Functional Algebra: Monoids Applied

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In functional programming, words from Category Theory are thrown around, but how useful are they really?
This session looks at applications of monoids specifically and how using their algebraic properties offers a solid foundation of reasoning in many types of business domains and reduces developer error as computational context complexity increases.
This will provide a tiny peak at Category Theory's practical uses in software development and modeling. Code examples will be in Haskell and Scala, but monoids could be constructed in almost any language by software craftsmen and women utilizing higher orders of reasoning to their code.

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Functional Algebra: Monoids Applied

  1. 1. Functional Algebra Monoids Applied Susan Potter Nov 10, 2012
  2. 2. OMG, Algebra? FML.. . .
  3. 3. OMG, Algebra? FML.. . . Figure: Chill! Algebra is just a domain specific language
  4. 4. % whoami Figure: From OO patterns to FP abstractions
  5. 5. Monoids: What are they? An abstraction (laws) not a design pattern (forces, context, . . . ) Algebraic structure . . . over a set with a binary operator and an identity element Accumulator . . . Its sole purpose Special case category . . . with only one object
  6. 6. Monoids: What are they? An abstraction (laws) not a design pattern (forces, context, . . . ) Algebraic structure . . . over a set with a binary operator and an identity element Accumulator . . . Its sole purpose Special case category . . . with only one object
  7. 7. Monoids: What are they? An abstraction (laws) not a design pattern (forces, context, . . . ) Algebraic structure . . . over a set with a binary operator and an identity element Accumulator . . . Its sole purpose Special case category . . . with only one object
  8. 8. Monoids: What are they? An abstraction (laws) not a design pattern (forces, context, . . . ) Algebraic structure . . . over a set with a binary operator and an identity element Accumulator . . . Its sole purpose Special case category . . . with only one object
  9. 9. Monoids: Typeclasses Listing 1: Haskell Monoid Typeclass Definition1 -- | In Haskell Prelude Data. Monoid2 class Monoid a where3 mempty :: a -- identity4 mappend :: a -> a -> a -- binary op5 mconcat :: [a] -> a -- helper Listing 2: Scalaz Monoid Trait Definition1 // Scalaz 7’s Monoid typeclass definition , kinda2 trait Monoid [A] extends Semigroup [A] { self =>3 def zero: A /* identity */4 def append(x: A, y: => A): A /* binary op */5 }6 // from SemigroupOps [A] ...7 final def |+|(other: => A): A = A.append(self, other)
  10. 10. Monoids: Typeclasses Listing 3: Haskell Monoid Typeclass Definition1 -- | In Haskell Prelude Data. Monoid2 class Monoid a where3 mempty :: a -- identity4 mappend :: a -> a -> a -- binary op5 mconcat :: [a] -> a -- helper Listing 4: Scalaz Monoid Trait Definition1 // Scalaz 7’s Monoid typeclass definition , kinda2 trait Monoid [A] extends Semigroup [A] { self =>3 def zero: A /* identity */4 def append(x: A, y: => A): A /* binary op */5 }6 // from SemigroupOps [A] ...7 final def |+|(other: => A): A = A.append(self, other)
  11. 11. Monoids: Typeclasses Listing 5: Haskell Monoid Typeclass Definition1 -- | In Haskell Prelude Data. Monoid2 class Monoid a where3 mempty :: a -- identity4 mappend :: a -> a -> a -- binary op5 mconcat :: [a] -> a -- helper Listing 6: Scalaz Monoid Trait Definition1 // Scalaz 7’s Monoid typeclass definition , kinda2 trait Monoid [A] extends Semigroup [A] { self =>3 def zero: A /* identity */4 def append(x: A, y: => A): A /* binary op */5 }6 // from SemigroupOps [A] ...7 final def |+|(other: => A): A = A.append(self, other)
