All Elephants are Bigger than All Mice

Transcription

1 All Elephants are Bigger than All Mice Description Logic Rules Sebastian Rudolph Markus Krötzsch Pascal Hitzler Institut AIFB, Universität Karlsruhe (TH) 21st Description Logic Workshop Dresden, Germany May 2008 Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 1 / 10

2 Concept Products Some simple statements: Alkaline solutions neutralise acid solutions. Antihistamines alleviate allergies. Oppositely charged bodies attract each other. Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 2 / 10

3 Concept Products Some simple statements: Alkaline solutions neutralise acid solutions. Antihistamines alleviate allergies. Oppositely charged bodies attract each other. Concept products The relation R holds between all objects of A and all objects of B: A B R How can these be modelled in DL? Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 2 / 10

4 Concept Products Some simple statements: Alkaline solutions neutralise acid solutions. Antihistamines alleviate allergies. Oppositely charged bodies attract each other. Concept products The relation R holds between all objects of A and all objects of B: A B R How can these be modelled in DL? (In OWL DL and SHOIQ they cannot!) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 2 / 10

5 Concept Products in SROIQ Concept product A B R: Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

6 Concept Products in SROIQ Concept product A B R: A R 1.{a} Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

7 Concept Products in SROIQ Concept product A B R: A R 1.{a} Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

8 Concept Products in SROIQ Concept product A B R: A R 1.{a} B R 2.{a} Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

9 Concept Products in SROIQ Concept product A B R: A R 1.{a} B R 2.{a} R 1 R 2 R Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

10 Concept Products in SROIQ Concept product A B R: A R 1.{a} B R 2.{a} R 1 R 2 R Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

11 Concept Products in SROIQ Concept product A B R: A R 1.{a} B R 2.{a} R 1 R 2 R Polynomial transformation, no complexity issue Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 3 / 10

12 SHOIQ, SHOI, and EL ++ What about less expressive DLs? Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

13 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

14 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete proof by reduction to FOL with 2 variables Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

15 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete proof by reduction to FOL with 2 variables also possible with concept products: A(x) B(y) R(x, y) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

16 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete proof by reduction to FOL with 2 variables also possible with concept products: A(x) B(y) R(x, y) SHOIQ is NEXPTIME-complete Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

17 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

18 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete Transformation of concept products possible Roles R 1, R 2 as in SROIQ, but replace subconcepts: R.C R.C R 1. R 2.C, R.C R.C R 1. R 2.C Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

19 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete Transformation of concept products possible Roles R 1, R 2 as in SROIQ, but replace subconcepts: R.C R.C R 1. R 2.C, R.C R.C R 1. R 2.C SHOI is EXPTIME-complete Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

20 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete EL ++ is P-complete Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

21 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete EL ++ is P-complete extend reasoning algorithm with rule for concept products Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

22 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete EL ++ is P-complete extend reasoning algorithm with rule for concept products EL ++ is P-complete Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

23 SHOIQ, SHOI, and EL ++ What about less expressive DLs? SHOIQ is NEXPTIME-complete SHOI is EXPTIME-complete EL ++ is P-complete Worst-case complexity for SHOIQ, SHOI, EL ++ unaffected Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 4 / 10

24 A Rule Perspective Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 5 / 10

25 A Rule Perspective Concept products can be expressed as rules: A(x) B(y) R(x, y) More examples of rules: Woman(x) haschild(x, y) motherof(x, y) Man(x) hasbrother(x, y) haschild(y, z) Uncle(x) marriedto(x, y) loves(x, y) Happy(x) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 5 / 10

26 Modelling rules in SROIQ Some rules can be expressed in SROIQ! Woman(x) haschild(x,y) motherof(x,y) Man(x) hasbrother(x,y) haschild(y,z) Uncle(x) Elephant(x) Mouse(y) biggerthan(x,y) marriedto(x,y) loves(x,y) Happy(x) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 6 / 10

27 Modelling rules in SROIQ Some rules can be expressed in SROIQ! Woman(x) haschild(x,y) motherof(x,y) new role woman, new axioms: Woman woman.self woman haschild motherof Man(x) hasbrother(x,y) haschild(y,z) Uncle(x) Elephant(x) Mouse(y) biggerthan(x,y) marriedto(x,y) loves(x,y) Happy(x) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 6 / 10

