Rules of Inference in Artificial Intelligence

Introduction

Rules of Inference In Artificial Intelligence enables machines to make intelligent decisions and predictions based on available data and knowledge.

In machine learning, for example, inference involves using a trained model to predict new data based on the patterns and relationships learned from the training data. In natural language processing, inference can be used to understand the meaning of a sentence based on the context and grammar rules. In computer vision, inference can recognize objects in an image based on their features and characteristics.

Rules of Inference

Rules of Inference In Artificial Intelligence refer to logical rules and principles used to derive new knowledge or conclusions from existing knowledge or facts. These rules are designed to mimic the way humans reason and make decisions.

They are considered templates for generating valid arguments.

Here are some terminologies related to inference rules:-

• Implication: This signifies a boolean expression, a logical connective that can be represented as P → Q. The right side of the expression implies the left side. Example: If it is raining outside ( p ) , then the ground is wet (q).
• Converse: It is the opposite or the converse of implication. The letters Q → P denotes it. Example: "If the ground is wet (q), then it is raining outside ( p )."
• Contrapositive: The negation of converse is called contrapositive, which can be represented as ¬ Q → ¬ P. Example: "If the ground is not wet (¬q), then it is not raining outside (¬p)."
• Inverse: The negation of implication is termed as inverse. It can be represented as (¬ P → ¬ Q). Example: "If it is not raining outside (¬p), then the ground is not wet (¬q)."

Types of inference rules

Modus Ponens

Modus Ponens is a basic logical inference rule that is commonly used in Artificial Intelligence (AI) systems. Modus Ponens is a rule of inference that states that if P implies Q, and P is true, then Q must also be true. In AI, Modus Ponens is used in various applications, including expert systems, decision support systems, and natural language processing.

The notation of Modus Ponens is:

where P → Q represents the premise "If P then Q", P represents the premise "P", and Q represents the conclusion "Q". The symbol |---| is a table separator that separates the premises from the conclusion. The symbol ∴ is the therefore symbol, which indicates the conclusion that follows from the premises.

Example

A simple example of Modus Ponens would be as follows:

Premise 1: The ground is wet if it is raining. (P implies Q) Premise 2: It is raining. (P is true) Conclusion: Therefore, the ground is wet. (Q must be true)

In this example, Modus Ponens is used to infer that the ground is wet based on the two premises. The first premise establishes the logical relationship between rain and wet ground, and the second premise provides the information that it is currently raining. By applying Modus Ponens, we can conclude that the ground is wet, as the logical relationship established in the first premise must hold when the second premise is also true.

Modus Tollens

Modus Tollens is another basic logical inference rule commonly used in Artificial Intelligence (AI) systems. Modus Tollens is a rule of inference that states that if P implies Q, and Q is false, then P must also be false.

The notation of Modus Tollens is:

where P → Q represents the premise "If P then Q", ¬Q represents the premise "not Q", and ¬P represents the conclusion "not P". The symbol |---| is a table separator that separates the premises from the conclusion. The symbol ∴ is the therefore symbol, which indicates the conclusion that follows from the premises.

Example

A simple example of Modus Tollens would be as follows:

Premise 1: The ground is wet if it is raining. (P implies Q) Premise 2: The ground is not wet. (Q is false) Conclusion: Therefore, it is not raining. (P must be false)

In this example, Modus Tollens is used to infer that it is not raining based on the two premises. The first premise establishes the logical relationship between rain and wet ground, and the second premise provides the information that the ground is not wet. Therefore, by applying Modus Tollens, we can conclude that it is not raining, as the logical relationship established in the first premise cannot hold when the second premise is also true.

Hypothetical Syllogism

Hypothetical Syllogism is a rule of inference that states that if P implies Q, and Q implies R, then P implies R.

The notation of Hypothetical Syllogism is:

where P → Q represents the premise "If P then Q", Q → R represents the premise "If Q then R", and P → R represents the conclusion "If P then R". The symbol |---| is a table separator that separates the premises from the conclusion. The symbol ∴ is the therefore symbol, which indicates the conclusion that follows from the premises.

