n Haskell n Covered syntax, lazy evaluation, static typing n Algebraic data types and pattern matching n Type classes n Monads and more n Types

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1 Aoucemets Exam 2 is graded, but I will eed some time to go over it I ll release grades this eveig (figers crossed!) Raibow grades: HW1-6, Exam 1-2, Quiz 1-5 Will post aswer key Still gradig: Quiz 6, HW7 ad HW8 We will grade HW8 over weeked to use i HW9 Today s Lecture Outlie Haskell Covered sytax, lazy evaluatio, static typig Algebraic data types ad patter matchig Type classes Moads ad more Types Type systems Type checkig Type equivalece Fall 18 CSCI 4430, A Milaova 1 2 Algebraic Data Types Types Read: Scott, Chapter 7 Algebraic data types are tagged uios (aka sums) of products (aka records) data Shape = Lie Poit Poit Triagle Poit Poit Poit Quad Poit Poit Poit Poit uio Haskell keyword the ew type ew costructors (a.k.a. tags, disjucts, summads) Lie is a biary costructor, Triagle is a terary 3 Fall 18 CSCI 4430, A Milaova (example from MIT 2015 Program Aalysis OCW) 4 Algebraic Data Types i HW9 Patter Matchig Type sigature of achorpt: takes a Shape ad returs a Poit. Defiig a lambda expressio type Name = Strig data Expr = Var Name Lambda Name Expr App Expr Expr > e1 = Var x // Lambda term x > e2 = Lambda x e1 // Lambda term λx.x Fall 18 CSCI 4430, A Milaova 5 Examie values of a algebraic data type achorpt :: Shape -> Poit achorpt s = case s of Lie p1 p2 -> p1 Triagle p3 p4 p5 -> p3 Quad p6 p7 p8 p9 -> p6 Two poits Test: does the give value match this patter? Bidig: if value matches, bid correspodig values of s ad patter Fall 18 CSCI 4430, A Milaova (from MIT 2015 Program Aalysis OCW) 6 1

2 Patter Matchig Patter matchig decostructs a term > let h:t = aa i t a > let (x,y) = (10, aa ) i x 10 Fall 18 CSCI 4430, A Milaova 7 Patter Matchig i HW9 isfree::name -> Expr -> Bool isfree v e = case e of Var -> if ( == v) the True else False Lambda Type sigature of isfree. I Haskell, all fuctios are curried, i.e., they take just oe argumet. (-> is right-associative, Name -> Expr -> Bool is Name -> (Expr -> Bool).) isfree takes a variable ame, ad returs a fuctio that takes a expressio ad returs a boolea. Of course, we ca iterpret isfree as a fuctio that takes a variable ame ame ad a expressio Fall 18 CSCI 4430, A Milaova E, ad returs true if variable ame is free i E. 8 Geeric Fuctios i Haskell We ca geeralize a fuctio whe a fuctio makes o assumptios about the type: cost :: a -> b -> a cost x y = x apply :: (a->b)->a->b apply g x = g x Fall 18 CSCI 4430, A Milaova (examples from MIT 2015 Program Aalysis OCW) 9 Geeric Fuctios -- List datatype data List a = Nil Cos a (List a) Ca we write fuctio sum over a list of a s? sum :: a -> List a -> a sum Nil = sum (Cos x xs) = sum (+x) xs No. a o loger ucostrait. Type ad fuctio defiitio imply that we ca apply + o a but + is ot defied o all types! Type error: No istace for (Num a) arisig from a use of + 10 Haskell Type Classes Geeric Fuctios with Type Class Not to be cofused with Java classes/iterfaces Defie a type class cotaiig the arithmetic operators class Num a where (==) :: a -> a -> Bool (+) :: a -> a -> a istace Num It where x == y = istace Num Float where Fall 18 CSCI 4430, A Milaova Read: A type a is a istace of the type class Num if it provides overloaded defiitios of operatios ==, +, Read: It ad Float are istaces of Num 11 sum :: (Num a) => a -> List a -> a sum Nil = sum (Cos x xs) = sum (+x) xs Oe view of type classes: predicates (Num a) is a predicate i type defiitios Costrais the types we ca istatiate a geeric fuctio to specific types A type class has associated laws Fall 18 CSCI 4430, A Milaova 12 2

