A SHORT GUIDE TO FUZZY DESCRIPTION LOGICS
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1 Institute of Theoretical Computer Science Chair of Automata Theory A SHORT GUIDE TO FUZZY DESCRIPTION LOGICS Stefan Borgwardt Oxford, December 9th, 2014
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3 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
4 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
5 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
6 Motivation hasdisease.flu hassymptom.headache hassymptom.fever Oxford, December 9th, 2014 Fuzzy Description Logics 1
7 Motivation hasdisease.flu hassymptom.headache hassymptom.fever Oxford, December 9th, 2014 Fuzzy Description Logics 1
8 Motivation hasdisease.flu hassymptom.headache hassymptom.fever Oxford, December 9th, 2014 Fuzzy Description Logics 1
9 Motivation hasdisease.flu hassymptom.headache hassymptom.fever 37.5 C Oxford, December 9th, 2014 Fuzzy Description Logics 1
10 Motivation hasdisease.flu hassymptom.headache hassymptom.fever 39.2 C Oxford, December 9th, 2014 Fuzzy Description Logics 1
11 Motivation hasdisease.flu hassymptom.headache hassymptom.fever C 0 Oxford, December 9th, 2014 Fuzzy Description Logics 1
12 Motivation hasdisease.flu hassymptom.headache hassymptom.fever 39.2 C Oxford, December 9th, 2014 Fuzzy Description Logics 1
13 Motivation hasdisease.flu hassymptom.headache hassymptom.fever 39.2 C HighConcentration (GALEN), LowFrequency (SNOMED CT) Oxford, December 9th, 2014 Fuzzy Description Logics 1 0
14 Motivation hasdisease.flu hassymptom.headache hassymptom.fever 39.2 C HighConcentration (GALEN), LowFrequency (SNOMED CT) 0 Oxford, December 9th, 2014 Fuzzy Description Logics 1
15 Motivation hasdisease.flu hassymptom.headache hassymptom.fever 39.2 C HighConcentration (GALEN), LowFrequency (SNOMED CT) 0 Oxford, December 9th, 2014 Fuzzy Description Logics 1
16 Uncertainty testresult.positive hasdisease.flu Oxford, December 9th, 2014 Fuzzy Description Logics 2
17 Uncertainty testresult.positive hasdisease.flu... with probability 0.5. Oxford, December 9th, 2014 Fuzzy Description Logics 2
18 Uncertainty testresult.positive hasdisease.flu... with probability 0.5. Probabilistic/Possibilistic DLs: probability/possibility distributions on possible worlds Oxford, December 9th, 2014 Fuzzy Description Logics 2
19 Uncertainty testresult.positive hasdisease.flu... with probability 0.5. Probabilistic/Possibilistic DLs: probability/possibility distributions on possible worlds ALCN with possibilistic axioms (Hollunder 1995) P-SHOIN(D) (Lukasiewicz 2008) Prob-ALC (Lutz and Schröder 2010) Bayesian DL-Lite and EL (Ceylan and Peñaloza 2014; d Amato, Fanizzi, and Lukasiewicz 2008) Oxford, December 9th, 2014 Fuzzy Description Logics 2
20 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
21 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation Oxford, December 9th, 2014 Fuzzy Description Logics 3
22 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation Oxford, December 9th, 2014 Fuzzy Description Logics 3
23 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation Oxford, December 9th, 2014 Fuzzy Description Logics 3
24 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation Oxford, December 9th, 2014 Fuzzy Description Logics 3
25 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation 1 0 Oxford, December 9th, 2014 Fuzzy Description Logics 3
26 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation 1 doctor textbook 0 newspaper neighbor Oxford, December 9th, 2014 Fuzzy Description Logics 3
27 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation 1 doctor textbook 0 newspaper neighbor Oxford, December 9th, 2014 Fuzzy Description Logics 3
