CS 515 Homework 3 problems 1-4 solutions.

PROBLEM 1

;; 1.1
(f g x v) w -->

((((f g) x) v) w)

;; 1.2
\xy.(\z.x(y z))v w -->

(\x.(\y.(((\z.(x (y z))) v) w)))

;; 1.3
\x.xx \x.xx --> 
;; Note! don't be decieved into grouping by spaces
(\x.(x (x (\x.(x x)))))
;; If we treated the spaces as grouping, we would get this:
((\x.(x x)) (\x.(x x)))

PROBLEM 2

======== reduction 1 ========

(\xyz.(x z) (y z)) (\xy.x) (\xy.x)
--------------------------        ;; only choice: \xy.x for x 

(\yz.((\xy.x) z) (y z)) (\xy.x)
     -----------------            ;; innermost: z for x

(\yz.(\y.z) (y z)) (\xy.x)        ;; applicative: (y z) for y
     ------------

(\z. (\y.z) ((\xy.x) z))   ;; only choice: z for x
            -----------

(\z. (\y.z) (\y.z)) 
     -------------
(\z. z)

============ reduction 2 =============

(\xyz.(x z) (y z)) (\xy.x) (\xy.x)
--------------------------        ;; only choice: \xy.x for x 

(\yz.((\xy.x) z) (y z)) (\xy.x)   ;; leftmost first: (\xy.x) for y
-------------------------------

(\z.((\xy.x) z) ((\xy.x) z))      ;; two choices: take left
    -----------

(\z. (\y.z) ((\xy.x) z))          ;; only choice: take right
            -----------

(\z. (\y.z) (\y.z)) 
     -------------

(\z. z)

============ reduction 3 (similar to 2) =============

(\xyz.(x z) (y z)) (\xy.x) (\xy.x)
--------------------------        ;; only choice: \xy.x for x 

(\yz.((\xy.x) z) (y z)) (\xy.x)   ;; normal: (\xy.x) for y
-------------------------------

(\z.((\xy.x) z) ((\xy.x) z))      ;; two choices: take right
                -----------

(\z.((\xy.x) z) (\y.z))  ;; now take left

(\z. (\y.z) (\y.z)) 
     -------------

(\z. z)

============ reduction 4 ===========

(\xyz.(x z) (y z)) (\xy.x) (\xy.x)
--------------------------        ;; only choice: \xy.x for x 

(\yz.((\xy.x) z) (y z)) (\xy.x)
     -----------------            ;; applicative: z for x

(\yz.(\y.z) (y z)) (\xy.x)        ;; normal: (\xy.x) for y
--------------------------

(\z.(\y.z) ((\xy.x) z))          ;; only choice: z for x
           -----------

(\z.(\y.z) (\y.z))
    -------------

(\z. z)

PROBLEM 3.1

or (not a) b -->
(\x.\y.(x true) y)  ((\x.(x false) true) a)  b --> (normal order)
(\y.(((\x.(x false) true) a) true) y) b -->
(((\x.(x false) true) a) true) b -->
(((a false) true) true) b


PROBLEM 3.2

(and (or a b) (not (and a b))) -->
(\x.\y.(x y) false)  ((\x.\y.(x true) y) a b)  ((\x.(x false) true) ((\x.\y.(x y) false) a b)) -->
(\y.(((\x.\y.(x true) y) a b) y) false) ((\x.(x false) true) ((\x.\y.(x y) false) a b)) -->
(((\x.\y.(x true) y) a b) ((\x.(x false) true) ((\x.\y.(x y) false) a b))) false -->
(((a true) b) ((((\x.\y.(x y) false) a b) false) true)) false
(((a true) b) ((((a b) false) false) true) false


PROBLEM 4.1
 (\x.xx) (\x.xx)  --> (\x.xx) (\x.xx)
 (\x.xxx) (\x.xxx) --> (\x.xxx) (\x.xxx) (\x.xxx)
 (\x.xxxx) (\x.xxxx)  --> (\x.xxxx) (\x.xxxx) (\x.xxxx) (\x.xxxx)

PROBLEM 4.2
 (\x.z) ((\x.xx) (\x.xx))
 (\x.\y.z) ((\x.xx) (\x.xx)) ((\x.xx) (\x.xx))

PROBLEM 4.3
 (a b) (((\x.x) c) ((\y.y) d))

 

PROBLEM 6

See ~uli/cs515/examples/Scheme/ValueDynamic.ss  on ilab cluster.