From 403a19865f1e9d03bffd870fbbd2ad86ab4bab24 Mon Sep 17 00:00:00 2001 From: Stefan Dresselhaus Date: Sun, 17 Apr 2016 17:18:38 +0200 Subject: [PATCH] =?UTF-8?q?added=20Vorlesung1=20and=20=C3=9Cbungsblatt=201?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Folien/Vorlesung1.pdf | Bin 0 -> 461858 bytes Übungen/Blatt1.hs | 115 ++++++++++++++++++++++++++++++++++++++++++ Übungen/Blatt1.lhs | 112 ++++++++++++++++++++++++++++++++++++++++ Übungen/Blatt1.md | 93 ++++++++++++++++++++++++++++++++++ Übungen/Blatt1.pdf | Bin 0 -> 145796 bytes 5 files changed, 320 insertions(+) create mode 100644 Folien/Vorlesung1.pdf create mode 100644 Übungen/Blatt1.hs create mode 100644 Übungen/Blatt1.lhs create mode 100644 Übungen/Blatt1.md create mode 100644 Übungen/Blatt1.pdf diff --git a/Folien/Vorlesung1.pdf b/Folien/Vorlesung1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..66bd6bbc62f980b007684933084e619c09ead017 GIT binary patch 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zov0i7=zBkem-;O|Ps8||t{L3V={LZY|2&`L*+SnRc1ZQ}e9C81eH)k!A4kJtFKh@g zh=1F^R+ zXkaX^jSwIo^VxqgKN9#qqZ-s=##rZ`D LUcQvqR3P{d!dV1G literal 0 HcmV?d00001 diff --git a/Übungen/Blatt1.hs b/Übungen/Blatt1.hs new file mode 100644 index 0000000..2397685 --- /dev/null +++ b/Übungen/Blatt1.hs @@ -0,0 +1,115 @@ +{- +Übungsblatt 1 +============= + +Typtheorie +---------- + +Schreiben sie alle **möglichen** Implementationen der folgenden +Funktionen. Wozu könnte `fun2` nützlich sein? +-} + +fun1 :: a -> a +fun1 = _fun1 + +fun2 :: a -> b -> a +fun2 = _fun2 + +fun3 :: (Eq a) => a -> a -> Bool +fun3 = _fun3 + +{- Wir haben in der Vorlesung parametrisierte Typen kennengelernt. Der +simpelste hiervon ist `Identity`, der genau gar nichts macht, sondern +nur einen anderen Typen einpackt.-} + +data Identity a = Identity a + +{- Diese Definition stellt uns automatisch den Konstruktor +`Identity :: a -> Identity a` zur Verfügung, der ein `a` einpackt. +Schreiben sie die Funktion -} + +unIdentity :: Identity a -> a +unIdentity = _unIdentity + +{- welche diesen Vorgang wieder rückgängig macht. + +Angenommen, sie hätten nun ein Wert vom Typen `Identity a` und eine +Funktion mit dem Typen `a -> b`. Wie wenden sie diese auf das `a` +"innerhalb" des `Identity` an um ein `Identity b` herzustellen? +Schreiben sie also eine Funktion-} + +mapIdentity :: (a -> b) -> Identity a -> Identity b +mapIdentity = _mapIdentity + +{- **Hinweis:** Es gibt *zwei* prinzipielle Vorgehen dieses zu +implementieren. Kommen sie auf beide? + +Funktionen sind auch nur Typen +------------------------------ + +Datentypen können auch Funktionen enthalten. Sehen sie sich einmal den +Datentypen-} + +data Pred a = Pred (a -> Bool) + +{- an. Hier wird ein Prädikat definiert, welches (gegeben einen Datentyp +`a`) eine Funktion gespeichert hat, die `a` in einen `Bool` umwandeln +kann (etwa um irgendwas zu filtern/selektieren/löschen/..., wenn man +dies an eine weitere Funktion übergibt). + +Auch hier können sei eine Funktion schreiben, die das `Pred a` wieder +"auspackt". Definieren sie-} + +unPred :: Pred a -> (a -> Bool) +unPred = _unPred + +{- Da Haskell-Funktionen aber "gecurried" sind, können sie die Klammern +hinten in der Signatur auch weglassen und erhalten +`unPred :: Pred a -> a -> Bool`, was man zugleich als "wende `Pred a` +an, wenn du ein `a` bekommst" lesen kann. In der Tat sind beide +Funktionen identisch (wieso?). + +Bonus + +Was für eine Funktion bräuchten sie um ein `Pred a` in ein `Pred b` +umzuwandeln? Können sie diese Implementieren?