28 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation 1 doctor textbook 0 newspaper neighbor Oxford, December 9th, 2014 Fuzzy Description Logics 3
29 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation 1 doctor textbook 0 newspaper neighbor Oxford, December 9th, 2014 Fuzzy Description Logics 3
30 Degrees of Truth (L,,,, ) complete lattice : L L L for conjunction disjunction implication negation 1 doctor textbook 0 newspaper neighbor Oxford, December 9th, 2014 Fuzzy Description Logics 3
31 Degrees of Truth (L,,,, ) complete lattice 1 : L L L for conjunction disjunction implication negation t doctor textbook i u 0 newspaper neighbor f (Belnap 1977; Maier, Ma, and Hitzler 2013) Oxford, December 9th, 2014 Fuzzy Description Logics 3
32 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) stefan:headache p hasdisease.flu hassymptom.headache p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
33 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) stefan:headache p hasdisease.flu hassymptom.headache p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
34 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) ( hassymptom.psychosomatic) I (x) = inf hassymptomi (x, y) Psychosomatic I (y) y I stefan:headache p hasdisease.flu hassymptom.headache p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
35 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) ( hassymptom.psychosomatic) I (x) = inf hassymptomi (x, y) Psychosomatic I (y) y I stefan:headache p hasdisease.flu hassymptom.headache p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
36 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) ( hassymptom.psychosomatic) I (x) = inf hassymptomi (x, y) Psychosomatic I (y) y I stefan:headache p hasdisease.flu hassymptom.headache p ( hasdisease.flu) I (x) ( hassymptom.headache) I (x) p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
37 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) ( hassymptom.psychosomatic) I (x) = inf hassymptomi (x, y) Psychosomatic I (y) y I stefan:headache p hasdisease.flu hassymptom.headache p ( hasdisease.flu) I (x) ( hassymptom.headache) I (x) p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
38 Fuzzy ALC Fever I : I L hassymptom I : I I L stefan I I (Headache Psychosomatic) I (x) = Headache I (x) Psychosomatic I (x) ( Flu) I (x) = Flu I (x) ( hassymptom.psychosomatic) I (x) = inf hassymptomi (x, y) Psychosomatic I (y) y I stefan:headache p hasdisease.flu hassymptom.headache p ( hasdisease.flu) I (x) ( hassymptom.headache) I (x) p {...} = 1, a:c p, C D p? best degree? Oxford, December 9th, 2014 Fuzzy Description Logics 4
39 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) Oxford, December 9th, 2014 Fuzzy Description Logics 5
40 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) hassymptom.fever: 0.8 hassymptom hassymptom hassymptom Fever: 0.8 Fever: 0.79 witnessed interpretation (Hájek 2005). Fever: Oxford, December 9th, 2014 Fuzzy Description Logics 5
41 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) hassymptom.fever: 0.8 hassymptom hassymptom hassymptom Fever: 0.8 Fever: 0.79 witnessed interpretation (Hájek 2005). Fever: Oxford, December 9th, 2014 Fuzzy Description Logics 5
42 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) hassymptom.fever: 0.8 hassymptom hassymptom hassymptom Fever: 0.8 Fever: 0.79 witnessed interpretation (Hájek 2005). Fever: Bulldog haschild.bulldog 0.5 haschild haschild haschild Bulldog: 1 Bulldog: 0.5 Bulldog: 0.25 Bulldog: Oxford, December 9th, 2014 Fuzzy Description Logics 5
