Knowledge Graph - Study Notes 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

First-order logic (FOL), also known as predicate logic, is a formal system used in mathematics, philosophy, computer science, and linguistics to represent and reason about propositions and their relationships.

Key Features of First-Order Logic:

1. Variables and Quantifiers:

   • Variables (e.g., x, y) stand for objects in a domain.

   • Quantifiers allow statements about "all" or "some" objects:

     • Universal quantifier (∀x): "for all x"

     • Existential quantifier (∃x): "there exists an x"

2. Predicates:

   • Predicates represent properties or relationships between objects.

   • For example, Loves(x, y) might mean "x loves y".

3. Logical Connectives:

   • ¬ (not), ∧ (and), ∨ (or), → (implies), ↔ (if and only if)

4. Constants and Functions:

   • Constants represent specific objects (e.g., a, b, or John).

   • Functions map objects to objects (e.g., fatherOf(x) returns the father of x).

5. Domain of Discourse:

   • The set of objects that variables refer to. All quantifiers range over this domain.

Example in FOL:

"Everyone loves someone":

∀x ∃y Loves(x,y)

This means: for every person x, there exists a person y such that x loves y.

Why It's Important:

• First-order logic is powerful enough to express most of classical mathematics.

• It underpins many areas in artificial intelligence, especially knowledge representation and automated reasoning.

 

 

Horn logic is a special subset of first-order logic (and propositional logic) that's widely used in logic programming, databases (like Datalog), and artificial intelligence.

Key Concept:

A Horn clause is a clause (a disjunction of literals) with at most one positive literal.

Types of Horn Clauses:

1. Definite clause (exactly one positive literal):

   • Example:

     A←B1∧B2∧⋯∧Bn​

     This means: If B₁ and B₂ and ... and Bₙ are true, then A is true.

     • This is commonly used as a rule in logic programming.

2. Fact (a definite clause with no conditions):

   • Example:

     AAA

     Simply states that A is true.

3. Goal clause or query (no positive literal):

   • Example:

     ←B1∧B2∧⋯∧Bn​

     Used in proving or querying (i.e., "is this condition derivable?").

Example in Prolog-like Syntax:

loves(mary, john).            % Fact
happy(X) :- loves(X, john).   % Rule (Horn clause)
?- happy(mary).               % Query (goal)

Why Horn Logic Is Important:

• It enables efficient reasoning and deduction (e.g., via forward chaining or backward chaining).

• It forms the foundation of logic programming languages, such as Prolog.

• Unlike general first-order logic, reasoning with Horn clauses is decidable and efficient in many cases.

 

 

 

The expression:

Bachelor ≡ ¬∃married.⊤ ⊓ Man

is written in Description Logic (DL), a formalism used in knowledge representation (like in OWL ontologies). Let's break it down:

Symbols and Meaning

• Bachelor: A class (concept) we're defining.

• ≡: Equivalence — the left-hand side (Bachelor) is equivalent to the right-hand side.

• ¬: Negation (NOT).

• ∃married.⊤: There exists a relation married to anything (⊤ means "top", i.e., any individual).

• ⊓: Conjunction (AND).

• Man: A class (concept) representing men.

Step-by-step Interpretation

• ∃married.⊤: The individual is married to someone (exists a "married" relation with any individual).

• ¬∃married.⊤: The individual is not married to anyone.

• ¬∃married.⊤ ⊓ Man: The individual is a man AND not married to anyone.

Full Interpretation

Bachelor ≡ ¬∃married.⊤ ⊓ Man

A Bachelor is exactly a man who is not married to anyone.

This is a formal way of defining the concept of a bachelor in terms of logic: a person who is a man and has no marriage relationship.

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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