No matter what happens, the U.S. Navy is not going to be caught napping.-Secretary of the Navy Frank Knox
December 5, 1941
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(let ((n 0)) (or (= n 0) (= (/ 12 n) 4))) ; returns #t (let ((n 0)) (or* (= n 0) (= (/ 12 n) 4))) ; gets an ERROR - division by zero
Observe that and* and or* must handle a variable number of arguments.
Examples:
(or* #f #f 'a 'b) ; a (and* #f #f 'a 'b) ; #f (or* 'a 'b 'c 'd) ; a (and* 'a 'b 'c 'd) ; d (or* 'a 'b 'c #f) ; a (and* 'a 'b 'c #f) ; #f (or* 'whatever) ; whatever (or* #f) ; #f (and* 'anything) ; anything (and* #f) ; #f (and*) ; #t (or*) ; #f
An ``arithmetic expression'' is an s-expression which is either: a number, or a list of length three, of which the first element is an ``arithmetic operator,'' one of the symbols +, *, - or /, and the remaining two elements are themselves arithmetic expressions.
Examples:
(expr1 92) ; 92 (expr1 '(+ 3 4)) ; 7 (expr1 '(* 2 (+ 1 3))) ; 8 (expr1 '(+ (* (/ 6 2) 5) (- 100 97))) ; 18Note: the functions expr1 and expr2 do not need to do any error checking, ie don't worry about the case where the arithmetic expression they're expecting really isn't one. Also note that all operators take two arguments. Including -. So you do not have to worry about an expression like (- 3). (You can get the intended effect with (- 0 3) or (* -1 3) anyway.)
Note: you may not use eval in your definition of expr1 or expr2.
An ``arithmetic expression'' is here extended to also consider symbols (like x, y or blorf) to be arithmetic expressions.
The ``environment'' is a mapping from symbols to numbers, represented as an association list.
Examples:
(expr2 92 '((a 1)(b 2))) ; 92 (expr2 'a '((a 1)(b 2))) ; 1 (expr2 '(+ b 3) '((a 1)(b 2))) ; 5 (expr2 '(* b (+ a 3)) '((a 1)(b 2))) ; 8 (expr2 '(+ (* (/ 6 blorf) (+ blorf (+ blorf 1))) (- c d)) '((c 100) (blorf 2) (a 1) (d 97))) ; 18Note: you can assume that all symbols appearing in the expression are listed in the environment.
Examples:
(simplify '(+ (* 1 (+ 0 (* (/ a a) (+ x 2)))) 10)) ; (+ x 12)(or some different but equivalent result.)
Note: Don't try to get too fancy for this or the next problem. It is sufficient to do a rough job, so that in the output there is no arithmetic operation on two numbers, a symbol is not divided by itself, things are not added to zero, zero isn't subtracted from things, things aren't multiplied or divided by zero or one. Doing a better job than just these minimal requirements will get some extra credit, and more importantly the inner peace that only comes from a Zen programming experience.
Examples:
(sym+ 'a 'b) ; (+ a b) (sym+ 'a 'a) ; (* 2 a) (sym+ 1 2) ; 3 (sym+ 1 '(+ a 19)) ; (+ a 20) (sym/ 'c '(/ a b)) ; (/ (* c b) a)(or different but equivalent results.)
Note: you probably want to define simplify in terms of these.
Examples:
(diff 57 'x) ; 0 (diff 'a 'x) ; 0 (diff '(+ a b) 'x) ; 0 (diff 'x 'x) ; 1 (diff '(* a (* x x)) 'x) ; (* a (* 2 x)) (diff '(/ x a) 'x) ; (/ 1 a) (diff '(/ a x) 'x) ; (* a (/ -1 (* x x))) (diff '(+ a (* x x)) 'x) ; (* 2 x) (diff '(* (+ x 1) (+ x 2)) 'x) ; (+ (+ x 2) (+ x 1))(or different but equivalent results.)
An SRE is defined to be an expression that is either (a) an SRE polynomial, or (b) the quotient of two SRE polynomials. An SRE polynomial is an expression that is either a single term, or the sum of some terms. And a term is either a number, a variable, or the product of a number (optional) with one or more variables. Note that an SRE cannot contain the subtraction operator. Examples:
;;; These are some "terms". (They are also valid albeit degenerate ;;; SREs and "SRE polynomials.") (* x x 13 y z x) 24 z (* -1 z) ;;; These are some "SRE polynomials" (They are also valid albeit ;;; degenerate SREs.) (+ (* x x 13 y z x) y 24 (* -2 z)) (+ (* y y y) (* 6 y y) (* 4 y) 7) (* -1 z) 92 (+ 92 (* -1 z)) ; this is like (- 92 x) except that SREs have no -'s ;;; This is an SRE (/ (+ (* x x 13 y z x) y 24 (* -2 z)) (+ (* y y y) (* 6 y y) (* 4 y) 7))
The file should be legal scheme code, with comments indicating which fragment of code corresponds to which problem.
You are allowed to make corrections, because the last submit command supersedes all previous submissions.