Institute of Theoretical Computer Science Chair of Automata Theory POSITIVE SUBSUMPTION IN FUZZY EL WITH GENERAL T-NORMS
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1 Institute of Theoretical Computer Science Chair of Automata Theory POSITIVE SUBSUMPTION IN FUZZY EL WITH GENERAL T-NORMS Stefan Borgwardt Rafael Peñaloza 北京, August 7, 2013
2 Vagueness in SNOMED CT EL is used in SNOMED CT PerinatalCyanoticAttack CardiovascularDisorder occurrence.perinatalperiod manifestation.cyanosis 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 2
3 Vagueness in SNOMED CT EL is used in SNOMED CT PerinatalCyanoticAttack CardiovascularDisorder compute all inferences in polynomial time highly optimized reasoners occurrence.perinatalperiod manifestation.cyanosis 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 2
4 Vagueness in SNOMED CT EL is used in SNOMED CT PerinatalCyanoticAttack CardiovascularDisorder occurrence.perinatalperiod manifestation.cyanosis compute all inferences in polynomial time highly optimized reasoners vague concepts: perinatal period = period of time around birth cyanosis = bluish discoloration of the skin 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 2
5 Vagueness in SNOMED CT EL is used in SNOMED CT PerinatalCyanoticAttack CardiovascularDisorder occurrence.perinatalperiod manifestation.cyanosis compute all inferences in polynomial time highly optimized reasoners vague concepts: perinatal period = period of time around birth cyanosis = bluish discoloration of the skin 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 2
6 Vagueness in SNOMED CT EL is used in SNOMED CT PerinatalCyanoticAttack CardiovascularDisorder occurrence.perinatalperiod manifestation.cyanosis compute all inferences in polynomial time highly optimized reasoners vague concepts: perinatal period = period of time around birth cyanosis = bluish discoloration of the skin model this using values from [0, 1] 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 2
7 Vagueness in SNOMED CT EL is used in SNOMED CT PerinatalCyanoticAttack CardiovascularDisorder occurrence.perinatalperiod manifestation.cyanosis compute all inferences in polynomial time highly optimized reasoners vague concepts: perinatal period = period of time around birth cyanosis = bluish discoloration of the skin model this using values from [0, 1] does reasoning stay tractable? 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 2
8 Mathematical fuzzy logic all statements hold to a degree from [0, 1] t-norm : [0, 1] [0, 1] [0, 1] generalizes : {0, 1} {0, 1} {0, 1}: associative commutative monotone unit 1 (continuous) 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 3
9 Mathematical fuzzy logic all statements hold to a degree from [0, 1] t-norm : [0, 1] [0, 1] [0, 1] generalizes : {0, 1} {0, 1} {0, 1}: associative commutative monotone unit 1 (continuous) residuum generalizes implication: (x y) z iff y (x z) unique if is continuous propositional: (x y) z iff y (x z) 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 3
10 Mathematical fuzzy logic all statements hold to a degree from [0, 1] t-norm : [0, 1] [0, 1] [0, 1] generalizes : {0, 1} {0, 1} {0, 1}: associative commutative monotone unit 1 (continuous) residuum generalizes implication: (x y) z iff y (x z) unique if is continuous propositional: (x y) z iff y (x z) [Hájek 2001, 2005] 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 3
11 Fundamental continuous t-norms Gödel (G): x y = min(x, y) 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 4
12 Fundamental continuous t-norms Gödel (G): x y = min(x, y) Product (Π): x y = x y 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 4
13 Fundamental continuous t-norms Gödel (G): x y = min(x, y) Product (Π): x y = x y Łukasiewicz (Ł): x y = max(0, x + y 1) 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 4
14 Fundamental continuous t-norms Gödel (G): x y = min(x, y) Product (Π): x y = x y Łukasiewicz (Ł): x y = max(0, x + y 1) perinatal cyanotic attack: A cardiovascular disease in the perinatal period causing cyanosis. 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 4
15 Fundamental continuous t-norms Gödel (G): x y = min(x, y) Product (Π): x y = x y Łukasiewicz (Ł): x y = max(0, x + y 1) perinatal cyanotic attack: A cardiovascular disease in the perinatal period causing cyanosis. 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 4
16 Fundamental continuous t-norms Gödel (G): x y = min(x, y) Product (Π): x y = x y Łukasiewicz (Ł): x y = max(0, x + y 1) perinatal cyanotic attack: A cardiovascular disease in the perinatal period causing cyanosis. 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 4
17 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms. x 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
18 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms. 1 y x 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
