CS 363 Comparative Programming Languages Functional Languages: Scheme

CS 363 Spring 2005 GMU2 Fundamentals of Functional Programming Languages The objective of the design of a functional programming language (FPL) is to mimic mathematical functions to the greatest extent possible The basic process of computation is fundamentally different in a FPL than in an imperative language –In an imperative language, operations are done and the results are stored in variables for later use –Management of variables is a constant concern and source of complexity for imperative programming In an FPL, variables are not necessary, as is the case in mathematics

CS 363 Spring 2005 GMU3 Fundamentals of Functional Programming Languages In an FPL, the evaluation of a function always produces the same result given the same parameters –This is called referential transparency

CS 363 Spring 2005 GMU4 Functions A function is a rule that associates to each x from some set X of values a unique y from another set Y of values: –y = f(x) or f: X  Y Functional Forms –Def: A higher-order function, or functional form, is one that either takes functions as parameters, yields a function as its result, or both domain range

CS 363 Spring 2005 GMU5 Lisp Lisp – based on lambda calculus (Church) –Uniform representation of programs and data using single general data structure (list) –Interpreter based (written in Lisp) –Automatic memory management –Evolved over the years –Dialects: COMMON LISP, Scheme

CS 363 Spring 2005 GMU6 Scheme (define (gcd u v) (if ( = v 0) u (gcd v (remainder u v)) ) (define (reverse l) (if (null? l) l (append (reverse (cdr l))(list (car l))) )

CS 363 Spring 2005 GMU7 Introduction to Scheme A mid-1970s dialect of LISP, designed to be a cleaner, more modern, and simpler version than the contemporary dialects of LISP Uses only static scoping Functions are first-class entities –They can be the values of expressions and elements of lists –They can be assigned to variables and passed as parameters

CS 363 Spring 2005 GMU8 Scheme (define (gcd u v) (if ( = v 0) u (gcd v (remainder u v)) ) Once defined in the interpreter:  (gcd 25 10)  5

CS 363 Spring 2005 GMU9 Scheme Syntax expression  atom | list atom  number | string | identifier | character | boolean list  ‘(‘ expression-sequence ‘)’ expression-sequence  expression | expression-sequence expression simplest elements

CS 363 Spring 2005 GMU10 Scheme atoms Constants: –numbers, strings, #T = True, … Identifier names: Function/operator names –pre-defined & user defined

CS 363 Spring 2005 GMU11 Scheme Expression vs. C In Scheme: (+ 3 (* 4 5 )) (and (= a b)(not (= a 0))) (gcd 10 35) In C: * 5 (a = = b) && (a != 0) gcd(10,35)

CS 363 Spring 2005 GMU12 Evaluation Rules for Scheme Expressions 1.Constant atoms (numbers, strings) evaluate to themselves 2.Identifiers are looked up in the current environment and replaced by the value found there (using dynamically maintained symbol table) 3.A list is evaluated by recursively evaluating each element in the list as an expression; the first expression must evaluate to a function. The function is applied to the evaluated values of the rest of the list.

CS 363 Spring 2005 GMU13 Scheme Evaluation To evaluate (* (+ 2 3)(+ 4 5)) 1.* is the function – must evaluate the two expressions (+ 2 3) and (+ 4 5) 2.To evaluate (+ 2 3) 1.+ is the function – must evaluation the two expressions 2 and evaluates to the integer evaluates to the integer = 5 3.To evaluate (+ 4 5) follow similar steps 4.* 5 9 = 45 *

CS 363 Spring 2005 GMU14 Scheme Conditionals If statement: (if ( = v 0) u (gcd v (remainder u v)) ) (if (= a 0) 0 (/ 1 a)) Cond statement: (cond (( = a 0) 0) ((= a 1) 1) (else (/ 1 a)) ) Both if and cond functions use delayed evaluation for the expressions in their bodies (i.e. (/ 1 a) is not evaluated unless the appropriate branch is chosen).