  12. 12. Monoids: Typeclasses Listing 7: Haskell Monoid Typeclass Definition1 -- | In Haskell Prelude Data. Monoid2 class Monoid a where3 mempty :: a -- identity4 mappend :: a -> a -> a -- binary op5 mconcat :: [a] -> a -- helper Listing 8: Scalaz Monoid Trait Definition1 // Scalaz 7’s Monoid typeclass definition , kinda2 trait Monoid [A] extends Semigroup [A] { self =>3 def zero: A /* identity */4 def append(x: A, y: => A): A /* binary op */5 }6 // from SemigroupOps [A] ...7 final def |+|(other: => A): A = A.append(self, other)
  13. 13. Monoids: Laws Closure: ∀a, b ∈ S : ab ∈ S for all a and b in set S, the result of a and b given to the binary operator is also in set S. Associativity: ∀a, b, c ∈ S : (ab)c = a(bc) for all a, b, and c in set S, either binary operator can be evaluated first to produce same result. Identity: ∃e ∈ S : ∀a ∈ S : ea = a = ae there exists an e in set S such that for all a in set S ea evaluates to a and is equal to ae
  14. 14. Monoids: Laws Closure: ∀a, b ∈ S : ab ∈ S for all a and b in set S, the result of a and b given to the binary operator is also in set S. Associativity: ∀a, b, c ∈ S : (ab)c = a(bc) for all a, b, and c in set S, either binary operator can be evaluated first to produce same result. Identity: ∃e ∈ S : ∀a ∈ S : ea = a = ae there exists an e in set S such that for all a in set S ea evaluates to a and is equal to ae
  15. 15. Monoids: Laws Closure: ∀a, b ∈ S : ab ∈ S for all a and b in set S, the result of a and b given to the binary operator is also in set S. Associativity: ∀a, b, c ∈ S : (ab)c = a(bc) for all a, b, and c in set S, either binary operator can be evaluated first to produce same result. Identity: ∃e ∈ S : ∀a ∈ S : ea = a = ae there exists an e in set S such that for all a in set S ea evaluates to a and is equal to ae
  16. 16. Monoids: Properties (Haskell) 1 -- property based tests for monoid "laws" 2 -- does not compile yet; must specify type a 3 module Tests where 4 import Test. QuickCheck ( quickCheck ) 5 6 -- closure law verified by type system 7 8 propMonoidAssoc :: Monoid a => a -> a -> a -> Bool 9 propMonoidAssoc x y z =10 mappend ( mappend x y) z == mappend x ( mappend y z)1112 propMonoidIdent :: Monoid a => a -> Bool13 propMonoidIdent x =14 mappend mempty x == x && mappend x mempty == x
  17. 17. Monoids: Simple "Natural" Examples Listing 9: Haskell1 -- OP ID List2 foldr (+) 0 [1 ,2 ,3 ,4] -- 103 foldr (*) 1 [1 ,2 ,3 ,4] -- 244 foldr (++) [] [[1] , [1 ,2]] -- [1 ,1 ,2]5 foldr (&&) True [True , False , True] -- False6 foldr (||) False [True , False , True] -- True7
  18. 18. Monoids: Simple "Natural" Examples Listing 11: Haskell1 -- OP ID List2 foldr (+) 0 [1 ,2 ,3 ,4] -- 103 foldr (*) 1 [1 ,2 ,3 ,4] -- 244 foldr (++) [] [[1] , [1 ,2]] -- [1 ,1 ,2]5 foldr (&&) True [True , False , True] -- False6 foldr (||) False [True , False , True] -- True7
  19. 19. Monoids: Simple "Natural" Examples Listing 13: Haskell1 -- OP ID List2 foldr (+) 0 [1 ,2 ,3 ,4] -- 103 foldr (*) 1 [1 ,2 ,3 ,4] -- 244 foldr (++) [] [[1] , [1 ,2]] -- [1 ,1 ,2]5 foldr (&&) True [True , False , True] -- False6 foldr (||) False [True , False , True] -- True7
  20. 20. Monoids: Simple "Natural" Examples Listing 15: Haskell1 -- OP ID List2 foldr (+) 0 [1 ,2 ,3 ,4] -- 103 foldr (*) 1 [1 ,2 ,3 ,4] -- 244 foldr (++) [] [[1] , [1 ,2]] -- [1 ,1 ,2]5 foldr (&&) True [True , False , True] -- False6 foldr (||) False [True , False , True] -- True7 mconcat = foldr mappend mempty
