functions and recursion

Dr. Philip Cannata 1

Functions and Recursion


10

Java (Object Oriented)Java (Object Oriented)

ASPASP

RDF (Horn Clause Deduction, RDF (Horn Clause Deduction, Semantic Web)Semantic Web)

RelationRelation

Jython in JavaJython in Java

This CourseThis Course

High LevelHigh LevelLanguagesLanguages


“let” transformation, differed substitution and closures, and interpretation in FAE

“let” transformation:

(let ((A B)) C) == ((lambda (A) C) B)

A B ---------------- C -------------------

A ---------- B -------- C

(let (( x 3)) (let ((f (lambda (y) (+ x y)))) (f 4))

((lambda (x) ((lambda (f) (f 4)) (lambda (y) (+ x y)))) 3)

(app (fun 'x [app (fun 'f [app (id 'f) (num 4)]) (fun 'y (add (id 'x) (id 'y)))]) (num 3))

(app ------- arg1 ------------------ ------------arg2---------------

(app --------------------------------------- arg1------------------------------------- --arg2—

Differed substitution and closures:

(aSub 'f (closureV 'y (add (id 'x) (id 'y)) (aSub 'x (numV 3) (mtSub))) (aSub 'x (numV 3) (mtSub)))

Interpretation:

(interp (app (id 'f) (num 4)) (aSub 'f (closureV 'y (add (id 'x) (id 'y)) (aSub 'x (numV 3) (mtSub))) (aSub 'x (numV 3) (mtSub))))

(numV 7)


Eduardo Saenz (anon. to classmates) (1 day ago) - If our AST does not have let class nodes, then when our interpreter visits every node of the AST, making the environment of differed substitutions along the way, our environment will only have closures?You only do: (aSub x 3 (mtSub)) when you encounter a let, and since our parser converts lets to lambdas we'll never see this type of differed substitution in our environment; only closures in our environment.Is this logic correct?

Philip Cannata (Instructor) (Just now) - This is a good observation but try to understand the following three cases that can occur and see if you can distinguish when deferred substitution should be done in each of them and when function application should be done:

> (parse '(with (f (fun (x) x)) 5))(app (fun 'f (num 5)) (fun 'x (id 'x)))> (interp (parse '(with (f (fun (x) x)) 5)) (mtSub))(numV 5)

> (parse '(with (f (fun (x) x)) (f 5)))(app (fun 'f (app (id 'f) (num 5))) (fun 'x (id 'x)))> (interp (parse '(with (f (fun (x) x)) (f 5))) (mtSub))(numV 5)

> (parse '(with (f (fun (x) x)) ((fun (y) (+ y 2)) 5)))(app (fun 'f (app (fun 'y (add (id 'y) (num 2))) (num 5))) (fun 'x (id 'x)))> (interp (parse '(with (f (fun (x) x)) ((fun (y) (+ y 2)) 5))) (mtSub))(numV 7)


Static and Dynamic Scoping

Static scoping:

(interp (parse '{with {x 5} {f 4}}) (aSub 'f (closureV 'y (add (id 'x) (id 'y)) (aSub 'x (numV 3) (mtSub))) (aSub 'x (numV 3) (mtSub))))

(numV 7)

Dynamic Scoping:

(interp (parse '{with {x 5} {f 4}}) (aSub 'f (closureV 'y (add (id 'x) (id 'y)) (aSub 'x (numV 3) (mtSub))) (aSub 'x (numV 3) (mtSub))))

(numV 9)

Think about this expression for both Static and Dynamic Scoping:

(let ((z 3)) (let ((d 3) (f (lambda (x) x))) (let ((z 27))

(let ((z 3) (a 5) (x (lambda (x y) (+ x (+ y z))))) (let ((z 9) (a 7)) (x z a))))))


PLAI Chapters 4, 5 and 6

Page 27 - "

Chapter 6, Pages 41 & 42 – “first-order Functions are not values in the language. They can only be defined in a designated portion of the program, where they must be given names for use in the remainder of the program. The functionsin F1WAE are of this nature, which explains the 1 in the name of the language.

higher-order Functions can return other functions as values.

first-class Functions are values with all the rights of other values. In particular, they can be supplied as the value of arguments to functions, returned by functions as answers, and stored in data structures.