Example

A simple example of Hypothetical Syllogism would be as follows:

Premise 1: If it rains, then the ground will be wet. (P implies Q) Premise 2: The grass will be slippery if the ground is wet. (Q implies R) Conclusion: Therefore, if it rains, the grass will be slippery. (P implies R)

In this example, Hypothetical Syllogism combines two rules and draws a more complex conclusion. The first premise establishes the logical relationship between rain and wet ground, and the second establishes the logical relationship between wet ground and slippery grass. Therefore, by applying Hypothetical Syllogism, we can infer that the grass will be slippery if it rains, as the logical relationships established in the two premises can be chained together to draw this conclusion.

Disjunctive Syllogism

Disjunctive Syllogism is a rule of inference that states that if either P or Q is true, and P is false, then Q must be true.

Notation:

where P ∨ Q represents the disjunction and ¬P represents the negation of one of the disjuncts. The conclusion Q follows as a logical consequence of the disjunction and the negation of one of the disjuncts.

Example

A simple example of Disjunctive Syllogism would be as follows:

Premise 1: John or Sarah will be the next manager. (P or Q) Premise 2: John will not be the next manager. (not P) Conclusion: Therefore, Sarah will be the next manager. (Q)

In this example, Disjunctive Syllogism is used to infer that Sarah will be the next manager based on the two premises. The first premise establishes that either John or Sarah will be the next manager, and the second premise provides the information that John will not be the next manager. Therefore, by applying Disjunctive Syllogism, we can conclude that Sarah will be the next manager, as the logical relationship established in the first premise cannot hold when the second premise is also true. Therefore, the only remaining possibility is that Sarah will be the next manager.

Addition

The Addition rule is one of the common inference rules, stating that If P is true, then P or Q will be true.

Notation:-

|P

P ∨ Q

where P represents a proposition, and P ∨ Q represents the disjunction of P and an arbitrary proposition Q. The conclusion P ∨ Q follows as a logical consequence of the premise P.

Example

A simple example of Addition would be as follows:

Premise 1: John is a doctor (P). Conclusion: Therefore, John is a doctor or a musician. (P or Q)

In this example, Addition is used to infer that John is either a doctor or a musician based on the premise given. The premise establishes that John is a doctor, and by applying the Addition rule of inference, we can conclude that John is also a doctor or a musician. The rule states that if P is true, P or Q is true. Since P is true, we can add any proposition Q and still have a true statement. Therefore, the conclusion that John is a doctor or a musician is valid.

Simplification

The simplification rule state that if P and Q are true, then Q or P will also be true.

Notation:

|P ∧ Q

P

where P ∧ Q represents the conjunction of propositions P and Q. The conclusion P follows as a logical consequence of the premise P ∧ Q.

Example

A simple example of Simplification would be as follows:

Premise 1: John is a doctor. (P) Premise 2: John is also a musician. (Q) Conclusion: Therefore, John is a doctor. (P)

In this example, Simplification is used to infer that John is a doctor based on the two premises. The premises establish that John is both a doctor and a musician, and by applying the Simplification rule of inference, we can conclude that John is also a doctor. This is because the rule states that if P and Q are true, then P is true. Therefore, the conclusion that John is a doctor is valid.

Resolution

The Resolution rule state that if P or Q and ¬ P and R are true, then Q R will also be true.

Notation:

where P ∨ Q and ¬Q ∨ R represent two clauses that contain complementary literals (i.e., Q and ¬Q). The conclusion P ∨ R follows as a logical consequence of resolving the two clauses by removing the complementary literals and joining the remaining literals with the disjunction operator.

Example

A simple example of a Resolution would be as follows:

Premise 1: All dogs are animals. (A -> B) Premise 2: All cats are animals. (C -> B) Premise 3: No dogs are cats. (not(A and C)) Conclusion: Therefore, dogs and cats are mutually exclusive. (not(A and C))

In this example, Resolution is used to infer that dogs and cats are mutually exclusive based on the three premises. The first two premises establish that dogs and cats are animals, and the third premise states that no dogs are cats. Therefore, by negating the statement "Dogs are cats" and applying the Resolution rule of inference, we can conclude that dogs and cats are mutually exclusive. This is because the negation of "Dogs are cats" is equivalent to "No dogs are cats," which is already a premise given. Therefore, the conclusion that dogs and cats are mutually exclusive is valid.