Monad laws
From HaskellWiki
(Why is it an "associative law"?) 
(possible halfway point?) 

(11 intermediate revisions by 5 users not shown)  
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−  All instances of the [[Monad]] class should obey: 
+  All instances of the [[Monad]] typeclass should obey the three '''monad laws''': 
−  # "Left identity": <hask> return a >>= f = f a </hask> 
+  { 
−  # "Right identity": <hask> m >>= return = m </hask> 
+   
−  # "Associativity": <hask> (m >>= f) >>= g = m >>= (\x > f x >>= g) </hask> 
+   '''Left identity:''' 
+   <haskell>return a >>= f</haskell> 

+   ≡ 

+   <haskell>f a</haskell> 

+   

+   '''Right identity:''' 

+   <haskell>m >>= return</haskell> 

+   ≡ 

+   <haskell>m</haskell> 

+   

+   '''Associativity:''' 

+   <haskell>(m >>= f) >>= g</haskell> 

+   ≡ 

+   <haskell>m >>= (\x > f x >>= g)</haskell> 

+  } 

+  
+  Here, ''p'' ≡ ''q'' simply means that you can replace ''p'' with ''q'' and viceversa, and the behaviour of your program will not change: ''p'' and ''q'' are equivalent. 

== What is the practical meaning of the monad laws? == 
== What is the practical meaning of the monad laws? == 

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Let us rewrite the laws in donotation: 
Let us rewrite the laws in donotation: 

−  <haskell> 
+  { 
−  1. do { x' < return x do { f x 
+   
−  ; f x' == } 
+   '''Left identity:''' 
−  } 
+   <haskell> 
−  +  do { x' < return x; 

−  2. do { x < m == do { m 
+  f x' 
−  ; return x } } 
+  } 
−  +  </haskell> 

−  3. do { y < do { x < m do { x < m 
+   ≡ 
−  ; f x ; do { y < f x 
+   <haskell> 
−  } == ; g y 
+  do { f x } 
−  ; g y } 
+  </haskell> 
−  } } 
+   
−  +   '''Right identity:''' 

−  do { x < m 
+   <haskell> 
−  using 3.14 ; y < f x 
+  do { x < m; 
−  == ; g y 
+  return x 
−  } 
+  } 
+  </haskell> 

+   ≡ 

+   <haskell> 

+  do { m } 

+  </haskell> 

+   

+   '''Associativity:''' 

+   <haskell> 

+  do { y < do { x < m; 

+  f x 

+  } 

+  g y 

+  } 

+  </haskell> 

+   ≡ 

+   <haskell> 

+  do { x < m; 

+  do { y < f x; 

+  g y 

+  } 

+  } 

+  </haskell> 

+   ≡ 

+   <haskell> 

+  do { x < m; 

+  y < f x; 

+  g y 

+  } 

</haskell> 
</haskell> 

+  } 

−  In this notation the laws appear as plain common sense. 
+  In this notation the laws appear as plain commonsense transformations of imperative programs. 
== But why should monads obey these laws? == 
== But why should monads obey these laws? == 

−  When we see a program written in a form on the LHS, we expect it to do 
+  When we see a program written in a form on the lefthand side, we expect it to do 
−  the same thing as the corresponding RHS; and vice versa. And in 
+  the same thing as the corresponding righthand side; and vice versa. And in 
−  practice, people do write like the lengthier LHS once in a while. 
+  practice, people do write like the lengthier lefthand side once in a while. 
First example: beginners tend to write 
First example: beginners tend to write 

<haskell> 
<haskell> 

−  skip_and_get = do { unused < getLine 
+  skip_and_get = do 
−  ; line < getLine 
+  unused < getLine 
−  ; return line 
+  line < getLine 
−  } 
+  return line 
</haskell> 
</haskell> 

and it would really throw off both beginners and veterans if that did 
and it would really throw off both beginners and veterans if that did 

−  not act like (by law #2) 
+  not act like (by right identity) 
<haskell> 
<haskell> 

−  skip_and_get = do { unused < getLine 
+  skip_and_get = do 
−  ; getLine 
+  unused < getLine 
−  } 
+  getLine 
</haskell> 
</haskell> 

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<haskell> 
<haskell> 

−  main = do { answer < skip_and_get 
+  main = do 
−  ; putStrLn answer 
+  answer < skip_and_get 
−  } 
+  putStrLn answer 
</haskell> 
</haskell> 

Line 66:  Line 67:  
<haskell> 
<haskell> 

−  main = do { answer < do { unused < getLine 
+  main = do 
−  ; getLine 
+  answer < do 
−  } 
+  unused < getLine 
−  ; putStrLn answer 
+  getLine 
−  } 
+  putStrLn answer 
</haskell> 
</haskell> 

−  and applying law #3 so you can pretend it is 
+  and applying associativity so you can pretend it is 
<haskell> 
<haskell> 

−  main = do { unused < getLine 
+  main = do 
−  ; answer < getLine 
+  unused < getLine 
−  ; putStrLn answer 
+  answer < getLine 
−  } 
+  putStrLn answer 
</haskell> 
</haskell> 

−  Law #3 is amazingly pervasive: you have always assumed it, and you 
+  The associativity law is amazingly pervasive: you have always assumed it, and you 
have never noticed it. 
have never noticed it. 