$ java crono.Crono (let ((z 17)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z a)))) Evaluating: (let ((z 17)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z a)))) Environment: empty Evaluating: (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z a))) Environment: z: 17 Evaluating: (\ (x y) (- x (+ y z))) Environment: z: 17 Result: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (let ((z 10) (a 5)) (x z a)) Environment: a: 5 z: 3 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (x z a) Environment: a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (- x (+ y z)) Environment: y: 5 z: 17 x: 10 Evaluating: (+ y z) Environment: y: 5 z: 17 x: 10 Result: 22 Result: -12 Result: -12 Result: -12 Result: -12 Result: -12

In Crono, “\” mean lambda


In CronoOptions.java set public static boolean ENVIRONMENT_DYNAMIC = true; Run ant to rebuild crono. $ java crono.Crono (let ((z 17)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z a)))) Evaluating: (let ((z 17)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z a)))) Environment: empty Evaluating: (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z a))) Environment: z: 17 Evaluating: (\ (x y) (- x (+ y z))) Environment: z: 17 Result: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (let ((z 10) (a 5)) (x z a)) Environment: a: 5 z: 3 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (x z a) Environment: a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (- x (+ y z)) Environment: a: 5 y: 5 z: 10 x: 10 Evaluating: (+ y z) Environment: a: 5 y: 5 z: 10 x: 10 Result: 15 Result: -5 Result: -5 Result: -5 Result: -5 Result: -5


(let ((z 17)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z (x 0 0))))) Evaluating: (let ((z 17)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z (x 0 0))))) Environment: empty Evaluating: (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z (x 0 0)))) Environment: z: 17 Evaluating: (\ (x y) (- x (+ y z))) Environment: z: 17 Result: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (let ((z 10) (a 5)) (x z (x 0 0))) Environment: a: 5 z: 3 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (x z (x 0 0)) Environment: a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (x 0 0) Environment: a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [z: 17] Evaluating: (- x (+ y z)) Environment: y: 0 z: 17 x: 0 Evaluating: (+ y z) Environment: y: 0 z: 17 x: 0 Result: 17 Result: -17 Result: -17 Evaluating: (- x (+ y z)) Environment: y: -17 z: 17 x: 10 Evaluating: (+ y z) Environment: y: -17 z: 17 x: 10 Result: 0 Result: 10 Result: 10 Result: 10 Result: 10 Result: 10

Notice function application here.


(let ((z 17)(i 22)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z (x 0 ((\ (x) x) i)))))) Evaluating: (let ((z 17) (i 22)) (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z (x 0 ((\ (x) x) i)))))) Environment: empty Evaluating: (let ((z 3) (a 5) (x (\ (x y) (- x (+ y z))))) (let ((z 10) (a 5)) (x z (x 0 ((\ (x) x) i))))) Environment: i: 22 z: 17 Evaluating: (\ (x y) (- x (+ y z))) Environment: i: 22 z: 17 Result: (\ (x y) (- x (+ y z))) [i: 22, z: 17] Evaluating: (let ((z 10) (a 5)) (x z (x 0 ((\ (x) x) i)))) Environment: i: 22 a: 5 z: 3 x: (\ (x y) (- x (+ y z))) [i: 22, z: 17] Evaluating: (x z (x 0 ((\ (x) x) i))) Environment: i: 22 a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [i: 22, z: 17] Evaluating: (x 0 ((\ (x) x) i)) Environment: i: 22 a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [i: 22, z: 17] Evaluating: ((\ (x) x) i) Environment: i: 22 a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [i: 22, z: 17] Evaluating: (\ (x) x) Environment: i: 22 a: 5 z: 10 x: (\ (x y) (- x (+ y z))) [i: 22, z: 17] Result: (\ (x) x) [i: 22, a: 5, z: 10, x: (\ (x y) (- x (+ y z)))]

Result: 22 Evaluating: (- x (+ y z)) Environment: i: 22 y: 22 z: 17 x: 0 Evaluating: (+ y z) Environment: i: 22 y: 22 z: 17 x: 0 Result: 39 Result: -39 Result: -39 Evaluating: (- x (+ y z)) Environment: i: 22 y: -39 z: 17 x: 10 Evaluating: (+ y z) Environment: i: 22 y: -39 z: 17 x: 10 Result: -22 Result: 32 Result: 32 Result: 32 Result: 32 Result: 32


