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monad_._haskell [2014/09/11 23:06] nikolaj |
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| Here is what makes the use of predicates possible: Note that 'join [ [1,2],[],[3],[] ]' is '[1,2,3]' and this is part of the working of the list monads >>=. Since for x=2, y=2, the Bool 'x/=y' isn't 'True', the value '[]' is returned and consequently dropped from the resulting list. | Here is what makes the use of predicates possible: Note that 'join [ [1,2],[],[3],[] ]' is '[1,2,3]' and this is part of the working of the list monads >>=. Since for x=2, y=2, the Bool 'x/=y' isn't 'True', the value '[]' is returned and consequently dropped from the resulting list. | ||
| - | == The List monad == | + | === Examples === |
| + | == Example: The List monad == | ||
| The list-functors $\text{fmap}$ maps functions $f :: a \to b$ to functions $T(f)$ defined by concatenation with Haskells $\text{map}$ function, i.e. $\text{fmap}\ f\ xs\ =\ [\ f\ a\ |\ a \leftarrow xs\ ]$, e.g. $\text{map}\ (x\mapsto x^2)\ [2,3,4]$ evaluates to $[4,9,16]$. $\text{return}$ then maps terms $x$ of type $a$ to terms $[x]$ (list with one member) of type $[a]$. $\text{join}$ is flatten, e.g. $\text{join}$ applied to the list of lists $[[7,9,3],[2],[5,10]]$ evaluates to $[7,9,3,2,5,10]$. | The list-functors $\text{fmap}$ maps functions $f :: a \to b$ to functions $T(f)$ defined by concatenation with Haskells $\text{map}$ function, i.e. $\text{fmap}\ f\ xs\ =\ [\ f\ a\ |\ a \leftarrow xs\ ]$, e.g. $\text{map}\ (x\mapsto x^2)\ [2,3,4]$ evaluates to $[4,9,16]$. $\text{return}$ then maps terms $x$ of type $a$ to terms $[x]$ (list with one member) of type $[a]$. $\text{join}$ is flatten, e.g. $\text{join}$ applied to the list of lists $[[7,9,3],[2],[5,10]]$ evaluates to $[7,9,3,2,5,10]$. | ||
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| * $\text{map}\ f\ (\text{join}\ [[7,3],[2]])$ is $\text{map}\ f\ [7,3,2]$ is $[f\ 7,f\ 3,f\ 2]$. | * $\text{map}\ f\ (\text{join}\ [[7,3],[2]])$ is $\text{map}\ f\ [7,3,2]$ is $[f\ 7,f\ 3,f\ 2]$. | ||
| * $\text{join}\ (\text{map}\ (\text{map}\ f)\ [[7,3],[2]])$ is $\text{join}\ [(\text{map}\ f)\ [7,3],(\text{map}\ f)\ [2]]$ is $\text{join}\ [[f\ 7,f\ 3],[f\ 2]]$ is $[f\ 7,f\ 3,f\ 2]$ as well. | * $\text{join}\ (\text{map}\ (\text{map}\ f)\ [[7,3],[2]])$ is $\text{join}\ [(\text{map}\ f)\ [7,3],(\text{map}\ f)\ [2]]$ is $\text{join}\ [[f\ 7,f\ 3],[f\ 2]]$ is $[f\ 7,f\ 3,f\ 2]$ as well. | ||
| + | |||
| + | == Example: The Maybe monad == | ||
| + | Types $a$ get mapped to $\text{Maybe}\ a$. For each type $a$, the type $\text{Maybe}\ a$ contains a copy of all of $a$'s terms $x,y,\dots$, there written $\text{Just}\ x,\text{Just}\ y,\dots$, plus an additional term called $\text{Nothing}$. I.e. the maybe functor effectively adds an „exception“ term to all types of $\text{Hask}$. If $f:: a\to b$ and $x::a$, then the arrow image of the functor on $f$ is just the function which maps $\text{Just}\ x$ to $\text{Just}\ f(x)$ and | ||
| + | |||
| + | Return maps $x$ in $a$ to $\text{Just}\ x$ in $\text{Maybe}\ a$ and join maps $\text{Just}\ (\text{Just}\ x)$ to $\text{Just}\ x$ and in particular, $\text{Just}\ \text{Nothing}$ to $\text{Nothing}$. | ||
| == Lifting == | == Lifting == | ||
| Line 81: | Line 87: | ||
| (<*>) :: m (a -> b) -> m a -> m b | (<*>) :: m (a -> b) -> m a -> m b | ||
| (<*>) mg ma = liftM2 id mg my | (<*>) mg ma = liftM2 id mg my | ||
| + | -- or = join $ fmap (\g -> fmap g mx) mg | ||
| </code> | </code> | ||
| + | |||
| Note that conversely, '<*>' doesn't suffice to define a '>>='. | Note that conversely, '<*>' doesn't suffice to define a '>>='. | ||
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| === Reference === | === Reference === | ||
| + | [[http://en.wikipedia.org/wiki/Monad_%28category_theory%29|Monad (category theory)]], | ||
| + | [[http://de.wikipedia.org/wiki/Monade_%28Informatik%29|Monade (Informatik)]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| === Subset of === | === Subset of === | ||
| [[Applicative]] | [[Applicative]] | ||
