Artificial Intelligence: First-order Logic

First-order Logic (FOL) is a fundamental topic in Artificial Intelligence (AI), particularly in the field of knowledge representation and reasoning. Here's a structured overview to help you understand it better:


What is First-Order Logic (FOL)?

First-order logic (also known as predicate logic or first-order predicate calculus) is a formal system used to express statements and reason about objects and their relationships.


Components of First-Order Logic

  1. Constants
    Represent specific objects in the domain.
    Example: John, Paris, 3

  2. Variables
    Stand for objects that are not specified.
    Example: x, y, z

  3. Predicates
    Represent properties or relationships among objects.
    Example: Loves(John, x), GreaterThan(x, y)

  4. Functions
    Map a tuple of objects to another object.
    Example: MotherOf(x), Add(x, y)

  5. Quantifiers

    • Universal Quantifier (∀): “For all”
      Example: ∀x Human(x) → Mortal(x)

    • Existential Quantifier (∃): “There exists”
      Example: ∃x Cat(x) ∧ Black(x)

  6. Connectives

    • Conjunction ( ∧ ) → and

    • Disjunction ( ∨ ) → or

    • Negation ( ¬ ) → not

    • Implication ( → )

    • Biconditional ( ↔ )

  7. Equality (=)
    Tests whether two terms refer to the same object.


Example Statements in FOL

  • "All humans are mortal":

    ∀x (Human(x) → Mortal(x))

  • "Socrates is a human":

    Human(Socrates)

  • "There exists a cat that is black":

    ∃x (Cat(x) ∧ Black(x))


Why FOL is Important in AI

  • Expressiveness: Can represent a wide range of knowledge.

  • Inference: Allows reasoning through deduction, resolution, etc.

  • Semantic precision: Well-defined formal semantics.

  • Used in: Expert systems, automated theorem proving, knowledge bases.


Difference Between Propositional Logic and First-Order Logic

Feature Propositional Logic First-Order Logic
Objects/Entities Not supported Supported
Predicates Not used Used
Quantifiers Not available Available (∀, ∃)
Expressiveness Limited More expressive
Example P ∧ Q ∀x (Cat(x) → Animal(x))


Inference in FOL

Common inference methods include:

  • Unification

  • Resolution

  • Forward chaining

  • Backward chaining


Applications in AI

  • Expert systems

  • Natural language understanding

  • Automated reasoning

  • Planning systems

  • Semantic web

Sentences in First-Order Logic


In First-Order Logic (FOL), sentences are used to represent facts and relationships. These sentences can be classified into atomic sentences and complex sentences, as explained below −

Atomic Sentences

An atomic sentence is the most basic form of a statement in First-Order Logic. It consists of a predicate and arguments, which can be either constants or variables. An atomic statement asserts a fact about things without using logical connectives.
Structure of Atomic Sentences: Predicate(Argument 1,Argument 2,...,Argument n)​

where -

* The predicate represents a property or relationship.

* The arguments represent objects (constants or variables).

Example

The following example demonstrate the use of atomic sentences in First-Order Logic to represent facts and relationships −

* Teacher(John, Mathematics) This atomic sentence states that John is a teacher of Mathematics.

Complex sentence

Two or more atomic statements connected by logical connectives like AND (), OR (), NOT (¬), Implies (→), and Bi-conditional () create a complex sentence. These sentences allow us to express more complex logical statements.

Example

The following examples demonstrate the use of complex sentences in First-Order Logic to represent facts and relationships −

* (y Teacher(y) Teaches(y, Math)) "There exists at least one teacher who teaches Mathematics."

* (x Customer(x) → Buys(x, TutorialsPointCourses)) "For all customers, if x is a customer, then x buys courses from Tutorials Point."

* (y Employee(y) WorksFor(y, TutorialsPoint)) "There exists at least one employee who works for Tutorials Point."

Free and Bound Variables in First-Order Logic


Variables in FOL can be either free or bounded depending on the fact that a variable is controlled by any quantifier.

Free Variable

A free variable cannot be quantified using or . It can hold any value but has no fixed meaning unless assigned.

Example

Let us consider a statement "y is the price of a product."

* The FOL representation for the above statement will be P(y) (where P(y) means y is the price of a product).

* Here, y is free because we don't specify which products price we are referring to.

Bound Variable

A bound variable is within the scope of a quantifier and is dependent on it, so its value is limited by the quantifier.

Example

Let the sentence be "There exists a product whose price is less than $10."

* The FOL representation of the above sentence is yP(y) (where P(y) means y is less than $10).

* Here, the existential quantifier () constrains the variable y, indicating that it is applicable to a specific product in the context.

Challenges and limitations of First Order Logic


When utilizing First-Order Logic (FOL) for knowledge representation and reasoning, several key challenges and limitations arise, as follows −

* Challenges in Handling Uncertainty: FOL requires facts to be definitively true or false, making it difficult to represent probabilistic or uncertain information.

* High Computational Demands: When dealing with extensive knowledge bases, logical inference in FOL can become slow and resource-intensive.

* Limited Expressive Power: Concepts from the real world that involve fuzzy or probabilistic logic, such as "most people" or "approximately," are difficult for FOL to articulate.

* Rigid Rule-Based Framework: The necessity for facts and rules to be explicitly defined results in a structure that is inflexible and struggles to incorporate new information.

* Common-Sense thinking Difficulty: FOL finds it difficult to manage common sense thinking that incorporates assumptions, exceptions, or insufficient information.