10

Java (Object Oriented)Java (Object Oriented)

ASPASP

RDF (Horn Clause Deduction, RDF (Horn Clause Deduction, Semantic Web)Semantic Web)

RelationRelation

Jython in JavaJython in Java

This CourseThis Course

High LevelHigh LevelLanguagesLanguages


int h, i;void B(int w) { int j = 1, k = 2; i = 2*w; w = w+1; printf("In B - w, j, k, h, i: %d, %d, %d, %d, %d\n",

w, j, k, h, i);}void A(int x, int y) { float i = 1.1, j = 2.2; B(h); printf("In A - x, y, i, j, h: %d, %d, %f, %f, %d\n", x, y, i, j, h);}int main() { int a, b; h = 5; a = 3; b = 2; A(a, b); printf("In Main a, b, h, i: %d, %d, %d, %d\n",

a, b, h, i);}

Parameter and Arguments

DefinitionsAn argument is an expression that appears in a function application/call.A parameter is an identifier that appears in a function definition/declaration.

In the code on the right the call A(a, b) has arguments a and b.

The function declaration A has parameters x and y.


• By value - Compute the value of the argument at the time of the call and assign that value to the parameter. So passing by value doesn’t normally allow the called function to modify an argument’s value.

• By reference - Compute the address of the argument at the time of the call and assign it to the parameter. Passing by value allows the called function to modify an argument’s value.

• By value-result

• By name

Parameter Passing Mechanisms


int h, i;void B(int w) { int j = 1, k = 2; i = 2*w; w = w+1; printf("In B - w, j, k, h, i: %d, %d, %d, %d,

%d\n", w, j, k, h, i);}void A(int x, int y) { float i = 1.1, j = 2.2; B(h); printf("In A - x, y, i, j, h: %d, %d, %f, %f,

%d\n", x, y, i, j, h);}int main() { int a, b; h = 5; a = 3; b = 2; A(a, b); printf("In Main a, b, h, i: %d, %d, %d, %d\n", a, b, h, i);}

$ ./aIn B - w, j, k, h, i: 6, 1, 2, 5, 10In A - x, y, i, j, h: 3, 2, 1.100000, 2.200000, 5In Main a, b, h, i: 3, 2, 5, 10

int h, i;void B(int *w) { int j = 1, k = 2; i = 2*(*w); *w = *w + 1; printf("In B - w, j, k, h, i: %d, %d, %d, %d, %d\n",

w, j, k, h, i);}void A(int *x, int *y) { float i = 1.1, j = 2.2; B(&h); printf("In A - x, y, i, j, h: %d, %d, %f, %f, %d\n", x, y, i, j, h);}int main() { int a, b; h = 5; a = 3; b = 2; A(&a, &b ); printf("In Main a, b, h, i: %d, %d, %d, %d\n",

a, b, h, i);}

$ ./aIn B - w, j, k, h, i: 4206640, 1, 2, 6, 10In A - x, y, i, j, h: 2280676, 2280672, 1.100000, 2.200000, 6In Main a, b, h, i: 3, 2, 6, 10

Pass by Value Pass by Reference


Pass by value at the time of the call and/or copy the result back to the argument at the end of the call – also called copy-in-copy-out.

Pass by Value-Results

Textually substitute the argument for every instance of its corresponding parameter in the function body. Originated with Algol 60 (Jensen’s device), but was dropped by Algol’s

successors -- Pascal, Ada, Modula.

Exemplifies late binding, since evaluation of the argument is delayed until its occurrence in the function body is actually executed.Associated with lazy evaluation in functional languages (e.g., Haskell).

real procedure Sum(j, lo, hi, Ej); value lo, hi; integer j, lo, hi; real Ej;begin real S; S := 0; for j := lo step 1 until hi do S := S + Ej; Sum := S end;

x[j]*j

Pass by Name


(letrec ((f (lambda (l) (if (null? l)

'()

(cons (car l) (f (cdr l))))))) (f '(1 2 3 4 5 6)))

'(1 2 3 4 5 6)

(letrec ((f (lambda (v l) (if (null? l)

v

(cons (car l) (f v (cdr l))))))) (f '() '(1 2 3 4 5 6)))

'(1 2 3 4 5 6)

(letrec ((f (lambda (f1 v l) (if (null? l)

v

(f1 (car l) (f f1 v (cdr l))))))) (f cons '() '(1 2 3 4 5 6)))

'(1 2 3 4 5 6)

f == foldr

If f1 == cons, foldr is the identity function for a list.

