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NCCU Programming Languages 程式語言原理 Spring 2006 Lecture 7: Lisp, Functional Programming in Scheme 1 Outline • McCarthy’s Original Paper of Lisp: – Recursive Functions of Symbolic Expressions and Their computation by Machine, Part 1,” 1960 課程網站 資源區下載閱讀 (Part 2 was never published) • Functional Programming in Scheme, I – eBook,課程網站 軟體區 2 Recursive Functions of Symbolic Expressions and Their Application, Part I J OHN M C C ARTHY Review: Amit Kirschenbaum Seminar in Programming Languages Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 1/43 Historical Background LISP ( LISt Processor ) is the second oldest programming language and is still in widespread use today. Defined by John McCarthy from M.I.T. Development began in the 1950s at IBM as FLPL Fortran List Processing Language. Implementation developped for the IBM 704 computer by the A.I. group at M.I.T. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 2/43 Historical Backgroung (Cont’d) “ The main requirement was a programming system for manipulating expressions representing formalized declerative and imperative sentences so that the Advice Taker could make deductions.” Many dialects have been developed from LISP: Franz Lisp, MacLisp, ZetaLisp . . . Two important dialects Common Lisp - ANSI Standard Scheme - A simple and clean dialect. Will be used in our examples. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 3/43 Imperative Programming Program relies on modfying a state, using a sequence of commands. State is mainly modified by assignment Commands can be executed one after another by writing them sequentially. Commands can be executed conditinonally using if and repeatedly using while Program is a series of instructions on how to modify the state. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 4/43 Imperative Prog. (Cont’d) Execution of program can be considered, abstractly as: s0 → s1 · · · → sn Program starts at state s0 including inputs Program passes through a finite sequence of state changes,by the commands, to get from s0 to sn Program finishes in sn containing the outputs. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 5/43 Functional Programming A functional program is an expression, and executing a program means evaluating the expression. There is no state, meaning there are no variables. No assignments, since there is nothing to assign to. No sequencing. No repetition but recursive functions instead. Functions can be used more flexibly. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 6/43 Why use it? At first glance, a language without variables, assignments and sequencing seems very impractical Imperative languages have been developed as an abstraction of hardware from machine-code to assembler to FORTRAN and so on. Maybe a different approach is needed i.e, from human side. Perhaps functional languages are more suitable to people. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 7/43 Advantages of functional programming Clearer sematics. Programs correspond more directly to mathematical objects. More freedom in implementation e.g, parallel programs come for free. The flexible use of functions we gain elegance and better modularity of programs. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 8/43 Some Mathematical Concepts Partial Function - function that is defined only of part of its domain. Propositonal Expressions and Predicates Expressions whose possible values are T (truth) and F (false). Conditional Expressions - Expressing the dependence of quantties on propositional quantities. Have the form (p1 → e1 , · · · , pn → en ) Equivalent to “If p1 then e1 , else if p2 then e2 , · · · else if pn then en ” Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 9/43 Mathematical Concepts (Cont’d) Conditional expression can define noncommutative propositional connectives: p ∧ q = (p → q, T → F ) p ∨ q = (p → T, T → q) ¬p = (p → F, T → T ) Recursive function definitions - Using conditional expressions, we can define recursive functions n! = (n = 0 → 1, T → n · (n − 1)!) Functions are defined and used, using λ-notation. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 10/43 Brief intro to λ-calculus A formal system designed to investigate function definition function application recursion Can be called the smallest universal programming language. It is universal in the sense that any computable function can be expressed within this formalism. Thus, it is equivalent in expressive power to Turing machines. λ-calculus was developed by Alonzo Church in the 1930s Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 11/43 λ-notation Defining a function in mathematics means: Giving it a name. The value of the function is an expression in the formal arguments of the function. e.g., f (x) = x + 1 Using λ-notation we express it as a λ-expression λx . (+ x 1) It has no name. prefix notation is used. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 12/43 λ-notation (Cont’d) The function f may be applied to the argument 1 : f (1) Similarly, the λ-expression may be applied to the argument 1 (λx . (+ x 1))1 Application here means Subtitue 1 for x: (+ x 1) ⇒ (+ 1 1) Evaluate the function body: make the addition operation . Return the result: 2 Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 13/43 Syntax of λ-calculus Pure λ-calculus contains just three kinds of expressions variables (identifiers) function applications λ-abstractions (function defintions) It is convinient to add predefined constants (e.g., numbers) and operations (e.g., arithmetic operators) hexpi ::= | | | var const (hexpi hexpi) (λ var . hexpi) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 14/43 Function Application Application is of the form (E1 E2 ) E1 is expected to be evaluated to a function. The function may be either a predefined one or one defined by a λ-abstraction. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 15/43 λ-abstractions The expression (λx . (∗ x 2)) is the function of x which multiplies x by 2 The part of the expression that occurs after λx is called the body of the expression. When application of λ-abstraction occurs, we return the result of the body evaluation. The body can be any λ-expression, therefore it may be a λ-abstraction. The parameter of λ-abstraction can be a function itself Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 16/43 λ-abstractions In mathematics there are also functions which return functions as values and have function arguments. Usually they are called operators or f unctionals For example: the differentiation operator d 2 x = 2x dx . Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 17/43 Constants Pure λ-calculus doesn’t have any constants like 0, 1, 2, . . . or built in functions like +, −, ∗, . . ., since they can be defined by λ-expressions. For the purpose of this discussion we’ll assume we have them. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 18/43 Naming Expressions Expressions can be given names, for later reference: square ≡ (λx . (∗ x x)) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 19/43 Free and bound variables Consider the expression (λx . (∗ x y))2 x is bound: it is just the formal parameter of the function. y is free: we have to know its value in advance. A variable v is called bound in an expression E if there is some use of v in E that is bound by a decleration λv of v in E . A variable v is called f ree in an expression E if there is some use of v in E that is not bound by any decleration λv of v in E . Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 20/43 Reduction rule The main rule for simplifiying expressions in λ-calculus is called β -reduction. Applying a λ-abstraction to an argument is an instance of its body in which f ree occurences of the formal parameter are substituted by the argument. parameter may occur multiple times in the body (λx . (∗ x x))4 → (∗ 4 4) → 16 Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 21/43 Reduction rule Functions may be arguments (λf .(f 3))(λx .(− x 1)) (λf .(f 3))(λx .(− x 1)) → (λx .(− x 1))3 → (− 3 1) →2 Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 22/43 Expressions for Recursive Functions The λ-notation is inadequte for defining functions recursively the function n! = (n = 0 → 1, T → n · (n − 1)!) should be converted into ! = λ((n)(n = 0 → 1, T → n · (n − 1)!)) There is no clear reference from ‘!’ inside the λ-clause, to the expression as a whole. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 23/43 Expressions for Recursive Functions A new notation: label(a, E) denotes the expression E , provided that occurences of a within E are to be referred as a whole. For example, for the latter function the notation would be label(!, λ((n)(n = 0 → 1, T → n · (n − 1)!) (There is a way to describe recursion in λ-calculus, using Y-combinator, but McCarthy doesn’t use it.) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 24/43 S-Expressions A new class of Symbolic expressions. S-Expression are composed of the special characters ( - start of composed expression ) - end of composed expression • - composition and “an infinite set of distinguishable atomic symbols”. e.g., A ABA APPLE-PIE-NUMBER-3 Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 25/43 S-Expression : Definition Atomic symbols are S-expression. if e1 and e2 are S-expressions then so is (e1 · e2 ) examples AB (A· B) ((AB· C) · D) S-expression is then simply an ordered pair. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 26/43 S-Expressions: Lists • 基本資料結構:cons cells/dotted pairs “cons cell” A Dotted pair Cons[3,4] Æ (3.4) Constructor: cons cons[3, 4] cons[3, cons[4,5]] 3 4 5 (3.(4.5)) S-Expressions: Lists •一般資料結構中的串列由許多cons cells組成, 最後的元素為empty list •Empty list: null, nil, ‘() The empty list (a.k.a. null or nil) cons[1, cons[3,cons[2, nil]]] = (1.(3.(2.nil))) 1 3 2 S-Expression : Lists The list (m1 , m2 , . . . , mn ) is represented by the S-expression (m1 · (m2 · (· · · (mn · N IL) · · · ))) N IL is an atomic symbol, used to terminate lists, also known as the empty list. (m) stands for (m · N IL) (m1 , m2 , . . . , mn · x) stands for (m1 · (m2 · (· · · (mn · x) · · · ))) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 27/43 M-expressions Meta-expressions are functions of S-expressions, also called S-functions. Written in conventional functional notation. There are some elementry S-functions and predicates Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 28/43 M-expressions atom - atom[x] has the value T or F according to whether x is atomic symbol. atom[X] = T . atom[(X·A)] = F . eq - eq[x;y] is defined iff both x and y are symbols. eq[x;y] = T if x and y are the same symbol and eq[x;y] = F otherwise eq[X;X] = T . eq[X;A] = F . eq[X;(A · B)] is undefined Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 29/43 M-expressions car - car[x] is defined iff x is not atomic. car[(e1 · e2 )]=e1 car[(X·A)] = X. car[(X·A)· Y)]= (X·A). cdr - cdr[x] is also defined iff x is not atomic. cdr[(e1 · e2 )]=e2 cdr[(X·A)] = A. cdr[(X·A)· Y)]= Y. cons - cons[x;y] is defined for any x and y. It is the list constructor cons[(e1 ;e2 )]=(e1 · e2 ) cons[X;A] = (X·A). cons[(X·A);Y]= ((X·A)· Y). Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 30/43 M-expressions Compositions of car and cdr arise very frequently. Many expressions can be written more concisely if we abbreviate. cadr[x] ≡ car[cdr[x]] caddr[x] ≡ car[cdr[cdr[x]]] cdadr[x] ≡ cdr[car[cdr[x]]] expressions are not defined for every x. depends on the list structure. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 31/43 Recursive S-functions Forming new functions of S-expression by conditional expression and recursive definition gives us much larger class of functions. In fact all computable functions. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 32/43 Recursive S-function examples ff[x] - returns the first atomic symbol of the S-expression x, ignoring the parentheses. ff[x] = [atom[x]→x ; T → ff[car[x]]] ff[(A·B)] = [atom[(A·B)]→ (A·B) ; T → ff[car[(A·B)]]] = [F →(A·B);T → ff[car[(A·B)]]] = ff[car[(A·B)]] = ff[A] = ff[atom[A]→A ; T →ff[car[A]]] = [T →A ; T →ff[car[A]]] =A Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 33/43 Transform M-expressions to S-expressions There is a transformation mechanism that translate an M-expression E into S-expression E ∗ if E is an S-expression, E ∗ is (QUOTE E ). M-expression f [e1 ; . . . ; en ] is translated to (f ∗ e∗1 . . . e∗n ). Thus, {cons[A; B]}∗ is (CONS (QUOTE A) (QUOTE B)) {[p1 → e1 ]; . . . ; [pn → en ]}∗ is (COND (p∗1 e∗1 ) . . . (p∗n e∗n )) {λ[x1 ; . . . ; xn ]E}∗ is (LAMBDA(x1 . . . xn ) E ∗ . {label[a; E]}∗ is (LABEL a E ∗ ) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 34/43 What do we gain? Unifying Symbol-level and Meta-level, gives us a way to treat expressions over symbols exactly the same as symbols. Functions and data are the same. Thus we can write a program, which write another program and evaluating it. This is useful in AI. Furthermore, we can expand the language with new features. LISP interpreters are easily implemented in LISP. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 35/43 S-function apply “Plays the theoretical role of a universal Turing machine and the practical role of an interpreter”. Formally, If f is an S-expression for an S-function f 0 and args is a list of arguments of the form (arg1 , . . . , argn ) where arg1 , . . . , argn are S-expressions, Then apply[f ; args] and f 0 [arg1 , . . . , argn ] are defined for the same values of arg1 , . . . , argn and are equal when defined. example: λ[[x; y]; cons[car[x]; y]] [(A, B); (C, D)] ≡ apply[(LAMBDA, (X, Y )(CONS(CARX)Y ))((A B)(C D))] = (A C D) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 36/43 S-function eval serves both as a formal definition of the language and as an interpreter Before apply applies the function f on the list of arguments (arg1 , . . . , argn ), it sends them to eval for evaluating the S-expressions which represents them. > (eval ’(lambda (x) (+ x 1))) #<procedure> ’(lambda (x) (+ x 1))) is an S-expression which repersents a function. eval evalutes it and return its value, which is indeed a function Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 37/43 Implementing eval (define (eval exp env) (cond ((self-evaluating? exp) exp) ((variable? exp) (lookup-variable-value exp env)) ((quoted? exp) (text-of-quotation exp)) ((assignment? exp) (eval-assignment exp env)) ((definition? exp) (eval-definition exp env)) ((if? exp) (eval-if exp env)) ((lambda? exp) (make-procedure (lambda-parameters exp) (lambda-body exp) env)) ((begin? exp) (eval-sequence (begin-actions exp) env)) ((cond? exp) (eval (cond->if exp) env)) ((application? exp) (apply (eval (operator exp) env) (list-of-values (operands exp) env))) (else (error "Unknown expression type -- EVAL" exp)))) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 38/43 Strength of the mechanism Extending the language is done easily by adding required forms to eval. Just add syntax and evaluation rules. Paraphrasing Oscar Wilde: LISP programmers know the value of everything but the cost of nothing. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 39/43 The cost Performance of LISP systems became a growing issue Garbage Collection. Representation of internal structures. Became difficult to run on the memory-limited hardware of that time. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 40/43 LISP Machines The solution was LISP machine - a computer which has been optimized to run LISP efficiently and provide a good environment for programming in it. Typical optimizations to LISP machines Fast function calls. Efficient representation of lists. Hardware garbage collection. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 41/43 LISP in the real world de-facto standard in AI NLP Modelling speech and vision Some more AutoCAD Yahoo Store Emacs Mirai, the 3d animation package was used to create Gollum in Lord Of The Rings. Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 42/43 The End ((lambda(x)(x x)) (lambda(x)(x x))) Recursive Functions of Symbolic Expressions and Their Application, Part I – p. 43/43 Scheme • • • • • • • : an Interpreter for Extended Lambda Calculus Developed in 1975 by G. Sussman and G. Steele A version of LISP Simple syntax, small language Closer to initial semantics of LISP Provides basic list processing tools Allows functions to be first-class values Provides some support for lazy evaluation 3 The Structure of a Scheme Program • • • • All programs and data are expressions (S-expr) Expressions can be atoms or lists Atom: number, string, identifier, character, boolean List: sequence of expressions separated by spaces, between parentheses • Syntax: expression → atom | list atom → number | string | identifier | character | boolean list → ( expr_seq ) expr_seq → expression expr_seq | expression 4 S-expressions • In LISP/Scheme, data & programs are all of the same form: S-expressions (Symbolic-expressions) – an S-expression is either an atom or a list (ignore dotted pairs) Atoms numbers characters strings Booleans symbols 4 3.14 1/2 #xA2 #\a #\Q #\space #\tab "foo" “Hello Hi" #t #f Dave num123 #b1001 "@%!?