  21. 21. Monoids: Simple "Natural" Examples Listing 17: Haskell1 -- OP ID List2 foldr (+) 0 [1 ,2 ,3 ,4] -- 103 foldr (*) 1 [1 ,2 ,3 ,4] -- 244 foldr (++) [] [[1] , [1 ,2]] -- [1 ,1 ,2]5 foldr (&&) True [True , False , True] -- False6 foldr (||) False [True , False , True] -- True7 mconcat = foldr mappend mempty Listing 18: Same in Scala: WTF?1 List (1 ,2 ,3 ,4). foldRight (0)(_+_) // 102 List (1 ,2 ,3 ,4). foldRight (1)(_*_) // 243 List(List (1), List (1 ,2)). foldRight (List[Int ]())( _++_)4 List(true ,false ,true ). foldRight (true )(_&&_)5 List(true ,false ,true ). foldRight ( false )(_||_)
  22. 22. Monoids: Define Your Own (Haskell) Listing 19: Haskell Monoid Definition 1 import Data. Monoid 2 3 data Asset = Cash Int 4 | Receivables Int ... 5 data Liability = NotesPayable Int 6 | AccountsPayable Int ... 7 -- naive , but illustrative 8 data BalSheet = BalSheet [ Asset ] [ Liability ] 910 instance Monoid BalSheet where11 mempty :: m12 mempty = BalSheet [] []13 mappend :: m -> m -> m14 mappend ( BalSheet a1 l1) ( BalSheet a2 l2) =15 BalSheet ( mappend a1 a2) ( mappend l1 l2)
  23. 23. Monoids: Define Your Own (Scala) Listing 20: Scalaz Monoid Definition 1 import scalaz ._; import Scalaz ._; 2 3 // naive , but illustrative 4 case class Portfolio ( positions : Seq[ Position ]) 5 object Portfolio { 6 implicit val portfolioMonoid = 7 new Monoid [ Portfolio ] { 8 def append (p1: Portfolio , p2: Portfolio ) = 9 Portfolio ( append (p1.positions , p2. positions ))10 def zero = Portfolio (Seq. empty )11 }12 }
  24. 24. Monoids: So what? Properties "Interface" Once you understand one monoid, you understand them all; simpler layers => simpler tests Type Safe & Type Expressive Can mappend A s but not a A and a B where A ! = B and myCalc :: Monoid a => a -> b Generic Functions e.g. consolidate = foldr mappend mempty Highly Applicable Look around your domain. Do you see Monoids Everywhere™ yet?
  25. 25. Monoids: So what? Properties "Interface" Once you understand one monoid, you understand them all; simpler layers => simpler tests Type Safe & Type Expressive Can mappend A s but not a A and a B where A ! = B and myCalc :: Monoid a => a -> b Generic Functions e.g. consolidate = foldr mappend mempty Highly Applicable Look around your domain. Do you see Monoids Everywhere™ yet?
  26. 26. Monoids: So what? Properties "Interface" Once you understand one monoid, you understand them all; simpler layers => simpler tests Type Safe & Type Expressive Can mappend A s but not a A and a B where A ! = B and myCalc :: Monoid a => a -> b Generic Functions e.g. consolidate = foldr mappend mempty Highly Applicable Look around your domain. Do you see Monoids Everywhere™ yet?
  27. 27. Monoids: So what? Properties "Interface" Once you understand one monoid, you understand them all; simpler layers => simpler tests Type Safe & Type Expressive Can mappend A s but not a A and a B where A ! = B and myCalc :: Monoid a => a -> b Generic Functions e.g. consolidate = foldr mappend mempty Highly Applicable Look around your domain. Do you see Monoids Everywhere™ yet?
  28. 28. Monoids: But . . . Types With Multiple Monoids More boilerplate though usually manageable. e.g. Listing 21: Haskell Monoid Typeclass Definition 1 import Data. Monoid 2 toSums = map Sum 3 mconcat $ toSums [1 ,2 ,3 ,4] -- 10 4 5 toAlls = map All 6 getAll $ mconcat $ toAlls [True , False , True] Think! Does it make sense to declare Vector as a Monoid in Haskell?
  29. 29. Monoids: But . . . Types With Multiple Monoids More boilerplate though usually manageable. e.g. Listing 22: Haskell Monoid Typeclass Definition 1 import Data. Monoid 2 toSums = map Sum 3 mconcat $ toSums [1 ,2 ,3 ,4] -- 10 4 5 toAlls = map All 6 getAll $ mconcat $ toAlls [True , False , True] Think! Does it make sense to declare Vector as a Monoid in Haskell?
