Williams (1988) presented a so-called qualitative algebraic reasoning that mixes qualitative with quantitative information. His approach is briefly introduced below.
Let
be real with operators ‘+’ and ‘×’. Let
be a qualitative algebra with operators
and
shown below.
A unary operator ⊝ is defined as
Element ‘?’ corresponds to the entire real axis
.
Qualitative algebra
has the following properties.
An integral power of element
s is defined as
, where
n is an integer.
Williams constructed a qualitative reasoning system with numbers based on hybrid algebra
. Its reasoning procedure is the following.
The advantage of the above qualitative reasoning procedure is the following. If the sign of some quantities is known rather than the precise value, the quantities cannot be operated
on real space
R using standard real operators. But they can be operated on
using qualitative operators. Moreover, since on
for any element
t,
, any polynomial can be transformed into a quadratic polynomial. This is a specific property that does not occur in real space
R. But the weakness of operation on
is that the projection of
on
is not homomorphism. In our terms, it means that operator
is not a quotient operation of
, i.e., the operation
on space
does not necessarily have a unique result. For example,
does not have a unique result on
S.Now, we analyze Williams’s hybrid algebra from the multi-granular computing viewpoint (
Zhang and Zhang, 1989c, 1990b).
Let
and
. Therefore,
is a quotient space of
.
p is a projection from
R to
S. Operators
and
of
S are the projections of operators
and
of
R, respectively.
It is easy to know that operator
is a quotient operation of
S but
is not. As stated before, the upper bound space of
S with respect to operator
is
R itself.
Thus, in our terms, Williams’s qualitative reasoning with numbers is a reasoning on quotient space
S with respect to the projections of operators
and
. Since
is not a quotient operator on
S, in order to get a unique result from the operation, the approximate approach for finding upper bound space as shown in the above section can be used for solving the problem.