Why is a Monad Like a Writing Desk?

@carinmeier Monday, March 12, 12

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Monad

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Monad

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“Shall I be pure or impure?” - Wadler

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Pure Lazy Evaluation Equational Reasoning Shall I be pure or impure?

Impure State Error Handling Input/ Output

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Pure Haskell

Impure Clojure Scheme Monday, March 12, 12

“Is there a way to combine the indulgences of impurity and the blessings of purity?” - Wadler

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Engenio Moggi

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Engenio Moggi Proposed way of structuring semantics to describe state, exceptions, etc in programming

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Engenio Moggi Proposed way of structuring semantics to describe state, exceptions, etc in programming Category Theory: Monads

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Monad (Category Theory) Also known as a Kleisi triple or triple Is an endofunctor, together with two natural transformations Used in theory of pairs of adjoint functions

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You Are Late

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Monad Symposium in 15 min

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TextA monad is just a

monoid in the category of endofunctors, What's the problem?

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A monad is just a Purity has its regrets. monoid in the category of endofunctors, what's the problem?

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Is it true that this could be a monadic return operation? Is

(defn return [v] (fn [] v))

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Is it true that this could be a monadic return operation? Is

(defn return [v] (fn [] v)) (return “jelly”)Text ;=> #

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Is it true that this could be a monadic return operation? Is

(defn return [v] (fn [] v)) (return “jelly”)Text ;=> # ((return “jelly”)) ;=> “jelly”

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Yes, because the return object is a function Is (container/ monadic value) that when executed is “jelly” Text (defn return [v] (fn [] v)) ((return “jelly”)) ;=> “jelly”

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Is it true that this could be a monadic return operation? Is

(defn return [v] v)

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Is it true that this could be a monadic return operation? Is

(defn return [v] v) (return “jelly”) ;=> jelly

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No, because the return object “jelly” is not a function Is (container/ monadic value) (defn return [v] v) (return “jelly”) ;=> jelly

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Is it true that this could be a monadic bind operation?

Is (defn bind [mv f] (f (mv)))

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Is it true that this could be a monadic bind operation?

Is (defn bind [mv f] (f (mv)))

(defn with-toast [s] (return (str "toast & " s)))

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Is it true that this could be a monadic bind operation?

Is (defn bind [mv f] (f (mv)))

(defn with-toast [s] (return (str "toast & " s))) (bind (return "jelly") with-toast) ;=> container/ monadic value

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Is it true that this could be a monadic bind operation?

Is (defn bind [mv f] (f (mv)))

(defn with-toast [s] (return (str "toast & " s))) (bind (return "jelly") with-toast) ;=> container/ monadic value ((bind (return "jelly") with-toast)) ;=> “toast & jelly”

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Yes, because it takes a monadic value, and applies a function and returns a monadic value

Is (defn bind [mv f] (f (mv)))

(defn with-toast [s] (return (str "toast & " s))) (bind (return "jelly") with-toast) ;=> container/ monadic value ((bind (return "jelly") with-toast)) ;=> “toast & jelly”

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Can you write a monadic return function for yourself? Is

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Can you write a monadic return function for yourself? Is (return "me") ;=> monadic value ((return "me")) ;=> “me”

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(defn return [v] (fn [] v))

Can you write a function that makes you grow and returns Is a monadic value?

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Can you write a function that makes you grow and returns Is a monadic value? (defn grow [s] (return (str s (last s)))) (grow "me") ((grow "me"))

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;=> monadic value ;=> “mee"

Now, can you use a monadic bind operation to grow? Is

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Now, can you use a monadic bind operation to grow? Is (defn bind [mv f] (f (mv))) (defn m-grow [mv] (bind mv grow)) (m-grow (return "me")) ;=> m. val ((m-grow (return "me"))) ;=> “mee”

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Grow Bigger! Is

((m-grow (m-grow (m-grow (return "me"))))) ;=> “meeee”

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Grow Bigger! Is

((m-grow (m-grow (m-grow (return "me"))))) ;=> “meeee” ((-> (return "me") m-grow m-grow m-grow)) ;=> “meeee”

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Where you are Destination Directions

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(defn directions [start] (return (.concat start (if (> 0.5 (rand 1)) ": right" ": left"))))

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(defn directions [start] (return (.concat start (if (> 0.5 (rand 1)) ": right" ": left")))) (defn m-directions [mv] (bind mv directions))

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(defn directions [start] (return (.concat start (if (> 0.5 (rand 1)) ": right" ": left")))) (defn m-directions [mv] (bind mv directions)) ((-> (return “here”) m-directions m-directions)) ;=> "here: right: left"

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(defn directions [start] (return (.concat start (if (> 0.5 (rand 1)) ": right" ": left")))) (defn m-directions [mv] (bind mv directions)) ((-> (return “here”) m-directions m-directions)) ;=> "here: right: left" ((-> (return nil) m-directions m-directions)) ;=> NullPointerException!

