Extending ALC with the power-set construct Laura Giordano 1 Alberto - PowerPoint PPT Presentation

Extending ALC with the power-set construct Laura Giordano 1 Alberto Policriti 2 1 DiSIT, Universit` a del Piemonte Orientale “Amedeo Avogadro”, Italy 2 Dipartimento di Scienze Matematiche, Informatiche e Fisiche Universit` a di Udine, Italy This work was presented in JELIA 2019

Aim of the talk ALC and Ω The description logic ALC Ω A set-theoretic translation of ALC Ω

Aim of the talk We explore the relationships between Description Logics and Set Theory. ◮ On the set-theoretic side, we consider a very rudimentary axiomatic set theory Ω , consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. ◮ We consider an extension of the description logic ALC , ALC Ω [ICTCS 2018], in which concepts are naturally interpreted as sets living in Ω -models: membership between concepts and power-set construct to add metamodeling capabilities. In previous work we defined a polynomial translation of ALC Ω in the DL ALCOI (showing that concept satisfiability in ALC Ω is E XP T IME -complete) ◮ In this paper we develop a set-theoretic translation of the description logic ALC Ω in the set theory Ω

Motivations: metamodeling capabilities The idea of enhancing the language of description logics with statements of the form C ∈ D , with C and D concepts is not new: similar assertions are allowed in OWL-Full. Example [Welty1994,Motik05] One can represent the fact that eagles are in the red list of endangered species , by the axiom Eagle ∈ RedListSpecies and that Harry is an eagle , by the assertion harry ∈ Eagle . The power-set concept , Pow ( C ) , allows to capture in a natural way the interactions between concepts and metaconcepts. RedListSpecies ⊑ Pow ( CannotHunt ) , means that: “all the instances of the species in the Red List are not allowed to be hunted”

The theory Ω ◮ The first-order theory Ω consists of the four axioms x ∈ y ∪ z ↔ x ∈ y ∨ x ∈ z ; x ∈ y \ z ↔ x ∈ y ∧ x �∈ z ; x ⊆ y ↔ ∀ z ( z ∈ x → z ∈ y ); x ∈ Pow ( y ) ↔ x ⊆ y . ◮ In any Ω -model everything is supposed to be a set, and circular definition of sets are not forbidden ◮ no extensionality axiom: there are Ω -models in which different sets have equal collection of elements. ◮ The most natural Ω -model is the collection of well-founded sets HF = HF 0 = � n ∈ N HF n , where: HF 0 = ∅ and HF n + 1 = Pow ( HF n ) .

The description logic ALC Ω [Ictcs 2018] The set of ALC Ω concepts are defined inductively as follows: ◮ A ∈ N C , ⊤ and ⊥ are ALC Ω concepts ; ◮ if C , D are ALC Ω concepts and R ∈ N R , then the following are ALC Ω concepts : C ⊓ D , C ⊔ D , ¬ C , C \ D , Pow ( C ) , ∀ R . C , ∃ R . C New membership axioms: C ∈ D and ( C , D ) ∈ R besides the standard assertions D ( a ) and R ( c , d ) General concepts (and not only concept names) can be instances of other concepts, e.g., polar bears are in the red list of endangered species , Polar ⊓ Bear ∈ RedListSpecies and polar bears are more endangered than eagles by the role membership axiom ( Polar ⊓ Bear , Eagle ) ∈ moreEndangered

Semantics of ALC Ω An interpretation for ALC Ω is a pair I = � ∆ , · I � over a set of atoms A where: ◮ the non-empty domain ∆ is a transitive set (i.e., ( ∀ y ∈ ∆)( y ⊆ ∆) ) chosen in the universe U of a model of Ω over the atoms in A ◮ the extension function · I maps each concept name A ∈ N C to an element A I ⊆ ∆ ; each role name R ∈ N R to a binary relation R I ⊆ ∆ × ∆ ; and each individual name a ∈ N I to an element a I ∈ A ∩ ∆ . The function · I is extended to complex concepts of ALC Ω as follows: ⊤ I = ∆ ⊥ I = ∅ ( ¬ C ) I = ∆ \ C I ( C \ D ) I = ( C I \ D I ) ( Pow ( C )) I = Pow ( C I ) ∩ ∆ ( C ⊓ D ) I = C I ∩ D I ( C ⊔ D ) I = C I ∪ D I ( ∀ R . C ) I = { x ∈ ∆ | ∀ y (( x , y ) ∈ R I → y ∈ C I ) } ( ∃ R . C ) I = { x ∈ ∆ | ∃ y (( x , y ) ∈ R I ∧ y ∈ C I ) }

Semantics of ALC Ω Observe that ◮ ∆ is not guaranteed to be closed under union, intersection, etc., the interpretation C I of a concept C is a set in U , but not necessarily an element of ∆ . ◮ However, C I ⊆ ∆ , as the interpretation of the power-set concept ( Pow ( C )) I = ( Pow ( C I )) ∩ ∆ is the portion of the (set-theoretic) power-set visible in ∆ .

Example Let K = ( T , A ) be the set of inclusions and assertions: (1) ReadingGroup ⊑ Pow ( Person ) (2) Meeting ⊑ Pow ( ReadingGroup ) (3) Meeting ⊑ Pow ( ∃ has leader . Person ) (4) SummerMeeting ⊑ Pow ( ∃ has paid . Fee ) HistoryGroup , FantasyGroup , ScienceGroup ∈ ReadingGroup ; SummerMeeting , WinterMeeting ∈ Meeting ; ScienceGroup , FantasyGroup ∈ SummerMeeting ; bob ∈ FantasyGroup ; alice , bob ∈ ScienceGroup ; carl ∈ HistoryG Each reading group is a set of persons (1). The history, fantasy and science groups are reading groups. Each meeting is a set of reading groups (2). The SummerMeeting and the WinterMeeting are meetings. Both the Science group and the Fantasy group participate to the SummerMeeting . Each reading group in a meeting has a leader, who is a person (3). All participants to the SummerMeeting have paid the fee (4).

