Knowledge Representations:

2 Knowledge Representations:
The process of acquisition of knowledge could be carryout manually or automatically. In manual mode, knowledge receives from one or more domain experts. Whereas in automatic mode, a machine learning system is used for autonomous learning and refining knowledge from the external world. Now knowledge after acquisition can be represented by various tools and techniques such:
Logic Representation
? Propositional Calculus.
? Predicate Calculus.
Production Rules Systems.
Networks Representation
? Semantic Nets

2.1 PROPOSITIONAL CALCULUS
2.1.1 Syntax of Propositional Calculus:
The propositional calculus and predicate calculus are first of all languages. Using their words, phrases, and sentences, we can represent and reason about properties and relationships in the world. The first step in describing a language is to produce the pieces that make it up: its set of symbols.
DEFINITION: Propositional Calculus Symbols
The symbols of propositional calculus are: {P, Q, R, S, …}
Truth symbols: {True, false}
Connectives: {? ,? , ? , ? , ?}
Propositional symbols denote propositions, or statements about the world that may be either true or false, such as "the car is red" or "water is wet." Propositions are denoted by uppercase letters near the end of the English alphabet Sentences in the propositional calculus are fanned from these atomic symbols according to the following rules:


DEFINITIONS: Propositional Calculus Sentence
Every propositional symbol and truth symbol is a sentence.
For example: true, P, Q, and R are sentences.
The negation of a sentence is a sentence.
For example: ?P and ?false are sentences.
The conjunction, AND, of two sentences is a sentence.
For example: P ? ?P is a sentence.
The disjunction, OR of two sentences is a sentence.
For example: P ? ?P is a sentence.
The implication of one sentence from another is a sentence.
For example: P ? Q is a sentence.
The equivalence of two sentences is a sentence.
For example: P ? Q ? R is a sentence.
Legal sentences are also called well-formed formulas or WFFs.
In expressions of the form P ? Q, P and Q are called the conjuncts. In P ? Q, P and Q are referred to as disjuncts. In an implication, P ? Q, P is the premise and Q, the conclusion or consequent.
In propositional calculus sentences, the symbols ( ) and [ ] are used to group symbols into sub-expressions and so to control their order of evaluation and meaning.
For Example: (P ? Q) ? R is quite different from P ? (Q ? R) as can be demonstrated using truth tables. An expression is a sentence, or well-formed formula, of the propositional calculus if and only if it can be formed of legal symbols through some sequence of these rules.
For Example: ((P ? Q) ? R) ? ?P ? ?Q ? R
is a well-formed sentence in the propositional calculus because:
P, Q, and R are propositions and thus sentences.
P ? Q, the conjunction of two sentences, is a sentence.
(P ? Q) ? R, the implication of a sentence for another, is a sentence.
?P and ?Q, the negations of sentences, are sentences.
?P ? ?Q the disjunction of two sentences, is a sentence.
?P ? ?Q ? R, the disjunction of two sentences, is a sentence.
((P ? Q) ? R) ? ?P ? ?Q ? R, the equivalence of two sentences, is a sentence.

2.1.2 The Semantics of the Propositional Calculus:
In this section we formally define the semantics or "meaning" of these sentences. Because AI programs must reason with their representational structures, it is important to demonstrate that the truth of their conclusions depends only on the truth of their initial knowledge, i.e., that logical errors are not introduced by the inference procedures. A precise treatment of semantics is essential to this goal.
A proposition symbol corresponds to a statement about the world. For example, P may denote the statement "it is raining" or Q, the statement "I live in a brown house." A proposition may be either true or false, given some state of the world. The truth value assignment to propositional sentences is called an interpretation, an assertion about their truth in some possible world.
Formally, an interpretation is a mapping from the propositional symbols into the set {T, F}. As mentioned in the previous section, the symbols true and false are part of the set well-formed sentences of the propositional calculus; i.e., they are distinct from the truth value assigned to a sentence. To enforce this distinction, the symbols T and F are used for truth value assignment.
Each possible mapping of truth value onto propositions corresponds to a possible world of interpretation. For example, if P denotes the proposition "it is raining" and Q denotes "I am at work" then the set of propositions {P, Q} has four different functional mappings into the truth values {T, F}. These mappings correspond to four different interpretations