Rules of Inference in Artificial intelligence

In this page we will learn about Rules of Inference in Artificial intelligence| Rules of Inference in Artificial intelligence | Types of Inference rules | Modus Ponens | Modus Tollens | Hypothetical Syllogism | Disjunctive Syllogism | Addition | Simplification | Resolution


Inference:

We need intelligent computers in artificial intelligence to construct new logic from old logic or evidence, therefore inference is the process of drawing conclusions from data and facts.

Inference rules:

The templates for creating valid arguments are known as inference rules. In artificial intelligence, inference rules are used to generate proofs, and a proof is a series of conclusions that leads to the intended outcome. The implication among all the connectives is vital in inference rules. Some terms relating to inference rules are as follows:

  1. Implication: It's one of the logical connectives, denoted by the letters P → Q. It's a Boolean expression, to be precise.
  2. Converse: The converse of implication is when the right-hand side statement is applied to the left-hand side, and vice versa. It is denoted by the letters Q → P.
  3. Contrapositive: Contrapositive is the negation of converse, and it can be expressed as ¬ Q → ¬ P.
  4. Inverse: Inverse is the antithesis of implication. ¬ P → ¬ Q can be used to symbolize it.

Some of the compound statements in the above term are equivalent to each other, which we can verify using the truth table:

P Q P → Q Q → P ¬ Q → ¬ P ¬ P → ¬ Q
T T T T T T
T F F T F T
F T T F T F
F F T T T T

As a result from the above truth table, we can prove that P → Q is equivalent to ¬ Q → ¬ P, and Q→ P is equivalent to ¬ P → ¬ Q.

Types of Inference rules:

1. Modus Ponens:

One of the most essential laws of inference is the Modus Ponens rule, which asserts that if P and P → Q are both true, we can infer that Q will be true as well. It's written like this:

Rules of Inference  in AI 2 in Artificial Intelligence (AI)

Example:
Statement-1: "If I am sleepy then I go to bed" ==> P → Q
Statement-2: ""I am sleepy" ==> P"
Conclusion: "I go to bed." ==> Q.
Hence, we can say that, if P → Q is true and P is true then Q will be true.

Proof by Truth table:

Rules of Inference  in AI3 in Artificial Intelligence (AI)

2.Modus Tollens:

According to the Modus Tollens rule if P→ Q is true and ¬ Q is true, then ¬ P will also true. It can be represented as:

Rules of Inference  in AI4 in Artificial Intelligence (AI)

Example:
Statement-1: "If I am sleepy then I go to bed" ==> P→ Q
Statement-2: "I do not go to the bed."==> ~Q
Statement-3: Which infers that "I am not sleepy" => ~P

Proof by Truth table:
Rules of Inference  in AI5 in Artificial Intelligence (AI)

3. Hypothetical Syllogism:

According to the Hypothetical Syllogism rule if P→R is true whenever P→Q is true, and Q→R is true. It can be represented as the following notation:
Example:
Statement-1: Statement-1: If you have my home key then you can unlock my home. P→Q
Statement-2: Statement-2: If you can unlock my home then you can take my money. Q→R
Statement-3: Conclusion: If you have my home key then you can take my money. P→R

Proof by Truth table:
Rules of Inference  in AI6 in Artificial Intelligence (AI)

4. Disjunctive Syllogism:

According to the Disjunctive syllogism rule if P∨Q is true, and ¬P is true, then Q will be true. It can be represented as:

Rules of Inference  in AI7 in Artificial Intelligence (AI)

Example:
Statement-1:Today is Sunday or Monday. ==>P∨Q
Statement-2:Today is not Sunday. ==> ¬P
Conclusion: Today is Monday. ==> Q

Proof by Truth table:
Rules of Inference  in AI8 in Artificial Intelligence (AI)

5. Addition:

According to the Addition rule which is one of the common inference rule, If P is true, then P∨Q will be true.

Rules of Inference  in AI9 in Artificial Intelligence (AI)

Example:
Statement-1: I have a vanilla ice-cream. ==> P
Statement-2: I have Chocolate ice-cream.
Conclusion: I have vanilla or chocolate ice-cream. ==> (P∨Q)

Proof by Truth table:
Rules of Inference  in AI10 in Artificial Intelligence (AI)

6. Simplification:

According to the simplification rule if P∧ Q is true, then Q or P will also be true. It can be represented as:

Rules of Inference  in AI11 in Artificial Intelligence (AI)
Proof by Truth table:
Rules of Inference  in AI12 in Artificial Intelligence (AI)

7. Resolution:

According to the Resolution rule if P∨Q and ¬ P∧R is true, then Q∨R will also be true. It can be represented as

Rules of Inference  in AI13 in Artificial Intelligence (AI)
Proof by Truth table:
Rules of Inference  in AI14 in Artificial Intelligence (AI)
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