3 Type Class Hierarchy class Eq a where (==), (/=) :: a -> a -> Bool class (Eq a) => Ord where (<), (<=), (>), (>=) :: a -> a -> Bool mi, max :: a -> a -> a Each type class correspods to oe cocept Class costraits give rise to a hierarchy Eq is a superclass of Ord Ord iherits specificatio of (==) ad (/=) Notio of true subtypig Fall 18 CSCI 4430, A Milaova (modified from MIT 2015 Program Aalysis OCW) 13 Today s Lecture Outlie Haskell Algebraic data types ad patter matchig Type classes Moads ad more Types Type systems Type checkig Type equivalece 14 Moads Oe source: All About Moads (haskell.org) Aother source: Scott s book A way to clealy compose computatios E.g., f may retur a value of type a or Nothig Composig computatios becomes tedious: case (f s) of Nothig à Nothig Just m à case (f m) I Haskell, moads ecapsulate IO ad other imperative features Fall 18 CSCI 4430, A Milaova 15 A Example: Cloed Sheep type Sheep = father :: Sheep à Maybe Sheep father =... mother :: Sheep à Maybe Sheep mother = (Note: a cloed sheep may have both parets, or ot...) materalgradfather :: Sheep à Maybe Sheep materalgradfather s = case (mother s) of Nothig à Nothig Just m à father m Fall 18 CSCI 4430, A Milaova (Example from All About Moads Tutorial) 16 A Example The Moad Type Class motherspateralgradfather :: Sheep à Maybe Sheep motherspateralgradfather s = case (mother s) of Nothig à Nothig Just m à case (father m) of Nothig à Nothig Just gf à father gf Tedious, ureadable, difficult to maitai Moads help! Fall 18 CSCI 4430, A Milaova (Example from All About Moads Tutorial) 17 Haskell s Moad class requires 2 operatios, >>= (bid) ad retur class Moad m where // >>= (the bid operatio) takes a moad // m a, ad a fuctio that takes a ad turs // it ito a moad m b (>>=) :: m a à (a à m b) à m b // retur ecapsulates a value ito the moad retur :: a à m a 18 3

4 The Maybe Moad data Maybe a = Nothig Just a istace Moad Maybe where Nothig >>= f = Nothig (Just x) >>= f = f x retur = Just Cloed Sheep example: motherspateralgradfather s = (retur s) >>= mother >>= father >>= father (Note: if at ay poit, some fuctio returs Nothig, Nothig gets clealy propagated.) 19 The List Moad The List type is a moad! li >>= f = cocat (map f li) retur x = [x] Note: cocat::[[a]] à [a] e.g., cocat [[1,2],[3,4],[5,6]] yields [1,2,3,4,5,6] Use ay f s.t. f::aà[b]. f may yield a list of 0,1,2, elemets of type b, e.g., > f x = [x+1] > [1,2,3] >>= f --- yields? 20 The List Moad parets :: Sheep à [Sheep] parets s = MaybeToList (mother s) ++ MaybeToList (father s) gradparets :: Sheep à [Sheep] gradparets s = (parets s) >>= parets The do Notatio do otatio is sytactic sugar for moadic bid > f x = x+1 > g x = x*5 > [1,2,3] >>= (retur. f) >>= (retur. g) Or > [1,2,3] >>= \x->[x+1] >>= \y->[y*5] Or, make ecapsulated elemet explicit with do > do { x <- [1,2,3]; y <- (\x->[x+1]) x; (\y->[y*5]) y } Fall 18 CSCI 4430, A Milaova List Comprehesios List Comprehesios > [ x x <- [1,2,3,4] ] [1,2,3,4] > [ x x <- [1,2,3,4], x `mod` 2 == 0 ] [2,4] > [ [x,y] x <- [1,2,3], y <- [6,5,4] ] [[1,6],[1,5],[1,4],[2,6],[2,5],[2,4],[3,6],[3,5],[3,4]] --- Willy s all-pairs fuctio from test 23 List comprehesios are sytactic sugar o top of the do otatio! [ x x <- [1,2,3,4] ] is sytactic sugar for do { x <- [1,2,3,4]; retur x } [ [x,y] x <- [1,2,3], y <- [6,5,4] ] is sytactic sugar for do { x <- [1,2,3]; y<-[6,5,4]; retur [x,y] } Which i tur, we ca traslate ito moadic bid 24 4

