# Imperative languages

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1 Imperative languages Von Neumann model: store with addressable locations machine code: effect achieved by changing contents of store locations instructions executed in sequence, flow of control altered by jumps imperative language: variable corresponds to store location instructions executed in sequence, flow of control altered by conditional and loop statements efficient implementation since close to design of conventional computers 10.1

2 Functional languages computational model: lambda calculus mathematical functions: domain, range functional languages achieve effect by applying functions functional vs. imperative languages store location assignment statement vs. application of a function side-effects aliasing referential transparency (you can replace function calls with their results without changing the functionality) (allows for memoization: basically caching for the sake of future function calls) 10.2

3 Program structure what, not how primitive functions mechanism to define new functions from old max a b = if a < b then b else a min a b = if a < b then a else b difference a b c = max a (max b c) min a (min b c) considerations: synchronization simple solutions symbolic manipulation list-processing 10.3

4 Features of functional languages higher-order functions can accept functions as parameters can return functions as results comp f g = \x -> f (g x) recursion as a basic principle application of rewrite rule: function call replaced by code body run-time overhead garbage collection 10.4

5 Lists Lisp: atoms, lists, functions atoms symbolic numeric lists each element is an atom or another list (ALPHA BETA GAMMA) ( ) ((A B) (HELLO THERE) 94) 10.5

6 ALPHA 12 BETA 14 GAMMA nil 16 nil A B nil HELLO 94 nil THERE nil 10.6

7 Operations on lists list operations head ALPHA is the head of (ALPHA BETA GAMMA) tail (BETA GAMMA) is the tail of (ALPHA BETA GAMMA) cons cons(head(x), tail(x)) = x head(cons(a,x)) = a tail(cons(a,x)) = x recursion ismember x a = if a==[] then False else if x==head a then True else ismember x (tail a) 10.7

8 Higher order functions function map (f, x) is begin if x = nil then return nil; else return cons(f(head(x)), map (f, tail(x))); endif; end map; function timesten (x) is begin return 10 * x; end timesten; alist ( ) map(timesten, alist) ( ) 10.8

9 Lisp - functional core function application symbolic expressions (S-expressions) lambda expressions naming new functions: DEF interactive system prefix notation all expressions are parenthesised 10.9

10 predefined functions CAR, CDR QUOTE short form ATOM NULL MAPCAR special forms NIL, T COND 10.10

11 (TIMES 10 2) (LAMBDA (X) (TIMES 10 X)) (DEF TIMESTEN (LAMBDA (X) (TIMES 10 X))) (TIMESTEN 2) ((LAMBDA (X) (TIMES 10 X)) 2) (TIMES 10 2) 20 (MAPCAR TIMESTEN ( )) (DEF ISMEMBER (LAMBDA (X A) (COND ((NULL A) NIL) ((EQ X (CAR A)) T) (T (ISMEMBER X (CDR A))) ) )) 10.11

12 Lisp - Example Derivative: Given an S-expression such as: (TIMES 4 (POWER X 3)) the aim is to produce the S-expression that represent its derivative with respect to X, that is, in this case, to produce the list: (TIMES 12 (POWER X 2)) (DEF THIRD (LAMBDA (Y) )) (CAR (CDR (CDR Y))) 10.12

13 (DEF DERIVE (LAMBDA (Y) (COND ((EQ (CAR Y) POWER) (LIST TIMES (THIRD Y) (LIST POWER X (DIFFERENCE (THIRD Y) 1)) )) ((EQ (CAR Y) TIMES) (LIST TIMES (TIMES (CAR (CDR Y)) (THIRD (THIRD Y))) (THIRD (DERIVE (THIRD Y))) )) (T ERROR) ) )) 10.13

14 Scope rules dynamic scope: identifier bound to most recent definition free variable example: (DEF BASE (LAMBDA (F A) (PLUS (F 10) A) )) (DEF TWODIGIT (LAMBDA (A B) (BASE (LAMBDA (C) (TIMES A C)) B) )) The result of (TWODIGIT 2 3) is not

