tani-ro-hei / logic-nor

Demonstrates that the truth function logical NOR is functionally complete.

nor.scm

This Scheme (Racket) program demonstrates that the truth function logical NOR(↓) is functionally complete, namely, that any binary truth function f: {0,1}×{0,1}→{0,1} is given by a composition just of NOR.

Description

The higher-order function truth-func, constructed with only NOR, take six arguments map-11, map-10, map-01, map-00, x, y, where the function map-xy is composed just of NOR and x, y are truth values, and return the binary truth function which value of arguments (x, y) is equal to the (map-xy x y).

So, if a func-imply is defined as below:

(define func-imply
  (lambda (x y)
    (truth-func
      map-11-1
      map-10-0
      map-01-1
      map-00-1
    x y)
  )
)

where the map-xy-z maps truth values (x, y) to z, then the func-imply turns out to be the logical IMP(⇒) itself. In fact,

(func-imply #t #t)
(func-imply #t #f)
(func-imply #f #t)
(func-imply #f #f)

outputs

#t
#f
#t
#t