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Monoid? Da war doch was… – Home
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<h1 class='flex items-end justify-center mb-4 p-3 bg-purple-100 text-5xl font-extrabold text-black rounded'>
<a class='z-40 tracking-tighter '>
Monoid? Da war doch was…
</a>
</h1>
<article class='overflow-auto'>
<!-- What goes in this file will appear on top of note body-->
<p class='mb-3'>
Stellen wir uns vor, dass wir eine Funktion schreiben, die einen String bekommt (mehrere Lines mit ACSII-Text) und dieses Wort-für-Wort rückwärts ausgeben soll. Das ist ein einfacher Einzeiler:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>module Main where
import System.Environment (getArgs)
import Data.Monoid (mconcat)
import Data.Functor ((&lt;$&gt;))
main = do
ls &lt;- readFile =&lt;&lt; head &lt;$&gt; getArgs
mconcat &lt;$&gt; mapM (putStrLn . unwords . reverse . words) (lines ls) --die eigentliche Funktion, ls ist das argument.</code></pre></div>
<p class='mb-3'>
Was passiert hier an Vodoo? Und was machen die ganzen wilden Zeichen da?
</p>
<p class='mb-3'>
Gehen wir die Main zeilenweise durch: Wir lesen die Datei, die im ersten Kommandozeilen-Argument gegeben wird. getArgs hat folgende Signatur:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>getArgs :: IO [String]</code></pre></div>
<p class='mb-3'>
Wir bekommen als eine Liste der Argumente. Wir wollen nur das erste. Also machen wir head getArgs. Allerdings fliegt uns dann ein Fehler. head sieht nämlich so aus:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>head :: [a] -&gt; a</code></pre></div>
<p class='mb-3'>
Irgendwie müssen wird as <strong>in</strong> das IO bekommen. Hierzu gibt es fmap. Somit ist
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>fmap head :: IO [a] -&gt; IO a</code></pre></div>
<p class='mb-3'>
Ein inline-Alias (um die Funktion links und das Argument rechts zu schreiben und sich ne Menge Klammern zu sparen) ist &lt;$&gt;. Somit ist schlussendlich der Inhalt der Datei aus dem ersten Argument (lazy) in ls.
</p>
<p class='mb-3'>
Eine andere Möglichkeit sich das (in diesem Fall) zu merken, bzw. drauf zu kommen ist, dass [] AUCH ein Funktor (sogar eine Monade) ist. Man könnte das also auch so schreiben:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>head :: [] a -&gt; a
head :: Functor f =&gt; [] (f a) -&gt; f a -- das "a" geschickt ersetzt zur Verdeutlichung
getArgs :: IO [] String
fmap head :: Functor f =&gt; f [] a -&gt; f a</code></pre></div>
<p class='mb-3'>
fmap “packt” die Funktion quasi 1 Umgebung (Funktor, Monade, ..) weiter rein - Sei es nun in Maybe, Either oder irgendwas anderes.
</p>
<p class='mb-3'>
Alternatives (ausführliches) Beispiel am Ende.
</p>
<p class='mb-3'>
Wenn wir uns die Signatur ansehen, dann haben wir nun
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>head &lt;$&gt; getArgs :: IO String</code></pre></div>
<p class='mb-3'>
readFile will aber nun ein String haben. Man kann nun
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>f &lt;- head &lt;$&gt; getArgs
ls &lt;- readFile f</code></pre></div>
<p class='mb-3'>
kann man auch “inline” mit =&lt;&lt; die Sachen “auspacken”.
