X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=313db95633e36c857b5e6afe417a9ffb03c33b9c;hp=4980cc2714311f140e0d465b374e7e494d0263c5;hb=da04ac1d11456ea14ab5c07a1ba8a16e45ffcd58;hpb=a1bb9458275d3da01448d6ad4a7b4a8d502b27cb diff --git a/assignment1.mdwn b/assignment1.mdwn index 4980cc27..313db956 100644 --- a/assignment1.mdwn +++ b/assignment1.mdwn @@ -1,18 +1,18 @@ -*Reduction* +**Reduction** Find "normal forms" for the following (that is, reduce them as far as it's possible to reduce them): -1. (\x \y. y x) z -2. (\x (x x)) z -3. (\x (\x x)) z -4. (\x (\z x)) z -5. (\x (x (\y y))) (\z (z z)) -6. (\x (x x)) (\x (x x)) -7. (\x (x x x)) (\x (x x x)) + 1. (\x \y. y x) z + 2. (\x (x x)) z + 3. (\x (\x x)) z + 4. (\x (\z x)) z + 5. (\x (x (\y y))) (\z (z z)) + 6. (\x (x x)) (\x (x x)) + 7. (\x (x x x)) (\x (x x x)) -*Booleans* +**Booleans** Recall our definitions of true and false. @@ -24,14 +24,14 @@ In Racket, these can be defined like this: (define true (lambda (t) (lambda (f) t))) (define false (lambda (t) (lambda (f) f))) - -8. Define a "neg" operator that negates "true" and "false". -Expeceted behavior: (((neg true) 10) 20) evaluates to 20, +(8). Define a "neg" operator that negates "true" and "false". +Expected behavior: (((neg true) 10) 20) evaluates to 20, (((neg false) 10) 20) evaluates to 10. -9. Define an "and" operator. +(9). Define an "and" operator. 10. Define an "xor" operator. (If you haven't seen this term before, here's a truth table: + true xor true = false true xor false = true false xor true = true @@ -44,6 +44,7 @@ one of those values, call it a black-or-white-value, we should be able to write: the-black-or-white-value if-black if-white + (where if-black and if-white are anything), and get back one of if-black or if-white, depending on which of the black-or-white values we started with. Give a definition for each of "black" and "white". (Do it in both lambda calculus