#" miles->km !_^_! symbols are sequences of letters, digits, and "extended alphabetic characters" + - . * / < > = ! ? : $ % + & ~ ^ can't start with a digit, case insensitive 5 S-expressions (cont.) • Lists () (L1 L2 . . . Ln) is a list is a list, where each Li is either an atom or a list for example: • () • (a b c d) • (((((a))))) (a) ((a b) c (d e)) 6 A List is a recursive structure! • (cdr a-list) evaluates also to a list! Data Definition of a list: Self-referential data structure! A list consists of: 1) The empty list, empty, or 2) (cons z loz) where z is a value and loz is a list. … <x1> first (car) <x2> <xn> Rest (cdr) is also a list 7 List manipulation in Lisp & Scheme • Three primitives and one constant – get head of list: car – get rest of list: cdr – add an element to a list: cons – null list: nil or () • Add equality ( = or eq?), conditional expression, and recursion, and this is a universal model of computation • “quote” for distinguishing data from programs. 8 Interacting with Scheme: Evaluation • Interpretor: "read-eval-print" loop >1 1 Reads 1, evaluates it (1 evaluates to itself), then prints its value > (+ 2 3) 5 + => function + (or called procedure) 2 => 2 3 => 3 Applies function + on operands 2 and 3 => 5 9 Evaluation • Constant atoms - evaluate to themselves 42 3.14 "hello" #/a #t - a number - another number - a string - character 'a' - boolean value "true" > "hello world" "hello world" 10 Evaluation • Identifiers (symbols) - evaluate to the value bound to them (define a 7) >a 7 >+ #<procedure +> 11 Evaluation • Lists - evaluate as "function calls": (function arg1 arg2 arg3 ...) • First element must evaluate to a function (procedure) • Recursively evaluate each argument • Apply the function on the evaluated arguments > (- 7 1) 6 > (* (+ 2 3) (/ 6 2)) 15 12 Operators - More Examples • Prefix notation • Any number of arguments > (+) 0 > (+ 2 3 4) 9 > (+ 2) 2 > (- 10 7 2) 1 > (+ 2 3) 5 > (/ 20 5 2) 2 13 Preventing Evaluation (quote) • Evaluate the following: > (1 2 3) Error: attempt to apply non-procedure 1. • Use the quote to prevent evaluation: > (quote (1 2 3)) (1 2 3) • Short-hand notation for quote: > '(1 2 3) (1 2 3) 14 More Examples (define a 7) a 'a => 7 => a (+ 2 3) '(+ 2 3) ((+ 2 3)) => 5 => (+ 2 3) => Error: attempt to apply non-procedure 5. '(her 3 "sons") => (her 3 "sons") • Make a list: (list 'her (+ 2 1) "sons") (list '(+ 2 1) (+ 2 1)) => => (her 3 "sons") ((+ 2 1) 3) 15 Forcing Evaluation (eval) (+ 1 2 3) '(+ 1 2 3) (eval '(+ 1 2 3)) => 6 => (+ 1 2 3) => 6 (list + 1 2) (eval (list + 1 2)) => (+ 1 2) => 3 • Eval evaluates its single argument • Eval is implicitly called by the interpretor to evaluate each expression entered: “read-eval-print” loop 16 Special Forms • Have different rules regarding whether/how arguments are evaluated (define a 5) ; binds a to 5, does not evaluate a a 5 (quote (1 2 3)) ; does not evaluate (1 2 3) • There are a few other special forms – discussed later 17 Using Scheme: DrScheme an IDE for Scheme How to Design Programs (methodology) 課程網站下載 Windows 版 Scheme (language) DrScheme (environment) 18 Scheme Programming Environment DrScheme http://www.drscheme.org/ 定義框 求值框 19 DrScheme : Languages->Standard Scheme 20 List Operations • cons – returns a list built from head and tail (cons 'a '(b c d)) (cons 'a '()) (cons '(a b) '(c d)) (cons 'a (cons 'b '())) => (a b c d) => (a) => ((a b) c d) => (a b) (cons 'a 'b) => (a . b) ; dotted pair, improper list 21 List Operations • car – returns first member of a list (head) (car (car (car (car '(a b c d)) => a '(a)) => a (a b) => '((a b) c d)) '(this (is no) more difficult)) => this • cdr – returns the list without its first member (tail) (cdr '(a b c d)) (cdr '(a b)) (cdr '(a)) (cdr '(a (b c))) => => => => (b c d) (b) () ((b c)) (car (cdr (cdr '(a