A single axiom that is satisfied only by NAND or NOR must be of the form "something equals 
," since otherwise constant functions would satisfy the
equation. With up to six NANDs and two variables, none of the 
possible axiom systems of this kind work even up to 3-value
operators. But with 6 NANDS and 3 variables, 296 of the 
possible axiom systems work up to 3-value operators,
and 100 work up to 4-value operators (Wolfram 2002, p. 809).
Of the 25 of these that are not trivially equivalent, it then turns out that only the Wolfram axiom
 |
and the axiom
 |
where 
denotes the NAND operator, are equivalent to the axioms
of Boolean algebra (Wolfram 2002, pp. 808-811 and 1174).
These candidate axioms were identified by S. Wolfram in 2000, who also proved
that there were no smaller candidates.