It‘s the same as

(cons 1 (cons 2 (cons 3( cons 4 (cons 5 (cons 6 '()))))))

Recursive Functions Exemplified by foldr in lisp

(con

s 1

|| (c

ons

2 ||

(con

s 3

|| (c

ons

4 ||

(con

s 5

|| (c

ons

5 '()

))))

)) (cons 1 || (cons 2 || (cons 3 || (cons 4 || (cons 5 || (cons 5 '()))))))T

his

can

be th

ough

t of

as a

sta

ck w

ith

“con

s”s

on it

.H

ere the stack is upside down



v

(f f1 (car l) (cdr l)))))) (f cons '() '(1 2 3 4 5 6)))

6


v

(f f1 (f1 (car l) v) (cdr l)))))) (f cons '() '(1 2 3 4 5 6)))

'(6 5 4 3 2 1)

f == foldl

If f1 == cons, foldl reverses the list.

foldl is tail-recursive because nothing goes on the stack.

It‘s the same as

(cons 6 (cons 5 (cons 4 ( cons 3 (cons 2 (cons 1 '()))))))

Recursive Functions Exemplified by foldl in lispN

othi

ng g

oes

on th

e st

ack

in th

is c

ase.


$ cat test.cint factorial (int n) { int i; if (n < 2) { printf("factorial returning 1\n"); return 1; } else { i = n * factorial(n-1); printf("factorial returning %d\n", i); return i; }}

int main() { printf("factorial(3) returns: %d\n", factorial(3));}

$ ./afactorial returning 1factorial returning 2factorial returning 6factorial(3) returns: 6

A function that can call itself, either directly or indirectly, is a recursive function.

Recursive Functions


A stack of activation records:•An activation record is a block of information associated with each function call, which includes:

• parameters and local variables• Return address

• Each new call pushes an activation record, and each completing call pops the topmost one.

• So, the topmost record is the most recent call, and the stack has all active calls at any run-time moment.

Runtime Stack


int h, i;void B(int w) { int j, k; i = 2*w; w = w+1;} void A(int x, int y) { bool i, j; B(h);}int main() { int a, b; h = 5; a = 3; b = 2; A(a, b);}

• parameters and local variables

• Return address

• Saved registers

• Temporary variables

• Return value

Runtime Stack for Functions Program


Calling function: factorialBasePtr = 3, printing runtime stacknull: nulln: 3null: nullanswer: nullnumber: 3

Calling function: factorialBasePtr = 5, printing runtime stacknull: nulln: 2null: nulln: 3null: nullanswer: nullnumber: 3

Calling function: factorialBasePtr = 7, printing runtime stacknull: nulln: 1null: nulln: 2null: nulln: 3null: nullanswer: nullnumber: 3

Calling function: factorialBasePtr = 9, printing runtime stacknull: nulln: 0null: nulln: 1null: nulln: 2null: nulln: 3null: nullanswer: nullnumber: 3

Exiting function: factorialBasePtr = 9, printing runtime stacknull: nulln: 0return#factorial: 1n: 1null: nulln: 2null: nulln: 3null: nullanswer: nullnumber: 3 Exiting function: factorialBasePtr = 7, printing runtime stackreturn#factorial: 1n: 1return#factorial: 1n: 2null: nulln: 3null: nullanswer: nullnumber: 3 Exiting function: factorialBasePtr = 5, printing runtime stackreturn#factorial: 1n: 2return#factorial: 2n: 3null: nullanswer: nullnumber: 3 Exiting function: factorialBasePtr = 3, printing runtime stackreturn#factorial: 2n: 3return#factorial: 6answer: nullnumber: 3

int factorial(int n) { if(n < 1) { return 1; } else { return n * factorial(n - 1); }}

int main() { int number, answer; number = 3; answer = factorial(number); print(answer);}

Hmm Runtime Stack for Factorial 3

functions and recursion

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