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

Extending ALC with the power-set construct

Laura Giordano1 Alberto Policriti2

1DiSIT, Universit`

a del Piemonte Orientale “Amedeo Avogadro”, Italy

2Dipartimento 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 ALCΩ A set-theoretic translation of ALCΩ

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,

ALCΩ [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 ALCΩ in the DL ALCOI (showing that concept satisfiability in ALCΩ is EXPTIME-complete)

◮ In this paper we develop a set-theoretic translation of the

description logic ALCΩ 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 = HF0 =

n∈N HFn, where: HF0 = ∅ and

HFn+1 = Pow(HFn).

The description logic ALCΩ [Ictcs 2018]

The set of ALCΩ concepts are defined inductively as follows:

◮ A ∈ NC, ⊤ and ⊥ are ALCΩ concepts; ◮ if C, D are ALCΩ concepts and R ∈ NR, then the following

are ALCΩ 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

• f endangered species,

Polar ⊓ Bear ∈ RedListSpecies and polar bears are more endangered than eagles by the role membership axiom (Polar ⊓ Bear, Eagle) ∈ moreEndangered

Semantics of ALCΩ

An interpretation for ALCΩ 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

• ver the atoms in A

◮ the extension function ·I maps each concept name A ∈ NC

to an element AI ⊆ ∆; each role name R ∈ NR to a binary relation RI ⊆ ∆ × ∆; and each individual name a ∈ NI to an element aI ∈ A ∩ ∆. The function ·I is extended to complex concepts of ALCΩ as follows: ⊤I = ∆ ⊥I = ∅ (¬C)I = ∆\CI (C\D)I = (CI\DI) (Pow(C))I = Pow(CI) ∩ ∆ (C ⊓ D)I = CI ∩ DI (C ⊔ D)I = CI ∪ DI (∀R.C)I = {x ∈ ∆ | ∀y((x, y) ∈ RI → y ∈ CI)} (∃R.C)I = {x ∈ ∆ | ∃y((x, y) ∈ RI ∧ y ∈ CI)}

Semantics of ALCΩ

Observe that

◮ ∆ is not guaranteed to be closed under union, intersection,

etc., the interpretation CI of a concept C is a set in U, but not necessarily an element of ∆.

◮ However, CI ⊆ ∆, as the interpretation of the power-set

concept (Pow(C))I = (Pow(CI)) ∩ ∆ 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

• f 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 ALCΩ into ALCOI

◮ each concept C of ALCΩ is translated to a concept CT of

ALCOI by replacing all occurrences of the power-set concept Pow(C) with ∀e.C;

◮ a new individual name eC 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 DT(eC) (similarly for role membership axioms);

◮ the role e relates eC with all the instances of concept C, by

axiom CT ≡ ∃e−.{eC}

◮ for each (standard) individual name a ∈ NI, the assertion

(¬∃e.⊤)(a) is added. Soundness and completeness of the polynomial translation in ALCOI provide, besides decidability, an EXPTIME upper bound for satisfiability in ALCΩ.

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−.{eEagle} Polar ⊓ Bear ≡ ∃e−.{ePolar⊓Bear} and ABox AT: Eagle(harry), RedListSpecies(eEagle), (¬∃e.⊤)(harry), RedListSpecies(ePolar⊓Bear)

Set-theoretic translation of ALCΩ in the set theory Ω

◮ Our translation of ALCΩ 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 LCΩ of ALCΩ without roles

and individual names.

◮ An encoding of ALCΩ into the fragment LCΩ

Set-theoretic translation of LCΩ in the set theory Ω

◮ A ∈ NC, ⊤ and ⊥ are LCΩ concepts; ◮ if C, D are LCΩ concepts, the following are LCΩ concepts:

C ⊓ D, C ⊔ D, ¬C, C\D, Pow(C) ⊤S = x ⊥S = ∅ AS

i = xi , for Ai in K

(¬C)S = x\CS (C ⊓ D)S = CS ∩ DS (C ⊔ D)S = CS ∪ DS (C\D)S = CS\DS (Pow(C))S = Pow(CS) C1 ⊑ C2 in TBox is translated: CS

1 ∩ x ⊆ CS 2

C1 ∈ C2 in ABox is translated: CS

1 ∈ CS 2

K | =LC C ⊑ D if and only if Ω ⊢ ∀x(Trans(x) → ∀x1, . . . , ∀xn( ABoxA ∧ TBoxT → CS ∩ x ⊆ DS))

Set-theoretic translation of ALC in the set theory Ω

⊤S = x; ⊥S = ∅; AS

i = xi , for Ai in K;

(¬C)S = x\CS; (C ⊓ D)S = CS ∩ DS; (C ⊔ D)S = CS ∪ DS; (∀Ri.C)S = Pow(((x ∪ y1 ∪ . . . ∪ yk)\yi) ∪ Pow(CS)) A set Ui (represented by the variable yi) is used to translate role Ri: (v, v′) ∈ RI

i iff there is some ui ∈ Ui such that v′ ∈ ui ∈ v.

