Weighted Tree Automata in Description Logics Pavlos Marantidis Technische Universit¨ at Dresden pavlos.marantidis@tu-dresden.de March 15, 2017 Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 1 / 32
Overview Description Logics 1 Representation of languages 2 Weighted Tree Automata 3 Future Work 4 Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 2 / 32
Description Logics Description Logics are a family of knowledge representation languages with a formal, logic-based semantics. Description comes from concept description, i.e. a formal expression that determines a set of individuals with common properties. Logics comes from the fact that the semantics of concept descriptions can be defined using logic. Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 3 / 32
Description Logics Concepts Constructors for building concept descriptions out of atomic concepts (unary predicates) and roles (binary predicates). Interpretation I assigns sets C I to concept descriptions C according to the semantics of the constructors. Syntax Semantics ⊤ I = ∆ I top concept ⊤ A I ⊆ ∆ I concept name A ( C ⊓ D ) I = C I ∩ D I conjunction C ⊓ D ( C ⊔ D ) I = C I ∪ D I disjunction C ⊔ D ( ∃ r . C ) I = { d ∈ ∆ I | existential ∃ r . C ∃ e ∈ ∆ I . ( d , e ) ∈ r I ∧ e ∈ C I } restriction ( ∀ r . C ) I = { d ∈ ∆ I | value restriction ∀ r . C ∀ e ∈ ∆ I . ( d , e ) ∈ r I → e ∈ C I } Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 4 / 32
Subsumption and Equivalence Subsumption and equivalence C ⊑ D iff C I ⊆ D I for all interpretations C ≡ D iff C ⊑ D and D ⊑ C Examples C = Human ⊓ ∀ hasChild . Smart ⊓ ∀ hasChild . Strong D = Human ⊓ ∀ hasChild . ( Smart ⊓ Strong ) It can be deduced that C ⊑ Human C ≡ D Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 5 / 32
Unification Check whether two concepts can be made equivalent Woman ⊓ ∀ hasChild . Woman Person ⊓ Female ⊓ ∀ hasChild . ( Person ⊓ Female ) Woman �→ Person ⊓ Female Human ⊓ Male ⊓∃ loves . ExtremeSports Man ⊓∃ loves . ( Sport ⊓ Dangerous ) Man �→ Human ⊓ Male ExtremeSports �→ Sport ⊓ Dangerous Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 6 / 32
Unification Using some concept names as variables we can define concept patterns. An assignment is a mapping σ : variable → concept It can be extended to a mapping σ : pattern → concept . The unification problem Given concept patterns C , D , Check whether ∃ σ , s.t. σ ( C ) ≡ σ ( D ). Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 7 / 32
Approximate Unification Suppose there is no solution. What is the “best” we can do? • Define best: An assignment that is least “bad” wrt some measure. A concept distance measure (CDM) is a mapping m : concept × concept → [0 , ∞ ] s.t. • m ( C , D ) = 0 iff C ≡ D , • m ( C , D ) = m ( D , C ), • C ≡ D = ⇒ m ( C , E ) = m ( D , E ). The approximate unification problem Given concept patterns C , D , CDM m , threshold t ∈ [0 , ∞ ], Check whether ∃ σ , s.t. m ( σ ( C ) , σ ( D )) < t . Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 8 / 32
Knowledge base Knowledge base K A knowledge base K consists of a TBox T , a finite set of terminological axioms, i.e. properties of concepts of the form C ⊑ D and C ≡ D , an ABox A , a finite set of factual assertions of the form C ( d ) and r ( d , e ). Talk ≡∃ has - subject . ⊤ Speaker ≡ Participant ⊓ ∃ gives . Talk Speaker ( Pavlos ) Logic ⊑ Mathematics ∃ has - subject . Logic ( wtA - in - DL ) BoringTalk ≡ Talk ⊓ ∀ attended - by . Bored gives ( Pavlos , wtA - in - DL ) BoringSpeaker ≡ Speaker ⊓ ∀ gives . BoringTalk Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 9 / 32
Knowledge base Knowledge base K A knowledge base K consists of a TBox T , a finite set of terminological axioms, i.e. properties of concepts of the form C ⊑ D and C ≡ D , an ABox A , a finite set of factual assertions of the form C ( d ) and r ( d , e ). An interpetation I is a model of K if C I ⊆ D I for every C ⊑ D ∈ T , d I ∈ C I for every C ( d ) ∈ A , ( d I , e I ) ∈ r I for every r ( d , e ) ∈ A . Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 10 / 32