5 So What is the Poit of the Moad Coveietly chais (builds) computatio Ecapsulates mutable state. E.g., IO: opefile :: FilePath -> IOMode -> IO Hadle hclose :: Hadle -> IO () -- void hiseof :: Hadle -> IO Bool hgetchar :: Hadle -> IO Char These operatios break referetially trasparecy. For example, hgetchar typically returs differet value whe called twice i a row! Fall 18 CSCI 4430, A Milaova 25 Today s Lecture Outlie Haskell Algebraic data types ad patter matchig Type classes Moads ad more Types Type systems Type checkig Type equivalece 26 What is a type? A set of values ad the valid operatios o those values Itegers: + - * / < <= = >= >... Arrays: lookup(<array>,<idex>) assig(<array>,<idex>,<value>) iitialize(<array>), setbouds(<array>) User-defied types: Java iterfaces What is the role of types? What is the role of types i programmig laguages? Sematic correctess Data abstractio Abstract Data Types Documetatio (static types oly) Fall 18 CSCI 4430, A Milaova/BG Ryder 27 Fall 18 CSCI 4430, A Milaova/BG Ryder 28 3 Views of Types Deotatioal (or set) poit of view: A type is simply a set of values. A value has a give type if it belogs to the set. E.g. it = {,0,1,2,... } char = { a, b,... } bool = { true, false } Abstractio-based poit of view: A type is a iterface cosistig of a set of operatios with well-defied meaig Fall 18 CSCI 4430, A Milaova/BG Ryder 29 3 Views of Types Costructive poit of view: Primitive/simple/built-i types: e.g., it, char, bool Composite/costructed types: Costructed by applyig type costructors poiter e.g., poiterto(it) array e.g., arrayof(char) or arrayof(char,20) or... record/struct e.g., record(age:it, ame:strig) uio e.g. uio(it, poiterto(char)) list e.g., list(...) fuctio e.g., float it CAN BE NESTED! poiterto(arrayof(poiterto(char))) For most of us, types are a mixture of these 3 views Fall 18 CSCI 4430, A Milaova/BG Ryder 30 5

6 What is a Type System? A mechaism to defie types ad associate them with programmig laguage costructs What is Type Checkig? The process of esurig that the program obeys the type rules of the laguage Additioal rules for type equivalece, type compatibility Importat from pragmatic poit of view Fall 18 CSCI 4430, A Milaova/BG Ryder 31 Type checkig ca be doe statically At compile-time, i.e., before executio Statically typed (or statically checked) laguage Type checkig ca be doe dyamically At rutime, i.e., durig executio Dyamically typed (or dyamically checked) laguage Fall 18 CSCI 4430, A Milaova/BG Ryder 32 What is Type Checkig? Statically typed (better term: statically checked) laguages Typically require type aotatios (e.g., A a, List<A> list) Typically have a complex type system, ad most of type checkig is performed statically (at compile-time) Ada, Pascal, Java, C++, Haskell, OCaml A form of early bidig Dyamically typed (better term: dyamically checked) laguages. Also kow as Duck typed Typically require o type aotatios! All type checkig is performed dyamically (at rutime) Smalltalk, Lisp ad Scheme, Pytho, JavaScript What is Type Checkig? The process of esurig that the program obeys the type rules of the laguage Textbook defies term prohibited applicatio (also kow as forbidde error): ituitively, a prohibited applicatio is a applicatio of a operatio o values of the wrog type is the property that o operatio ever applies to values of the wrog type at rutime. I.e., o prohibited applicatio (forbidde error) ever occurs Fall 18 CSCI 4430, A Milaova/BG Ryder 33 Fall 18 CSCI 4430, A Milaova/BG Ryder 34 Laguage Desig Choices Desig choice: what is the set of forbidde errors? Obviously, we caot forbid all possible sematic errors Defie a set of forbidde errors Desig choice: Oce we ve chose the set of forbidde errors, how does the type system prevet them? Static checks oly? Dyamic checks oly? A combiatio of both? Furthermore, are we goig to absolutely disallow forbidde errors (be type safe), or are we goig to allow for programs to circumvet the system ad exhibit forbidde errors (i.e., be type usafe)? Forbidde Errors Example: idexig a array out of bouds a[i], a is of size Boud, i<0 or Boud i I C, C++, this is ot a forbidde error 0 i ad i<boud is ot checked (bouds are ot part of type) What are the tradeoffs here? I Pascal, this is a forbidde error. Preveted with static checks 0 i ad i<boud must be checked at compile time What are the tradeoffs here? I Java, this is a forbidde error. It is preveted with dyamic checks 0 i ad i<boud must be checked at rutime What are the tradeoffs here? Fall 18 CSCI 4430, A Milaova 35 Fall 18 CSCI 4430, A Milaova/BG Ryder 36 6

