Categories of Knowledge Representation

We now consider the main categories of knowledge representation schemes such as rules, graphs, and networks.

2.1. Rules

In their most generic format, rules are conditional statements of the form

IF condition THEN conclusion (action) (1)

where the conditionandconclusion are descriptors of pieces of knowledge about the domain, while the rule itself expresses the relationship between these descriptors. namely, For instance, the rule “IF the temperature ishighTHEN the electricity demand ishigh” captures a piece of domain knowledge that is essential to planning the activities of an electric company. Notably, rules of this sort are quite qualitative yet highly expressive. We are perhaps not so much concerned with detailed numeric quantification of the descriptors occurring in such rules since we appreciate that the rule exhibits an interesting relationship that is pertinent to the problem. We note that both the condition and the conclusion are formed as information granules-conceptual entities that are semantically sound abstractions. The operational context within which information granules are formalized and used can be established by considering any of the available formal frameworks such as sets, fuzzy sets, and rough sets.

In practice, domain knowledge is typically structured into a family of rules, with each of these assuming the same format or a similar format, e.g.,

IF condition is A_i THEN conclusion is B_i (2) where A_i and B_i are information granules. The rules articulate a collection of meaningful relationships existing within the problem.

We may envision more complex rules whose left-hand side may include several conditions of the form

IF condition₁andcondition₂ and andcondition_nTHEN conclusion (3) with multidimensional input space composed as a Cartesian product of the input variables. Note that the individual conditions are aggregated together by theandlogic connective. Rules give rise to the highly modular form of a granular model, and expansion of the model requires addition of some new rules while the existing ones are left intact.

There are two important points regarding the development of this form of knowledge represen-tation. First, information granules can be expressed within various formal frameworks. Second, rules exhibit various architecture and a richness of possible extensions.

72 2. Categories of Knowledge Representation

2.1.1. Gradual Rules

In gradual (graded) rules, rather than expressing an association between condition and conclusion, we capture the trend within information granules and hence the condition and conclusion parts will each contain a term referring to that trend. For instance, the gradual rules rules of the form:

“the higher the values of the condition, the higher the values of the conclusion” (4) or

“the lower the values of the condition, the higher the values of the conclusion” (5) represent knowledge about the relationships between changes in the condition and the conclusion, as in “the higher the income, the higher the taxes”. Using the previous notation, we can rewrite such rules in the form

“IFA_iTHENB_i” (6)

where and are modifiers acting upon the values of A_i and B_i and causing their shift in a certain direction (hence capturing the existing increasing or decreasing trends in their values).

2.1.2. Quantified Rules

It is quite common to quantify the relevance (confidence) of a specific rule. For instance, a rule of the form

the likelihood thathighfluctuations in real estate prices lead to a

significant migrationof population within the province isquite moderate (7) is an example of a quantified rule whose likelihood of satisfaction is quite moderate. Again, we may have several schemes for the realization of the quantification effect itself. In some expert systems, these take the form of confidence factors, that is numeric values in the range of [–1, 1].

In other cases, we may use linguistic characters of quantification captured in the language of fuzzy sets.

2.1.3. Analogical Rules

Here the rules focus on levels of similarity (closeness, resemblance, etc.) between pairs of items standing in the condition and the conclusion. The analogical rules assume the form

IF similarityA_iA_j THEN similarityB_iB_j (8) Present in this form, the rules constitute a framework for analogical reasoning. In a nutshell, we are provided with an information granule A in the condition space and wish to infer a corresponding conclusion B in the conclusion space.

2.1.4. Rules with Regression Local Models

In such rules the conclusion comes in the form of a “local” regression models whose scope is narrowed down to the condition portion of the rule, that is,

IF condition is A_i THEN y=f_ixa_i (9)

The regression modelf_icould be linear or nonlinear and applies only to the inputsx∈Rⁿthat belong to the information granule represented by A_i.

A B

C

D

Figure 5.4. An example graph representing dependencies between concepts.

Let us reiterate two general and highly essential observations that hold despite the significant diversity of formats of the rules themselves and of their applications. First, the rules give rise to highly modular architectures, and this an organization becomes crucial to the efficient realization of the mechanisms of model formation, reasoning, and maintenance. Secondly, rules are always formulated in the language of information granules; hence, they constitute an abstract reflection of the problem or of problem solving. Since we never reason in terms of numbers, the explicit use of numbers in the rules is highly unjustifiable.

