1. Functional programming: LISP Originally developed for symbolic computing Main motivation: include recursion (see McCarthy biographical excerpt on web site). First interactive, interpreted language Dynamic typing: values have types, variables do not Garbage-collected Ability to form program fragments and execute. Extensible: multiple dialects, implementations. Convenient testbed for language design experiments Main descendants: Scheme (compact) Common Lisp (massive). Statically typed functional languages: ML

2. 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))))

3. 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

4. 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

5. Quoted data If every list is a computation, how do we describe data? Another primitive: quote (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)

6. 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)) => () ; the empty list (car ‘()) ; run time error

7. 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 ‘())))))

8. Control structures Conditional (if condition expr1 expr2) Generalized form (cond (pred1 expr1) (pred2 expr2) … (else exprn) Needs special rule: evaluate only the successful entry if and cond are not regular functions

9. 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

10. Lambda or no lambda (define (sqr n) (* n n)) And (define sqr (lambda (n) (* n n))) Mean the same thing. The advantage of the lambda formulation is one can have nameless functions, e.g. ((lambda (x y) (+ x y)) 5 6) The lambda brackets the formal parameters.

11. Eval McCarthy had the idea: what if I could formulate some data and then execute it; (eval ‘(+ 5 7)) (eval '(define x 15)) ; creates the binding (x 15)

12. Recursion (define (add1 x) (+ x 1)) ; the beginnings of Peano arithmetic (define (sub1 x) (- x 1)) (define (add x y) (if (= y 0) x (add ( (add1 x) (sub1 y)))) (define (times x y) (cond ((= y 0) 0) ((= y 1) x) (else (add x (times (x (sub1 y)))))))

13. Recursion (ii) (define (exp x y) (cond ((eq y 0) 1) (else (times x (exp x (sub1 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

14. 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

15. 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...

16. Functional arguments (functional = a function that takes functions as arguments) (define (map fun lis) (cond ((null? lis) ‘()) (cons (fun (car lis)) (map fun (cdr lis))))) (map sqr (map sqr ‘( 1 2 3 4)) => (1 16 81 256)

17. 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: ((s1 v1) (s2 v2)…) 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)

18. 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 (f1 f2..) 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

19. Scoping Rules Consider the following examples: (define x 5) (define sqr (lambda (z) (* z z ))) (sqr x) ; yields 25 (define addto (lambda (y) (define x 6) (+ x y))) (addto 9) ; yields 15 given definition of x as 6 (sqr x) ; what should this give?

20. Scoping Rules 2 Because Scheme is “lexically scoped”, the answer is still 25. The scope of the definition of x within “addto” is just that function. Lexical scoping: scope of variable is defined by text of program. A “dynamically scoped” Lisp variant would simply take the last definition of x.

21. Now Consider this (define x 5) (define sqr (lambda (z) (* z z ))) (sqr x) ; yields 25 (define x 8) (sqr x)

22. x has been redefined in the same scope as in its first definition, so x changes Therefore answer is 64.

23. This notion naturally extends to functions (look up “closure” in Scott) (define (compose f1 f2) (lambda (x) (f1 (f2 x)))) (define (sqr x) (* x x)) (define fourth (lambda (x) ((compose sqr sqr) x))) (fourth 4) ; yields 256 When fourth is called, we need to fetch the latest definition of sqr at the right level of scoping.

24. What happens here? (define (compose f1 f2) (lambda (x) (f1 (f2 x)))) (define (sqr x) (* x x)) (define fourth (lambda (x) ((compose sqr sqr) x))) (define (sqr x) (+ x x)) (fourth 4) ; yields 16 (based on latest def of ; sqr at proper scope level

25. And here? (define (compose f1 f2) (lambda (x) (f1 (f2 x)))) (define (sqr x) (* x x)) (define fourth (lambda (x) ((compose sqr sqr) x))) (define (foo x) (define (sqr x) (+ x x)) (* 2 (sqr x))) (foo 15) ; yields 60 (fourth 4) ; yields 256

26. 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.

27. 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)) ;; Which N? ; n undefined if static binding. ; Bound in sum-powers if dynamic binding.

28. 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) ; apply function (eval-list (cdr exp) env) ; to arguments env)) ; in current env Apply: (eval (procedure-body procedure) (extend-environment ; add bindings (parameters procedure) ; of formals arguments env))) ; to current env

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

30. 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)

31. Self-definition (advanced) (define (eval exp env) ; the lisp interpreter (cond ((number? exp) 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 (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

32. Function application defined (advanced) (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 procedure definition (closure)

33. 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)))))

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

35. 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))) ; stack grows to size n define factorial n) (fact-iter 1 1 n) ; alternate definition (define (fact-iter prod count var) (if (> count var) prod (fact-iter (* count prod) ; tail recursion (+ count 1) ; implemented as loop var))))

36. 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 (cdr lis)))) ; can be done with loop (define (length lis) (if (null? lis) 0) (+ 1 (length (cdr lis)))) ; not tail recursive!