Aportar una fuente de suministro de alimentos de calidad, estable y altamente saludable, que

This semantic net represents the following knowledge:

(1) The controller has a servo amp, which uses AC power.

(2) The controller is in a plant, which is located in Vancouver.

(3) It is a manufacturing plant.

(4) The servo amplifier of the controller belongs to a specific model type.

Note that M is a generic model number. Based on some new information (say, a specified model number), the semantic network may be searched (processed) to generate new knowledge (e.g., to which plant the servo amp belongs). A semantic net takes a hierarchical structure when the “Is_A” relationship alone is used in its representation. This is just a special case of a general semantic net.

1.4.2 Crisp logic

Logic is a useful technique of representing and processing knowledge, and is applicable in knowledge-based systems. In logic, knowledge is represented by statements called propositions, which may be joined together using connect-ives. The knowledge may be processed through reasoning, by the application of various laws of logic including an appropriate rule of inference. We shall limit our discussion to crisp, binary (i.e., two-valued) logic. Crisp sets and binary logic are analogous. Now we will present an introduction to crisp sets, and highlight the isomorphism that exists between crisp sets and binary (crisp) logic.

1.4.2.1 Crisp sets

A crisp set is a collection of elements within a crisp boundary. Since there cannot be any elements on the boundary, this is called a “crisp” set. A set of this type may be graphically represented by a Venn diagram, as shown in Figure 1.12(a). Here A denotes a set. The universal set (or universe of dis-course), as denoted by X, is the largest set that one could consider in a par-ticular problem domain. It contains all possible elements, and there cannot be any elements outside it. The null set is the empty set, and is denoted by Ø.

It does not have any elements. If an element x is inside set A, then x is called a member of A. This is represented by the notation x∈ A. Of course, by definition, x∈ X. If x is outside A (i.e., x is not a member of A), we write x∉ A. An example of a set is the group of people who live in Cincinnati who are over 50 years old. As another example,

A= {a1, a2, . . . , a_n} (1.2)

represents the set containing the n elements a1, a2, . . . , a_n. As yet another example, the set of numbers greater than 50 may be denoted by

1.4 Knowledg e represent ation and proc es s ing

A= {x|x > 50} (1.3)

Here, the expression that follows the symbol “|” gives the condition (or property) to which the elements of the set are subjected.

1.4.2.2 Operations of sets

The basic operations involving sets are complement, union, and intersection.

Complement: A complement of a set A is the set formed by all the elements outside A. It is denoted by A′ and is shown by the shaded area in Figure Figure 1.12: Some concepts of crisp sets: (a) membership, (b) complement, (c) union, (d) intersection, (e) subset (proper)

1 Introduction to intel lig ent sy s tems and sof t c omputing

30 1.12(b). In particular, the universal set X is the complement of the null set Ø, and vice versa.

Union: The union of two sets A and B is the set formed by all the elements in A and B. This is denoted by A ∪ B, and is shown by the shaded area in Figure 1.12(c).

Intersection: The intersection of two sets A and B is the set formed by all the elements that are common to both A and B. This is denoted by A ∩ B, and is shown by the shaded area in Figure 1.12(d).

Subset: A set A is a subset of another set B if all the elements in A are com-mon to (i.e., contained in) B. This is denoted by A⊆ B. This is equivalent to B⊇ A. If A is a subset of B but B is not a subset of A, then A is said to be a

“proper subset” of B, and is denoted by A⊂ B. This is equivalent to B ⊃ A, and is shown by Figure 1.12(e).

The definitions given above assume that the sets involved belong to the same universe (X).

1.4.2.3 Logic

Conventional logic deals with statements called “propositions.” In binary (or two-valued) logic, a proposition can assume one of only two truth values: true(T), false(F). An example of a proposition would be “John is over 50 years old.” Now consider the following propositions:

(1) Charcoal is white.

(2) Snow is cold.

(3) Temperature is above 60°C.

Here proposition 1 has the truth value F, and proposition 2 has the truth value T. But, for proposition 3 the truth value depends on the actual value of the temperature. Specifically, if the temperature is above 60°C the truth value is T and otherwise it is F.

In logic, knowledge is represented by propositions. A simple proposition does not usually make a knowledge base. Many propositions connected/

modified by logical connectives such as AND, OR, NOT, EQUALS, and IMPLIES may be needed. These basic logical operations are defined next.

The truth tables of these connectives are shown in Table 1.1. Note that a truth table gives the truth values of a combined proposition in terms of the truth values of its individual components.

Negation: The negation of a proposition A is “NOT A” and may be denoted as A (also ~A). It is clear that when A is TRUE, then NOT A is FALSE and vice versa. These properties of negation may be expressed by a truth table, as shown in Table 1.1(a).

1.4 Knowledg e represent ation and proc es s ing

As an example, consider the proposition “John is over 50 years old.” If John’s age is actually over 50 years, then this proposition is true. Otherwise it is false. Now consider the set of people who are more than 50 years old. If John is a member of this set, then the above proposition is true. If John is a member of the complement of the set, the above proposition is false. Then the negated proposition “John is not over 50 years old” becomes true. This shows that there is a correspondence between the “complement” operation in set theory and the “negation” operation in logic.