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(defn bind [mv f] (f (mv)) (defn bind [mv f] (let [v (mv)] (if (nil? v) (return nil) (f v))))

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(defn bind [mv f] (f (mv)) (defn bind [mv f] (let [v (mv)] (if (nil? v) (return nil) (f v)))) ((-> (return nil) m-directions m-directions)) ;=> nil

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That way ->

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(

That way ->

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)

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(defn return [v] (fn [] v)) (defn bind [mv f] (f (mv))) (defn m-tea [mv name] (bind mv (fn [v] (return (str v " and " name)))))

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(defn return [v] (fn [] v)) (defn bind [mv f] (f (mv))) (defn m-tea [mv name] (bind mv (fn [v] (return (str v " and " name))))) ((-> (return "me") (m-tea "you"))) ;=> "me and you"

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(defn return [v] (fn [s] [v s]))

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(defn return [v] (fn [s] [v s])) (defn bind [mv f] (fn [s] (let [[v sn] (mv s)] ((f v) sn))))

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(defn return [v] (fn [s] [v s])) (defn bind [mv f] (fn [s] (let [[v sn] (mv s)] ((f v) sn)))) (defn m-tea [mv name] (bind mv (fn [v] (return (str v " and " name))))) ((-> (return "me") (m-tea "you")) 10) ;=> ["me and you" 10]

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(defn take-sugar [mv] (bind mv (fn [v] (fn [s] [v (dec s)]))))

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(defn take-sugar [mv] (bind mv (fn [v] (fn [s] [v (dec s)])))) ((-> (return "me") (take-sugar) (m-tea "you")) 10) ;=> ["me and you" 9] ((-> (return "me") (take-sugar) (take-sugar) (m-tea "you")) 10) ;=> ["me and you" 8]

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Identity Monad Is

(defn return [v] (fn[] v)) (defn bind [mv f] (f (mv)))

(defn m-grow [w] (bind w grow)) ((-> (return "me") m-grow m-grow m-grow)) ;=> “meeee”

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Maybe Monad (defn return [v](fn[] v))

Is (defn bind [mv f] (let [v (mv)] (if (nil? v) (return nil) (f v)))) ((-> (return nil) m-directions m-directions)) ;=> nil

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State Monad (defn return [v] (fn [s] [v s]))

Is

(defn bind [mv f] (fn [s] (let [[v sn] (mv s)] ((f v) sn)))) (defn take-sugar [mv] (bind mv (fn [v] (fn [s] [v (dec s)])))) ((-> (return "me")(take-sugar) (m-tea "you")) 10) ;=> ["me and you" 9]

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Three Monad Laws 1) Left Unit - "return" acts as a neutral element of bind (bind (return v) f) ≡ (f v)

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Three Monad Laws 1) Left Unit - "return" acts as a neutral element of bind (bind (return v) f) ≡ (f v) (defn return [v] (fn [] v)) (defn bind [mv f] (f (mv))) (defn grow [s](return (str s (str (last s))))) ((bind (return "me") grow))

;=> “mee”

((grow "me"))

;=> “mee”

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Three Monad Laws 2) Right Unit - "return" acts as a neutral element of bind (bind mv return) ≡ mv

((bind (return "me") return)) ;=> "me" ((return "me"))

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;=> "me"

Three Monad Laws 2) Right Unit - "return" acts as a neutral element of bind (bind mv return) ≡ mv (defn return [v] (fn [] v)) (defn bind [mv f] (f (mv)) (defn grow [s](return (str s (str (last s))))) ((bind (return "me") return)) ;=> "me" ((return "me"))

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;=> "me"

Three Monad Laws 3) Associative - Binding two functions in succession is the same as binding one function that can be determined from them (bind (bind mv f) g) ≡ (bind mv (fn [x] (bind (f x) g))

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Three Monad Laws 3) Associative - Binding two functions in succession is the same as binding one function that can be determined from them (bind (bind mv f) g) ≡ (bind mv (fn [x] (bind (f x) g)) (defn return [v] (fn [] v)) (defn bind [mv f] (f (mv)) (defn grow [s](return (str s (str (last s))))) ((bind (bind (return "me") grow) grow)) ;=> "meee" ((bind (return "me") (fn [v] (bind (grow v) grow))))

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;=> "meee"

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Summary Many More Types of Monads

• IO • Collections • Continuation • Writer Etc ...

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Summary Have your cake and eat it too!

Inside functions pure and messy impure stuff on the outside

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Summary Practical Uses

• Sequential execution( parsing, fuzzing, web request processing)

• Reusable modular side effect processing (error handling/ logging)

• Keep your business logic clean in pure functions

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Summary Use Caution When Mixing Monads and Cheese

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Resources Philip Wadler: Monads for functional Programming Comprehending Monads Adam Smyczek: Introduction to Monads (Video) http://www.youtube.com/watch?v=ObR3qi4Guys Jim Duey: Monads in Clojure http://www.clojure.net/2012/02/02/Monads-in-Clojure/ Clojure Lib: clojure.algo.monads Konrad Hinsen Monday, March 12, 12

Photos http://www.flickr.com/photos/jepoirrier/1387560191/sizes/z/in/photostream/ - blocks http://www.flickr.com/photos/alspic/409365553/sizes/m/in/photostream/ - cake http://www.flickr.com/photos/moyogo/4884992/sizes/z/in/photostream/ - stairs http://www.flickr.com/photos/aeireono/369981290/ - forest path http://www.flickr.com/photos/gillpoole/238722066/ - tree branch http://www.flickr.com/photos/tofflerann/5936636472/sizes/m/in/photostream/ - high tea http://www.flickr.com/photos/gsfc/4399423264/ - astronaught http://www.flickr.com/photos/via/2115916059/ - sugar cubes

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carinmeier - GitHub

http://www.flickr.com/photos/jepoirrier/1387560191/sizes/z/in/photostream/ - blocks http://www.flickr.com/photos/alspic/409365553/sizes/m/in/photostream/ - cake.
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