1. Com2010 - Functional Programming Algebraic Data Types Marian Gheorghe Lecture 12 Module homepage Mole & http://www.dcs.shef.ac.uk/~marian ©University of Sheffieldcom2010

2. 19. Algebraic data types 19.1 What is an algebraic type? 19.2 Algebraic Types, More Systematically 19.2.1 Enumeration 19.2.2 Product 19.2.3 Nested 19.2.4 Recursive 19.2.5 Polymorphic 19.3 General syntax Summary ©University of Sheffieldcom2010

3. We have seen built-in data types: primitive data types: Int, Float, Bool, Char, …. composite data types (tuples, lists, strings etc) (Int, String), [Int], String … User-defined data types - these are called algebraic data types Ex: Maybe a Algebraic types: introduced by the keyword data, followed by the name_of_the_type, =, and then the constructor (s). The type name and the constructor(s) must start with upper case letters. Ex: data Pres = Result String | Fail Introduction ©University of Sheffieldcom2010

4. Consider the example data Pres = Result String | Fail Elements of this type: Fail :: Pres Result “Green”:: Pres Result “m.gheorghe”::Pres This type definition introduces the following constructors Result :: String -> Pres Fail :: Pres Result is like a function but with no equation definition. Result used like a function (!): map Result["a","b"]  [Result "a",Result "b"] Data type construction ©University of Sheffieldcom2010

5. Every element of Pres is built up from the constructors Result and Fail. This makes it possible to define functions f :: Pres -> X – structural decomposition To define f x, where x :: Pres, deconstruct x into its components, and define f x from these (simpler) components. print’ :: Pres -> String print’ (Result x) = "Result " ++ x -- equation 1 (pattern Result x) print’ Fail = "Fail" -- equation 2 (pattern Fail) When use print’ arg then arg is of type Pres and the result is of type String Structural decomposition ©University of Sheffieldcom2010

6. Patterns are used for deconstructing elements of an algebraic type. They are basic values: these are all constants of types String, Bool, Char, Int, Float or more complex or algebraic types variables: identifiers starting with lower case letters wildcard _ : this is an anonymous variable for a subexpression as-patterns: they occur in the form var@pattern Examples: Fail Result “m.gheorghe” Result x Result _ x v@(Result x) -- like aliases Patterns ©University of Sheffieldcom2010

7. Patterns may be 1. overlapping - they are evaluated in order. The first pattern that matches is taken: isOK :: Pres -> Bool isOK (Result _) = True -- matched first isOK _ = False -- only when pattern ‘Result _’ fails 2. non-exhaustive - if no pattern matches, then we get a run time error: prop :: Pres -> String prop (Result x) = x --no pattern for constructor ‘Fail’ prop Fail -- does not match ⇒ Program execution error: {prop Pres_Fail} More on Patterns ©University of Sheffieldcom2010

8. We will become familiar with alternative compound nested recursive polymorphic algebraic data types Algebraic data types - hierarchy ©University of Sheffieldcom2010

9. In an enumeration type all constructors are constants. Type definition data Temp = Cold | Hot data Season = Spring | Summer | Autumn | Winter Ex: weather :: Season -> Temp weather Summer = Hot weather _ = Cold -- ordering important! Alternatives – Enumeration types ©University of Sheffieldcom2010

10. In product types there is one constructor with many parameters. Type definition data People = Person String Int Int Ex: aPerson :: People aPerson = Person "M Gheorghe" 111 21843 Like for Pres type, the constructors introduced by an algebraic type definition can be used as functions. Person :: Name -> Office->TelNo->People An alternative definition of type People is given by the type synonym type People =(Name, Office, TelNo) Advantages and limitations of algebraic and synonym types Compound – Product type ©University of Sheffieldcom2010

11. Type definition data Employees = Employee Name Gender Dates type Name = String data Dates = Date Day Month Year type Day = Int data Month = Jan | Feb | Mar | Apr |May |Jun | Jul | Aug | Sep | Oct | Nov | Dec type Year = Int data Gender = Male | Female Nested algebraic data types ©University of Sheffieldcom2010