CS 363 Spring 2005 GMU15 Example of COND (DEFINE (compare x y) (COND ((> x y) (DISPLAY “x is greater than y”)) ((< x y) (DISPLAY “y is greater than x”)) (ELSE (DISPLAY “x and y are equal”)) )

CS 363 Spring 2005 GMU16 Predicate Functions 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () –Note that if EQ? is called with list parameters, the result is not reliable –EQ? does not work for numeric atoms (use = )

CS 363 Spring 2005 GMU17 Predicate Functions 2. LIST? takes one parameter; it returns #T if the parameter is a list; otherwise () 3. NULL? takes one parameter; it returns #T if the parameter is the empty list; otherwise () Note that NULL? returns #T if the parameter is () 4. Numeric Predicate Functions =, <>, >, =, <=, EVEN?, ODD?, ZERO?, NEGATIVE?

CS 363 Spring 2005 GMU18 let function Allows values to be given temporary names within an expression –(let ((a 2 ) (b 3)) ( + a b)) –5 Semantics: Evaluate all expressions, then bind the values to the names; evaluate the body

CS 363 Spring 2005 GMU19 Quote (‘) function A list that is preceeded by QUOTE or a quote mark (‘) is NOT evaluated. QUOTE is required because the Scheme interpreter, named EVAL, always evaluates parameters to function applications before applying the function. QUOTE is used to avoid parameter evaluation when it is not appropriate –QUOTE can be abbreviated with the apostrophe prefix operator Can be used to provide function arguments –(myfunct ‘(a b) ‘(c d))

CS 363 Spring 2005 GMU20 Output functions Output Utility Functions: (DISPLAY expression) (NEWLINE)

CS 363 Spring 2005 GMU21 define function Form 1: Bind a symbol to a expression: (define a 2) (define emptylist ‘( )) (define pi )

CS 363 Spring 2005 GMU22 define function Form 2: To bind names to lambda expressions define (cube x) (* x (* x x )) ) (define (gcd u v) (if ( = v 0) u (gcd v (remainder u v)) ) function name and parameters function body – lambda expression

CS 363 Spring 2005 GMU23 Function Evaluation Evaluation process (for normal functions): 1. Parameters are evaluated, in no particular order 2. The values of the parameters are substituted into the function body 3. The function body is evaluated 4. The value of the last expression in the body is the value of the function (Special forms use a different evaluation process)

CS 363 Spring 2005 GMU24 Data Structures in Scheme: Box Notation for Lists first element (car)rest of list (cdr) 1 List manipulation is typically written using ‘car’ and ‘cdr’

CS 363 Spring 2005 GMU25 Data Structures in Scheme 1 23 (1,2,3) c da b ((a b) c (d))

CS 363 Spring 2005 GMU26 Basic List Manipulation ( car L) – returns the first element of L ( cdr L) – returns L minus the first element (car ‘(1 2 3)) = 1 (car ‘((a b)(c d))) = (a b) (cdr ‘(1 2 3)) = (2 3) (cdr ‘((a b)(c d))) = ((c d))

CS 363 Spring 2005 GMU27 Basic List Manipulation ( list e1 … en) – return the list created from the individual elements ( cons e L) – returns the list created by adding expression e to the beginning of list L (list 2 3 4) = (2 3 4) (list ‘(a b) x ‘(c d) ) = ((a b)x(c d)) (cons 2 ‘(3 4)) = (2 3 4) (cons ‘((a b)) ‘(c)) = (((a b)) c)

CS 363 Spring 2005 GMU28 Example Functions 1. member - takes an atom and a simple list; returns #T if the atom is in the list; () otherwise (DEFINE (member atm lis) (COND ((NULL? lis) '()) ((EQ? atm (CAR lis)) #T) ((ELSE (member atm (CDR lis))) ))

CS 363 Spring 2005 GMU29 Example Functions 2. equalsimp - takes two simple lists as parameters; returns #T if the two simple lists are equal; () otherwise (DEFINE (equalsimp lis1 lis2) (COND ((NULL? lis1) (NULL? lis2)) ((NULL? lis2) '()) ((EQ? (CAR lis1) (CAR lis2)) (equalsimp(CDR lis1)(CDR lis2))) (ELSE '()) ))

CS 363 Spring 2005 GMU30 Example Functions 3. equal - takes two general lists as parameters; returns #T if the two lists are equal; () otherwise (DEFINE (equal lis1 lis2) (COND ((NOT (LIST? lis1))(EQ? lis1 lis2)) ((NOT (LIST? lis2)) '()) ((NULL? lis1) (NULL? lis2)) ((NULL? lis2) '()) ((equal (CAR lis1) (CAR lis2)) (equal (CDR lis1) (CDR lis2))) (ELSE '()) ))