C1 ⊑ C2 is translated: CS

1 ∩ x ⊆ CS 2

K | =ALC C ⊑ D if and only if Ω ⊢ ∀x∀y1 . . . ∀yk(Trans2(x) ∧ AxiomH(x, y1, . . . , yk) → ∀x1, . . . , ∀xn( TBoxT → CS ∩ x ⊆ DS)) 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 ALCΩ in LCΩ

Given an ALCΩ knowledge base K, we define the encoding K E

• f K in LCΩ:

CE ⊓ ¬(U1 ⊔ . . . ⊔ Uk) ⊑ DE, C ⊑ D ∈ K CE ∈ DE C ∈ D in K; aE

i ∈ CE

C(ai) in K; aE

j ∈ F i h,j ∈ aE h and F i h,j ∈ Ui

Ri(ah, aj); CE

j ∈ Gi Ch,Cj ∈ CE h and Gi Ch,Cj ∈ Ui

Ri(Ch, Cj). The following additional axioms are also needed in K E: Ai ⊑ ¬(U1 ⊔ . . . ⊔ Uk), one for each concept name Ai in K; Bi ∈ ¬(U1 ⊔ . . . ⊔ Uk), one for each individual name ai in K; CE ∈ ¬(U1 ⊔ . . . ⊔ Uk), one for each C ∈ D in K ¬(U1 ⊔. . . ⊔Uk) ⊑ Pow(¬(U1 ⊔. . . ⊔Uk)⊔Pow(¬(U1 ⊔. . . ⊔Uk)))

Set-theoretic translation of ALCΩ in the set theory Ω

LCΩ has the same expressive power as ALCΩ: universal and existential restrictions of the language ALCΩ, as well as all assertions, can be encoded into LCΩ. The encoding, together with the set-theoretic translation of LCΩ given in the previous section, determines a set-theoretic translation for ALCΩ, ⊤∗ = x ⊥∗ = ∅ A∗

i = xi

(¬C)∗ = x\C∗ (C ⊓ D)∗ = C∗ ∩ D∗ (C ⊔ D)∗ = C∗ ∪ D∗ (∀Ri.C)∗ = Pow(((x ∪ y1 ∪ . . . ∪ yk)\yi) ∪ Pow(C∗)) Pow(C)∗ = Pow((y1 ∪ . . . ∪ yk ∪ C∗) C1 ⊑ C2 is translated: CS

1 ∩(x\(y1 ∪ . . . ∪ yk)) ⊆ CS 2

C1 ∈ C2 is translated: CS

1 ∈ CS 2 ∩(x\(y1 ∪ . . . ∪ yk))

Discussion and future work

◮ The complementary problem to subsumption corresponds

to the satisfiability of a formula in the existential fragment

• f Ω.

◮ The decidability of subsumption ALCΩ comes from the

translation into ALCOI.

◮ The problem of deciding the satisfiability of an existential

formula of a set theory with power-set is decidable under the extensionality and well-foundedness assumptions [Cantone et al, 1985]

◮ Possibility of defining, through a set-theoretic translation as

the one above, a variant of ALCΩ with well-founded sets.

◮ is there a translation of ALCΩ with extensionality and

well-foundedness in DLs? Which complexity?

◮ Can the set-theoretic translation of ALCΩ be extended to

expressive DLs? As the finite model property does not hold for them, alternative proof techniques will be needed.

Related work

◮ Motik [ISWC, 2005] proved that metamodelling in

ALC-Full is undecidable due to free mixing of logical and metalogical symbols. ⇒ two decidable semantics, a contextual π semantics and a Hilog ν-semantics.

◮ [DeGiacomo et al.,11] Hi(SHIQ) and [Homola et al., 14]

T H(SROIQ) employ an Hilog-style semantics:

◮ [Motik,05] and [DeGiacomo et al.,11]: untyped higher-order

languages (a concept can be an instance of itself); polynomial translation of Hi(SHIQ) into SHIQ

◮ [Homola et al., 14]: typed higher-order extension of

SROIQ allowing for a hierarchy of concepts; translation with axioms A′ ≡ ∃instanceOf.{cA′}, for each atomic concept A′

Related work (contd.)

◮ Kubincova et al. (2015) propose a Hylog-style semantics

by dropping the ordering requirement in [Homola et al., 14] and using the instanceOf role in axioms as any other role.

◮ Pan and Horrocks (2005) and Motz et al. (2015) define

extensions of OWL DL and of SHIQ (respectively), based

• n semantics interpreting concepts as well-founded sets.

◮ Gu (2016) introduces the language Hi(Horn-SROIQ), an

extension of Horn-SROIQ which allows classes and roles to be used as individuals based on the ν-semantics. Reduction to Horn-SROIQ.

Thank you!!!