43 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) hassymptom.fever: 0.8 hassymptom hassymptom hassymptom Fever: 0.8 Fever: 0.79 witnessed interpretation (Hájek 2005). Fever: Bulldog haschild.bulldog 0.5 haschild haschild haschild Bulldog: 1 Bulldog: 0.5 Bulldog: 0.25 Bulldog: Oxford, December 9th, 2014 Fuzzy Description Logics 5
44 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) hassymptom.fever: 0.8 hassymptom hassymptom hassymptom Fever: 0.8 Fever: 0.79 witnessed interpretation (Hájek 2005). Fever: Bulldog haschild.bulldog 0.5 haschild haschild haschild Bulldog: 1 Bulldog: 0.5 Bulldog: 0.25 Bulldog: Oxford, December 9th, 2014 Fuzzy Description Logics 5
45 Infinity ( hassymptom.fever) I (x) = sup y I hassymptom I (x, y) Fever I (y) hassymptom.fever: 0.8 hassymptom hassymptom hassymptom Fever: 0.8 Fever: 0.79 witnessed interpretation (Hájek 2005). Fever: Bulldog haschild.bulldog 0.5 haschild haschild haschild Bulldog: 1 Bulldog: 0.5 Bulldog: 0.25 Bulldog: Oxford, December 9th, 2014 Fuzzy Description Logics 5
46 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
47 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
48 Zadeh semantics (L,,,, ) (Zadeh 1965) [0, 1] min(x, y) max(x, y) max(1 x, y) Kleene-Dienes-implication 1 x involutive negation Oxford, December 9th, 2014 Fuzzy Description Logics 6
49 Zadeh semantics (L,,,, ) (Zadeh 1965) [0, 1] min(x, y) max(x, y) max(1 x, y) Kleene-Dienes-implication 1 x involutive negation crisp inclusion C D : C I (x) D I (x) Oxford, December 9th, 2014 Fuzzy Description Logics 6
50 Zadeh semantics (L,,,, ) (Zadeh 1965) [0, 1] min(x, y) max(x, y) max(1 x, y) Kleene-Dienes-implication 1 x involutive negation crisp inclusion C D : C I (x) D I (x) problem with Kleene-Dienes-implication: C C 0.5 Oxford, December 9th, 2014 Fuzzy Description Logics 6
51 Zadeh semantics (L,,,, ) (Zadeh 1965) [0, 1] min(x, y) max(x, y) max(1 x, y) Kleene-Dienes-implication 1 x involutive negation crisp inclusion C D : C I (x) D I (x) problem with Kleene-Dienes-implication: C C 0.5 tableaux algorithms for f KD -SHIN : FiRE, fuzzydl (Stoilos et al. 2006; Straccia 2001) Oxford, December 9th, 2014 Fuzzy Description Logics 6
52 Zadeh semantics (L,,,, ) (Zadeh 1965) [0, 1] min(x, y) max(x, y) max(1 x, y) Kleene-Dienes-implication 1 x involutive negation crisp inclusion C D : C I (x) D I (x) problem with Kleene-Dienes-implication: C C 0.5 tableaux algorithms for f KD -SHIN : FiRE, fuzzydl (Stoilos et al. 2006; Straccia 2001) x :(C D) p x :C p, x :D p x : r.c p, (x, y):r > p y :C p Oxford, December 9th, 2014 Fuzzy Description Logics 6
53 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
54 Triangular Norms (L,,,, ) (Hájek 2001) [0, 1] t-norm t-conorm: 1 ((1 x) (1 y)) residuum: (x y) z iff y (x z) residual negation: x 0 t-norm: associative, commutative, monotone, unit 1, (continuous) Oxford, December 9th, 2014 Fuzzy Description Logics 7
55 Triangular Norms (L,,,, ) (Hájek 2001) [0, 1] t-norm t-conorm: 1 ((1 x) (1 y)) residuum: (x y) z iff y (x z) residual negation: x 0 t-norm: associative, commutative, monotone, unit 1, (continuous) Gödel (G): min{x, y} G y x Oxford, December 9th, 2014 Fuzzy Description Logics 7
56 Triangular Norms (L,,,, ) (Hájek 2001) [0, 1] t-norm t-conorm: 1 ((1 x) (1 y)) residuum: (x y) z iff y (x z) residual negation: x 0 t-norm: associative, commutative, monotone, unit 1, (continuous) Gödel (G): min{x, y} 1 Product (Π): x y Π y x Oxford, December 9th, 2014 Fuzzy Description Logics 7
57 Triangular Norms (L,,,, ) (Hájek 2001) [0, 1] t-norm t-conorm: 1 ((1 x) (1 y)) residuum: (x y) z iff y (x z) residual negation: x 0 t-norm: associative, commutative, monotone, unit 1, (continuous) Gödel (G): min{x, y} 1 Product (Π): x y Łukasiewicz (Ł): max(0, x + y 1) y x Oxford, December 9th, 2014 Fuzzy Description Logics 7 Ł