Instance Queries The instance query problem Given knowledge base K = ( T , A ), element d , concept C , Check whether K | = C ( d ). Talk ≡∃ has - subject . ⊤ Speaker ≡ Participant ⊓ ∃ gives . Talk Speaker ( Peter ) Logic ⊑ Mathematics ∃ has - subject . Logic ( wtA - in - DL ) BoringTalk ≡ Talk ⊓ ∀ attended - by . Bored gives ( Pavlos , wtA - in - DL ) BoringSpeaker ≡ Speaker ⊓ ∀ gives . BoringTalk K | = Talk ( wtA - in - DL ) K | = ∃ has - subject . Mathematics ( wtA - in - DL ) K | = Participant ( Pavlos ) K �| = BoringTalk ( wtA - in - DL ) K �| = BoringSpeaker ( Pavlos ) Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 11 / 32
Relaxed Instance Queries What if d is not an element of C but of a concept “similar” to C ? A concept similarity measure (CSM) is a mapping m : concept × concept → [0 , 1] s.t. • m ( C , C ) = 1 for every C ∈ C , • C ≡ D = ⇒ m ( C , E ) = m ( D , E ). The relaxed instance query problem Given knowledge base K = ( T , A ), element d , concept C , CSM m , Given threshold t ∈ [0 , 1], Check whether ∃ D ∈ C , s.t. K | = D ( d ) and m ( C , D ) > t . Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 12 / 32
Concept Comparison Measures A concept distance measure (CDM) is a mapping m : C × C → [0 , ∞ ] s.t. • m ( C , D ) = 0 iff C = D , • m ( C , D ) = m ( D , C ), • C ≡ D = ⇒ m ( C , E ) = m ( D , E ). A concept similarity measure (CSM) is a mapping m : C × C → [0 , 1] s.t. • m ( C , C ) = 1 for every C ∈ C , • C ≡ D = ⇒ m ( C , E ) = m ( D , E ). A concept comparison measure (CSM) is a mapping m : C × C → ( S , ≤ ) s.t. C ≡ D = ⇒ m ( C , E ) = m ( D , E ). Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 13 / 32
The Description Logic FL 0 Constructors value restriction ∀ r . C , conjunction ⊓ , top concept ⊤ Value restriction sets L T ( C , A ) = { r 1 . . . r n ∈ N ∗ R | C ⊑ ∀ r 1 . . . ∀ r n . A } Example C := A ⊓ ∀ s . A D := ∀ s . A T = { A ⊑ ∀ r . A } L T ( C , A ) = r ∗ ∪ sr ∗ L T ( D , A ) = sr ∗ Theorem C ≡ D iff L T ( C , A ) = L T ( D , A ) for every A in C , D , T . Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 14 / 32
CCMs for FL 0 ≡ C D ( L 1 , . . . , L n ) � � � � = L T ( C , A 1 ) , . . . , L T ( C , A n ) L T ( D , A 1 ) , . . . , L T ( D , A n ) = ⇒ To obtain equivalence invariant CCMs for FL 0 , it is sufficient to define measures that compare tuples of formal languages. Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 15 / 32
From languages to ??? How does one finitely represent an infinite language? Using a finite automaton! Can we represent all of them? No...But luckily! Theorem (Pensel, 2015) Given concept C, TBox T , and concept name A, the value restriction set L T ( C , A ) is a regular language. How does one finitely represent a tuple of infinite languages? Using a tuple of finite automata! Using a single object? Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 16 / 32
Σ ∗ as an infinite tree Σ = { a , b } (1 , 1) ε (1 , 0) (0 , 0) a b (0 , 1) (1 , 1) (0 , 0) (1 , 0) aa ab ba bb (0 , 0) (0 , 0) (0 , 1) (0 , 1) (0 , 1) (1 , 0) (1 , 1) (1 , 1) aaa aab aba abb baa bab bba bbb . . . . . . . . . . . . . . . . . . . . . . . . L 1 = { ε, a , ab , bb , abb , bab , bbb , . . . } L 2 = { ε, aa , ab , aab , baa , bab , bba , bbb , . . . } Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 17 / 32
Regular trees L = { 0 , 1 } × { 0 , 1 } 0 (1 , 1) 1 0 (1 , 0) (0 , 0) 1 0 0 1 (0 , 1) (1 , 1) (0 , 0) (1 , 0) 0 0 1 1 1 1 0 0 (0 , 0) (0 , 1) (0 , 0) (1 , 0) (0 , 1) (1 , 1) (0 , 1) (1 , 1) . . . . . . . . . . . . . . . . . . . . . . . . Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 18 / 32
Looping Tree Automata Automata Automata on words on infinite words Automata Automata on trees on infinite trees see blackboard! Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 19 / 32
Weighted Looping Tree Automata Weighted Weighted automata automata on words on infinite words Weighted Weighted automata automata on trees on infinite trees see blackboard! Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 20 / 32
Semirings Semiring S = ( S , ⊕ , ⊗ , O , 1 ) where: 1 ( S , ⊕ , O ) a commutative monoid, 2 ( S , ⊗ , 1 ) a monoid, 3 multiplication distributes over addition from left and right, 4 O ⊗ a = a ⊗ O = O for all a ∈ S . A semiring is called commutative if a ⊗ b = b ⊗ a for all a , b ∈ S totally complete commutative if ⊕ → � and ⊗ → � i ∈ I i ≥ 0 Pavlos Marantidis (TU Dresden) Weighted Tree Automata in DLs March 15, 2017 21 / 32
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