Problem Statement

"Simulate" a four-bit adder. This design can be realized using four 1-bit full adders. Each of these 1-bit full adders can be built with two half adders and an   or   gate. ; Finally a half adder can be made using an   xor   gate and an   and   gate. The   xor   gate can be made using two   nots,   two   ands   and one   or. Not,   or   and   and,   the only allowed "gates" for the task, can be "imitated" by using the bitwise operators of your language. If there is not a bit type in your language, to be sure that the   not   does not "invert" all the other bits of the basic type   (e.g. a byte)   we are not interested in,   you can use an extra   nand   (and   then   not)   with the constant   1   on one input. Instead of optimizing and reducing the number of gates used for the final 4-bit adder,   build it in the most straightforward way,   connecting the other "constructive blocks",   in turn made of "simpler" and "smaller" ones.

Solutions should try to be as descriptive as possible, making it as easy as possible to identify "connections" between higher-order "blocks". It is not mandatory to replicate the syntax of higher-order blocks in the atomic "gate" blocks, i.e. basic "gate" operations can be performed as usual bitwise operations, or they can be "wrapped" in a block in order to expose the same syntax of higher-order blocks, at implementers' choice. To test the implementation, show the sum of two four-bit numbers (in binary).

Let's start with the solution:

;; returns a list of bits: '(sum carry)
(defun half-adder (a b)
  (list (logxor a b) (logand a b)))

;; returns a list of bits: '(sum, carry)
(defun full-adder (a b c-in)
  (let*
    ((h1 (half-adder c-in a))
    (h2 (half-adder (first h1) b)))
    (list (first h2) (logior (second h1) (second h2)))))

;; a and b are lists of 4 bits each
(defun 4-bit-adder (a b)
  (let*
    ((add-1 (full-adder (fourth a) (fourth b) 0))
      (add-2 (full-adder (third a) (third b) (second add-1)))
      (add-3 (full-adder (second a) (second b) (second add-2)))
      (add-4 (full-adder (first a) (first b) (second add-3))))
    (list
      (list (first add-4) (first add-3) (first add-2) (first add-1))
      (second add-4))))

(defun main ()
  (print (4-bit-adder (list 0 0 0 0) (list 0 0 0 0)))   ;; '(0 0 0 0) and 0
  (print (4-bit-adder (list 0 0 0 0) (list 1 1 1 1)))   ;; '(1 1 1 1) and 0
  (print (4-bit-adder (list 1 1 1 1) (list 0 0 0 0)))   ;; '(1 1 1 1) and 0
  (print (4-bit-adder (list 0 1 0 1) (list 1 1 0 0)))   ;; '(0 0 0 1) and 1
  (print (4-bit-adder (list 1 1 1 1) (list 1 1 1 1)))   ;; '(1 1 1 0) and 1
  (print (4-bit-adder (list 1 0 1 0) (list 0 1 0 1)))   ;; '(1 1 1 1) and 0
 )

(main)