CS 363 Spring 2005 GMU31 Example Functions (define (count L) (if (null? L) 0 (+ 1 (count (cdr L))) ) (count ‘( a b c d)) = (+ 1 (count ‘(b c d))) = (+ 1 (+ 1(count ‘(c d)))) (+ 1 (+ 1 (+ 1 (count ‘(d)))))= (+ 1 (+ 1 (+ 1 (+ 1 (count ‘())))))= (+ 1 (+ 1 (+ 1 (+ 1 0))))= 4

CS 363 Spring 2005 GMU32 Scheme Functions Now define (define (count1 L) ?? ) so that (count1 ‘(a (b c d) e)) = 5

CS 363 Spring 2005 GMU33 Scheme Functions This function counts the individual elements: (define (count1 L) (cond ( (null? L) 0 ) ( (list? (car L)) (+ (count1 (car L))(count1 (cdr L)))) (else (+ 1 (count (cdr L)))) ) so that (count1 ‘(a (b c d) e)) = 5

CS 363 Spring 2005 GMU34 Example Functions (define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M)) ) (append ‘(a b) ‘(c d)) = (a b c d)

CS 363 Spring 2005 GMU35 How does append do its job? (define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M)))) (append ‘(a b) ‘(c d)) = (cons a (append ‘(b) ‘(c d))) =

CS 363 Spring 2005 GMU36 How does append do its job? (define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M)))) (append ‘(a b) ‘(c d)) = (cons a (append ‘(b) ‘(c d))) = (cons a (cons b (append ‘() ‘(c d)))) =

CS 363 Spring 2005 GMU37 How does append do its job? (define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M)))) (append ‘(a b) ‘(c d)) = (cons a (append ‘(b) ‘(c d))) = (cons a (cons b (append ‘() ‘(c d)))) = (cons a (cons b ‘(c d))) =

CS 363 Spring 2005 GMU38 How does append do its job? (define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M)))) (append ‘(a b) ‘(c d)) = (cons a (append ‘(b) ‘(c d))) = (cons a (cons b (append ‘() ‘(c d)))) = (cons a (cons b ‘(c d))) = (cons a ‘(b c d)) =

CS 363 Spring 2005 GMU39 How does append do its job? (define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M)))) (append ‘(a b) ‘(c d)) = (cons a (append ‘(b) ‘(c d))) = (cons a (cons b (append ‘() ‘(c d)))) = (cons a (cons b ‘(c d))) = (cons a ‘(b c d)) = (a b c d)

CS 363 Spring 2005 GMU40 Reverse Write a function that takes a list of elements and reverses it: (reverse ‘( )) = ( )

CS 363 Spring 2005 GMU41 Reverse (define (reverse L) (if (null? L) ‘() (append (reverse (cdr L))(list (car L))) )

CS 363 Spring 2005 GMU42 Selection Sort in Scheme Let’s define a few useful functions first: (DEFINE (findsmallest lis small) (COND ((NULL? lis) small) ((< (CAR lis) small) (findsmallest (CDR lis) (CAR lis))) (ELSE (findsmallest (CDR lis) small)) )

CS 363 Spring 2005 GMU43 Selection Sort in Scheme (DEFINE (remove lis item) (COND ((NULL? lis) ‘() ) ((= (CAR lis) item) lis) (ELSE (CONS (CAR lis) (remove (CDR lis) item))) ) Cautious programming! Assuming integers

CS 363 Spring 2005 GMU44 Selection Sort in Scheme (DEFINE (selectionsort lis) (IF (NULL? lis) lis (LET ((s (findsmallest (CDR lis) (CAR lis)))) (CONS s (selectionsort (remove lis s)) ) )

CS 363 Spring 2005 GMU45 Higher order functions Def: A higher-order function, or functional form, is one that either takes functions as parameters, yields a function as its result, or both Mapcar Eval

CS 363 Spring 2005 GMU46 Higher-Order Functions: mapcar Apply to All - mapcar - Applies the given function to all elements of the given list; result is a list of the results (DEFINE (mapcar fun lis) (COND ((NULL? lis) '()) (ELSE (CONS (fun (CAR lis)) (mapcar fun (CDR lis)))) ))

