TDDC74 Programmering: Abstraktion och modellering VT 2015 Johannes Schmidt
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TDDC74 Programmering: Abstraktion och modellering VT 2015 Johannes Schmidt Institutionen för datavetenskap Linköpings universitet 1 Lecture 3 Higher order procedures More on syntactic sugar Procedures as arguments Procedures as return values SICP 1, Del 3 2 More on syntactic sugar let, let* is called syntactic sugar because they just hide the equivalent and more difficult to read lambda constructs There is more syntactic sugar to hide lambda constructs: define a proceure via (define (<name> <parameter list>) <body>) instead of (define <name> (lambda (<parameter list>) <body>)) 3 More on syntactic sugar (define square (lambda (x) (* x x))) is simpler written as (define (square x) (* x x)) From now on we will primarily use this syntactic sugar version to define procedures. But keep in mind that it hides a lambda construct. 4 Exercise Rewrite without syntactic sugar (define (distance x1 y1 x2 y2) (let ((dx (- x2 x1)) (dy (- y2 y1))) (sqrt (+ (square dx) (square dy))))) 5 (procedure param1 param2 …) Until now: parameters = primitive objects (integers, constants) What about procedures as parameters for procedures? (define (sum-of-some-function f x y) (+ (f x) (f y))) >(sum-of-some-function (lambda (x) (* x x)) 3 4) 25 6 Different sums (define (sum-ints a b) (if (> a b) 0 (+ a (sum-ints (+ a 1) b)))) >(sum-ints 3 6) 18 (= (+ 3 (+ 4 (+ 5 (+ 6 0))))) 7 Different sums (define (sum-ints a b) (if (> a b) 0 (+ a (sum-ints (+ a 1) b)))) (define (sum-squares a b) (if (> a b) 0 (+ (square a) (sum-squares (+ a 1) b)))) (define (sum-cubes a b) (if (> a b) 0 (+ (cube a) (sum-cubes (+ a 1) b)))) 8 Common structure and code (define (sum-ints a b) (if (> a b) 0 (+ a (sum-ints (+ a 1) b)))) (define (sum-squares a b) (if (> a b) 0 (+ (square a) (sum-squares (+ a 1) b)))) (define (sum-cubes a b) (if (> a b) 0 (+ (cube a) (sum-cubes (+ a 1) b)))) 9 General sum (define (sum-cubes a b) (if (> a b) 0 (+ (cube a) (sum-cubes (+ a 1) b)))) (define (sum a b term) (if (> a b) 0 (+ (term a) (sum (+ a 1) b term)))) 10 General Sum (define (sum-squares a b) (sum a b square)) (define (sum-cubes a b) (sum a b cube)) How to write sum-ints? 11 Two Kinds of Accumulation (define (sum a b term next) (if (> a b) 0 (+ (term a) (sum (next a) b term next)))) (define (mult a b term next) (if (> a b) 1 (* (term a) (mult (next a) b term next)))) 12 Similarities (define (sum a b term next) (if (> a b) 0 (+ (term a) (sum (next a) b term next)))) (define (mult a b term next) (if (> a b) 1 (* (term a) (mult (next a) b term next)))) 13 Accumulation in general (define (acc a b term next init op) (if (> a b) init (op (term a) (acc (next a) b term next init op)))) 14 Accumulation in general (define (acc a b term next init op) (if (> a b) init (op (term a) (acc (next a) b term next init op)))) (define (sum-ints a b) (acc a b id inc 0 +)) where id stands for (lambda (x) x) and inc for (lambda (x) (+ x 1)) 15 Accumulation in general (define (acc a b term next init op) (if (> a b) init (op (term a) (acc (next a) b term next init op)))) (define (sum-cubes a b) (acc a b cube inc 0 +)) where cube stands for (lambda (x) (* x x x)) and inc for (lambda (x) (+ x 1)) 16 Procedures as return values A procedure that returns a procedure? (define (f x) (lambda (y) (* x y))) (define (get-incrementer inc-id) (cond ((= inc-id 0) (lambda (x) (+ x 1))) ((= inc-id 1) (lambda (x) (* x 2))) (else (lambda (x) x)))) 17 Procedures as return values (define (make-greeter name) (lambda () (display "Hello ") (display name) (display "\n"))) (define greet-annika (make-greeter "Annika")) (greet-annika) Hello Annika ((make-greeter "Greta")) Hello Greta 18 Procedures as return values The composition of two functions f and g is defined as f(g(x)) Define a procedure compose that implements composition (define (compose f g) (lambda (x) (f (g x)))) 19 Higher order procedures A procedure is of higher order if it either has procedures as parameters or as return values (or both) 20 Iterative Fibonacci 0, 1, 1, 2, 3, 5, 8, 13, ... (define fib-iter (lambda (n) (fib2 n 0 1))) Scope of fib2: global (define fib2 (lambda (n a b) (cond ((= n 0) a) (else (fib2 (- n 1) b (+ a b)))))) 21 Iterative Fibonacci, nicer (define fib-iter (lambda (n) (define fib-help Scope of fib-help: (lambda (n a b) local (cond ((= n 0) a) (else (fib-help (- n 1) b (+ a b)))))) (fib-help n 0 1))) 22 Iterative Fibonacci, more compact (define (fib-iter n) (define (fib-help n a b) (cond ((= n 0) a) (else (fib-help (- n 1) b (+ a b))))) (fib-help n 0 1)) 23 Where Do Algorithms Come From Thinking and Problem Solving Already the ancient Egyptions had kind of „algorithms“ for solving geometry problems of the pyramids Fortunately many problems are solved already. ”Just” translate the solutions to algorithms Many mathematical formulas can easily be translated to algorithmic descriptions Some existing solutions can be modified to solve other problems 24 Example Equation: C/100 = (F - 32)/180 can be formulated as two algorithms that compute C or F (define get-celsius (lambda (f) (* 100 (/ (- f 32) 180)))) (define get-fahrenheit (lambda (c) (+ (/ (* c 180) 100) 32))) 25 Errors Syntactic Error Expression impossible to evaluate e.g. (5 + 4) Semantic Error evaluable (compilable), but not solving the task you want e.g. (* 4 5) instead of (+ 4 5) 26 Summary Expressions Primitive Constants Names Compound Function applications Special forms If Cond Logical operators: and, or, not Abstraction Lambda Define 27 Summary Evaluation and substitution model Procedures and processes Recursive procedures Recursive processes Iterative processes Tree-recursive processes Scope: local and global variables Let, Let* Syntactic sugar Higher order functions Functions as arguments Functions as returned values 28 Exercise 1.11 SICP f(n) = n if n<3 f(n) = f(n-1)+2f(n-2)+3f(n-3) otherwise Write procedures that solve f(n) both recursively and iteratively See if you can follow the patterns used in similar solutions for comuting the Fibonnaci numbers 29 Hints Read the course book! Follow the coding guidelines (labs may be refused otherwise) 30 Dugga 1 Contents: Lecture 1, 2, 3 When: Thursday, February 12th (in 2 and a half weeks) Registration in Studentportalen Last day for registration: February 2nd (in 1 week) 31