19 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms. 1 y x 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
20 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms y Gödel Łukasiewicz 0.3 Product 0 x 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
21 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms. 1 y Gödel Gödel 0.7 Łukasiewicz 0.3 Product Gödel 0 x 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
22 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms. 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
23 Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the product and Łukasiewicz t-norms. starts with product contains Łukasiewicz 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 5
24 -EL concept names: Patient, Disease 1 Young(x) Tall(x) 0 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 6
25 -EL concept names: Patient, Disease 1 Young(x) Tall(x) 0 role names: hasseverity, hassymptom top: 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 6
26 -EL concept names: Patient, Disease 1 Young(x) Tall(x) 0 role names: hasseverity, hassymptom top: conjunction: Young Tall 1 0 Young(x) Tall(x) 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 6
27 -EL (2) existential restriction: hassymptom.cough sup hassymptom(x, y) Cough(y) y 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 7
28 -EL (2) existential restriction: hassymptom.cough sup hassymptom(x, y) Cough(y) y y : hassymptom(x, y) Cough(y) 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 7
29 -EL (2) existential restriction: hassymptom.cough sup hassymptom(x, y) Cough(y) y y : hassymptom(x, y) Cough(y) TBox T : set of general concept inclusions (GCIs): PerinatalCyanoticAttack hasmanifestation.cyanosis q PerinatalCyanoticAttack(x) ( hasmanifestation.cyanosis)(x) q 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 7
30 -EL (2) existential restriction: hassymptom.cough sup hassymptom(x, y) Cough(y) y y : hassymptom(x, y) Cough(y) TBox T : set of general concept inclusions (GCIs): PerinatalCyanoticAttack hasmanifestation.cyanosis q PerinatalCyanoticAttack(x) ( hasmanifestation.cyanosis)(x) q reasoning tasks: p-subsumption: does C D p follow from T? compute the best subsumption degree (bsd) of C and D! classification: compute all bsds between concept names! positive subsumption: does C(x) D(x) > 0 follow from T? 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 7
31 Results related work: classification in classical EL can be done in polynomial time [Baader, Brandt, Lutz 2005] several highly optimized reasoners (CEL, jcel, ELK) classification in G-EL can be done in polynomial time [Mailis, Stoilos, Simou, Stamou, Kollias 2012] 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 8
32 Results related work: classification in classical EL can be done in polynomial time [Baader, Brandt, Lutz 2005] several highly optimized reasoners (CEL, jcel, ELK) classification in G-EL can be done in polynomial time [Mailis, Stoilos, Simou, Stamou, Kollias 2012] positive subsumption in -EL: CO-NP-hard if starts with Łukasiewicz in P if does not start with Łukasiewicz 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 8
33 Results related work: classification in classical EL can be done in polynomial time [Baader, Brandt, Lutz 2005] several highly optimized reasoners (CEL, jcel, ELK) classification in G-EL can be done in polynomial time [Mailis, Stoilos, Simou, Stamou, Kollias 2012] positive subsumption in -EL: CO-NP-hard if starts with Łukasiewicz in P if does not start with Łukasiewicz p-subsumption: CO-NP-hard if contains Łukasiewicz 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 8
34 Results related work: classification in classical EL can be done in polynomial time [Baader, Brandt, Lutz 2005] several highly optimized reasoners (CEL, jcel, ELK) classification in G-EL can be done in polynomial time [Mailis, Stoilos, Simou, Stamou, Kollias 2012] positive subsumption in -EL: CO-NP-hard if starts with Łukasiewicz in P if does not start with Łukasiewicz p-subsumption: CO-NP-hard if contains Łukasiewicz 1-subsumption is in P if does not start with Łukasiewicz, all roles are crisp, and all GCIs are in normal form [DL 2013] 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 8
35 Positive subsumption in Ł-EL is CO-NP-hard vertex cover problem V = (V, E, k): V = {v 1,..., v m} E V V 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 9
36 Positive subsumption in Ł-EL is CO-NP-hard vertex cover problem V = (V, E, k): V = {v 1,..., v m} E V V 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 9
37 Positive subsumption in Ł-EL is CO-NP-hard vertex cover problem V = (V, E, k): V = {v 1,..., v m} E V V vertex cover of size k? 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 9
38 Positive subsumption in Ł-EL is CO-NP-hard vertex cover problem V = (V, E, k): V = {v 1,..., v m} E V V vertex cover of size k? T V := { V m k i V m k+1 i 1, V i m k 1 1 i m} m k 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 9
