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is obtained by the following strict separable system:
[k −1/2; k,k −1,k −2,...,1,1,...,1
| {z }
n−k
]
and such a tolerance is τ = 1/(2n + k
2
+ k −1).
Again, any other strict linearly (natural or not)
separating system (of non null weights) with tolerance
τ
0
fulfills τ
0
≤ τ.
In the same vein, we establish the corresponding
conjecture for the G-tolerance:
Conjecture 3.6 The maximum G-tolerance for n
variables and k types (k > 1) of distinguished vari-
ables is given by the following strict separable sys-
tem:
[
p
k(k −1);k,k −1,k −2,..., 1, 1, . . . , 1
| {z }
n−k
].
and such a G-tolerance is χ =
√
k(k−1)−(k−1)
√
k(k−1)+(k−1)
.
In general, given a strict separating system for an
arbitrary separable switching function, we are able to
find an equivalent strict separating system (with nat-
ural weights having minimum integer sum) achiev-
ing the maximum tolerance and the maximum G-
tolerance. Moreover, it is known, see (Freixas and
Molinero, 2008b), that for less than 8 variables all
linearly separable switching functions have, up to iso-
morphism, a unique strict separating system with nat-
ural weights having minimum sum. So, we only
need to adjust the threshold,
A+B
2
for the tolerance
and
√
AB for the G-tolerance, for these systems to
achieve the maximum tolerance and the maximum
G-tolerance, respectively. Note that for 8 variables
the situation changes: There are 154 linearly separa-
ble switching functions, up to isomorphism, with two
strict separating systems having natural weights and
minimum sum.
4 FUTURE WORK
Generating a huge number of random strict separating
natural systems and then to conjecturate what distri-
bution follows the tolerance, the maximum tolerance,
the G-tolerance and the maximum G-tolerance.
It is also interesting to provide tables of all, up
to isomorphism, linearly separable switching func-
tions of a reasonable high number of variables with:
a strict separating system achieving its tolerance; its
tolerance; a strict separating system achieving its G-
tolerance; and its G-tolerance.
Although the computational limitation (complex-
ity) of finding the tolerance and the G-tolerance of a
given strict separating system (both are NP-hard), it
will be of interest to develop an efficient algorithm
able to calculate the tolerance and the greatest toler-
ance for strict separating systems with a reasonable
high number of variables.
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