7 Type Safety C++ is type usafe Java vs. C++: Java: Duck q; ; q.quack()class Duck has quack C++: Duck *q; ; q->quack()class Duck has quack Ca we ed up with a program that calls quack()o a object that is t a Duck? I Java? I C++? q->foo(66) is a prohibited applicatio (i.e., applicatio of a operatio o a value of the wrog type, i.e., forbidde error). Static type B* q promises the programmer that q will poit to a to be type usafe B object. However, laguage does ot hoor this promise 37 Fall 18 CSCI 4430, A Milaova/BG Ryder 38 Java is said to be type safe while C++ is said //#1 void* x = (void *) ew A; A B* q = (B*) x; //a safe dowcast? it case1 = q->foo(); //what happes? B //#2 void* x = (void *) ew A; B* q = (B *) x; //a safe dowcast? it case2 = q->foo(66); //what happes? virtual foo() virtual foo() vritual foo(it) What is Type Checkig What is Type Checkig? type safe type usafe statically ot statically typed typed (i.e., dyamically typed) Static typig vs. dyamic typig What are the advatages of static typig? What are the advatages of dyamic typig? Fall 18 CSCI 4430, A Milaova/BG Ryder 39 Fall 18 CSCI 4430, A Milaova/BG Ryder 40 Today s Lecture Outlie Haskell Algebraic data types ad patter matchig Type classes Moads ad more Types Type systems Type checkig Type Equivalece ad Type Compatibility Discussio ceters o o-object-orieted vo Neuma laguages that use the value model for variables: Fortra, C, Pascal Questios: e := expressio þ or ý Are e ad expressio of same type? a + b þ or ý Are a ad b of same type ad type supports +? foo(arg1, arg2,,argn) þ or ý Do the types of the argumets match the types of the formal parameters? Type equivalece

8 Type Equivalece Two ways of defiig type equivalece Structural equivalece: based o shape Roughly, two types are the same if they cosists of the same compoets, put together i the same way Name equivalece: based o lexical occurrece of the type defiitios Roughly, each type defiitio itroduces a ew type Strict ame equivalece Loose ame equivalece Fall 18 CSCI 4430, A Milaova/BG Ryder 43 Structural Equivalece A type ame is structurally equivalet to itself Two types are structurally equivalet if they are formed by applyig the same type costructor to structurally equivalet argumets After type declaratio type = T the type ame is structurally equivalet to T E.g., i Haskell, we saw type Name = Strig Declaratio makes a alias of T. ad T are said to be aliased types Fall 18 CSCI 4430, A Milaova/BG Ryder 44 Structural Equivalece Structural Equivalece Example, Pascal-like laguage: type S = array [0..99] of char type T = array [0..99] of char Example, C: typedef struct { it j, it k, it *ptr } cell; typedef struct { it, it m, it *p } elemet; Show by isomorphism of correspodig type trees Are these types equivalet? struct cell struct elemet { char data; { char c; it a[3]; it a[5]; struct cell *ext; struct elemet *ptr; } } Equivalet types: are field ames part of the struct costructed type? are array bouds part of the array costructed type? Fall 18 CSCI 4430, A Milaova/BG Ryder 45 Fall 18 CSCI 4430, A Milaova/BG Ryder 46 Name Equivalece Name equivalece Two types are ame equivalet if they correspod to the same type defiitio T: type array [1..20] of it; x,y: array [1..20] of it; w,z: T; v: T; x ad y are of same type, w, z,v are of same type, but x ad w are of differet types! A applicatio of a type costructor is a type defiitio. Red array is ONE TYPE DEFINITION. Blue array is ANOTHER TYPE DEFINITION. 47 Name Equivalece A subtlety arises with aliased types (e.g., type = T, typedef it Age i C) Strict ame equivalece A laguage i which aliased types are cosidered distict, is said to have strict ame equivalece (e.g., it ad Age above would be distict types) Loose ame equivalece A laguage i which aliased types are cosidered the same, is said to have loose ame equivalece (e.g., it ad Age would be same) 48 8

9 Name Equivalece type cell = // record/struct type type alik = poiter to cell type blik = alik p,q : poiter to cell r : alik s : blik t : poiter to cell u : alik Fall 18 CSCI 4430, A Milaova/BG Ryder 49 Fall 18 CSCI 4430, A Milaova/BG Ryder 50 9