58 Triangular Norms (L,,,, ) (Hájek 2001) [0, 1] t-norm t-conorm: 1 ((1 x) (1 y)) residuum: (x y) z iff y (x z) residual negation: x 0 t-norm: associative, commutative, monotone, unit 1, (continuous) Gödel (G): min{x, y} 1 Product (Π): x y Łukasiewicz (Ł): max(0, x + y 1) 0.8 Ordinal sums y x Oxford, December 9th, 2014 Fuzzy Description Logics 7
59 Triangular Norms (L,,,, ) (Hájek 2001) [0, 1] t-norm t-conorm: 1 ((1 x) (1 y)) residuum: (x y) z iff y (x z) residual negation: x 0 t-norm: associative, commutative, monotone, unit 1, (continuous) Gödel (G): min{x, y} 1 Product (Π): x y Łukasiewicz (Ł): max(0, x + y 1) 0.8 Ordinal sums ends with Łukasiewicz starts with Product y x Oxford, December 9th, 2014 Fuzzy Description Logics 7
60 Undecidable Consistency tableaux algorithm (fuzzydl): (Bobillo and Straccia 2008) x :(C D) p x :C = v x :C, x :D = v x :D, p v x :C v x :D x : r.c = p, (x, y):r = q y :C = v y :C, q v y :C p Oxford, December 9th, 2014 Fuzzy Description Logics 8
61 Undecidable Consistency tableaux algorithm (fuzzydl): (Bobillo and Straccia 2008) x :(C D) p x :C = v x :C, x :D = v x :D, p v x :C v x :D x : r.c = p, (x, y):r = q y :C = v y :C, q v y :C p not sound: GCIs can generate infinitely many values blocking condition only on constraints Oxford, December 9th, 2014 Fuzzy Description Logics 8
62 Undecidable Consistency tableaux algorithm (fuzzydl): (Bobillo and Straccia 2008) x :(C D) p x :C = v x :C, x :D = v x :D, p v x :C v x :D x : r.c = p, (x, y):r = q y :C = v y :C, q v y :C p not sound: GCIs can generate infinitely many values blocking condition only on constraints simple fuzzy DLs are undecidable (Borgwardt and Peñaloza 2012) Ł-ALC with crisp ontologies Π-ALC with crisp GCIs Oxford, December 9th, 2014 Fuzzy Description Logics 8
63 Undecidable Consistency tableaux algorithm (fuzzydl): (Bobillo and Straccia 2008) x :(C D) p x :C = v x :C, x :D = v x :D, p v x :C v x :D x : r.c = p, (x, y):r = q y :C = v y :C, q v y :C p not sound: GCIs can generate infinitely many values blocking condition only on constraints simple fuzzy DLs are undecidable (Borgwardt and Peñaloza 2012) Ł-ALC with crisp ontologies Π-ALC with crisp GCIs any of the following is bad: t-norms starting with Ł assertions a:a p with involutive negation 1 x Oxford, December 9th, 2014 Fuzzy Description Logics 8
64 Trivial Consistency fuzzy SROIQ is decidable (Borgwardt, Distel, and Peñaloza 2012) Oxford, December 9th, 2014 Fuzzy Description Logics 9
65 Trivial Consistency fuzzy SROIQ is decidable (Borgwardt, Distel, and Peñaloza 2012) t-norms not starting with Ł only -assertions only residual negation x = x 0 Oxford, December 9th, 2014 Fuzzy Description Logics 9
66 Trivial Consistency fuzzy SROIQ is decidable (Borgwardt, Distel, and Peñaloza 2012) t-norms not starting with Ł only -assertions only residual negation x = x 0 Gödel negation is crisp: x = { 1 if x = 0 0 otherwise Oxford, December 9th, 2014 Fuzzy Description Logics 9
67 Trivial Consistency fuzzy SROIQ is decidable (Borgwardt, Distel, and Peñaloza 2012) t-norms not starting with Ł only -assertions only residual negation x = x 0 Gödel negation is crisp: x = { 1 if x = 0 0 otherwise { C D 0.7, (a, b):r 0.3, trans(r) 0.9,... } is consistent iff {C D, (a, b):r, trans(r),... } is (classically) consistent Oxford, December 9th, 2014 Fuzzy Description Logics 9
68 Gödel T-norm finitely-valued model property decidable? Oxford, December 9th, 2014 Fuzzy Description Logics 10
