Hassan Aït-Kaci, “Description Logic vs. Order-Sorted Feature Logic”, Proceedings of the 20th Workshop on Description Logics, Brixen-Bressanone, Italy, June 2007. http://hassan-ait-kaci.net/pdf/dl07.pdf

1. DL vs. OSF
Hassan Aït-Kaci
ILOG, Inc.

2. Common Aspects
• Object description formalism
• Based on logical semantics
• Enjoys an effective proof theory
• Constraint-solving process
• Recursive concept definitions

3. Distinguishing Aspects
• Roles vs. features
(relations vs. functions)
• Sets vs. individuals
(extensional sorts)
• Cardinality constraints

4. Reconcilable Differences?
In fact all the three forecited distinguishing
aspects are cosmetic differences:
• Roles are set-valued features
trivially expressed as aggregates
• Individuals are singleton sets
• Cardinality constraints are also
trivially expressed as (lazy)
aggregates

5. Unreconcilable Difference?
ONE essential difference is explained by…
lattice-theoretic DUALITY:
• DL works by computing a least fixed
point inductively
• OSF works by computing a greatest fixed
point coinductively

6. T.
T.
Complete lattice L : v , t , u , ? , >
F : L g L
monotonic function:
x vx’ ) F(x)v F(x’)
(Alfred Tarski)
F(>F ) = >F
F(?F ) = ?F ?F
>F
.
.
h
5
(T)h
FF = tn¸0 Fn#
i
6
(T)F i
F = un¸0 Fn$
{ x 2 L | x = F(x) }
is also a complete lattice for same order
Fixpoints of F :
subset of L
GFP
LFP

7. Duality: Induction vs. Coinduction
Induction: (DL)
• Eager
• Finite sets (decidable)
• Bottom Up (LFP)
• From leaves to root
Coinduction: (OSF)
• Lazy
• R.E. sets (semi-decidable)
• Top Down (GFP)
• From root to … whatever!
* +

8. Conclusion
•DL and OSF are formally dual:
– The former is expansive, expensive,
and can only describe finitely
computable sets
– The latter is contractive, efficient,
and can also describe R.E. sets