G meredith scala

Monadic Design
Patterns for the Web
Session 1

Monday, October 17, 2011

Agenda
• What to expect
• Who is the audience
• Why monads
• The monadic toolbox
• A simple example
• Another simple example
Lucius Gregory Meredith, Managing Partner, Biosimilarity LLC -- SDEC 2011

Agenda (cont)

• Hinting at a more complex example
• Where we might go from here
• Questions?


What to expect
• Fun!
• Simplicity!
• Engagement!
• Ok... and some challenge
• Occasionally, people’s brains do melt out of
their ears...


What to expect

Service

Domain Logic

Browser

Container

Resource


What to expect

Service

Domain Logic

Browser

Container

Resource

Monads for
managing streams


What to expect
Monads for
managing requests

Service

Domain Logic

Browser

Container

Resource

Monads for
managing streams


What to expect
Monads for
domain model
Monads for
managing requests

Service

Domain Logic

Browser

Container

Resource

Monads for
managing streams


What to expect
Monads for
domain model
Monads for
managing requests

Service

Domain Logic

Browser
Monads for
accessing store
Container

Resource

Monads for
managing streams


What to expect
Algebraic data types

Parsing
combinators

Service

Domain Logic

Browser LINQ
XQuery
SQL
Container

Resource

streams and
continuations


What to expect
• Monads provide a practical abstraction for
everyday sorts of control ﬂow and data
structure in modern web app
• If objects were supposed to be the
packaging of control and structure that
provided optimal reuse, then monads seem
to be a better object than object


Who is the audience
• People who write code that serves a
purpose
• Monads will feel natural to those with 2yrs
functional programming
• Or, people who have 2yrs with distributed
and network programming
• Nota bene: all code examples in Scala!

Who is the audience

• But, really, monads are everywhere!


Why monads

• To manage complexity
• Monads help your code scale
• Abstract structure and control
• Engage the compiler more
• Compositional

The monadic toolbox
• What is a monad?
• Shape, wrap and roll, baby!
• Monads in a larger context
• Comprehensive comprehensions!


Shapely monads

• Three key ingredients in a monad
• Shape
• wrap
• roll


Embracing shape
“monads”
Shape
“eh?”

roll
“eh?”

wrap
“monads,”
“eh?”
“eh?”


Embracing shape

• Monads may be canonically thought of as a
way to structure an interface to “colored”
braces

• Shape -- the color of the braces [ ]

• wrap -- putting things between braces [ “eh?” ]

• roll -- ﬂattening nested braces (i.e.
rolling them up) [ [ “monads” ] [ “eh?” ] ]
[ “monads” , “eh?” ]

Embracing shape

Shape : <red> ... </red>
wrap : queen => <red>queen</red>
roll : <red><red>knight</red><red>queen</red></red>
=>
<red>knight queen</red>


Data structures as “colored” braces

Shape : <list> ... </list>
wrap : queen => <list>queen</list>
roll : <list><list>knight</list><list>queen</list></list>
=>
<list>knight queen</list>


Data structures as “colored” braces

Shape : <set> ... </set>
wrap : queen => <set>queen</set>
roll : <set><set>knight</set><set>queen</set></set>
=>
<set>knight queen</set>


Monads refactor container code

• List and Set are actually the same
structure up to a certain point -- the
difference can be captured by a set of
equations that Sets obey and Lists don’t

• a + b = b + a

• a + a = a

• How do we relate these equations to
Shape, wrap and roll?


a + b could just as easily be written +( a, b )

which could just as easily be written <+> a b </+>

which could just as easily be written
roll( <+><+>a</+><+>b</+></+> )

which could just as easily be written
roll( <+> wrap( a ) wrap( b ) </+> )

or
roll( wrap( a ) + wrap( b ) )



wrap( a ) + wrap( b )
in the case of both List and Set
as long as we have safely ‘wrapped’ everything, then ‘+’ is the natural
polymorphic interpretation in the ‘container’
Set1 + Set2 := Set1 U Set2
List1 + List2 := List1 append List2



Set1 U Set2 == Set2 U Set1 (not true of Lists)
Set1 U Set1 == Set1 (not true of Lists)



wrap( a ) + wrap( b )
<set><set>a</set> <set>b</set></set>
if this is really Set then this has to be the same as
<set><set>b</set> <set>a</set></set>
aka
wrap( b ) + wrap( a )


• So List and Set can actually be thought of as factored into
two pieces of information

• ( M, equations )
where

• M is the colored brace structure (Shape-wrap-n-roll)
• the equations give additional constraints about how
braces behave.

• Lists have no additional equations -- such structures are
called free.