Polynomial encoding of ALC Ω into ALCOI ◮ each concept C of ALC Ω is translated to a concept C T of ALCOI by replacing all occurrences of the power-set concept Pow ( C ) with ∀ e . C ; ◮ a new individual name e C is added, for each concept name C occurring on the left hand side of a membership axiom C ∈ D , which is translated to an assertion D T ( e C ) (similarly for role membership axioms); ◮ the role e relates e C with all the instances of concept C , by axiom C T ≡ ∃ e − . { e C } ◮ for each (standard) individual name a ∈ N I , the assertion ( ¬∃ e . ⊤ )( a ) is added. Soundness and completeness of the polynomial translation in ALCOI provide, besides decidability, an E XP T IME upper bound for satisfiability in ALC Ω .

Example: Translation in ALCOI Let K = ( T , A ) be the knowledge base with TBox T RedListSpecies ⊑ Pow ( CannotHunt ) and ABox A Eagle ( harry ) , Eagle ∈ RedListSpecies , Polar ⊓ Bear ∈ RedListSpecies K is translated into K T = ( T T , A ) T with TBox T T : RedListSpecies ⊑ ∀ e . CannotHunt , Eagle ≡ ∃ e − . { e Eagle } Polar ⊓ Bear ≡ ∃ e − . { e Polar ⊓ Bear } and ABox A T : Eagle ( harry ) , RedListSpecies ( e Eagle ) , ( ¬∃ e . ⊤ )( harry ) , RedListSpecies ( e Polar ⊓ Bear )

Set-theoretic translation of ALC Ω in the set theory Ω ◮ Our translation of ALC Ω into Ω , exploits the correspondence between membership ∈ and the accessibility relation of a normal modality R explored in [D’Agostino et al.1995]. ◮ Step by step ◮ A set-theoretic translation of ALC based on Schild’s correspondence with polymodal logics . ◮ A translation of the fragment LC Ω of ALC Ω without roles and individual names. ◮ An encoding of ALC Ω into the fragment LC Ω

Set-theoretic translation of LC Ω in the set theory Ω ◮ A ∈ N C , ⊤ and ⊥ are LC Ω concepts ; ◮ if C , D are LC Ω concepts, the following are LC Ω concepts : C ⊓ D , C ⊔ D , ¬ C , C \ D , Pow ( C ) ⊤ S = x ⊥ S = ∅ ( ¬ C ) S = x \ C S A S i = x i , for A i in K ( C ⊓ D ) S = C S ∩ D S ( C ⊔ D ) S = C S ∪ D S ( C \ D ) S = C S \ D S ( Pow ( C )) S = Pow ( C S ) C S 1 ∩ x ⊆ C S C 1 ⊑ C 2 in TBox is translated: 2 C S 1 ∈ C S C 1 ∈ C 2 in ABox is translated: 2 K | = LC C ⊑ D if and only if Ω ⊢ ∀ x ( Trans ( x ) → ∀ x 1 , . . . , ∀ x n ( � ABox A ∧ � TBox T → C S ∩ x ⊆ D S ))

Set-theoretic translation of ALC in the set theory Ω ⊤ S = x; ⊥ S = ∅ ; ( ¬ C ) S = x \ C S ; A S i = x i , for A i in K; ( C ⊓ D ) S = C S ∩ D S ; ( C ⊔ D ) S = C S ∪ D S ; ( ∀ R i . C ) S = Pow ((( x ∪ y 1 ∪ . . . ∪ y k ) \ y i ) ∪ Pow ( C S )) A set U i (represented by the variable y i ) is used to translate role i iff there is some u i ∈ U i such that v ′ ∈ u i ∈ v . R i : ( v , v ′ ) ∈ R I C S 1 ∩ x ⊆ C S C 1 ⊑ C 2 is translated: 2 K | = ALC C ⊑ D if and only if Ω ⊢ ∀ x ∀ y 1 . . . ∀ y k ( Trans 2 ( x ) ∧ Axiom H ( x , y 1 , . . . , y k ) → ∀ x 1 , . . . , ∀ x n ( � TBox T → C S ∩ x ⊆ D S )) This set-theoretic translation of ALC is based on Schild’s correspondence result [Schild91]and on the set-theoretic translation for normal polymodal logics in [DAgostino1995].

Set-theoretic translation of ALC Ω in LC Ω Given an ALC Ω knowledge base K , we define the encoding K E of K in LC Ω : C E ⊓ ¬ ( U 1 ⊔ . . . ⊔ U k ) ⊑ D E , C ⊑ D ∈ K C E ∈ D E C ∈ D in K ; a E i ∈ C E C ( a i ) in K ; a E j ∈ F i h , j ∈ a E h and F i h , j ∈ U i R i ( a h , a j ) ; C E j ∈ G i C h , C j ∈ C E h and G i C h , C j ∈ U i R i ( C h , C j ) . The following additional axioms are also needed in K E : A i ⊑ ¬ ( U 1 ⊔ . . . ⊔ U k ) , one for each concept name A i in K ; B i ∈ ¬ ( U 1 ⊔ . . . ⊔ U k ) , one for each individual name a i in K ; C E ∈ ¬ ( U 1 ⊔ . . . ⊔ U k ) , one for each C ∈ D in K ¬ ( U 1 ⊔ . . . ⊔ U k ) ⊑ Pow ( ¬ ( U 1 ⊔ . . . ⊔ U k ) ⊔ Pow ( ¬ ( U 1 ⊔ . . . ⊔ U k )))