39 Positive subsumption in Ł-EL is CO-NP-hard vertex cover problem V = (V, E, k): V = {v 1,..., v m} E V V vertex cover of size k? T V := { V m k i V m k+1 i 1, V i m k 1 1 i m} m k { V j1 V j2 m k 1 m k {v j1, v j2 } E} 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 9
40 Positive subsumption in Ł-EL is CO-NP-hard vertex cover problem V = (V, E, k): V = {v 1,..., v m} E V V vertex cover of size k? T V := { V m k i V m k+1 i 1, V i m k 1 1 i m} m k { V j1 V j2 m k 1 m k {v j1, v j2 } E} There is no vertex cover of (V, E) of size k iff (V 1... V m)(x) > 0 (w.r.t. T V ). 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 9
41 Positive subsumption in Π-EL is in P idea: transform TBox into classical TBox encoding all positive subsumptions reason over crisp interpretations 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 10
42 Positive subsumption in Π-EL is in P idea: transform TBox into classical TBox encoding all positive subsumptions reason over crisp interpretations T >0 := {C D C D q T, q > 0} 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 10
43 Positive subsumption in Π-EL is in P idea: transform TBox into classical TBox encoding all positive subsumptions reason over crisp interpretations T >0 := {C D C D q T, q > 0} does not work for the Łukasiewicz t-norm: A B 0.4, B C 0.4 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 10
44 Positive subsumption in Π-EL is in P idea: transform TBox into classical TBox encoding all positive subsumptions reason over crisp interpretations T >0 := {C D C D q T, q > 0} does not work for the Łukasiewicz t-norm: A B 0.4, B C 0.4 A(x) C(x) = 0, but not necessarily > 0 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 10
45 p-subsumption without Łukasiewicz idea: generalize the completion algorithm for classical EL A r.b, B C A r.c 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 11
46 p-subsumption without Łukasiewicz idea: generalize the completion algorithm for classical EL keep track of degrees A r.b, B C A r.c A r.b q, B C p A r.c p q 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 11
47 p-subsumption without Łukasiewicz idea: generalize the completion algorithm for classical EL keep track of degrees A r.b, B C A r.c A r.b q, B C p A r.c p q restricted to GCIs in normal form: A B C q, A r.b q, r.a B q 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 11
48 p-subsumption without Łukasiewicz idea: generalize the completion algorithm for classical EL keep track of degrees A r.b, B C A r.c A r.b q, B C p A r.c p q restricted to GCIs in normal form: A B C q, A r.b q, r.a B q problem: A B q, A C p, B C D o classical EL and G-EL A D o p q 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 11
49 p-subsumption without Łukasiewicz idea: generalize the completion algorithm for classical EL keep track of degrees A r.b, B C A r.c A r.b q, B C p A r.c p q restricted to GCIs in normal form: A B C q, A r.b q, r.a B q problem: A B q, A C p, B C D o classical EL and G-EL A D o p q Π-EL and Ł-EL A A D o p q need to distinguish A, A A, A 3,... 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 11
50 p-subsumption without Łukasiewicz idea: generalize the completion algorithm for classical EL keep track of degrees A r.b, B C A r.c A r.b q, B C p A r.c p q restricted to GCIs in normal form: A B C q, A r.b q, r.a B q problem: A B q, A C p, B C D o classical EL and G-EL A D o p q Π-EL and Ł-EL A A D o p q need to distinguish A, A A, A 3,... polynomial algorithm for 1-subsumption with only crisp roles [DL 2013] 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 11
51 Conclusions Summary: fuzzy variants of the light-weight DL EL positive subsumption either CO-NP-hard or in P p-subsumption often CO-NP-hard 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 12
52 Conclusions Summary: fuzzy variants of the light-weight DL EL positive subsumption either CO-NP-hard or in P p-subsumption often CO-NP-hard Future work: completion algorithm for p-subsumption upper bounds for CO-NP-hard cases 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 12
53 Thank You Franz Baader, Sebastian Brandt, and Carsten Lutz. Pushing the EL envelope. In Proc. IJCAI 05, pages , Stefan Borgwardt and Rafael Peñaloza. About subsumption in fuzzy EL. In Thomas Eiter, Birte Glimm, Yevgeny Kazakov, and Markus Krötzsch, editors, Proc. DL 13, volume 1014 of CEUR-WS, Poster paper. Petr Hájek. Metamathematics of Fuzzy Logic (Trends in Logic). Springer, Petr Hájek. Making fuzzy description logic more general. Fuzzy Set. Syst., 154(1):1 15, Theofilos Mailis, Giorgos Stoilos, Nikolaos Simou, Giorgos B. Stamou, and Stefanos Kollias. Tractable reasoning with vague knowledge using fuzzy EL ++. J. Intell. Inf. Syst., 39(2): , 北京, August 7, 2013 Positive Subsumption in Fuzzy EL 13
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