b c d)))) (caddr '(a b c d)) => c => c 22 List Operations • null? – returns #t if the list is null () #f otherwise • list – returns a list built from its arguments (list 'a 'b 'c) (list 'a) (list '(a b c)) (list '(a b) 'c) (list '(a b) '(c => => => => d)) (a b c) (a) ((a b c)) ((a b) c) => ((a b) (c d)) 23 List Operations • length – returns the length of a list (length '(1 3 5 7)) (length '((a b) c)) 2 => 4 => • reverse – returns the list reversed (reverse '(1 3 5 7)) (reverse '((a b) c)) => (7 5 3 1) => (c (a b)) • append – returns the concatenation of the lists received as arguments (append '(1 3 5) '(7 9)) => (1 3 5 7 9) (append '(a) '()) => (a) (append '(a b) '((c d) e)) => (a b (c d) e) 24 Type Predicates • Check the type of the argument and return #t or #f (boolean? x) (char? x) (string? x) (symbol? x) (number? x) (list? x) (procedure? x) ; is x a boolean? ; is x a char? ; is x a string? ; is x a symbol? ; is x a number? ; is x a list? ; is x a procedure (function)? 25 Boolean Expressions (< 1 2) (>= 3 4) (= 4 4) (eq? '(a b) '(a b)) (equal? '(a b) '(a b)) (not (> 5 6)) (and (< 3 4) (= 2 3)) (or (< 3 4) (= 2 3)) => #t => #f => #t => #f => #t ; same object? By reference ; recursively equivalent ; structure? By value => #t => #f => #t • and, or are special forms - evaluate arguments only while needed (short-circuited operations) – (or (> 2 1) (/ 5 0) => #t 26 Conditional Expressions • if – has the form: (if <test_exp> <then_exp> <else_exp>) (if (< 5 6) 1 2) => 1 (if (< 4 3) 1 2) => 2 • Anything other than #f is treated as true: (if 3 4 5) (if '() 4 5) => 4 => 4 ; as opposed to Lisp!! • if is a special form - evaluates its arguments only when needed: (if (= 3 4) 1 (2)) (if (= 3 3) 1 (2)) Error: => attempt to apply non-procedure 27 2. => 1 Conditional Expressions • cond – has the form: (cond (<test_exp1> <exp1> ...) (<test_exp2> <exp2> ...) ... (else <exp> ...)) (define n -5) (cond ((< n 0) "negative") ((> n 0) "positive") (else "zero")) => "negative" • cond is a special form - evaluates its arguments only 28 when needed Syntax (C vs. Scheme) C Scheme 1+2+3 3+4*5 factorial (9) (a == b) && (c != 0) (low < x) && (x < high) f (g(2,-1), 7) (+ 1 2 3) (+ 3 (* 4 5)) (factorial 9) (and (= a b) (not (= c 0))) (< low x high) (f (g 2 -1) 7) 29 Functions • Create a function by evaluating a lambda expression: (lambda (id1 id2 ...) exp1 exp2 ...) – – – – id1 id2 ... - formal parameters exp1 exp1 ... - body of the function return value of function - last expression in body return value of lambda expression - the (un-named) function (lambda (x) (* x x)) => #<procedure> – Returns an un-named function that takes a parameter and returns its square 30 Functions • Call a function by applying the evaluated lambda expression on its actual parameters: ((lambda (x) (* x x)) 3) => 9 the function the actual parameter • How can you reuse the function? – You can’t! • Why is it then useful? – Return a function from another function • What if you REALLY want to reuse it? 31 Functions • Bind a name to a function: (define square (lambda (x) (* x x))) square #<procedure> • Equivalent short-hand notation (typical way to use it): (define (square x) (* x x)) • Now call the function: (square 3) => 9 32 Functions • Functions vs. variables: (define f 3) f (f) (define (f) 3) f (f) => 3 => Error: attempt to apply non-procedure 3. => #<procedure> => 3 • Last definition is equivalent to: (define f (lambda () 3)); a function that takes no parameters and ; returns 3 33 Functions C Scheme if (a == 0) return f(x,y) ; else return g(x,y) ; (if (= a 0) (f x y) (g x y)) 34 Functions are First-Class Values C if (a == 0) return f(x,y) ; else return g(x,y) ; repeated arguments (x,y) • Can we write it better in Scheme? Scheme ((if (= a 0) f g) x y) evaluates to either #<procedure f> or #<procedure g> it is then applied on arguments x and 35 y Recursion • Recursion plays a greater role in Scheme than in other languages • Why? • Functional programming – avoid side effects (assignments) and iterations • Recursive data structures – a list is either empty, or has a car and a cdr; the cdr is (again) a list • Elegance – recursive algorithms are considered more elegant than iterative ones (just ask a Scheme or Lisp programmer!) 