CH-232 Problem Sheet #8 Solution

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Problem 8.1: digital circuit analysis (1+1+2 = 4 points)

You are given the following digital circuit. The circuit may as well be found online at http:// (but there is no guarantee that it persists).
a) Write down the truth table defining the outputs y0, y1, and y.
b) Write down the boolean expressions defining y0, y1, and y.
c) Describe in your own words what the circuit is doing and how it might be used.

Problem 8.2: fold function duality theorems (2+2+2 = 6 points)

The fold functions compute a value over a list (or some other type that is foldable) by applying an
operator to the list elements and a neutral element. The foldl function assumes that the operator
is left associative, the foldr function assumes that the operatore is right associative. For example,
the function application
1 foldl (+) 0 [3,5,2,1]
results in the computation of ((((0+3)+5)+2)+1) and the function application
1 foldr (+) 0 [3,5,2,1]
results in the computation of (3+(5+(2+(1+0)))). The value computed by the fold functions may be
more complex than a simple scalar. It is very well possible to construct a new list as part of the
fold. For example:
1 map’ :: (a -> b) -> [a] -> [b]
2 map’ f xs = foldr ((:) . f) [] xs
The evaluation of map’ succ [1,2,3] results in the list [2,3,4]. There are several duality theorems that can be stated for fold functions. Prove the following three duality theorems:
a) Let op be an associative operation with e as the neutral element:
op is associative: (x op y) op z = x op (y op z)
e is neutral element: e op x = x and x op e = x
Then the following holds for finite lists xs:
foldr op e xs = foldl op e xs
b) Let op1 and op2 be two operations for which
x `op1` (y `op2` z) = (x `op1` y) `op2` z
x `op1` e = e `op2` x
holds. Then the following holds for finite lists xs:
foldr op1 e xs = foldl op2 e xs
c) Let op be an associative operation and xs a finite list. Then
foldr op a xs = foldl op’ a (reverse xs)
holds with
x op’ y = y op x