The Wolfram axiom is a single Boolean algebra axiom expressed using NAND. 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).
The Wolfram axiom is written
 |
Another proved single axiom is
 |
Here 
denotes the NAND operator.
Both displayed axioms have been proved equivalent to the
axioms of Boolean algebra
(Wolfram 2002, pp. 808-811
and 1174).
Wolfram (2002) reports finding these candidates in 2000 and ruling out smaller candidates.
Of the 25 inequivalent candidates passing the finite tests,
two were proved equivalent to Boolean algebra,
four others were ruled out, and the remaining 19 were unresolved in that account.
Finite tests alone do not prove equivalence.