-} + +mapPred :: _fun -> Pred a -> Pred b +mapPred = _mapPred + +{- +Neue Typen erfinden +------------------- + +In Haskell ist ein zentraler Vorgehenspunkt das Definieren und Verwenden +von eigenen Datentypen. Zur Erinnerung; es gibt 2 Möglichkeiten, die man +miteinander kombinieren kann: `data Prod a b c = Prod a b c` benötigt +sowohl `a`, `b` als auch `c` um einen Wert zu erzeugen, +`data Sum a b = Sum1 a | Sum2 b` braucht entweder ein `a` um durch den +Konstruktor `Sum1` ein `Sum a b` zu erzeugen oder ein `b` um durch den +Konstruktor `Sum2` ein `Sum a b` zu erzeugen. + +Definieren sie einen Datentypen `Vielleicht a`, der zwei Konstruktoren +besitzt: Einen Konstruktor, der durch ein `a` ein `Vielleicht a` +konstruiert wird und ein zweiter Konstruktor, der keinen Wert nimmt, +sondern die "Abwesenheit eines `a`" symbolisieren soll.-} + +data Vielleicht a = Exercise + +{- Können sie hier eine Funktion schreiben, die das `a` extrahiert? Wenn +ja, implementieren sie diese; wenn nein, geben sie eine kurze +Begründung. + +Wie würden sie mittels einer Funktion `a -> b` ein `Vielleicht a` in ein +`Vielleicht b` wandeln? Implementieren sie-} + +mapVielleicht :: (a -> b) -> Vielleicht a -> Vielleicht b +mapVielleicht = _mapVielleicht + +{- +Bonus + +Man kann Typen natürlich auch Schachteln. Worin liegt eigentlich der +Unterschied zwischen einem `Pred (Vielleicht a)` und einem +`Vielleicht (Pred a)`? Oder sind diese Identisch? +-} diff --git a/Übungen/Blatt1.lhs b/Übungen/Blatt1.lhs new file mode 100644 index 0000000..981f582 --- /dev/null +++ b/Übungen/Blatt1.lhs @@ -0,0 +1,112 @@ +Übungsblatt 1 +============= + +Typtheorie +---------- + +Schreiben sie alle **möglichen** Implementationen der folgenden +Funktionen. Wozu könnte `fun2` nützlich sein? + +> fun1 :: a -> a +> fun1 = _fun1 +> +> fun2 :: a -> b -> a +> fun2 = _fun2 +> +> fun3 :: (Eq a) => a -> a -> Bool +> fun3 = _fun3 + +Wir haben in der Vorlesung parametrisierte Typen kennengelernt. Der +simpelste hiervon ist `Identity`, der genau gar nichts macht, sondern +nur einen anderen Typen einpackt. + +> data Identity a = Identity a + +Diese Definition stellt uns automatisch den Konstruktor +`Identity :: a -> Identity a` zur Verfügung, der ein `a` einpackt. +Schreiben sie die Funktion + +> unIdentity :: Identity a -> a +> unIdentity = _unIdentity + +welche diesen Vorgang wieder rückgängig macht. + +Angenommen, sie hätten nun ein Wert vom Typen `Identity a` und eine +Funktion mit dem Typen `a -> b`. Wie wenden sie diese auf das `a` +"innerhalb" des `Identity` an um ein `Identity b` herzustellen? +Schreiben sie also eine Funktion + +> mapIdentity :: (a -> b) -> Identity a -> Identity b +> mapIdentity = _mapIdentity + +**Hinweis:** Es gibt *zwei* prinzipielle Vorgehen dieses zu +implementieren. Kommen