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Description Logics Description Logics of typicality Reasoning in ALC + T min Theorem Proving A Multi-Engine Theorem Prover for a Description Logic of Typicality Laura Giordano 1 Valentina Gliozzi 2 Nicola Olivetti 3 Gian Luca Pozzato 2 Luca

  1. Description Logics Description Logics of typicality Reasoning in ALC + T min Theorem Proving A Multi-Engine Theorem Prover for a Description Logic of Typicality Laura Giordano 1 Valentina Gliozzi 2 Nicola Olivetti 3 Gian Luca Pozzato 2 Luca Violanti 4 1 DISIT - Universit´ a Piemonte Orientale - Alessandria, Italy 2 Dipartimento di Informatica, Universit´ a degli Studi di Torino, Italy 3 Aix Marseille Univ. - CNRS, ENSAM, Univ. de Toulon - LSIS UMR 7296, Marseille - France 4 NCR Edinburgh - United Kingdom AI*IA 2015 Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  2. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Theorem Proving Description Logics Description Logics Important formalisms of knowledge representation Two key advantages: well-defined semantics based on first-order logic good trade-o ff between expressivity and complexity at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components: TBox=inclusion relations among concepts Platypus v Mammal ABox= instances of concepts and roles = properties and relations among individuals Platypus ( perry ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  3. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Theorem Proving Description Logics Description Logics Important formalisms of knowledge representation Two key advantages: well-defined semantics based on first-order logic good trade-o ff between expressivity and complexity at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components: TBox=inclusion relations among concepts Platypus v Mammal ABox= instances of concepts and roles = properties and relations among individuals Platypus ( perry ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  4. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Theorem Proving Description Logics Description Logics Important formalisms of knowledge representation Two key advantages: well-defined semantics based on first-order logic good trade-o ff between expressivity and complexity at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components: TBox=inclusion relations among concepts Platypus v Mammal ABox= instances of concepts and roles = properties and relations among individuals Platypus ( perry ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  5. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Theorem Proving Description Logics Description Logics Important formalisms of knowledge representation Two key advantages: well-defined semantics based on first-order logic good trade-o ff between expressivity and complexity at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components: TBox=inclusion relations among concepts Platypus v Mammal ABox= instances of concepts and roles = properties and relations among individuals Platypus ( perry ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  6. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Theorem Proving Description Logics Reasoning TBox = taxonomy of concepts need of representing prototypical properties and of reasoning about defeasible inheritance integration with nonmonotonic reasoning mechanism to handle defeasible inheritance [BH95, BLW06, DLN + 98, DNR02, ELST04, Str93] all these methods present some di ffi culties Our solution DLs + typicality operator T for defeasible reasoning in DLs [GGOP13] meaning of T : (for any concept C ) T ( C ) singles out the “typical” instances of C semantics of T defined by a set of postulates that are a restatement of Kraus-Lehmann-Magidor axioms of preferential logic P Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  7. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Theorem Proving Description Logics Reasoning TBox = taxonomy of concepts need of representing prototypical properties and of reasoning about defeasible inheritance integration with nonmonotonic reasoning mechanism to handle defeasible inheritance [BH95, BLW06, DLN + 98, DNR02, ELST04, Str93] all these methods present some di ffi culties Our solution DLs + typicality operator T for defeasible reasoning in DLs [GGOP13] meaning of T : (for any concept C ) T ( C ) singles out the “typical” instances of C semantics of T defined by a set of postulates that are a restatement of Kraus-Lehmann-Magidor axioms of preferential logic P Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  8. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Basic notions A KB comprises assertions T ( C ) v D T ( Student ) v FacebookUsers means “normally, students use Facebook” T is nonmonotonic C v D does not imply T ( C ) v T ( D ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  9. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Basic notions A KB comprises assertions T ( C ) v D T ( Student ) v FacebookUsers means “normally, students use Facebook” T is nonmonotonic C v D does not imply T ( C ) v T ( D ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  10. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Example T ( BasketballPlayer ) v ¬ Rich T ( BasketballPlayer u NBAMember ) v Rich Reasoning ABox: BasketballPlayer ( marco ) Expected conclusions: ¬ Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  11. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Example T ( BasketballPlayer ) v ¬ Rich T ( BasketballPlayer u NBAMember ) v Rich Reasoning ABox: BasketballPlayer ( marco ) Expected conclusions: ¬ Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  12. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Example T ( BasketballPlayer ) v ¬ Rich T ( BasketballPlayer u NBAMember ) v Rich Reasoning ABox: BasketballPlayer ( marco ) Expected conclusions: ¬ Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  13. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Example T ( BasketballPlayer ) v ¬ Rich T ( BasketballPlayer u NBAMember ) v Rich Reasoning ABox: BasketballPlayer ( marco ) NBAMember ( marco ) Expected conclusions: Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  14. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Example T ( BasketballPlayer ) v ¬ Rich T ( BasketballPlayer u NBAMember ) v Rich Reasoning ABox: BasketballPlayer ( marco ) NBAMember ( marco ) Expected conclusions: Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  15. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving The logic ALC + T min Example T ( BasketballPlayer ) v ¬ Rich T ( BasketballPlayer u NBAMember ) v Rich Reasoning ABox: BasketballPlayer ( marco ) NBAMember ( marco ) Expected conclusions: Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  16. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving Weakness of monotonic semantics Logic ALC + T The operator T is nonmonotonic, but... The logic is monotonic If KB | = F , then KB’ | = F for all KB’ ◆ KB Example in the KB of the previous slides: if BasketballPlayer ( marco ) 2 ABox, we are not able to: assume that T ( BasketballPlayer )( marco ) infer that ¬ Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

  17. Description Logics Description Logics of typicality Introduction Reasoning in ALC + T min Nonmonotonic semantics ALC + T min Theorem Proving Weakness of monotonic semantics Logic ALC + T The operator T is nonmonotonic, but... The logic is monotonic If KB | = F , then KB’ | = F for all KB’ ◆ KB Example in the KB of the previous slides: if BasketballPlayer ( marco ) 2 ABox, we are not able to: assume that T ( BasketballPlayer )( marco ) infer that ¬ Rich ( marco ) Luca Violanti A Multi-Engine Theorem Prover for a Description Logic of Typicality

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