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DL vs. OSF
Hassan Aït-Kaci
ILOG, Inc.
Common Aspects
• Object description formalism
• Based on logical semantics
• Enjoys an effective proof theory
• Constraint...
Distinguishing Aspects
• Roles vs. features
(relations vs. functions)
• Sets vs. individuals
(extensional sorts)
• Cardina...
Reconcilable Differences?
In fact all the three forecited distinguishing
aspects are cosmetic differences:
• Roles are set...
Unreconcilable Difference?
ONE essential difference is explained by…
lattice-theoretic DUALITY:
• DL works by computing a ...
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...
Duality: Induction vs. Coinduction
Induction: (DL)
• Eager
• Finite sets (decidable)
• Bottom Up (LFP)
• From leaves to ro...
Conclusion
•DL and OSF are formally dual:
– The former is expansive, expensive,
and can only describe finitely
computable ...
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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

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Hak dl07

  1. 1. DL vs. OSF Hassan Aït-Kaci ILOG, Inc.
  2. 2. Common Aspects • Object description formalism • Based on logical semantics • Enjoys an effective proof theory • Constraint-solving process • Recursive concept definitions
  3. 3. Distinguishing Aspects • Roles vs. features (relations vs. functions) • Sets vs. individuals (extensional sorts) • Cardinality constraints
  4. 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. 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. 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. 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. 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

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

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