36 Recursion • How do you solve a problem recursively? • Do not rush to implement it • Think of a recursive way to describe the problem: – Show how to solve the problem in the general case, by decomposing it into similar, but smaller problems – Show how to solve the smallest version of the problem (the base case) • Now the implementation should be straightforward (in ANY language) – But don't forget to handle base case first when implementing 37 Recursion • How do you THINK recursively? • Example: define factorial factorial(n) = 1 * 2 * 3 * ...(n-1) * n factorial (n-1) 1 if n=1 (the base case) factorial(n) = n * factorial(n-1) otherwise (inductive step) 38 Recursion • Implement factorial in Scheme: (define (factorial n) (if (= n 1) 1 (* n (factorial (- n 1))))) (factorial 4) => 24 39 Recursion • Fibonacci: 0 fib(n) = if n = 0 1 if n = 1 fib(n-1) + fib(n-2) otherwise • Implement in Scheme: (define (fib n) (cond ((= n 0) 0) ((= n 1) 1) (else (+ (fib (- n 1)) (fib (- n 2)))))) 40 Recursion • Length of a list: 0 len(lst) = if list is empty 1 + len ( lst-without-first-element ) otherwise • Implement in Scheme: (define (len lst) (if (null? lst) 0 (+ 1 (len (cdr lst))))) 41 Recursion • Sum of elements in a list of numbers: 0 if list is empty first-element + sum (lst-without-first-element) otherwise sum(lst) = • Implement in Scheme: (define (sum lst) (if (null? lst) 0 (+ (car lst) (sum (cdr lst))))) 42 Recursion • Check membership in a list: (define (member? x lst) (cond ((null? lst) #f) ((equal? x (car lst)) #t) (else (member? x (cdr lst))))) 43 More recursive functions: append ¾ (define (append x y) (cond ((null? x) y) ((null? y) x) (else (cons (car x) (append (cdr x) y))) ) cdr-recursion ) (append ‘() ‘()) value: () (append ‘() ‘(1 2 3)) value: (1 2 3) (append ‘(1 2) ‘(3 (4) 5) ) value: (1 2 3 (4) 5) 44 Reverse a List, version 1 ¾ (define (reverse l) (cond ((null? l) l) (else (append (reverse (cdr l)) (list (car l)) ) ) cdr-recursion ) ) ¾ (reverse ‘(a b c d) ) value: (d c b a) ¾ (reverse ‘(a (2 b c) d)) value: (d (2 b c) a) ;; Not what we want 45 Atom Predicate • Write a function that takes a parameter x and returns #t if x is an atom, and false otherwise. Using cond: (define (atom? x) (cond ((symbol? x) #t) ((number? x) #t) ((char? x) #t) ((string? x) #t) ((null? x) #t) (else #f) ) ) 46 Reverse a List, version 2 (define (reverse l) (cond ((null? l) l) ((atom? (car l)) (append (reverse (cdr l)) (list (car l)))) (else (append (reverse (cdr l)) (list (reverse (car l)))) car-cdr-recursion ) )) ¾ (reverse ‘(a b c d) ) value: (d c b a) ¾ (reverse ‘(a (2 b c) d)) value: (d (c b 2) a) 47 Recursion: Counting Atoms • Count the number of atoms in a general list • >(count-atoms ‘(1 2 3)) => 3 • >(count-atoms ‘(1 (2 3) (4 (5)))) => 5 • Implement in Scheme: (define (count-atoms lst) (cond ((null? lst) 0) ((atom? lst) 1) (else (+ (count-atoms (car lst)) (count-atoms (cdr lst)))) )) car-cdr-recursion 48 Equality checking The eq? function only works for atoms, not for lists ¾ (eq? ‘() ‘()) value: #t ¾ (eq? (cons 1 ‘()) (cons 1 ‘()) ) value: #f Why? • (cons 1 ‘()) produces a new list • (cons 1 ‘()) produces another new list • different structures in memory: (eq? (cons 'a '()) (cons 'a '())) evaluates to #f. • eq? checks if two arguments are the same object • consider pointers or objects vs object references in Java • atoms and symbols are stored uniquely 49 Scheme: Equality Test, Revisited • the operators: =, eq?, eqv?, equal? ; – ‘=‘ tests sameness of numbers ; – eq? tests sameness of symbols ; note: each application of cons constructs a new cell ; • (eq? (cons 1 2) (cons 1 2)) returns #f! ; – eqv? tests sameness of numbers, symbols and booleans ; (as well as vectors, strings, and chars) ; – equal? is a universal test for sameness ; (tests all of the above and lists as well) ; • note: (equal? (cons 1 2) (cons 1 2)) returns #t! ; the difference is mainly one of efficiency ; use the predicate designed for the task at hand 50 Logical Procedures • “true” stands for #t; “false” stands for #f (‘()) • “not object” { “false” object” } – These procedures return #t if object is false; otherwise they return #f. • “and object …” – This procedure