15 Scope rules (TWODIGIT 2 3) (BASE (LAMBDA (C) (TIMES A C)) 3) most recent A = 2 (PLUS ((LAMBDA (C) (TIMES A C)) 10) 3) most recent A = 3 (PLUS ((TIMES 3 10) 3) (PLUS 30 3) 33 This problem would not have arisen if the atom A had not been used in two different contexts; for example if BASE had been defined as: (DEF BASE (LAMBDA (F X) (PLUS (F 10) X) )) Be careful!!! 10.15

16 Lisp - miscellaneous No language standard Scheme popular dialect meaningful identifiers, reserved words block structure, static scope rules (DEFINE TIMESTEN (LAMBDA (X) (* 10 X))) (DEFINE ISMEMBER (LAMBDA (X A) (COND ((NULL? A) #F) ((EQ? X (CAR A)) #T) (ELSE (ISMEMBER X (CDR A))) ) )) 10.16

17 Lisp - miscellaneous Imperative features assignment: SET, SETQ (SETQ A B) = (SET A B) (SET A C) PROG, LOOP, GO (DEF SUMLIST (LAMBDA (X) (PROG (TOTAL) (SETQ TOTAL 0) LOOP (COND ((NULL X) (RETURN TOTAL)) (T (SETQ TOTAL (PLUS TOTAL (CAR X))) )) (SETQ X (CDR X)) (GO LOOP) ))) 10.17

18 FP systems purely functional data values: atoms or sequences each element of a sequence is an atom or a sequence no variables functions can have only one parameter (may be a sequence) application of function G:X functional forms APPLYTOALL composition: o condition: (P -> H;G) construction: [F1, F1,, FN] : X 10.18

19 FP systems - example DEF ISMEMBER = (NULL SEC -> F; (EQ [HEAD, HEAD SEC] -> T; ISMEMBER [HEAD, TAIL SEC])) ISMEMBER : <3,<7,3,9>> 10.19

20 NULL SEC : <3,<7,3,9>> NULL : (SEC : <3,<7,3,9>>) NULL : <7,3,9> false EQ [HEAD, HEAD SEC] : <3,<7,3,9>> EQ : ([HEAD, HEAD SEC] : <3,<7,3,9>>) EQ : <HEAD : <3,<7,3,9>>, HEAD SEC : <3,<7,3,9>>> EQ : <3, HEAD : <7,3,9>> EQ : <3, 7> false ISMEMBER [HEAD, TAIL SEC] : <3,<7,3,9>> ISMEMBER : ([HEAD, TAIL SEC] : <3,<7,3,9>>) ISMEMBER : <HEAD : <3,<7,3,9>>, TAIL SEC : <3,<7,3,9>>> ISMEMBER : <3, TAIL : <7,3,9>> ISMEMBER : <3, <3,9>> 10.20

22 Example factorial :: Integer -> Integer factorial 0 = 1 factorial (n+1) = (n+1)*(factorial n) Integer (unbounded) vs. Int (bounded, at least ) factorial :: Integer 10.22

23 Identity id :: a -> a id x = x Composition Polymorphic functions (.) :: (b -> c) -> (a -> b) -> a -> c f. g = \x -> f (g x) Curry/Uncurry curry :: ((a, b) -> c) -> a -> b -> c curry f x y = f (x,y) uncurry :: (a -> b -> c) -> ((a, b) -> c) uncurry f p = f (fst p) (snd p) 10.23

24 Lists [a] data type of lists with elements from a. head, tail [], (:) length :: [a] -> Int length [] = 0 length (_:l) = 1 + length l map :: (a -> b) -> [a] -> [b] map f [] = [] map f (x:xs) = f x : map f xs filter :: (a -> Bool) -> [a] -> [a] filter p [] = [] filter p (x:xs) p x = x : filter p xs otherwise = filter p xs 10.24

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