sie auf beide? + +Funktionen sind auch nur Typen +------------------------------ + +Datentypen können auch Funktionen enthalten. Sehen sie sich einmal den +Datentype + +> data Pred a = Pred (a -> Bool) + +an. Hier wird ein Prädikat definiert, welches (gegeben einen Datentyp +`a`) eine Funktion gespeichert hat, die `a` in einen `Bool` umwandeln +kann (etwa um irgendwas zu filtern/selektieren/löschen/..., wenn man +dies an eine weitere Funktion übergibt). + +Auch hier können sei eine Funktion schreiben, die das `Pred a` wieder +"auspackt". Definieren sie + +> unPred :: Pred a -> (a -> Bool) +> unPred = _unPred + +Da Haskell-Funktionen aber "gecurried" sind, können sie die Klammern +hinten in der Signatur auch weglassen und erhalten +`unPred :: Pred a -> a -> Bool`, was man zugleich als "wende `Pred a` +an, wenn du ein `a` bekommst" lesen kann. In der Tat sind beide +Funktionen identisch (wieso?). + +Bonus + +Was für eine Funktion bräuchten sie um ein `Pred a` in ein `Pred b` +umzuwandeln? Können sie diese Implementieren? + +> mapPred :: _fun -> Pred a -> Pred b +> mapPred = _mapPred + +Neue Typen erfinden +------------------- + +In Haskell ist ein zentraler Vorgehenspunkt das Definieren und Verwenden +von eigenen Datentypen. Zur Erinnerung; es gibt 2 Möglichkeiten, die man +miteinander kombinieren kann: `data Prod a b c = Prod a b c` benötigt +sowohl `a`, `b` als auch `c` um einen Wert zu erzeugen, +`data Sum a b = Sum1 a | Sum2 b` braucht entweder ein `a` um durch den +Konstruktor `Sum1` ein `Sum a b` zu erzeugen oder ein `b` um durch den +Konstruktor `Sum2` ein `Sum a b` zu erzeugen. + +Definieren sie einen Datentypen `Vielleicht a`, der zwei Konstruktoren +besitzt: Einen Konstruktor, der durch ein `a` ein `Vielleicht a` +konstruiert wird und ein zweiter Konstruktor, der keinen Wert nimmt, +sondern die "Abwesenheit eines `a`" symbolisieren soll. + +> data Vielleicht a = Exercise + +Können sie hier eine Funktion schreiben, die das `a` extrahiert? Wenn +ja, implementieren sie diese; wenn nein, geben sie eine kurze +Begründung. + +Wie würden sie mittels einer Funktion `a -> b` ein `Vielleicht a` in ein +`Vielleicht b` wandeln? Implementieren sie + +> mapVielleicht :: (a -> b) -> Vielleicht a -> Vielleicht b +> mapVielleicht = _mapVielleicht + +Bonus + +Man kann Typen natürlich auch Schachteln. Worin liegt eigentlich der +Unterschied zwischen einem `Pred (Vielleicht a)` und einem +`Vielleicht (Pred a)`? Oder sind diese Identisch? + +> main = putStrLn "compiles" diff --git a/Übungen/Blatt1.md b/Übungen/Blatt1.md new file mode 100644 index 0000000..39a5e33 --- /dev/null +++ b/Übungen/Blatt1.md @@ -0,0 +1,93 @@ +# Übungsblatt 1 + +## Typtheorie + +Schreiben sie alle **möglichen** Implementationen der folgenden Funktionen. Wozu könnte `fun2` nützlich sein? + +```haskell +fun1 :: a -> a +fun1 = _fun1 + +fun2 :: a -> b -> a +fun2 = _fun2 + +fun3 :: (Eq a) => a -> a -> Bool +fun3 = _fun3 +``` + +Wir haben in der Vorlesung parametrisierte Typen kennengelernt. Der simpelste hiervon ist `Identity`, der genau gar nichts macht, sondern nur einen anderen Typen einpackt. + +```haskell +data Identity a = Identity a +``` + +Diese Definition stellt uns automatisch den Konstruktor `Identity :: a -> Identity a` zur Verfügung, der ein `a` einpackt. Schreiben sie die Funktion + +```haskell +unIdentity :: Identity a -> a +unIdentity = _unIdentity +``` + +welche diesen Vorgang wieder rückgängig macht. + +Angenommen, sie hätten nun ein Wert vom Typen `Identity a` und eine Funktion mit dem Typen `a -> b`. Wie wenden sie diese auf das `a` "innerhalb" des `Identity` an um ein `Identity b` herzustellen? Schreiben sie also eine Funktion + +```haskell +mapIdentity :: (a -> b) -> Identity a -> Identity b +mapIdentity = _mapIdentity +``` + +**Hinweis:** Es gibt *zwei* prinzipielle Vorgehen dieses zu implementieren. Kommen sie auf beide? + + +## Funktionen sind auch nur Typen + +Datentypen können auch Funktionen enthalten. Sehen sie sich einmal den Datentype + +```haskell +data Pred a = Pred (a -> Bool) +``` + +an. Hier wird ein Prädikat definiert, welches (gegeben einen Datentyp `a`) eine Funktion gespeichert hat, die `a` in einen `Bool` umwandeln kann (etwa um irgendwas zu filtern/selektieren/löschen/..., wenn man dies an eine weitere Funktion übergibt). + +Auch hier können sei eine Funktion schreiben, die das `Pred a` wieder "auspackt". Definieren sie + +```haskell +unPred :: Pred a -> (a -> Bool) +unPred = _unPred +``` + +Da Haskell-Funktionen aber "gecurried" sind, können sie die Klammern hinten in der Signatur auch weglassen und erhalten `unPred :: Pred a -> a -> Bool`, was man zugleich als "wende `Pred a` an, wenn du ein `a` bekommst" lesen kann. +In der Tat sind beide Funktionen identisch (wieso?). + +### Bonus + +Was für eine Funktion bräuchten sie um ein `Pred a` in ein `Pred b` umzuwandeln? Können sie diese Implementieren? + +```haskell +mapPred :: _fun -> Pred a -> Pred b +mapPred = _mapPred +``` + +## Neue Typen erfinden + +In Haskell ist ein zentraler Vorgehenspunkt das Definieren und Verwenden von eigenen Datentypen. Zur Erinnerung; es gibt 2 Möglichkeiten, die man miteinander kombinieren kann: `data Prod a b c = Prod a b c` benötigt sowohl `a`, `b` als auch `c` um einen Wert zu erzeugen, `data Sum a b = Sum1 a | Sum2 b` braucht entweder ein `a` um durch den Konstruktor `Sum1` ein `Sum a b` zu erzeugen oder ein `b` um durch den Konstruktor `Sum2` ein `Sum a b` zu erzeugen. + +Definieren sie einen Datentypen `Vielleicht a`, der zwei Konstruktoren besitzt: Einen Konstruktor, der durch ein `a` ein `Vielleicht a` konstruiert wird und ein zweiter Konstruktor, der keinen Wert nimmt, sondern die "Abwesenheit eines `a`" symbolisieren soll. + +Können sie hier eine Funktion schreiben, die das `a` extrahiert? Wenn ja, implementieren sie diese; wenn nein, geben sie eine kurze Begründung. + +Wie würden sie mittels einer Funktion `a -> b` ein `Vielleicht a` in ein `Vielleicht b` wandeln? Implementieren sie +```haskell +mapVielleicht :: (a -> b) -> Vielleicht a -> Vielleicht b +mapVielleicht = _mapVielleicht +``` + +### Bonus + +Man kann Typen natürlich auch Schachteln. Worin liegt eigentlich der Unterschied zwischen einem `Pred (Vielleicht a)` und einem `Vielleicht (Pred a)`? 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