -5)

Variable assignment in monadic code

The second argument to >> is a function which takes a single, non-monad argument. Because of that, you can use lambda to assign values to a variable withing monadic code, like this:

from pymonad.Maybe import *

Just(9) >> (lambda x:                           # Here, 'x' takes the value '9'
Just(8) >> (lambda y:                           # And 'y' takes the value '8'
Just(x + y)))                                           # The final returned value is 'Just(9 + 8)', or 'Just(17)'

You can also simply ignore values if you wish:

Just(9) >> Just(8)                                      # The '9' is thrown away and the result of this computation is 'Just(8)'

Implementing Monads

Implementing other functors, applicatives, or monads is fairly straight-forward. There are three classes, serving as interfaces:

Monad --> Applicative --> Functor

To implement a new functor, create a new class which derives from Functor and override the fmap method.

To implement a new applicative functor, create a new class which derives from Applicative and override the amap and fmap methods.

To implement a new monad, create a new class which derives from Monad and override at least the bind method, and preferably the amap and fmap methods as well.

The operators, *, &, and >> are pre-defined to call the above methods so you shouldn’t need to touch them directly.

unit (aka return)

The previous version of pymonad didn’t include the method unit (called return in Haskell). unit takes a bare value, such as 8, and places it in a default context for that monad. Haskell allows polymorphism on return types as well as supporting type inference, so you (mostly) don’t have to tell return what types to expect, it just figures it out. We can’t do that in Python, so you always need to tell unit what type you’re expecting.

The unit method is implemented as a class method in Functor.py, so it can be used with any functor, applicative or monad. There is also a unit function which expects a functor type (though you can also give it an instance) and a value and invokes the corresponding unit method. It is provided to give a more “functional look” to code, but use whichever method you prefer. With the Maybe monad for example:

1. Maybe.unit(8) # returns Just(8)

2. unit(Maybe, 8) # also returns Just(8)

In either case all functors (and applicatives and monads) should implement the unit class method.

Creating New Monoids

To create a new monoids type create a class deriving from Monoid and override the mzero static method which takes no arguments and should return an instance of the class containing the identity value for the monoid. Also override the mplus method. For instance, numbers can be a monoid in two ways, one way with zero and addition as discussed above and the other way with one and multiplication. We could implement that like this:

class ProductMonoid(Monoid):
        @staticmethod
        def mzero():
                return ProductMonoid(1)

        def mplus(self, other):
                return ProductMonoid(self.getValue() * other.getValue())

The + operator (aka __add__()) is defined to call mplus on monoid instances, so you can simply “add” monoid values together rather than having to call mplus directly.

“Natural” Monoids

Similar to unit for monads, there is an mzero function which expects a type and can be used instead of the mzero method. Unlike unit however, the mzero function serves another purpose. Numbers, strings and lists can all be used as monoids and all already have an appropriate definition for +. What they don’t have is an mzero method. To allow numbers, strings and lists to be used as monoids without any extra work, the mzero function will return the appropriate value for these types and will attempt to call the mzero method on anything else. For instance:

mzero(int)                                      # returns 0, also works with float
mzero(str)                                      # returns ""
mzero(list)                                     # returns []
mzero(ProductMonoid)            # return ProductMonoid(1)
# etc...

If you write code involving monoids, and you’re not sure what type of monoid you might be handed, you should use the mzero function and not the mzero method.