Functional programming LISP Originally developed for symbolic computing

Functional programming: LISP • • • Originally developed for symbolic computing First interactive, interpreted language Dynamic typing: values have types, variables do not Garbage-collected Homogeneity of program and data Extensible: multiple dialects, implementations. Convenient testbed for language design experiments • Main descendants: Scheme (compact) Common Lisp (massive). Statically typed functional languages: ML

Uniform syntax: lists • • Expressions are either atoms or lists atoms are numeric or symbols lists nest, to form full trees Syntax is simple because programmer supplies what would otherwise be the internal representation of a program: (+ ( * 10, 12) (* 7, 11)) evaluates (10*12 + 7*11) • A program is a list: (define (factorial n) (if (eq n 0) 1 ( * n (factorial (- n 1))))

List manipulation • 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) and recursion, and this is a universal model of computation

Rule of evaluation • A number evaluates to itself • An atom evaluates to its current binding • A list is a computation. – Its first element must evaluate to an operation – the remaining elements are actual parameters – the result is the application of the operation to the evaluated actuals

Quoted data • If every list is a computation, how do we describe data? • Another primitive: quote (1 2 3 4)) => (1 2 3 4) (quote (this is a simple declarative sentence) => (this is a simple declarative sentence) ‘ (this also works) => (this also works)

Decomposing a list (car ‘(this is a list of symbols)) => this (cdr ‘(this is a list of symbols)) => (is a list of symbols) (cdr ‘(this that)) => (that) ; a list (cdr ‘(singleton)) => () (car ‘()) ; the empty list ; run time error

Building lists (cons ‘this ‘(that and the other)) => (this that and the other) (cons ‘a ‘()) => (a) useful shortcut: list (list ‘a ‘b ‘c ‘d ‘e) => (a b c d e) equivalent to (cons ‘a (cons ‘b (cons ‘c (cons ‘d (cons ‘e ‘())))))

Control structures • Conditional (if condition expr 1 expr 2) • Generalized form (cond (pred 1 expr 1) (pred 2 expr 2) … (else exprn) Needs special rule: evaluate only the successful entry if and cond are not regular functions

Function declarations (define (sqr n) (* n n)) • define is also special: body is not evaluated • defines produces a binding: sqr is bound to the body of the computation: (lambda (n) (* n n)) • define can produce value bindings as well: (define x 15) (sqr x) => 225

Recursion (define (add 1 x) (+ x 1)) ; the beginnings of Peano arithmetic (define (sub 1 x) (- x 1)) (define (add x y) (if (= y 0) x (add ( (add 1 x) (sub 1 y)))) (define (times x y) (cond ((= y 0) 0) ((= y 1) x) (else (add x (times (x (sub 1 y)))))))

Recursion (ii) (define (exp x y) (cond ((eq y 0) 1) (else (times x (exp x (sub 1 y)))))) better: (define (fast-exp x y) (cond (= y 0) 1) ( (even? y) (square (fast-exp x (/ y 2)))) (else (* x (fast-exp x (- y 1)))))) (define (even? n) (= (remainder n 2) 0)) ; defining a predicate

Recursion on lists (define (member elmt lis) (cond ((null? lis) ‘()) ((eq elmt (car lis)) lis) (else (member elmt (cdr lis))))) convention: return rest of list, starting from elmt, rather than #t or #f convention: every non-false value is true in a boolean context

Predicates • If variables are untyped, need run-time tests to determine kind: symbol? number? list? null? zero? Syntax conventions differ in different dialects: symbolp, numberp, listp, zerop. . .

Functional arguments (define (map fun lis) (cond ((null? lis) ‘()) (cons (fun (car lis)) (map fun (cdr lis))))) (map sqr ‘( 1 2 3 4)) => (1 16 81 256)

Self-definition (define (eval exp env) ; the lisp interpreter (cond ((number? exp) ; numbers are self-evaluating ((symbol? exp) (lookup exp env)) ; a symbol has a binding ((null? exp) ‘()) ((eq (car exp) ‘quote) (car (cdr exp))) ; could write cadr ((eq (car exp) ‘car) (car (cdr exp)))) ; caadr ((eq (car exp) ‘cdr) (cdr (car (cdr exp)))) : cdadr (else (apply (eval (car exp) env) ; apply function (eval-list (cdr exp) env))) ; to arguments

