Functional Programming and Operational Semantics for Scheme: Lecture 4 Summary - Prof. Jer, Papers of Programming Languages

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Programming Languages

(CS 550)

Lecture 4 Summary

Functional Programming and Operational

Semantics for Scheme

Jeremy R. Johnson

Starting Point Informal Scheme Semantics ™ To evaluate (E1 E2 ... En), recursively evaluateE1, E2,...,En - E1 should evaluate to a function -and then apply the function value of E1 to thearguments given by the values of E2,...,En. ™ In the base case, there are self evaluatingexpressions (e.g. numbers and symbols). Inaddition, various special forms such as quote andif must be handled separately.

Outline

Review scheme syntax and semantics ™ Functional programming ¾ Programs are functions – for every input there is aunique output ¾ No variables ⇒ no assignment and no loops ¾ Use recursion for control ¾ Functions are first class objects ƒ Pass as arguments and return as values ¾ Simple semantics [value semantics] (no side effects,referential transparency)

Outline

Substitution model of computation

Modeling state with assignment

Environments

Streams and delayed evaluation

Scheme interpreter (meta-circular)

Higher Order Functions

sort: (sort '(4 3 2 1) <) => (1 2 3 4) ™ map: (map sqr '(1 2 3 4)) => (1 4 9 16) ™ filter: (keep-matching-items '(1 2 3 4 5) odd?) => (1 3 5) ™ reduce: (reduce * 1 '(1 2 3 4)) => 24 ™ Functions that return functions ¾ (define (make-adder x) (lambda (y) (+ x y))) ¾ Function composition ƒ (define (compose g f) (lambda (x) (g (f x)))) ¾ Currying ƒ (define (curry f b) (lambda (a) (f a b)))

Substitution Model of Computation

function application corresponds tosubstituting the argument expressions intothe formal parameters of the function body

Order of evaluation

Applicative vs. normal order ¾ lambda calculus ¾ Church-Rosser

Order Matters

(define (p) (p)) (define (test x y) (if (= x 0) 0 y)) (test 0 (p))

Environments

When assignment is introduced thesubstitution model falls apart and a differentmodel, more complicated model, must beused.

Environments used to store variables andbindings.

Values can change ¾ assignment supported ¾ List of frames to support nested scope

Modeling State with Assignment

(define (make-account balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (define (dispatch m) (cond ((eq? m 'withdraw) withdraw) ((eq? m 'deposit) deposit) (else (error "Unknown request -- MAKE- ACCOUNT" m)))) dispatch) (define acc (make-account 100)) ((acc 'withdraw) 50) 50 ((acc 'withdraw) 60) "Insufficient funds" ((acc 'deposit) 40) 90 ((acc 'withdraw) 60) 30

Cost of Assignment

(define (make-simplified-withdraw balance) (lambda (amount) (set! balance (- balance amount)) balance)) (define W (make-simplified-withdraw 25)) (W 10) 15 (W 10) 5 (define (make-decrementer balance) (lambda (amount) (- balance amount))) (define D (make-decrementer 25)) (D 10) 15 (D 10) 15

Environment Model Solves Problem (define W1 (make-withdraw 100)) (W1 50) (define W2 (make-withdraw 100)) Global Env Balance = 50 Balance = 100 E E Parameters body W W Make-withdraw

Streams

Sequence of elements

(cons-stream x y) ¾ (stream-car s) ¾ (stream-cdr s) ¾ (stream-null? s) ¾ the-empty-stream

Streams

(define (sum-primes a b) (accumulate + 0 (filter prime? (enumerate-interval a b)))) Enumerate filter accumulate Prime? +, a,b

Streams are Not Lists

Consider the following example (car (cdr (filter prime? (enumerate-interval 10000 1000000)))) ™ This would be extremely inefficient ifimplemented with lists ¾ Do not build the entire stream of elements ¾ Get the next element from the stream when needed ¾ Necessary for potentially infinite streams