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<haskell> 
<haskell> 

(>=>) :: Monad m => (a > m b) > (b > m c) > a > m c 
(>=>) :: Monad m => (a > m b) > (b > m c) > a > m c 

−  m >=> n = \x > do { y < m x; n y } 
+  (m >=> n) x = do 
+  y < m x 

+  n y 

</haskell> 
</haskell> 

Using this operator, the three laws can be expressed like this: 
Using this operator, the three laws can be expressed like this: 

−  # "Left identity": <hask> return >=> g = g </hask> 
+  { 
−  # "Right identity": <hask> f >=> return = f </hask> 
+   '''Left identity:''' 
−  # "Associativity": <hask> (f >=> g) >=> h = f >=> (g >=> h) </hask> 
+   <haskell> return >=> g </haskell> 
+   ≡ 

+   <haskell> g </haskell> 

+   

+   '''Right identity:''' 

+   <haskell> f >=> return </haskell> 

+   ≡ 

+   <haskell> f </haskell> 

+   

+   '''Associativity:''' 

+   <haskell> (f >=> g) >=> h </haskell> 

+   ≡ 

+   <haskell> f >=> (g >=> h) </haskell> 

+  } 

It's now easy to see that monad composition is an associative operator with left and right identities. 
It's now easy to see that monad composition is an associative operator with left and right identities. 

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This is a very important way to express the three monad laws, because they are precisely the laws that are required for monads to form a mathematical [[Category theorycategory]]. So the monad laws can be summarised in convenient Haiku form: 
This is a very important way to express the three monad laws, because they are precisely the laws that are required for monads to form a mathematical [[Category theorycategory]]. So the monad laws can be summarised in convenient Haiku form: 

−  <blockquote> 
+  :Monad axioms: 
−  Monad axioms:<br/> 
+  :Kleisli composition forms 
−  Kleisli composition forms<br/> 
+  :a category. 
−  a category. 

−  </blockquote> 

[[Category:Standard_classes]] 
[[Category:Standard_classes]] 
Latest revision as of 18:00, 5 April 2012
All instances of the Monad typeclass should obey the three monad laws:
Left identity:  return a >>= f 
≡  f a 
Right identity:  m >>= return 
≡  m 
Associativity:  (m >>= f) >>= g 
≡  m >>= (\x > f x >>= g) 
Here, p ≡ q simply means that you can replace p with q and viceversa, and the behaviour of your program will not change: p and q are equivalent.
[edit] 1 What is the practical meaning of the monad laws?
Let us rewrite the laws in donotation:
Left identity:  do { x' < return x; f x' } 
≡  do { f x }  
Right identity:  do { x < m; return x } 
≡  do { m }  
Associativity:  do { y < do { x < m; f x } g y } 
≡  do { x < m; do { y < f x; g y } } 
≡  do { x < m; y < f x; g y } 
In this notation the laws appear as plain commonsense transformations of imperative programs.
[edit] 2 But why should monads obey these laws?
When we see a program written in a form on the lefthand side, we expect it to do the same thing as the corresponding righthand side; and vice versa. And in practice, people do write like the lengthier lefthand side once in a while. First example: beginners tend to write
skip_and_get = do unused < getLine line < getLine return line
and it would really throw off both beginners and veterans if that did not act like (by right identity)
skip_and_get = do unused < getLine getLine
Second example: Next, you go ahead to use skip_and_get:
main = do answer < skip_and_get putStrLn answer
The most popular way of comprehending this program is by inlining (whether the compiler does or not is an orthogonal issue):
main = do answer < do unused < getLine getLine putStrLn answer
and applying associativity so you can pretend it is
main = do unused < getLine answer < getLine putStrLn answer
The associativity law is amazingly pervasive: you have always assumed it, and you have never noticed it.
Whether compilers exploit the laws or not, you still want the laws for your own sake, just so you can avoid pulling your hair for counterintuitive program behaviour that brittlely depends on how many redundant "return"s you insert or how you nest your doblocks.
[edit] 3 But it doesn't look exactly like an "associative law"...
Not in this form, no. To see precisely why they're called "identity laws" and an "associative law", you have to change your notation slightly.
The monad composition operator (also known as the Kleisli composition operator) is defined in Control.Monad:
(>=>) :: Monad m => (a > m b) > (b > m c) > a > m c (m >=> n) x = do y < m x n y
Using this operator, the three laws can be expressed like this:
Left identity:  return >=> g 
≡  g 
Right identity:  f >=> return 
≡  f 
Associativity:  (f >=> g) >=> h 
≡  f >=> (g >=> h) 
It's now easy to see that monad composition is an associative operator with left and right identities.
This is a very important way to express the three monad laws, because they are precisely the laws that are required for monads to form a mathematical category. So the monad laws can be summarised in convenient Haiku form:
 Monad axioms:
 Kleisli composition forms
 a category.