69 Gödel T-norm fuzzy ALC over G: Gödel residuum: x y = finitely-valued model property decidable? no finitely-valued model property { 1 if x y y otherwise a:a = 0.5 r.a A 1 r. A 1 Oxford, December 9th, 2014 Fuzzy Description Logics 10
70 Gödel T-norm fuzzy ALC over G: Gödel residuum: x y = finitely-valued model property decidable? no finitely-valued model property { 1 if x y y otherwise a:a = 0.5 r.a A 1 r. A 1 A(x 0 ) = 0.5 r(x i, x i+1 ) > A(x i+1 ) r(x i, x i+1 ) A(x i ) Oxford, December 9th, 2014 Fuzzy Description Logics 10
71 Gödel T-norm fuzzy ALC over G: Gödel residuum: x y = finitely-valued model property decidable? no finitely-valued model property { 1 if x y y otherwise a:a = 0.5 r.a A 1 r. A 1 A(x 0 ) = 0.5 r(x i, x i+1 ) > A(x i+1 ) r(x i, x i+1 ) A(x i ) 0.5 = A(x 0 ) > A(x 1 ) > A(x 2 ) >... Oxford, December 9th, 2014 Fuzzy Description Logics 10
72 Gödel T-norm fuzzy ALC over G: Gödel residuum: x y = finitely-valued model property decidable? no finitely-valued model property { 1 if x y y otherwise a:a = 0.5 r.a A 1 r. A 1 A(x 0 ) = 0.5 r(x i, x i+1 ) > A(x i+1 ) r(x i, x i+1 ) A(x i ) 0.5 = A(x 0 ) > A(x 1 ) > A(x 2 ) >... too weak for undecidability EXPTIME-complete (Borgwardt, Distel, and Peñaloza 2014) Oxford, December 9th, 2014 Fuzzy Description Logics 10
73 Triangular Norms undecidable fuzzy DLs assertions residual negation involutive negation NEL IEL SROIQ ELC IALC crisp starts with Ł starts with Ł starts with Ł Π, Ł Π, starts with Ł starts with Ł starts with Ł starts with Ł not G not G = starts with Ł not G not G not G not G Oxford, December 9th, 2014 Fuzzy Description Logics 11
74 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
75 Finite Chains (L,,,, ) x finite chain t-norm t-conorm residuum residual negation Oxford, December 9th, 2014 Fuzzy Description Logics 12
76 Finite Chains (L,,,, ) x finite chain Ł 6 : x y = max(0, x + y 1) t-norm t-conorm residuum residual negation Oxford, December 9th, 2014 Fuzzy Description Logics 12
77 Finite Chains (L,,,, ) x finite chain Ł 6 : x y = max(0, x + y 1) t-norm t-conorm residuum residual negation crispification: cut concepts A p, cut roles r p recursive translation into crisp ontology Oxford, December 9th, 2014 Fuzzy Description Logics 12
78 Finite Chains (L,,,, ) x finite chain Ł 6 : x y = max(0, x + y 1) t-norm t-conorm residuum residual negation crispification: cut concepts A p, cut roles r p recursive translation into crisp ontology exponential blow-up (except for Zadeh semantics) implemented for f KD -SROIQ and G n-sroiq in DeLorean (Bobillo, Delgado, et al. 2012) Oxford, December 9th, 2014 Fuzzy Description Logics 12
79 Finite Lattices (L,,,, ) x finite lattice t-norm t-conorm residuum residual negation Oxford, December 9th, 2014 Fuzzy Description Logics 13
80 Finite Lattices (L,,,, ) x finite lattice t-norm t-conorm residuum residual negation fuzzy SHOI is PSpace/ExpTime-complete (Borgwardt 2014; Borgwardt and Peñaloza 2013b) automata for satisfiability (tree model property) pre-completion for ABox reasoning Oxford, December 9th, 2014 Fuzzy Description Logics 13
81 Finite Lattices (L,,,, ) x finite lattice t-norm t-conorm residuum residual negation fuzzy SHOI is PSpace/ExpTime-complete (Borgwardt 2014; Borgwardt and Peñaloza 2013b) automata for satisfiability (tree model property) pre-completion for ABox reasoning choice of operators is irrelevant Oxford, December 9th, 2014 Fuzzy Description Logics 13
82 Zadeh Introduction Trivial Fuzzy DLs Go del [0, 1] T-norms L Finite Chains Undecidable Uncertainty Applications? Finite Lattices
83 Fuzzy Ontologies hand-crafting fuzzy ontologies (FuzzyWine) Protegé plug-in using OWL annotations (Bobillo and Straccia 2011) support for fuzzy reasoners Oxford, December 9th, 2014 Fuzzy Description Logics 14