12. inJune :: [Employees]->[Dates] -- returns all male birthday dates in -- June inJune [] = [] inJune (Employee _ Male d@(Date _ Jun _):es = d:inJune es inJune _:es = inJune es Please note the use of as-pattern Caveat. To show the results obtained add deriving Show to both Dates and Month Question: is it possible to have a nested type with an inner type pointing to the current type? … Nested type - Example ©University of Sheffieldcom2010

13. Recursive type of simple expressions data Exp = Lit Int| Add Exp Exp| Sub Exp Exp Examples of expressions: 2 Lit 2 2+3 Add (Lit 2) (Lit 3) (3-1)+4 Add (Sub (Lit 3) (Lit 1)) (Lit 4) Do you remember any other recursive type? We define functions on type Exp by recursive pattern matching. eval :: Exp->Int –- evaluate expressions eval (Lit n) = n eval (Add e1 e2) = (eval e1) + (eval e2) eval (Sub e1 e2) = (eval e1) – (eval e2) eval e stops for e a finite expression; what if e is an infinite expression? …Yes: Recursive types ©University of Sheffieldcom2010

14. In Chapter 15 an important (recursive) polymorphic type was introduced, binary searching tree (look at the order of elements). Type definition of a binary searching tree (BSTree) data Tree a = Empty | Leaf a | Node a (Tree a) (Tree a) Specific instances are Tree Int, Tree String, or Tree (Tree Int). A tree of integers: intTree:: Tree Int intTree = Node 5 (Node 2 (Leaf 1) (Node 4 (Leaf 3) Empty)) (Node 8 (Node 6 Empty (Leaf 7)) Empty) Polymorphic types ©University of Sheffieldcom2010

15. BSTree of Integers ©University of Sheffieldcom2010 5 28 4 6 3 1 7

16. Binary searching tree application: traverse :: Tree a -> [a] -- traverse intTree = -- [1,2,3,4,5,6,7,8] ?? traverse Empty = [] traverse (Leaf x) = [x] traverse (Node x left right) = traverse left ++ [x] ++ traverse right Polymorphic functions on BSTrees (1) ©University of Sheffieldcom2010

17. Another polymorphic function on binary searching trees removeLast :: Tree a -> (a, Tree a) -- split off last element from a nonempty tree removeLast (Leaf x) = (x,Empty) removeLasT (Node y t_1 Empty)=(y,t_1) removeLast (Node y t_1 t_2) = (x, Node y t_1 t_3) where (x,t_3)=removeLast t_2 Polymorphic functions on BSTrees (2) ©University of Sheffieldcom2010

18. Remove the last element of a BSTree ©University of Sheffieldcom2010 2 =(8, 5 4 6 3 1 7)

19. Other polymorphic functions impose constraints on the type variable a. From Chapter 15 we have: tree_member :: Ord a => a->Tree a -> Bool tree_insert :: Ord a => a->Tree a -> Tree a For searching, the following polymorphic functions are useful: listToTree :: Ord a =>[a] -> Tree a -- turns a list into an ordered search tree listToTree [] = Empty listToTree (x:xs) = tree_insert x (listToTree xs) and treeSort :: Ord a => [a] -> [a] -- sorts a list via ordered tree treeSort xs = traverse(listToTree xs) Polymorphic functions on BSTrees (3) ©University of Sheffieldcom2010

20. Examples treeSort [2,91,7,35,28] ⇒ [2,7,28,35,91] treeSort['a','r','k',' ','9','i'] ⇒ [‘ ’,’9’,’a’,’i’,’k’,’r’] treeSort[[4,1],[3,9,5],[3],[9,1,0]] ⇒ [[3],[3,9,5],[4,1],[9,1,0]] Sorting using BSTrees ©University of Sheffieldcom2010

21. The general definition of an algebraic type has the form: data TypeName a_1 a_2 … a_n = ConstructName_1 T_(1,1) T_(1,2)…T_(1,k1) |ConstructName_2 T_(2,1) T_(2,2)…T_(2,k2) … |ConstructName_m T_(m,1) T_(m,2)…T_(m,km) Algebraic data type ©University of Sheffieldcom2010