11/1/20151 GC16/3011 Functional Programming Lecture 5 Miranda patterns, functions, recursion and lists

11/1/20152 Contents  Offside rule / where blocks / evaluation  Partial / polymorphic functions  Patterns and pattern-matching  Recursive functions  Lists  Functions using lists  Recursive functions using lists

11/1/20153 Functions  The offside rule and “where” blocks  Mapping from source to target type domains  Application associates to the left!  Function evaluation  normal order, lazy  main = fst (34, (23 / 0) )

11/1/20154 Functions  Partial functions  those which have no result (i.e. return an error) for some valid values of valid input type  f (x,y) = x / y  Polymorphic functions  fst, snd  g (x,y) = (-y, x)  three x = 3

11/1/20155 Patterns  Tuple patterns  (x,y,z) = (3, “hello”, (34,True,[3]))  as a test  as a definition  Patterns for function definitions not True = False not False = True  Top-down evaluation semantics f 3 = 45 f 489 = 3 f any = 345*219

11/1/20156 Patterns  Non-exhaustive patterns f True = False  Patterns can destroy laziness fst (x,0) = x fst (x,y) = x  Can a pattern contain a redex?  No! A pattern must be a constant expression  (special exception – “(n + 1)” in Miranda)  Duplicate parameter names (Miranda only)

11/1/20157 Recursive Functions  Recursion is the ONLY way to program loops in a functional language  Function calls itself inside its own body  Very powerful – very flexible  In imperative languages, can be slow  In functional languages, highly optimised

11/1/20158 Recursive Functions  Beware: loop_forever x = loop_forever x  Must have:  Terminating condition  Changing argument  …that converges on the terminating condition! f :: num -> [char] f 0 = “” f n = “X” ++ (f (n – 1))

11/1/20159 Recursive Functions  Stack recursion  f 3  “X” ++ (f (3 – 1))  “X” ++ (f 2)  “X” ++ (“X” ++ (f (2-1)))  “X” ++ (“X” ++ (f 1))  “X” ++ (“X” ++ (“X” ++ (f (1-1))))  “X” ++ (“X” ++ (“X” ++ “”))  “X” ++ (“X” ++ “X”)  “X” ++ “XX”  “XXX”

11/1/ Recursive Functions  Accumulative recursion plus :: (num, num) -> num plus :: (num, num) -> num plus (x, 0) = x plus (x, 0) = x plus (x, y) = plus (x+1, y-1) plus (x, y) = plus (x+1, y-1)

11/1/ Type Synonyms  f :: (([char],num,[char]),(num,num,num)) -> bool  str = = [char]  coord = = (num, num, num)  f :: ((str,num,str), coord) -> bool

11/1/ LISTS  Lists are another way to collect together related data  But lists are special - they are RECURSIVE  Data can be recursive, just like functions!

11/1/ LISTS  A list of type  is either:  Empty, or  An element of type  together with a list of elements of the same type  List of num:  []  (34: []) [34]  (34: (13: [])) [34, 13]

11/1/ Functions using lists bothempty :: ([*],[**]) -> bool bothempty ([], []) = True bothempty anything = False myhd :: [*] -> * myhd [] = error “take head of empty list” myhd (x : rest) = x

11/1/ Recursive functions using lists sumlist :: [num] -> num sumlist [] = 0 sumlist (x : rest) = x + (sumlist rest) length :: [*] -> num length [] = 0 length (x : rest) = 1 + (length rest)

11/1/ Exercise  Can you write a function “threes” which takes as input a list of whole numbers and produces as output a count of how many 3s occur in the input?

11/1/ Summary  Offside rule / where blocks / evaluation  Partial / polymorphic functions  Patterns and pattern-matching  Recursive functions  Lists  Functions using lists  Recursive functions using lists