CS 363 Spring 2005 GMU47 Higher-Order Functions: mapcar Using mapcar: (mapcar (LAMBDA (num) (* num num num)) ‘( )) returns ( )

CS 363 Spring 2005 GMU48 Higher Order Functions: EVAL It is possible in Scheme to define a function that builds Scheme code and requests its interpretation This is possible because the interpreter is a user-available function, EVAL

CS 363 Spring 2005 GMU49 Using EVAL for adding a List of Numbers Suppose we have a list of numbers that must be added: (DEFINE (adder lis) (COND((NULL? lis) 0) (ELSE (+ (CAR lis) (adder(CDR lis )))) )) Using Eval ((DEFINE (adder lis) (COND ((NULL? lis) 0) (ELSE (EVAL (CONS '+ lis))) )) (adder ‘( )) Returns 24

CS 363 Spring 2005 GMU50 Other Features of Scheme Scheme includes some imperative features: 1. SET! binds or rebinds a value to a name 2. SET-CAR! replaces the car of a list 3. SET-CDR! replaces the cdr part of a list

CS 363 Spring 2005 GMU51 COMMON LISP A combination of many of the features of the popular dialects of LISP around in the early 1980s A large and complex language – the opposite of Scheme

CS 363 Spring 2005 GMU52 COMMON LISP Includes: –records –arrays –complex numbers –character strings –powerful I/O capabilities –packages with access control –imperative features like those of Scheme –iterative control statements

CS 363 Spring 2005 GMU53 ML A static-scoped functional language with syntax that is closer to Pascal than to LISP Uses type declarations, but also does type inferencing to determine the types of undeclared variables It is strongly typed (whereas Scheme is essentially typeless) and has no type coercions Includes exception handling and a module facility for implementing abstract data types

CS 363 Spring 2005 GMU54 ML Includes lists and list operations The val statement binds a name to a value (similar to DEFINE in Scheme) Function declaration form: fun function_name (formal_parameters) = function_body_expression; e.g., fun cube (x : int) = x * x * x;

CS 363 Spring 2005 GMU55 Haskell Similar to ML (syntax, static scoped, strongly typed, type inferencing) Different from ML (and most other functional languages) in that it is purely functional (e.g., no variables, no assignment statements, and no side effects of any kind)

CS 363 Spring 2005 GMU56 Haskell Most Important Features –Uses lazy evaluation (evaluate no subexpression until the value is needed) –Has list comprehensions, which allow it to deal with infinite lists

CS 363 Spring 2005 GMU57 Haskell Examples 1. Fibonacci numbers (illustrates function definitions with different parameter forms) fib 0 = 1 fib 1 = 1 fib (n + 2) = fib (n + 1) + fib n

CS 363 Spring 2005 GMU58 Haskell Examples 2. Factorial (illustrates guards) fact n | n == 0 = 1 | n > 0 = n * fact (n - 1) The special word otherwise can appear as a guard

CS 363 Spring 2005 GMU59 Haskell Examples 3. List operations –List notation: Put elements in brackets e.g., directions = [“north”, “south”, “east”, “west”] –Length: # e.g., #directions is 4 –Arithmetic series with the.. operator e.g., [2, 4..10] is [2, 4, 6, 8, 10]

CS 363 Spring 2005 GMU60 Haskell Examples 3. List operations (cont) –Catenation is with ++ e.g., [1, 3] ++ [5, 7] results in [1, 3, 5, 7] –CONS, CAR, CDR via the colon operator (as in Prolog) e.g., 1:[3, 5, 7] results in [1, 3, 5, 7]

CS 363 Spring 2005 GMU61 Haskell Examples Quicksort: sort [] = [] sort (a:x) = sort [b | b ← x; b <= a] ++ [a] ++ sort [b | b ← x; b > a]

CS 363 Spring 2005 GMU62 Applications of Functional Languages LISP is used for artificial intelligence –Knowledge representation –Machine learning –Natural language processing –Modeling of speech and vision Scheme is used to teach introductory programming at a significant number of universities

CS 363 Spring 2005 GMU63 Comparing Functional and Imperative Languages Imperative Languages: –Efficient execution –Complex semantics –Complex syntax –Concurrency is programmer designed Functional Languages: –Simple semantics –Simple syntax –Inefficient execution –Programs can automatically be made concurrent