Study Notes on First-Order Logic

Readings

• Key Chapters:
  • Chapter 8, pp. 269-289, Russell and Norvig
  • Chapter 9, pp. 298-320, Russell and Norvig

Introduction to First-Order Logic (FOL)

• Limitations of Propositional Logic:
  • Examples highlight inadequacies in representing complex relationships.
  • Issues arise in succinctly expressing statements like "Pits cause breezes in adjacent squares."
  • Difficulty in articulating statements about groups, e.g., "All students know arithmetic" requires a richer logical representation.

Differences Between Propositional and First-Order Logic

• Components of FOL Missing in Propositional Logic:
  • Objects: Individual entities.
  • Quantifiers: Enables expressions of generality or existence (e.g., "for all" or "there exists").
  • Relations: Show how objects are connected.
  • Functions: Mappings from objects to objects.

Types of Logic Explained

• Ontological Commitments:
  • Logic types categorized by what entities they represent:
  • Propositional Logic: Facts
  • First-Order Logic: Facts + objects + relations
  • Temporal Logic: Adds time
  • Probability Theory: Incorporates degrees of belief
  • Fuzzy Logic: Allows for approximate truth values.
• Epistemological Commitments: Understanding how truth values are derived in different logics.

Examples of First-Order Logic

• Expressive Capabilities:
  • "There is some course that every student has taken."
  • "Every even integer greater than 2 can be expressed as the sum of two primes."
  • Logical implications like:
  • If a student takes a course and that course covers a concept, then the student understands that concept.

Syntax of First-Order Logic

• Components:
  • Constants: Objects that have fixed values.
  • Predicates: Relations among objects.
  • Functions: Map input objects to output objects.
  • Variables: Placeholders for objects.
  • Connectives: Logical connectors such as AND (∧), OR (∨), NOT (¬).
  • Equality: Allows comparison between objects.
  • Quantifiers:
  • Universal Quantifier (∀): Represents "for all" statements.
  • Existential Quantifier (∃): Represents "there exists" statements.

Atomic Sentences in FOL

• Definition:
  • Combination of predicates with terms or terms being equal.
  • Examples include:
  • Brother(John, Richard)
  • GreaterThan(Length(LeftLegOf(Richard)), Length(LeftLegOf(John)))

Complex Sentences

• Formation:
  • Built from atomic sentences using connectives.
  • Example: Sibling(King John, Richard) as a complex statement of relationships.

Quantifiers and Their Usage

• Universal Quantification (∀):
  • Example: "Every student knows arithmetic."
• Existential Quantification (∃):
  • Example: "Some student knows arithmetic."

Models in First-Order Logic

• Definition:
  • Models assign truth values to sentences.
  • More complex than propositional models due to object relationships.
• Important Assumptions:
  • Unique Names Assumption: No two constants refer to the same object.
  • Domain Closure: Every object is at least represented by a constant symbol.

Inference Mechanisms in First-Order Logic

• Inference Techniques:
  • Finding entailments from a knowledge base.
  • Techniques involve Propositionalization, Modus Ponens, Resolution, and may require additional tools for FOL.
• Substitution and Unification: Essential for resolving variables and establishing relationships.
  • Example: Subst[{x/alice}, P(x)] = P(alice) demonstrates how substitution works.

Advanced Inference Techniques

• Generalized Modus Ponens (GMP):
  • Utilizes definite clauses for sound and complete reasoning.
• Forward Chaining:
  • Build new conclusions from a knowledge base iteratively.
  • Example provided on criminal implications through resolution and deriving results from statements about weapons and individuals.
• Backward Chaining:
  • Start from a query to see if it can be inferred.
  • Follow through various substitutions to reach conclusions about criminal status.

Resolution Technique

• Resolution in First-Order Logic:
  • Combines sentences in conjunctive normal form (CNF) using unification to derive new conclusions.
  • Practical examples showcased how to reach desired outputs using clauses.
• Conversion to CNF: Process of converting sentences into a form manageable for resolution.
  • Involves eliminating implications and introducing Skolem functions for existential quantifiers.

Summary Points

• Key Distinctions:
  • Propositional Logic: Simpler but limited representation capabilities.
  • First-Order Logic: Allows for more compact and expressive knowledge representation.
• Mechanisms for Inference:
  • Employ resolution, unification and other strategies for complex reasoning tasks.

Conclusion

• First-Order Logic lends itself well to representing and reasoning about complex relationships in knowledge bases, increasing the possibilities for sophisticated reasoning applications.