  30. 30. Automatic Optimal Pipelininghttp://informatikr.com/2012/redis-pipelining.html Listing 23: Pipelining in Jedis 1 jedis . pipelined (new PipelineBlock () { 2 public void execute () { 3 incr(" hit_count "); 4 get(" mbbx6spp : repos_count "); 5 } 6 });
  31. 31. Automatic Optimal Pipelininghttp://informatikr.com/2012/redis-pipelining.html Listing 25: Pipelining in Jedis 1 jedis . pipelined (new PipelineBlock () { 2 public void execute () { 3 incr(" hit_count "); 4 get(" mbbx6spp : repos_count "); 5 } 6 }); Listing 26: Automatic Pipelining in Hedis 1 runRedis conn $ do 2 hits <- incr " hit_count " 3 repos <- get " mbbx6spp : repos_count " 4 liftIO $ print (hits , repos )
  32. 32. Monoids: Other Fun Examples Log Priorities / Filters in bittorrent http://jlouisramblings.blogspot.com/2010/02/how-logging-is-performed-in-haskell.html Associative Alpha Blending http://lukepalmer.wordpress.com/2010/02/05/associative-alpha-blending/ Writer Monad Accumulator factorial :: Integer -> Writer (Sum Integer) Integer Tree in Data.Git module of hit package
  33. 33. Monoids: Other Fun Examples Log Priorities / Filters in bittorrent http://jlouisramblings.blogspot.com/2010/02/how-logging-is-performed-in-haskell.html Associative Alpha Blending http://lukepalmer.wordpress.com/2010/02/05/associative-alpha-blending/ Writer Monad Accumulator factorial :: Integer -> Writer (Sum Integer) Integer Tree in Data.Git module of hit package
  34. 34. Monoids: Other Fun Examples Log Priorities / Filters in bittorrent http://jlouisramblings.blogspot.com/2010/02/how-logging-is-performed-in-haskell.html Associative Alpha Blending http://lukepalmer.wordpress.com/2010/02/05/associative-alpha-blending/ Writer Monad Accumulator factorial :: Integer -> Writer (Sum Integer) Integer Tree in Data.Git module of hit package
  35. 35. Monoids: Other Fun Examples Log Priorities / Filters in bittorrent http://jlouisramblings.blogspot.com/2010/02/how-logging-is-performed-in-haskell.html Associative Alpha Blending http://lukepalmer.wordpress.com/2010/02/05/associative-alpha-blending/ Writer Monad Accumulator factorial :: Integer -> Writer (Sum Integer) Integer Tree in Data.Git module of hit package
  36. 36. Monoids: Relationships Monoid v => Map k v also monoid All monoids are semigroups Semigroup is monoid minus identity requirement All groups are monoids Monoid is group minus inverse unary operator requirement Free Structures Get (money for nothing? and) monoids for free
  37. 37. Monoids: Relationships Monoid v => Map k v also monoid All monoids are semigroups Semigroup is monoid minus identity requirement All groups are monoids Monoid is group minus inverse unary operator requirement Free Structures Get (money for nothing? and) monoids for free
  38. 38. Monoids: Relationships Monoid v => Map k v also monoid All monoids are semigroups Semigroup is monoid minus identity requirement All groups are monoids Monoid is group minus inverse unary operator requirement Free Structures Get (money for nothing? and) monoids for free
  39. 39. Monoids: Relationships Monoid v => Map k v also monoid All monoids are semigroups Semigroup is monoid minus identity requirement All groups are monoids Monoid is group minus inverse unary operator requirement Free Structures Get (money for nothing? and) monoids for free
  40. 40. Monoids: RelationshipsMonads & Monoids .
  41. 41. Monoids: RelationshipsMonads & MonoidsA monad over X is the .
  42. 42. Monoids: RelationshipsMonads & MonoidsA monad over X is themonoid in category of endofunctors of X .
  43. 43. Monoids: RelationshipsMonads & MonoidsA monad over X is themonoid in category of endofunctors of Xwith binary operator as composition(of endofunctors) .
  44. 44. Monoids: RelationshipsMonads & MonoidsA monad over X is themonoid in category of endofunctors of Xwith binary operator as composition(of endofunctors)and identity being the identity endofunctor.
  45. 45. Homework What is an Endofunctor?
  46. 46. Questions? Figure: http://www.flickr.com/photos/42682395@N04/ @FunAlgebra @SusanPotter
  47. 47. Bonus: References / Resources Channel 9 Lectures (Erik Meijer) http://channel9.msdn.com/Shows/Going+Deep/ Lecture-Series-Erik-Meijer-Functional-Programming-Fundamentals-Chapter-1 The Catsters http://www.youtube.com/thecatsters Haskell Reddit http://www.reddit.com/r/haskell/ Haskell Cafe http://www.haskell.org/mailman/listinfo/haskell-cafe Scalaz Mailing List https://groups.google.com/forum/?fromgroups#!forum/scalaz

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