4.6. Qualitative Reasoning
An exactly quantitative description of the world being analyzed is not always available. For example, in the early stage of engineering design of a device, one may not know the exact value of all parameters of the device. In this case the incomplete knowledge can be used by representing the device in a qualitative form. By using the qualitative description we can get a better understanding of the device at the desired level. A reasoning based on the qualitative description of the world is called qualitative reasoning. The device’s behaviors can be predicted and the decisions can be made using the reasoning. Therefore, qualitative reasoning is to provide a broad picture of the functioning of the world by taking a step back from the detail. There is a great deal of interest in developing such techniques both by engineers and AI researchers.
In this section, the representation of qualitative reasoning in our multi-granular world model and its reasoning procedure are discussed.
4.6.1. Qualitative Reasoning Models
In common sense, if we say, ‘Someone is 185 cm tall’, it is regarded as a quantitative representation by describing an attribute with numbers. If we just say, ‘Someone is tall’, this is just a qualitative representation. In professional fields, for example, for a differential equation, if an analytical or a numerical solution is required, it needs a quantitative analysis. Meanwhile, only the existence and the properties of the solution are desired, it only needs a qualitative analysis. Qualitative analysis is widespread not only in common sense but also in professional fields.
In AI, researchers have been paying close attention to qualitative description and reasoning recently. They proposed several new fields such as qualitative physics, mechanics, etc. In these fields, people do not seek the precise solution of the problems but only the qualitative variation or variation trend of physical quantities is concerned.
In fact, ‘Tall’ is an uncertain description of ‘185 cm’ height of a person. A qualitative description of a variable is just a coarse-grained description of the variable with real number. Therefore, qualitative reasoning can be represented by a multi-granular world model. However, reasoning on some quotient space may be regarded as ‘quantitative’. But it may also be considered as ‘qualitative’ when the more precise analysis is required and vice versa. Therefore, quantitative and qualitative are relative.
Qualitative reasoning is the reasoning with qualitative representation. From the above observations, it is known that the projection and synthetic approaches of quotient spaces presented in the previous chapters can be applied to qualitative reasoning.
4.6.2. Examples
Example 4.20
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.
be real with operators ‘+’ and ‘×’. Let be a qualitative algebra with operators and shown below.A unary operator ⊝ is defined as
Define a projection 
:
as
: as
Element ‘?’ corresponds to the entire real axis 
.
.Qualitative algebra 
has the following properties.
has the following properties.Associative Law:
Identity Element:
Commutative Law:
Distributive Law:
An integral power of element s is defined as 
, where n is an integer.
, where n is an integer.Williams constructed a qualitative reasoning system with numbers based on hybrid algebra 
. Its reasoning procedure is the following.
. Its reasoning procedure is the following.(1) An equation with real number is simplified based on space 
.
.(2) The simplified equation is projected on space 
.
.(3) Then, the projected equation is operated using qualitative operators on space 
.
.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.
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.
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.
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.
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.Example 4.21
Murthy (1988) presented a qualitative reasoning at multiple resolutions. We briefly introduce this as follows.
(1) 
space denoted by 
is identical to S={-,0,+}.
space denoted by is identical to S={-,0,+}.The relationships 
between quantities on 
can be expressed as follows.
between quantities on can be expressed as follows.If 
, then 
. If 
, then 
. If 
, then 
.
, then . If , then . If , then .Where 
denotes the sign of a.
denotes the sign of a.It is noted that if 
and a and b have different signs, then 
is uncertain. This ambiguity can be solved by moving to the next 
space.
and a and b have different signs, then is uncertain. This ambiguity can be solved by moving to the next space.(2) 
space is denoted by 
(0, infinitesimality, large).
space is denoted by (0, infinitesimality, large).Refining 
space, first interval 
is divided into 
and 
, then interval 
is divided into 
and 
. Interval 
or 
is called ‘infinimality’. Interval 
or 
is said to be ‘large’. Therefore, on 
space in addition to relations 
and 
, the other two relations 
can be introduced as follows.
space, first interval is divided into and , then interval is divided into and . Interval or is called ‘infinimality’. Interval or is said to be ‘large’. Therefore, on space in addition to relations and , the other two relations can be introduced as follows.
Relation 
, if both a and b are infinitesimality or large.
, if both a and b are infinitesimality or large.On 
space, operations 
and 
are ambiguity. But on 
space, if a is large and b is infinitesimality, then 
, i.e., the signs of 
and a are the same. But if both a and b are infinitesimality or large, the sign of 
is still uncertain. These ambiguities can be solved by introducing 
space.
space, operations and are ambiguity. But on space, if a is large and b is infinitesimality, then , i.e., the signs of and a are the same. But if both a and b are infinitesimality or large, the sign of is still uncertain. These ambiguities can be solved by introducing space.(3) 
space is denoted by 
, where y is the base (e.g., 2 or 10 ) and z is an integer.
space is denoted by , where y is the base (e.g., 2 or 10 ) and z is an integer.The real number is divided by the logarithmic distance between two numbers, i.e., if 
, then 
. Therefore, when 
, then 
, in other words, when the order of magnitude of a is bigger than b, the orders of magnitude of a+b and a are the same. If 
and 
, then 
.
