Chapter 15 Functional Programming Languages

Copyright © 2007 Addison-Wesley. All rights reserved. 1–2 Introduction Design of imperative languages is based directly on the von Neumann architecture – Efficiency is the primary concern Design of the functional languages is based on mathematical functions – Solid theoretical basis – relatively unconcerned with the architecture of the machines on which programs will run

Copyright © 2007 Addison-Wesley. All rights reserved. 1–3 Mathematical Functions A mathematical function is a mapping of members of one set, called the domain set, to another set, called the range set A lambda expression specifies the parameter(s) and the mapping of a function in the following form (x) x * x * x for the function cube (x) = x * x * x

Copyright © 2007 Addison-Wesley. All rights reserved. 1–4 Fundamentals The objective is to mimic mathematical functions to the greatest extent possible The basic process of computation is fundamentally different than in an imperative language

Copyright © 2007 Addison-Wesley. All rights reserved. 1–5 Functional vs. Imperative 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 No assignment – The evaluation of a function always produces the same result given the same parameters (referential transparency)

Copyright © 2007 Addison-Wesley. All rights reserved. 1–6 LISP One of the first languages First functional language Designed for list processing applications The language wasn't standardized until very late so there are many dialects – We'll talk about a dialect called Scheme A mid-1970s dialect of LISP, designed to be a cleaner, more modern, and simpler version than the contemporary dialects of LISP Used for teaching – Common LISP is 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

Copyright © 2007 Addison-Wesley. All rights reserved. 1–7 Syntax Uses prefix notation Function applications and data have the same form. – If the list (A B C) is interpreted as data it is a simple list of three atoms, A, B, and C – If it is interpreted as a function application, it means that the function named A is applied to the two parameters, B and C Lambda notation is used to specify functions and function definitions.

Copyright © 2007 Addison-Wesley. All rights reserved. 1–8 Scheme 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 Dynamically typed Data object types: originally only atoms and lists List form: parenthesized collections of sublists and/or atoms (A B (C D) E) Lists are stored internally as single-linked lists

Copyright © 2007 Addison-Wesley. All rights reserved. 1–9 Running Scheme on onyx drscheme starts a GUI Scheme environment – Be sure to set Language to Standard (R5RS) scheme starts up MIT Scheme which runs in a command-line environment Both can be downloaded for free if you want them on your own computer

Copyright © 2007 Addison-Wesley. All rights reserved. 1–10 Scheme Program A sequence of expressions An expression can be either an atom (number or symbol) or a list – If it is a list, the first element is a function or a special form Anything following a semicolon is comment

Copyright © 2007 Addison-Wesley. All rights reserved. 1–11 Evaluation Parameters are evaluated, in no particular order The values of the parameters are substituted into the function body The function body is evaluated The value of the last expression in the body is the value of the function

Copyright © 2007 Addison-Wesley. All rights reserved. 1–12 Primitive Functions Arithmetic: +, -, *, /, abs, sqrt, remainder, min, max quote - takes one parameter; returns the parameter without evaluation –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 '(A B) is equivalent to (quote (A B))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–13 Special Forms Special forms are expressions that should not be evaluated in the usual order – function definitions – binding – control structures – and, or

Copyright © 2007 Addison-Wesley. All rights reserved. 1–14 Function Definition: lambda lambda expressions – Form is based on notation e.g., (lambda (x) (* x x)) x is called a bound variable lambda can be used to define anonymous functions lambda expressions can be applied ((lambda (x) (* x x)) 7)

Copyright © 2007 Addison-Wesley. All rights reserved. 1–15 Special Form Function: define A Function for Constructing Functions DEFINE - Two forms: 1.To bind a symbol to an expression (define pi ) Example use: (define two_pi (* 2 pi)) 1.To bind names to lambda expressions (define (square x) (* x x)) Example use: (square 5)

Copyright © 2007 Addison-Wesley. All rights reserved. 1–16 Output Functions (display expression) (newline)

Copyright © 2007 Addison-Wesley. All rights reserved. 1–17 Predicate Functions Predicate functions return either true or false –#T is true and #F or () is false Numeric =, <>, >, =, <= even?, odd?, zero?, negative? number? symbol? procedure? list? null?

Copyright © 2007 Addison-Wesley. All rights reserved. 1–18 Control Flow: if if for two-way selection needs a special form (if predicate then_exp else_exp) –Example (if (<> count 0) (/ sum count) 0)

Copyright © 2007 Addison-Wesley. All rights reserved. 1–19 Control Flow: cond cond for multiple Selection - the special form General form: (cond (predicate_1 expr {expr})... (predicate_1 expr {expr}) (else expr {expr})) Returns the value of the last expr in the first list whose predicate evaluates to true

Copyright © 2007 Addison-Wesley. All rights reserved. 1–20 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”)) )

Copyright © 2007 Addison-Wesley. All rights reserved. 1–21 List Building: cons and list cons takes two parameters First can be either an atom or a list The second should be a list Returns a new list with the first parameter inserted at the front of the second parameter (cons 'A '(B C)) returns (A B C) list takes any number of parameters; returns a list with the parameters as elements