Function application (define (apply procedure arguments) (eval (procedure-body procedure) (extend-environment (parameters procedure) arguments (environment procedure)))) In words: add actuals to environment, evaluate body of procedure in new environment Note: environment is in of procedure definition (closure)

Environments • An environment describes the current bindings of symbols • A binding is a pair: (symbol value) • A frame is a list of bindings (activation record), i. e. an association list: ((s 1 v 1) (s 2 v 2)…) • An environment is a list of frames • In some cases we can treat the environment as a single association list (e. g. with dynamic binding)

Association lists (define (add-assoc symb val env) (cons (list symb val) env)) ; add a binding (define (name binding) (car binding)) ; for readability (define (value binding) (car (cdr binding))); (define (lookup symb env) ; sequential search (cond ((null? env) ‘()) ; error detected later ((eq symb (name (car env))) (value (car env)) (else (lookup symb (cdr env)))))

Procedures and their environment • A function has three components: a list of formals, a body, and environment of definition. Formals and body are represented by a lambda expression: (lambda (f 1 f 2. . ) expr) • A function is evaluated in its environment of definition (lexical scoping) after adding the current bindings of the actuals (activation record) • The definition must capture all three components (define (make-procedure spec body env) (list ‘procedure (make-lambda spec body) env)) make-procedure is called by eval to evaluate a define

Procedure components given the representation (procedure (lambda parameters body) env) the components of a procedure can be obtained as follows: (define (parameters proc) (cadr proc))) (define (procedure-body proc) (caddr (cadr proc))) (define (environment proc) (caddr proc))

Evaluating a function definition (define (name binding) (car binding)) • results in the binding: (name (procedure (lambda (binding) (car binding)) env)) • procedure and lambda are labels to indicate kind • Env is the current environment. At the top level it includes the binding for car (used in the body) as well as all other predefined symbols.

The need for closure (define (compose f 1 f 2) (lambda (x) (f 1 (f 2 x)))) (define (sqr x) (* x x)) (define fourth (compose (sqr sqr))) (fourth 4) ; yields 256 When fourth is called, the bindings of f 1 and f 2 have disappeared. The evaluation of the lambda captures the environment in which f 1 and f 2 are bound to sqr.

Binding formals to actuals (define (make-frame formals actuals) (cond (null? formals) ‘()) ; should check (null? actuals) also ( else (cons (add-assoc (car formals) (car actuals)) (make-frame (cdr formals) (cdr actuals)))))) (define (extend-environment formals actuals env) (cons (make-frame formals actuals) env)

Dynamic binding • Alternative definition for free variables: use most recent binding for that symbol. Useful in cases such as: (define (sum fun a next b) (if ( > a b) 0 (+ (fun a) (sum fun (next a) b)))) (define sum-powers a b n) (sum nth-power a incr b) ; could write (sum nth-power (lambda (x) (+ x 1)) b) (define (nth-power x) (expt x n)) ; ; WHAT ; n undefined if static binding. ; Bound in sum-powers if dynamic binding. N?

Implementing dynamic binding • Change eval to pass current environment to apply • change apply to use current environment rather than procedure environment: Eval: . . . (else (apply (eval (car exp) env) (eval-list (cdr exp) env)) Apply: (eval (procedure-body procedure) (extend-environment (parameters procedure) arguments env))) ; apply function ; to arguments ; in current env ; add bindings ; of formals ; to current env

Other binding constructs • Local variables (let ((x 5) (y 10)) (* x y)) (let* ((x 5) (y (expt x 2)) (* x y 3)) ; x and y bound simultaneously ; sequential bindings: y after x

Tail recursion • Tail recursive function needs single stack frame. mandated by semantics of Scheme: (define (factorial n) (if (zero? N) 1) (* n (factorial (- n 1))) define factorial n) (fact-iter 1 1 n) (define (fact-iter prod count var) (if (> count var) prod (fact-iter (* count prod) (+ count 1) var)))) ; stack grows to size n ; alternate definition ; tail recursion ; implemented as loop

Implementation of tail recursion • If last operation in function is recursive call, overwrite actuals and go to beginning of code: (define (last lis) (if (null? (cdr lis) (car lis)) (last (crd lis)))) ; can be done with loop (define (length lis) (if (null? lis) 0) (+ 1 (length (cdr lis)))) ; not tail recursive!