2.2. Graphs and Directed Graphs

Graphs are fundamental constructs representing relationships between concepts. Concepts are represented as nodes of a graph. The linkages (associations, dependencies, etc.) between the concepts are represented as the edges of the graph. An example of a graph is shown in Figure 5.4;

here we have four nodes (A, B, C, and D) with several links between them (A-B, A-C, B-C, and C-D).

Graphs are helpful in visualizing a collection of concepts and presenting key relationships (dependencies) between them. They are highly appealing to designers as well as to users of developed systems. When properly displayed and augmented by colors, graphs help provide a solid insight into the behavior of the phenomenon under discussion.

There are many variations and augmentations of generic graphs. First, graphs can be directed, that is, their links can indicate directional relationships between the nodes. For instance, an occurrence of concept (node) A triggers occurrence of concept (node) B. We can represent a chaining effect (A-B-C) or looping (C-D-E -C) as illustrated in Figure 5.5.

Second, graphs can also show numeric quantification of the links that indicate the strength of dependency between the concepts (nodes). This quantification is shown in Figure 5.5 as well.

Here, the strength of the relationship between E and C is far lower than that between B and C.

A B

C

D E

0.7

1.0

0.4

0.6 1.0

Figure 5.5. A directed graph showing a chaining effect (A-B-C) and a loop (C-D-E-C) in the graph.

74 2. Categories of Knowledge Representation

Figure 5.6. A hierarchy of graphs; note that the lower graph (presenting more details) expands a node of the structure at the higher level of abstraction.

To deal with a significant number of nodes, a graph can be structured as a hierarchy of subgraphs where the nodes on the upper level are expanded to a collection of nodes in the graph on the lower level. An example of a hierarchy of graphs is illustrated in Figure 5.6.

2.3. Trees

Trees are a special category of graphs in which there is a single root node and a collection of terminal nodes. There are no loops in trees. An example tree is shown in Figure 5.7.

As in the case of the graphs themselves, each node represents a certain concept or attribute while the edges of the graph present the relationships between the concepts.

Decision treesare one of the most commonly used tree structures. Here each node of the tree represents an attribute that assumes a finite number of discrete values while the edges originating from each node are labeled with the corresponding values. An example of a decision tree is shown in Figure 5.8.

We note that decision trees can be easily translated into a collection of rules. For each rule, we traverse the tree starting from its root and moving down to one of the terminal nodes. Along the way, we collect the attributes and the values that they assume. For instance, the rules obtained from the tree shown in Figure 5.8 read as follows:

IF A is c and B is w THEN IF A is c and B is z THEN IF A is a and C is k THEN

Figure 5.7. An example tree. Terminal nodes are shaded.

A a

b c

z w C B

k l

ω

ω

φ φ

Figure 5.8.A decision tree showing nodes with the edges labeled by the discrete values (a, b, c… ) of the attributes (A, B, C). The terminal nodes are shaded.

IF A is a and C is l THEN (10)

Note that the length of each rule is different since it is the result of the tree traversal and each path being traveled could be of a different length. It is important to observe that the traversal process sets up the order of the conditions (attributes) in the rule. This organization of conditions is not captured when the rules are formed on a basis of available domain knowledge.

2.4. Networks

Networkscan be regarded as generalized graphs in the sense that at each node of the graph we encounter some local processing capability. In other words, the network not only represents the knowledge itself but also contains the underlying processing being realized at the local level.

An example network, illustrated in Figure 5.9 shows the nodes at which we compute the logic expressions of the inputs. In essence, these are logic functions operating on the input variables through the use of the logic operations of conjunction, disjunction, and negation.

Networks can be built in a hierarchical structure where a node at a higher conceptual level unfolds into a collection of nodes at a lower level; the nodes existing there may be endowed with some limited processing capabilities or they may contain no processing at all. The hierarchical organization of the network is helpful in representation of structure in data.

z₁=γ (x1, x₂, x₃)

z₂=η (x1, x₂, x₃)

y=ρ (z1, z₂, x₄)

Figure 5.9. A network whose nodes provide logic computing (logic formulas ) over the variables associated with the incoming links.

76 3. Granularity of Data and Knowledge Representation Schemes