returns #t if none of its arguments are #f. Otherwise it returns #f. • “or object …” – This procedure returns #f if all of its arguments are #f. Otherwise it returns #t. ¾ The arguments of “or”, “and” are evaluated sequentially ¾ “or”: until a true value is found ¾ “and”: until a false value is found 51 Equality checking for lists For lists, need a comparison function that does a structural equality test: ¾ (define (equal? x y) (or (and (atom? x) (atom? y) (eq? x y) ) (and (and (not (atom? x)) (not (atom? y))) (equal? (car x) (car y)) (equal? (cdr x) (cdr y)) ) ) ) ¾ The arguments of “or”, “and” are evaluated sequentially ¾ “or”: until a true value is found ¾ “and”: until a false value is found 52 Repetition via recursion • pure LISP/Scheme does not have loops – repetition is performed via recursive functions (define (sum-1-to-N N) (if (< N 1) 0 (+ N (sum-1-to-N (- N 1))))) (define (my-member item lst) (cond ((null? lst) #f) ((equal? item (car lst)) lst) (else (my-member item (cdr lst))))) 53 Recursion vs. Iteration (Loops) Tail Recursion is as efficient as looping. 54 Just the factorial... (define (fact n) (if (= n 0) 1 (* n (fact (- n 1))) ) ) • Recursion and runtime stack 55 Time So to calculate... Stack (space) (fact 5) (* 5 (fact 4)) (* 5 (* 4 (fact 3))) (* 5 (* 4 (* 3 (fact 2)))) (* 5 (* 4 (* 3 (* 2 (fact 1))))) (* 5 (* 4 (* 3 (* 2 (* 1 (fact 0)))))) . . (define (fact n) (if (= n 0) . 1 (* n (fact (- n 1))) . ) 56 120 ) But we could use tail recursion •a tail-recursive function is one in which the recursive call occurs last (define (tailfact n result) // help function (if (= n 0) Accumulating parameter result (tailfact (- n 1) (* n result)))) (define (fact n) (tailfact n 1)) 57 Time Analysis (fact 5) (tailfact (tailfact (tailfact (tailfact (tailfact (tailfact 120 Stack 5 4 3 2 1 0 1) 5) 20) 60) 120) 120) 58 But how does it work? • Tail recursion requires two elements – The tail recursive module must terminate with a recursive call that leaves no work on the stack to finish up. Any storage then must be done in the parameter list as opposed to the stack – The interpreter or compiler must be designed to recognize tail recursion and handle it appropriately 59 Tail recursion is logically equivalent to a loop! Just put a goto in place of the recursive call!!! 60 Rewrite in C int tailfact(int n, int result) { if(n == 0) { Recall that this function return result; will be called like: } tailfact(5,1) else { return tailfact(n - 1, result * n); } 61 Put arg calcs into assignments int tailfact(int n, int result) { if(n == 0) return result; else { result = result * n; return tailfact(n - 1, result); } 62 Put arg calcs into assignments int tailfact(int n, int result) { if(n == 0) return result; else { result = result * n; n = n - 1; return tailfact(n, result); } 63 Substitute goto beginning in place of recursive call int tailfact(int n, int result) { beginning: if(n == 0) return result; else { result = result * n; n = n - 1; goto beginning; } 64 Eliminate the goto int tailfact(int n, int result) { while(1) { if(n == 0) return result; result = result * n; n = n - 1; } return ERROR; } 65 Nested Functions (define (factorial N) (define (factorial-help N value-so-far) (if (zero? N) value-so-far (factorial-help (- N 1) (* N value-so-far)) ) ) (factorial-help N 1) ) since factorial-help is defined inside of factorial, hidden to outside 66 Fibonacci numbers fib(n) = 0 =1 = fib(n-1) + fib(n-2) if n = 0 if n = 1 if n > 1 (define (fib n) (cond ((= n 0) 0) ((= n 1) 1) (else (+ (fib (- n 1)) (fib (- n 2)))))) 67 Tree Recursion (fib 5) (fib 3) (fib 1) Inefficient! (fib 4) (fib 2) (fib 2) (fib 3) (fib 0) (fib 1) (fib0)(fib 1) (fib 1) (fib 2) (fib 0)(fib 1) 68 A Tail-Recursive Fib Function (define (fib n) (ifib 1 0 n) ) (define (ifib next-fib cur-fib cnt) (if (= cnt 0) cur-fib (ifib (+ next-fib cur-fib) next-fib (- cnt 1) ) ) ) 69 Polymorphism Polymorphic functions can be applied to arguments of different types • function length is polymorphic: ¾ (length ‘(1 2 3)) value: 3 ¾ (length ‘(a b c)) value: 3 ¾ (length ‘((a) b (c d))) value: 3 • function zero? is not polymorphic (monomorphic): ¾ (zero? 10) value: #t ¾ (zero? ‘a) error: object a is not the correct type 70