NMR -Description Logic

Nonmonotonic Reasoning

Introduction to Description Logic (DL)

• Description Logic (DL) is a family of formal knowledge representation languages.
• It is used to describe the structure of a domain in terms of concepts (classes), roles (properties/relations), and individuals (objects).
• DL forms the logical foundation of ontology languages such as OWL (Web Ontology Language).

Basic Example:

• Concepts: Student, Professor
• Role: hasAdvisor
• Assertion:

ALCF in Description Logic

1. Recap: ALC
   • ALC is a core Description Logic that supports:
     • Boolean constructors: ⊓, ⊔, ¬
     • Quantifiers on roles: ∀R.C, ∃R.C
2. F: Functional Roles
   • The F extension adds functionality restrictions.
   • A role declared as functional can relate an individual to at most one filler

Number Restrictions in Description Logic

1. Unqualified Number Restrictions (N)
   • Restrict only the number of role fillers, without specifying their type.
   • Syntax: ≤ n R, ≥ n R
   • Example:

     Person ⊑ ≤ 2 hasChild
     

     → A person can have at most two children (regardless of what kind of children they are).

2. Qualified Number Restrictions (Q)
   • Restrict the number of role fillers of a specific concept.
   • Syntax: ≤ n R.C, ≥ n R.C
   • Example:

     Person ⊑ ≤ 2 hasChild.Doctor
     

     → A person can have at most two children who are doctors (other types of children are unrestricted).

3. Relationship between N and Q
   • N is a special case of Q:

     ≤ n R   ≡   ≤ n R.⊤
     

   • Because of this, many modern Description Logic systems (e.g., OWL 2 DL) directly support Q as the general form.

👉 In short:

• N = “how many?” (unqualified)
• Q = “how many of type X?” (qualified)
• Therefore, N ⊂ Q.

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Paper List for DL

A non-monotonic Description Logic for reasoning about typicality	Giodano, 2013
A Description Logic Framework for Commonsense Conceptual Combination Integrating Typicality, Probabilities and Cognitive Heuristics	Lieto, 2018