Copyright © 2007 Addison-Wesley. All rights reserved. 1–22 List Decomposition: car and cdr car takes a list parameter; returns the first element of that list e.g., (car '(A B C)) yields A (car '((A B) C D)) yields (A B) cdr takes a list parameter; returns the list that remains after the first element is removed e.g., (cdr '(A B C)) yields (B C) (cdr '((A B) C D)) yields (C D)

Copyright © 2007 Addison-Wesley. All rights reserved. 1–23 Equality Testing: eq? 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 'B) yields #f – Note that if eq? is called with list parameters, the result is not reliable – Also eq? does not work for numeric atoms use = for numbers use eqv? for lists

Copyright © 2007 Addison-Wesley. All rights reserved. 1–24 Example Function: member 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) #f) ((eq? atm (car lis)) #t) ((else (member? atm (cdr lis))) ))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–25 Example Function: equalsimp equalsimp takes two simple lists as parameters; returns #T if the two simple lists are equal; #f otherwise (define (equalsimp lis1 lis2) (cond ((null? lis1) (null? lis2)) ((null? lis2) #f) ((eq? (car lis1) (car lis2)) ((equalsimp(cdr lis1)(cdr lis2))) (else '#f) ))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–26 Example Function: equal 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)) #f) ((null? lis1) (null? lis2)) ((null? lis2) #f) ((equal (car lis1) (car lis2)) (equal (cdr lis1) (cdr lis2))) (else #f) ))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–27 Example Function: append append takes two lists as parameters; returns the first parameter list with the elements of the second parameter list appended at the end (define (append lis1 lis2) (cond ((null? lis1) lis2) (else (cons (car lis1) (append (cdr lis1) lis2))) ))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–28 Special Form: let General form: (let ( (name_1 expression_1) (name_2 expression_2)... (name_n expression_n)) body ) Evaluate all expressions, then bind the values to the names; evaluate the body

Copyright © 2007 Addison-Wesley. All rights reserved. 1–29 let Example (define (quadratic_roots a b c) (let ( (root_part_over_2a (/ (sqrt (- (* b b) (* 4 a c)))(* 2 a))) (minus_b_over_2a (/ (- 0 b) (* 2 a))) (display (+ minus_b_over_2a root_part_over_2a)) (newline) (display (- minus_b_over_2a root_part_over_2a)) ))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–30 Scheme Functional Forms Composition – Create a new function that applies two functions one after the other – Example cddr is the composition of cdr with itself –(cdr (cdr ‘(A B C))) returns (C) Apply to All - one form in Scheme is mapcar – Applies the given function to all elements of the given list; (define (mapcar fun lis) (cond ((null? lis) '()) (else (cons (fun (car lis)) (mapcar fun (cdr lis))))))

Copyright © 2007 Addison-Wesley. All rights reserved. 1–31 Functions That Build Code 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

Copyright © 2007 Addison-Wesley. All rights reserved. 1–32 Adding a List of Numbers ((define (adder lis) (cond ((null? lis) 0) (else (eval (cons '+ lis))) )) The parameter is a list of numbers to be added; adder inserts a + operator and evaluates the resulting list – Use cons to insert the atom + into the list of numbers. – Be sure that + is quoted to prevent evaluation – Submit the new list to eval for evaluation

Copyright © 2007 Addison-Wesley. All rights reserved. Currying

Copyright © 2007 Addison-Wesley. All rights reserved. apply apply takes two parameters a function a list that could be the argument list for the function The result is what you would get from an expression that had the function at the beginning of the given list (apply + '( )) (apply car '((a b) (c d) (e f)))

Copyright © 2007 Addison-Wesley. All rights reserved. variable length argument lists Use a special notation to indicate variable length argument lists (define (f x y. z) ) z will be a list that contains all the arguments except the first two (define (g. z) ) g can be called with 0 or more arguments

Copyright © 2007 Addison-Wesley. All rights reserved. map Apply a specified operation to each element of a list Procedure needs to take a single argument (map sqrt '( )) -> ( ) More generally, if the procedure has multiple arguments, the arguments need to include one list per argument (map (lambda (x y) (+ x (* 2 y))) '(1 2 3) '(4 5 6))

Copyright © 2007 Addison-Wesley. All rights reserved. map-reduce

Copyright © 2007 Addison-Wesley. All rights reserved. Streams A stream is a sequence (possibly infinite) of values which allows access to initial values before later ones have been generated. They are sometimes called lazy lists. Finite streams can be implemented as lists.

Copyright © 2007 Addison-Wesley. All rights reserved. Uses for Streams A good example of an infinite stream is the data that comes from the keyboard. Random numbers – generally use one number at a time? – Is there any reason the entire sequence needs to be generated at one time. You can make a sieve to generate the sequence of prime numbers using streams.

Copyright © 2007 Addison-Wesley. All rights reserved. Stream ADT stream-car - takes a non-empty stream and returns the first element in the stream stream-cdr - takes a non-empty stream and returns a stream consisting of all but the first element as the original make-stream - make a stream from an initial value and a function of no arguments that can be used to generate the rest of the stream the-null-stream - an empty stream stream-null? - tests to see if a stream is empty

Copyright © 2007 Addison-Wesley. All rights reserved. How to create a stream A stream can be represented as a first element plus a procedure to generate the next element – Using a procedure to generate the rest of the stream allows evaluation to be delayed until more stream elements are needed. The procedure used to generate the rest of the stream is called a thunk.