, then . Therefore, when , then , in other words, when the order of magnitude of a is bigger than b, the orders of magnitude of a+b and a are the same. If and , then .While 
and 
, the sign of 
is uncertain. To resolve the ambiguity a finer resolution is needed. The 
space is introduced.
and , the sign of is uncertain. To resolve the ambiguity a finer resolution is needed. The space is introduced.(4) 
-space is denoted by 
, where y and z as shown in (3) and x is a number with n significant digits. As n increases the accuracy of the description increases, while 
the Q-space approaches the real space R.
-space is denoted by , where y and z as shown in (3) and x is a number with n significant digits. As n increases the accuracy of the description increases, while the Q-space approaches the real space R.In order to solve the uncertainty of the sign of 
, Murthy gradually refines the Q-space so that the uncertainty of computational results reduces.
, Murthy gradually refines the Q-space so that the uncertainty of computational results reduces.From the viewpoint presented in this book, Murthy’s Q-spaces of multiple resolutions are quotient spaces of real number at different granularities. The successive refinement approach presented by Murthy is just an approximation method for constructing an operational space of S = {-, 0, +} with respect to real addition. The approximation method for constructing operational space we presented can be used for general quotient space and any binary-operator. So the method can be applied to Murthy’s qualitative reasoning as well.
Example 4.22
Kuipers (1988) proposed a qualitative reasoning with incomplete quantitative measures. His basic idea is the following.
Assume that a system has several parameters and the relations among parameters are represented by algebra formulas, differential equations, or functions. Now, only partial knowledge of the parameters is known, for example, the variation range of the parameters, and the variation range can be represented by intervals. The problem is how to narrow the variation ranges via the known variation ranges of parameters and relations among them. Kuipers called it the propagation of incomplete quantitative knowledge and divides it into four categories.
(1) Propagation via arithmetic constraints 
,
,(2) Propagation via monotonic function constraints,
(3) Propagation via number spaces,
(4) Propagation via temporal points 
, where 
indicates the differential operation on t.
, where indicates the differential operation on t.For example, 
is known. And the variation ranges of x and y are known to be 
and 
, respectively. It’s easy to find the variation range of z via the arithmetic constraint 
. It’s 
.
is known. And the variation ranges of x and y are known to be and , respectively. It’s easy to find the variation range of z via the arithmetic constraint . It’s .Again, constraint 
is known. The variation range of x in temporal interval 
is 
. Assume that 
at 
(
). Now find the variation range of 
. From the mean value theorem, there exists 
such that 
. If the variation range of 
is 
, then the variation range of 
is 
.
is known. The variation range of x in temporal interval is . Assume that at (). Now find the variation range of . From the mean value theorem, there exists such that . If the variation range of is , then the variation range of is .We next use the quotient space model to explain the above examples.
First, assume that 
and 
are three parameters and 
is a constraint. 
and 
are variation ranges of x and y, respectively. Find the variation range of z.
and are three parameters and is a constraint. and are variation ranges of x and y, respectively. Find the variation range of z.Let 
and 
(real sets) be spaces that 
and 
are located, respectively. Let 
and 
be quotient spaces of X and Y, respectively. 
is regarded as a constraint on space 
and Z, i.e., 
. 
is an element of 
.
and (real sets) be spaces that and are located, respectively. Let and be quotient spaces of X and Y, respectively. is regarded as a constraint on space and Z, i.e., . is an element of .Second, find the section 
of C on 
.
of C on .
is the variation range of z and an interval in Z denoted by 
. Let 
be a quotient space of Z.
. 
is just an element of 
.Therefore, the propagation of incomplete quantitative knowledge under different constraints is equivalent to finding the quotient constraint of a given quotient space.
4.6.3. The Procedure of Qualitative Reasoning
As viewed from different granularities, a qualitative reasoning is reasoning on some quotient space of the original space. The procedure of qualitative reasoning is summarized as follows.
(1) The variables, parameters and the constraints among these variables and parameters in the original problem space are analyzed and then simplified.
(2) All certain and uncertain information is represented in its proper quotient space.
(3) According to the analytical requirement, a proper qualitative space is constructed. The space is also a quotient space of the original one.
(4) All constraints and operators are projected on the qualitative space.
(5) The reasoning is made in that space.
The concepts of quotient operation, quotient constraint, the projection and synthetic method, and the approximation of upper space presented in the preceding sections can be used for making reasoning on quotient space.
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