Every monad can be seen as a version of a data structure
that has a ‘+’ and a unit for the ‘+’, 1.
We’ve seen how the brace metaphor gives a simple model of
how the extra structure of wrap and roll allow to extend
‘+’ over entities we put inside the monadic container
roll( wrap( a ) + wrap( b ) )
It also relates the unit, 1, of the ‘+’ to wrap (which is often
called the unit of the monad)



The unit of the ‘+’ has a canonical representation as the empty braces:
<red> </red>
roll( <red><red> </red> <red> queen </red></red> )
=
<red> queen </red>
=
roll( <red><red> queen </red><red> </red></red> )


A simple example
class MList[A]( elems : A* )

extends List[A]( elems ) {

def wrap ( a : A ) : MList[A] = {

new MList( a )

}

def roll ( mma : MList[MList[A]] ) : MList[A] = {

( Nil /: mma )( ( acc, ma ) => acc.append( ma ) )

}

}


A simple example (cont)

• What can we say about this code?
• What about legacy uses of List?
• What about future uses of List?


A simple example

trait Option[+A]

case class Some[+A]( a : A ) extends Option[A]

case object None extends Option[Nothing]

for( Some( a ) <- optA ) yield { f( a ) }


Another simple example

trait LeftRightTree[A]
case class Leaf[A]( a : A )
extends LeftRightTree[A]
case class Branch[A](
left : List[LeftRightTree[A]],
right : List[LeftRightTree[A]]
) extends LeftRightTree[A]


Another simple example (cont)

• We have a choice when we implement unit
def wrap( a : A ) : LeftRightTree[A] = {
Branch( List( a ), Nil )
}

or
def wrap( a : A ) : LeftRightTree[A] = {
Branch( Nil, List( a ) )
}

The Monadic API as a
View if you google for Shape, wrap
and roll, you’ll only ﬁnd my
presentations; if you use unit/
yield/return and mult/bind,
you’ll ﬁnd lots of hits
shape

trait Monad[M[_]] { wrap

def unit [A] ( a : A ) : M[A]
def bind [A,B] ( ma : M[A], fn : A => M[B] ) : M[B]
}
jelly roll

Lucius Gregory Meredith, Managing Partner, Biosimiliarity LLC


A word about
presentations
def bind [A,B] ( ma : M[A], f : A => M[B] ) : M[B] = {

mult( fmap( f )( ma ) )

}
def fmap [A,B] ( f : A => B ) : M[A] => M[B]

def mult [A] ( mma : M[M[A]] ) : M[A] = {

bind[M[A],A]( mma, ( ma ) => ma ) i confess! this is all a plot to
expose you subliminally to
} Category Theory



View
class ListM[A] extends Monad[List] {
override def unit [S] ( s : S ) : List[S] = { List[S]( s ) }
override def bind [S,T] ( ls : List[S], fn : S => List[T] ) = {
( ( Nil : List[T] ) /: ls )( { ( acc, e ) => { acc ++ fn( e ) } } )
}
}



Two kinds of collections

• Extensional
• We can explicitly give each element
of the collection
• Intensional
• We must specify the elements
programmatically -- such as by a rule


Two kinds of collections
• Extensional -- examples

• { a, b, c }

• ( 1, 2, 3 )

• f(1) + g(2) + h(3)

• Intensional

• { a in Nat | a % 2 == 0 }

• fibs = 0 :: 1 :: zipWith (+) fibs (tail fibs)

• ∑ fi ( i )


Situations and intensions

• Inﬁnite collections -- ﬁbs, primes, etc, and
also languages!

• Compressed collections -- individual
elements are more easily computed than
stored or otherwise expressed

• Collections calculated from data from
the environment -- such as user input


Intensionally lazy

• Inﬁnite collections require some form of laziness or
“just-in-time-ness” to avoid consuming all available
memory

• Compressed collections can make use of laziness to
avoid consuming compute cycles until it’s necessary

• Lazy also blurs the boundary between computing
something just in time and waiting on i/o


In the stream...

• More importantly the web 2.0 is teaching us that what
initially looks like a small thing
type TwitterMsg = Char [140] // not legal Scala

becomes pretty valuable when it is iterated into a
stream
abstract class TwitterStream

case class Twitterful( msg : TwitterMsg, strm : TwitterStream )
extends TwitterStream


Intensions and comprehensions
{ pattern | generator1, ..., generatorM, condition1, ..., conditionN }

≈
for( generator1; ...; generatorM; condition1;...; conditionN )
yield { pattern }

≈
SELECT pattern FROM generator1, ..., generatorN
WHERE condition1, ..., conditionN


Consider Set[A]
{ a, b, c, ... }
the language of extensional collections comes from the
constructors of the type, A, we said we’d put in the collection,
together with the Set operators
{ pattern | generator1, ..., generatorM, condition1, ..., conditionN }
where does the language of patterns and conditions come
from?



If monads are like containers, where’s the “take something out
of the container interface”?