84 Fuzzy Ontologies hand-crafting fuzzy ontologies (FuzzyWine) Protegé plug-in using OWL annotations (Bobillo and Straccia 2011) support for fuzzy reasoners learning fuzzy axioms improve learning algorithms by fuzzy information (Lisi and Straccia 2014) distance, user ranking Oxford, December 9th, 2014 Fuzzy Description Logics 14
85 Fuzzy Ontologies hand-crafting fuzzy ontologies (FuzzyWine) Protegé plug-in using OWL annotations (Bobillo and Straccia 2011) support for fuzzy reasoners learning fuzzy axioms improve learning algorithms by fuzzy information (Lisi and Straccia 2014) distance, user ranking using fuzzy degrees consistently: derive from other data use few truth values use order axioms: stefan: hasdisease.flu > rafael: hasdisease.flu Oxford, December 9th, 2014 Fuzzy Description Logics 14
86 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR Oxford, December 9th, 2014 Fuzzy Description Logics 15
87 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR 2. if possible, absorb GCIs fuzzydl (Bobillo and Straccia 2014) unknown complexity Oxford, December 9th, 2014 Fuzzy Description Logics 15
88 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR 2. if possible, absorb GCIs fuzzydl (Bobillo and Straccia 2014) unknown complexity 3. use less expressive DLs Oxford, December 9th, 2014 Fuzzy Description Logics 15
89 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR 2. if possible, absorb GCIs fuzzydl (Bobillo and Straccia 2014) unknown complexity 3. use less expressive DLs subsumption in fuzzy EL is polynomial for G (Mailis, Stoilos, Simou, et al. 2012) CO-NP-hard if contains Ł (Borgwardt and Peñaloza 2013a) Oxford, December 9th, 2014 Fuzzy Description Logics 15
90 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR 2. if possible, absorb GCIs fuzzydl (Bobillo and Straccia 2014) unknown complexity 3. use less expressive DLs subsumption in fuzzy EL is polynomial for G (Mailis, Stoilos, Simou, et al. 2012) CO-NP-hard if contains Ł (Borgwardt and Peñaloza 2013a) subsumption in fuzzy FL 0 is CO-NP-/PSPACE-/EXPTIME-complete for G (Borgwardt, Leyva Galano, and Peñaloza 2014) Oxford, December 9th, 2014 Fuzzy Description Logics 15
91 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR 2. if possible, absorb GCIs fuzzydl (Bobillo and Straccia 2014) unknown complexity 3. use less expressive DLs subsumption in fuzzy EL is polynomial for G (Mailis, Stoilos, Simou, et al. 2012) CO-NP-hard if contains Ł (Borgwardt and Peñaloza 2013a) subsumption in fuzzy FL 0 is CO-NP-/PSPACE-/EXPTIME-complete for G (Borgwardt, Leyva Galano, and Peñaloza 2014) CQ answering in fuzzy DL-Lite under Zadeh or finite-valued semantics (Mailis, Peñaloza, and Turhan 2014; Mailis, Stoilos, and Stamou 2010; Pan et al. 2008; Straccia 2006) Oxford, December 9th, 2014 Fuzzy Description Logics 15
92 Conclusions 1. use Zadeh/Gödel/finite-valued fuzzy DLs FiRE, fuzzydl, DeLorean, LiFR 2. if possible, absorb GCIs fuzzydl (Bobillo and Straccia 2014) unknown complexity 3. use less expressive DLs subsumption in fuzzy EL is polynomial for G (Mailis, Stoilos, Simou, et al. 2012) CO-NP-hard if contains Ł (Borgwardt and Peñaloza 2013a) subsumption in fuzzy FL 0 is CO-NP-/PSPACE-/EXPTIME-complete for G (Borgwardt, Leyva Galano, and Peñaloza 2014) CQ answering in fuzzy DL-Lite under Zadeh or finite-valued semantics (Mailis, Peñaloza, and Turhan 2014; Mailis, Stoilos, and Stamou 2010; Pan et al. 2008; Straccia 2006) Thank you Oxford, December 9th, 2014 Fuzzy Description Logics 15
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