Dr. Philip Cannata 1 Functions and Recursion

Dr. Philip Cannata 2 10 Java (Object Oriented) ASP RDF (Horn Clause Deduction, Semantic Web) Relation Jython in Java This Course High Level Languages

Dr. Philip Cannata 3 “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 arg arg (app arg 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)

Dr. Philip Cannata 4 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)

Dr. Philip Cannata 5 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))))))

Dr. Philip Cannata 6 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 functions in 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.

Dr. Philip Cannata 7 $ 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

Dr. Philip Cannata 8 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

Dr. Philip Cannata 9 (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.

Dr. Philip Cannata 10 (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

Dr. Philip Cannata Java (Object Oriented) ASP RDF (Horn Clause Deduction, Semantic Web) Relation Jython in Java This Course High Level Languages

Dr. Philip Cannata 12 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 Definitions An 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.

Dr. Philip Cannata 13 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

Dr. Philip Cannata 14 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); } $./a In B - w, j, k, h, i: 6, 1, 2, 5, 10 In A - x, y, i, j, h: 3, 2, , , 5 In 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); } $./a In B - w, j, k, h, i: , 1, 2, 6, 10 In A - x, y, i, j, h: , , , , 6 In Main a, b, h, i: 3, 2, 6, 10 Pass by Value Pass by Reference

Dr. Philip Cannata 15 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

Dr. Philip Cannata 16 (letrec ((f (lambda (l) (if (null? l) '() (cons (car l) (f (cdr l))))))) (f '( ))) '( ) (letrec ((f (lambda (v l) (if (null? l) v (cons (car l) (f v (cdr l))))))) (f '() '( ))) '( ) (letrec ((f (lambda (f1 v l) (if (null? l) v (f1 (car l) (f f1 v (cdr l))))))) (f cons '() '( ))) '( ) 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 (cons 1 || (cons 2 || (cons 3 || (cons 4 || (cons 5 || (cons 5 '())))))) This can be thought of as a stack with “cons”s on it. Here the stack is upside down

Dr. Philip Cannata 17 (letrec ((f (lambda (f1 v l) (if (null? l) v (f f1 (car l) (cdr l)))))) (f cons '() '( ))) 6 (letrec ((f (lambda (f1 v l) (if (null? l) v (f f1 (f1 (car l) v) (cdr l)))))) (f cons '() '( ))) '( ) 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 lisp Nothing goes on the stack in this case.

Dr. Philip Cannata 18 $ cat test.c int 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)); } $./a factorial returning 1 factorial returning 2 factorial returning 6 factorial(3) returns: 6 A function that can call itself, either directly or indirectly, is a recursive function. Recursive Functions

Dr. Philip Cannata 19 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

Dr. Philip Cannata 20 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

Dr. Philip Cannata 21 Calling function: factorial BasePtr = 3, printing runtime stack null: null n: 3 null: null answer: null number: 3 Calling function: factorial BasePtr = 5, printing runtime stack null: null n: 2 null: null n: 3 null: null answer: null number: 3 Calling function: factorial BasePtr = 7, printing runtime stack null: null n: 1 null: null n: 2 null: null n: 3 null: null answer: null number: 3 Calling function: factorial BasePtr = 9, printing runtime stack null: null n: 0 null: null n: 1 null: null n: 2 null: null n: 3 null: null answer: null number: 3 Exiting function: factorial BasePtr = 9, printing runtime stack null: null n: 0 return#factorial: 1 n: 1 null: null n: 2 null: null n: 3 null: null answer: null number: 3 Exiting function: factorial BasePtr = 7, printing runtime stack return#factorial: 1 n: 1 return#factorial: 1 n: 2 null: null n: 3 null: null answer: null number: 3 Exiting function: factorial BasePtr = 5, printing runtime stack return#factorial: 1 n: 2 return#factorial: 2 n: 3 null: null answer: null number: 3 Exiting function: factorial BasePtr = 3, printing runtime stack return#factorial: 2 n: 3 return#factorial: 6 answer: null number: 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