28 Modelling rules in SROIQ Some rules can be expressed in SROIQ! Woman(x) haschild(x,y) motherof(x,y) new role woman, new axioms: Woman woman.self woman haschild motherof Man(x) hasbrother(x,y) haschild(y,z) Uncle(x) axiom: Man hasbrother. haschild Uncle Elephant(x) Mouse(y) biggerthan(x,y) marriedto(x,y) loves(x,y) Happy(x) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 6 / 10

29 Modelling rules in SROIQ Some rules can be expressed in SROIQ! Woman(x) haschild(x,y) motherof(x,y) new role woman, new axioms: Woman woman.self woman haschild motherof Man(x) hasbrother(x,y) haschild(y,z) Uncle(x) axiom: Man hasbrother. haschild Uncle Elephant(x) Mouse(y) biggerthan(x,y) new roles elephant, mouse, new axioms: Elephant elephant.self Mouse mouse.self elephant U mouse biggerthan marriedto(x,y) loves(x,y) Happy(x) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 6 / 10

30 Modelling rules in SROIQ Some rules can be expressed in SROIQ! Woman(x) haschild(x,y) motherof(x,y) new role woman, new axioms: Woman woman.self woman haschild motherof Man(x) hasbrother(x,y) haschild(y,z) Uncle(x) axiom: Man hasbrother. haschild Uncle Elephant(x) Mouse(y) biggerthan(x,y) new roles elephant, mouse, new axioms: Elephant elephant.self Mouse mouse.self elephant U mouse biggerthan marriedto(x,y) loves(x,y) Happy(x) not possible in SROIQ, but doable (for simple roles) Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 6 / 10

31 DL Rules Can we allow rules in general? undecidable SWRL = DL + function-free Horn logic Which rules can be expressed in SROIQ? Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 7 / 10

32 DL Rules Can we allow rules in general? SWRL = DL + function-free Horn logic undecidable Which rules can be expressed in SROIQ? Description Logic Rules SWRL rules (with complex concepts) rule bodies tree- or forest-shaped R(x, x) and R(x, y) S(x, y) in bodies only if R, S simple Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 7 / 10

33 EL Rules and DLP Rules DL rules syntactic sugar only for SROIQ what about smaller DLs? Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 8 / 10

34 EL Rules and DLP Rules DL rules syntactic sugar only for SROIQ what about smaller DLs? DLP Description Logic Programs Horn-logic fragment of OWL DL (SHOIN ) Body concepts with R.C, head concepts with R.C Easy to extend to support according DL rules Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 8 / 10

35 EL Rules and DLP Rules DL rules syntactic sugar only for SROIQ what about smaller DLs? DLP Description Logic Programs Horn-logic fragment of OWL DL (SHOIN ) Body concepts with R.C, head concepts with R.C Easy to extend to support according DL rules EL ++ rules EL ++ admits R.C in body and head has no native support for R.Self, role conjunctions, U EL ++ rules still polynomial ELP: combination of EL ++ rules and safe DLP rules still polynomial Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 8 / 10

36 All Elephants are Bigger than All Mice Conclusion Concept products: useful, easily explained modelling construct DL rules: non-boring FOL rule fragment of SROIQ full reasoning support in any SROIQ reasoner no impact on worst-case complexity even in smaller DLs Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 9 / 10

37 Literature Rudolph, S.; Krötzsch, M.; and Hitzler, P. All elephants are bigger than all mice. In Proc. 21st Int. Workshop on Description Logics (DL-08) Krötzsch, M.; Rudolph, S.; and Hitzler, P. Description logic rules. In Proc. 18th European Conf. on Artificial Intelligence (ECAI-08), Gasse, F.; Sattler, U.; and Haarslev, V. Rewriting rules into SROIQ axioms. Poster at 21st Int. Workshop on Description Logics (DL-08) Krötzsch, M.; Rudolph, S.; and Hitzler, P. ELP: Tractable rules for OWL 2. Technical report, Universität Karlsruhe, Germany, Markus Krötzsch (AIFB Karlsruhe) All Elephants are Bigger than All Mice DL 2008 Dresden 10 / 10