Disjunction: The disjunction of the two propositions A and B is “A OR B”

and is denoted by the symbol A∨ B. Its truth table is given in Table 1.1(b).

In this case, the combined proposition is true if at least one of its constituents is true. Note that this is not the “Exclusive OR” where “A OR B” is false also when both A and B are true, and is true only when either A is true or B is true. It is easy to see that the “disjunction” operation in logic corresponds to the “union” operation in sets.

Conjunction: The conjunction of two propositions A and B is “A AND B”

and is denoted by a symbol A∧ B. Its truth table is given in Table 1.1(c).

In this case the combined proposition is true if and only if both constituents are true. There is correspondence between “conjunction” in logic and “inter-section” in set theory.

Implication: Consider two propositions A and B. The statement “A implies B” is the same as “IF A THEN B.” This may be denoted by A → B. Note that if both A and B are true then A→ B is true. If A is false, the statement “When A is true then B is also true” is not violated regardless of whether B is true or

Table 1.1: Truth tables of some logical connectives (a) Negation (NOT )

1 Introduction to intel lig ent sy s tems and sof t c omputing

32 false. But if A is true and B is false, then clearly the statement A→ B is false.

These facts are represented by the truth table shown in Table 1.1(d).

Example 1.5

Consider two propositions A and B. Form the truth table of the combined pro-position (NOT A) OR B. Show that, in accordance with two-valued crisp logic, this proposition is identical to the statement “IF A THEN B.”

Solution

The truth of the combined proposition is given below.

A 3 B 3∨ B

T F T T

T F F F

F T T T

F T F T

Here we have used Tables 1.1(a) and (b). Note that the result is identical to Table 1.1(d). This equivalence is commonly exploited in logic associated with knowledge-based decision-making.

It is appropriate to mention here that the two propositions A and B are equivalent if A → B and also B → A. This may be denoted by either A ↔ B or A ≡ B. Note that the statement “A ↔ B” is true either “if both A and B are true” or “if both A and B are false.”

In real life we commonly make use of many “shades” of truth. Binary logic may be extended to many-valued logic that can handle different grades of truth value in between T and F. In fact, fuzzy logic may be interpreted as a further generalization of this idea.

1.4.2.4 Correspondence between sets and logic

As indicated before, there is an isomorphism between set theory and logic.

For crisp sets and conventional two-valued logic, this correspondence is summarized in Table 1.2. Boolean algebra is the algebra of two-valued logic.

The two values used are 1 (corresponding to true) and 0 (corresponding to false). Accordingly there is also a correspondence between the operations of Boolean algebra and logic, as indicated in Table 1.2. Note that the two values in Boolean algebra may represent any type of two states (e.g., on or off state; presence or absence state; high or low state; two voltage levels in transistor-to-transistor logic) in practical applications.

1.4 Knowledg e represent ation and proc es s ing

1.4.2.5 Logic processing (reasoning and inference)

Knowledge may be processed through reasoning, by the application of vari-ous laws of logic including an appropriate rule of inference, subjected to a given set of data (measurements, observations, external commands, previous decisions, etc.) to arrive at new inferences or decisions. In intelligent control, for example, a knowledge base is processed through reasoning, subjected to a given set of data (measurements, observations, external commands, previous decisions, etc.) to arrive at new control decisions.

Previously we have discussed the representation of knowledge in logic.

Now let us address the processing of knowledge in logic. The typical end objective of knowledge processing here is to make inferences. This may involve the following steps:

(1) Simplify the knowledge base by applying various laws of logic.

(2) Substitute into the knowledge base any new information (including data and previous inferences).

(3) Apply an appropriate rule of inference.

Note that these steps may be followed in any order and repeated any number of times, depending on the problem.

1.4.2.6 Laws of logic

The laws of logic govern the truth-value equivalence of various types of lo-gical expressions consisting of the connectives AND, OR, NOT, etc. They are valuable in simplifying (reducing) a logical knowledge base. Consider three propositions A, B, and C whose truth values are not known. Also, sup-pose that X is a proposition that is always true (T) and φ is a proposition that is always false (F). With this notation the important laws of logic are summar-ized in Table 1.3. It is either obvious or may be easily verified, perhaps using Table 1.1, that in each equation given in Table 1.3, the truth value of the left-hand side is equal to that of the right-left-hand side.

Table 1.2: Isomorphism between set theory, logic, and Boolean algebra

Set theory Set theory Binary logic Binary logic Boolean algebra

concept notation concept notation notation

Universal set X (Always) true (Always) T 1

Null set Ø (Always) false (Always) F 0

Complement A′ Negation (NOT ) A or ~A A

Union A ∪ B Disjunction (OR) A ∨ B A+ B

Intersection A ∩ B Conjunction (AND) A ∧ B A · B

Subset A⊆ B Implication (if-then) A→ B A ≤ B

1 Introduction to intel lig ent sy s tems and sof t c omputing