</p>
<p class='mb-3'>
Die 2. Zeile lesen wir nun einfach “von hinten”, wie man das meistens tun sollte. Hier ist ein
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>lines ls :: [String]</code></pre></div>
<p class='mb-3'>
was uns den Inhalt der Datei zeilenweise gibt. Mit jeder Zeile möchten wir nun folgendes machen:
</p>
<ul class='my-3 ml-6 space-y-1 list-decimal list-inside'>
<li>
nach Wörtern trennen (words)
</li>
<li>
Wörter in der reihenfolge umkehren (reverse)
</li>
<li>
Wörter wider zu einer Zeile zusammensetzen (unwords)
</li>
<li>
diese Zeile ausgeben (putStrLn)
</li>
</ul>
<p class='mb-3'>
Wenn wir uns die Signatur ansehen:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>(putStrLn . unwords . reverse . words) :: String -&gt; IO ()</code></pre></div>
<p class='mb-3'>
Das mag im ersten Moment verwirren, daher noch die Signaturen der Einzelfunktionen:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>words :: String -&gt; [String]
reverse :: [a] -&gt; [a]
unwords :: [String] -&gt; String
putStrLn :: String -&gt; IO ()</code></pre></div>
<p class='mb-3'>
Da wir am Ende in der IO-Monade landen müssen wir das auf unsere Zeilen mit mapM statt map anwenden. Dies sorgt auch dafür, dass die Liste der reihe nach durchgegangen wird. mapM mit unserer Funktion schaut dann so aus:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>mapM (putStrLn . unwords . reverse . words) :: [String] -&gt; [IO ()]</code></pre></div>
<p class='mb-3'>
eek! Das [IO ()] sieht ekelig aus. Wir haben eine Liste von IO-gar nichts. Das können wir eigentlich entsorgen. Da wir innerhalb der main-Funktion in einer IO-Monade sind, wollen wir IO () anstatt [IO ()] zurück haben.
</p>
<p class='mb-3'>
Wenn wir uns jetzt erinnern, dass [] auch nur eine Monade ist und dass jede Monade ein Monoid ist, dann ist die Lösung einfach. Monoide haben eine “append”-funktion (mappend oder (&lt;&gt;) genannt). Wenn wir “nichts” an “nichts” anhängen, dann erhalten wir …. <em>Trommelwirbel</em> “nichts”! Wir müssen die [IO ()]-Liste also “nur noch” mit mappend falten. Hierzu gibt es schon eine vorgefertigte Funktion:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>mconcat :: [a] -&gt; a
mconcat = foldr mappend mempty</code></pre></div>
<p class='mb-3'>
Was genau die gewünschte Faltung macht. Wir müssen nun wieder fmap nehmen, da wir die Liste selbst falten wollen - und nicht map, welches auf den IO () innerhalb der Liste arbeiten würde. Durch die Faltung fällt die Liste nun auf IO () zusammen.
</p>
<p class='mb-3'>
Viel Voodoo in wenig Code, aber wenn man sich dran gewöhnt hat, sind Monaden in Monaden auch nicht schlimm. Man muss sich immer nur richtig “rein” fmap’en.
</p>
<hr class='mb-3' />
<p class='mb-3'>
Kleinen Tipp gab es noch: mapM_ macht genau das, was oben mit mconcat erreicht werden sollte. Somit kann man auch
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>mapM_ (putStrLn . unwords . reverse . words) (lines ls)</code></pre></div>
<p class='mb-3'>
schreiben. Ich hab es aber mal wegen der klarheit oben so gelassen.
</p>
<h2 id='alternatives-fmap-beispiel' class='mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Alternatives fmap-Beispiel</h2>
<p class='mb-3'>
Nehmen wir als alternatives Beispiel mal an:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>a :: IO Maybe State t</code></pre></div>
<p class='mb-3'>
Um Funktionen vom Typ
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>f :: IO a -&gt; IO a
f a -- valide</code></pre></div>
<p class='mb-3'>
zu nehmen, brauchen wir nichts machen. Bei
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>f' :: Maybe a -&gt; Maybe a</code></pre></div>
<p class='mb-3'>
brauchen wir 1 fmap, also ein
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>f' a -- error
f' &lt;$&gt; a</code></pre></div>
<p class='mb-3'>
um eine Funktion
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>f'' :: State t -&gt; State t</code></pre></div>
<p class='mb-3'>
zu benutzen folglich:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>f'' a -- error
f'' &lt;$&gt; a -- error
fmap f'' &lt;$&gt; a</code></pre></div>
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