View
shape

trait Monad[M[_]] { wrap

def unit [A] ( a : A ) : M[A]
def bind [A,B] ( ma : M[A], fn : A => M[B] ) : M[B]
}
jelly roll



Intensions and the Ruby Slippers

Q: When can you check an element out of a monad?
A: With map you can check out (anything) any time you want,
but you can never leave!
f : A => B
M[f] : M[A] => M[B]


Intensions and the Ruby Slippers
def transformResults(
normalizeResults : A => B
) : Option[B] = {
val intermediateRslts : Option[A] = getRawResults( ... )
for( rslt <- intermediateRslts ) yield {
normalizeResults( rslt )
}
}



for( ptn <- src ; if condition ) yield { e( ptn ) }

≈
src.map(

p => p match {

case ptn => if condition { e( ptn ) }

case _ => throw new Exception( “” )

}

)





≈
src.map(

p => p match {



}

)





≈ { pattern | generator1, ..., generatorM, condition1, ..., conditionN }

src.map(

p => p match {



}

)


Comprehension syntax
for( x <- expr1 ; y <- expr2 ; <stmts> ) yield expr3 =
expr1 flatMap(
x => for( y <- expr2; <stmts> ) yield expr3
)

for( x<-expr1 ; y = expr2 ;<stmts> ) yield expr3 =
for(
( x, y ) <-
for( x <- expr1 ) yield ( x , expr2 ); <stmts>
) yield expr3

for( x <- expr1 if pred ) yield expr2 =
expr1 filter(x => pred) map (x => expr2 )


Three kinds of monad?

• Stuff goes in but never comes out -- the IO-monad

• Stuff goes in and sometimes comes out -- nearly all standard
collections

• Every thing that goes in is matched by something coming out --
linearly typed collections

Notice that these correspond to transactional modes!


Towards a more complex
example

• Parsing
• Moral: the dog doesn’t bark
• Evaluation
• Moral: the dog barks quietly -- even in a
distributed setting
• Storage?

REPLs and Web Apps and DSLs,
oh my!

Service

Domain Logic

Browser

Container

Resource


oh my!

Service

Domain Logic

Browser

Container

Resource
Non-blocking
continuation-based
request handling


oh my!
User actions become
expressions
in a DSL

Service

Domain Logic

Browser

Container

Resource
Non-blocking
continuation-based
request handling


oh my!
Domain model =
User actions become abstract syntax of
expressions DSL
in a DSL

Service

Domain Logic

Browser

Container

Resource
Non-blocking
continuation-based
request handling


oh my!
Domain model =
User actions become abstract syntax of
expressions DSL
in a DSL

Service

Domain Logic

Browser
Evaluation as a
service
Container

Resource
Non-blocking
continuation-based
request handling


Parsing - compositional
at multiple levels
Syntax tree
Expressions
in a DSL Parser


Parsing
Concrete syntax Abstract syntax

Normalization


Parsing
Abstract syntax Domain value

Evaluation


Parsing (DSLs)

• The surface API for parsing doesn’t (have
to) change
• Grammars are already compositional:
grammars are algebras!
• The programmatic API changes: parser
combinators


Parsing (DSLs)

• Internal vs External
• Parsing combinators vs higher level
abstraction
• Domain requirements


Internal vs External
DSLs
• External

• independent, self-contained language with own
grammar and semantics

• Example: SQL is a DSL for accessing relational
storage


Internal vs External
DSLs
• Internal

• embedded or hosted DSL

• makes use of expressions and computations in
ambient language

• examples

• lex/yacc -- semantic actions in C

• LINQ -- internal DSL for accessing storage


Parsing combinators vs higher level
abstraction

• Parsing combinators
• provide compositional mechanism
• work well in embedded situation
• BNF and other higher level abstractions
• considerably more concise
• target multiple platforms

Example
class TermParser extends JavaTokenParsers {

def term : Parser[Any] =

application | list | ground | variable

def list : Parser[Any] =

"["~repsep( term, "," )~"]"

def ground : Parser[Any] =

stringLiteral | floatingPointNumber | "true" | "false"

def variable : Parser[Any] = ident

def application : Parser[Any] =

ident~"("~repsep( term, "," )~")"

}


Domain requirements
• When DSL supports service-based/
program-to-program computing
• language is “closed” -- no expression
sublanguage
• When DSL supports human-interaction
• language is often “open” -- containing
some expression sublanguage


DSLs and algebras

• An algebra is a domain model (and don’t
you forget it!)
• An algebra is the abstract syntax to the ...
• ... concrete syntax of a DSL


Algebras

• Algebras and types are closely related

• Functional types are algebraic data types

• Understanding how these are related and
languages for manipulating them will get you
far

• Most importantly -- algebras are monads!



• List and Set are actually the same
structure up to a certain point -- the
difference can be captured by a set of
equations that Sets obey and Lists don’t

• a + b = b + a

•
Remember this?
a + a = a

• How do we relate these equations to
Shape, wrap and roll?


• So List and Set can actually be thought of as factored into
two pieces of information

• ( M, equations ) And this?
where

• M is the colored brace structure (Shape-wrap-n-roll)
• the equations give additional constraints about how
braces behave.

• Lists have no additional equations -- such structures are
called free.


Monads and algebras
• An algebra is expressed entirely as a set of
generators

• ≈ constructors
• ≈ grammar
• and relations aka equations
• Generators and relations are 2/3’s of the
speciﬁcation of a DSL


Where we might go
from here

• We haven’t talked about control ﬂow
• So, let’s talk a little about it...


Why monads -- did we do what
we promised?

• To manage complexity
• Monads help your code scale
• Abstract structure and control
• Engage the compiler more
• Compositional Actually... monads
don’t necessarily
compose!


Questions?


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