A Decidable Very Expressive n-ary Description Logic for Database Applications (Extended Abstract)

A Decidable Very Expressive n-ary Description Logic for Database Applications (extended abstract)

Alessandro Artale, Enrico Franconi, Rafael Pe˜naloza, Francesco Sportelli KRDB Research Centre, Free University of Bozen-Bolzano, Italy {artale,franconi,penaloza,sportelli}@inf.unibz.it

Abstract. We introduceDLR^{`</sup>, an extension of then-ary propositionally closed description logicDLRto deal with attribute-labelled tuples (generalising the po- sitional notation), projections of relations, and global and local objectification of relations, able to express inclusion, functional, key, and external uniqueness de- pendencies. The logic is equipped with both TBox and ABox axioms forming a DLR<sup>`}knowledge base (KB). We show how a simple syntactic restriction on the appearance of projections sharing common attributes in the KB makes reason- ing in the language decidable with the same computational complexity asDLR.

The obtainedDLR^˘n-ary description logic is able to encode more thoroughly conceptual data models such as EER, UML, and ORM.

1 Introduction

We introduce the new description logic (DL)DLR^`extending then-ary DLDLR[Cal- vaneseet al., 2008], in order to capture more database oriented constraints. WhileDLR is an expressive logic, it lacks a number of expressive means that can be added without increasing the complexity of reasoning—when used in a carefully controlled way. The added expressivity is motivated by the increasingly use of DLs as an abstract conceptual layer over relational databases, both to reason over such conceptual models during the database design phase, and to answer ontology-mediated queries over databases.

ADLRTBox can express axioms involving (i) propositional combinations of con- cepts and (compatible)n-ary relations, (ii) concepts as unary projections ofn-ary re- lations, and (iii) relations with a component selected to be of a certain type. For ex- ample, if Pilot and RacingCar are concepts and DrivesCar, DrivesMotobike, DrivesVehicleare binary relations, we can write the following axioms:

PilotĎDr1sσ2:RacingCarDrivesCar

DrivesCar\DrivesMotobikeĎDrivesVehicle. DLR^`extendsDLRin the following way.

– WhileDLRinstances ofn-ary relations aren-tuples of objects—whose compo- nents are identified by their position in the tuple—instances of relations inDLR^` areattribute-labelled tuples, i.e., tuples where each component is identified by an attribute and not by its position in the tuple (see, e.g., [Kanellakis, 1990]). For ex- ample, the relationEmployeemay have the signature:

Employeepfirstname,lastname,dept,deptAddrq, and an instance ofEmployeecould be the tuple:

xfirstname:John,lastname:Doe,dept:Purchase,deptAddr:Londony. – Attributes can berenamed, also with the goal to recover the positional perspectives

on relations:firstname,lastname,dept,deptAddr Õ 1,2,3,4.

– Relation projectionsallow to form new relations by projecting a given relation on some of its attributes. For example, ifPersonis a relation with signaturePerson pname,surnameq, it could be related toEmployeeas follows:

Drfirstname,lastnamesEmployeeĎPerson firstname,lastname Õ name,surname.

– The objectificationof a relation (also known as reification) is a concept whose instances are unique identifiers of the tuples instantiating the relation. Those iden- tifiers could be unique only within an objectified relation (local objectification), or they could be uniquely identifying tuples independently on the relation they are instance of (global objectification). For example, the conceptEmployeeCcould be the global objectification of the relationEmployee, assuming that there is a global one-to-one correspondence between pairs of values of the attributesfirstname, lastnameandEmployeeCinstances:

EmployeeCӁ

Drfirstname,lastnamesEmployee.

As an example of local objectification, let us consider the two (ternary) relations OwnsCarpname,surname,carq andDrivesCarpname,surname,carq, and as- sume that anybody driving a car also owns it; that is,DrivesCarĎOwnsCar. The locally objectified events of driving a car and owning a car, defined by the ax- iomsCarDrivingEvent”Ä

DrivesCarandCarOwningEventӀ

OwnsCar, model the fact that a driving event by a person of a car is not necessarily the own- ing event by the same person and the same car. Indeed, they should be disjoint:

CarDrivingEvent[CarOwningEventĎKshould hold.

It turns out thatDLR^{`</sup> is an expressive description logic able to assert relevant constraints in the context of relational databases. In Section 3 we considerinclusion dependencies, functional and key dependencies, external uniqueness and identifica- tionaxioms. For example,DLR<sup>`}can express the fact that the attributesfirstname, lastnameplay the role of a multi-attribute key for the relationEmployee:

Drfirstname,lastnamesEmployeeĎD^ď1rfirstname,lastnamesEmployee, and that the attributedeptAddrfunctionally depends on the attributedeptwithin the relationEmployee:

DrdeptsEmployeeĎD^ď1rdepts pDrdept,deptAddrsEmployeeq .

WhileDLR^{`</sup> turns out to be undecidable, in this paper we show how a simple syntactic condition on the appearance of projections sharing common attributes in the knowledge base makes the language decidable. The result of this restriction is a new language calledDLR<sup>˘</sup>. We prove thatDLR<sup>˘</sup>, while preserving most of theDLR<sup>`} expressivity, has a reasoning problem whose complexity does not increase w.r.t. the computational complexity of the basicDLRlanguage. We also present in Section 5 the implementation of an API for the reasoning services inDLR^˘.

2 The Description Logic DLR

ADLR^`signatureis a tupleL“ pC,R,O,U, τqwhereC,R,OandU are finite, mu- tually disjoint sets ofconcept names,relation names,individual names, andattributes, respectively, andτis arelation signaturefunction, associating a set of attributes to each relation nameτpRNq “ tU1, . . . , Unu ĎU withně2.

The syntax of conceptsC, relationsR, formulasϕ, and attribute renaming axiomsϑ is given in Figure 1, whereCNPC,RNPR,UPU,oPO,qis a positive integer and2ď kăARITYpRq. Thearityof a relationRis the number of the attributes in its signature;

i.e.,ARITYpRq “ |τpRq|, where we extend the signature functionτto arbitrary relations

C Ñ J | K |CN | C |C1[C2 |C1\C2 | D^ijqrUisR | Å R | Ä

RN R Ñ RN | R1zR2 | R1[R2| R1\R2|σU_i:CR | D^ijqrU1, . . . , UksR

ϕ Ñ C1ĎC2 |R1ĎR2|CNpoq |RNpU1:o1, . . . , Un:onq |o1“o2|o1‰o2

ϑ Ñ U1ÕU2

Fig. 1.Syntax ofDLR^`.

τpR1zR2q “ τpR1q ifτpR1q “τpR2q τpR1[R2q “ τpR1q ifτpR1q “τpR2q τpR1\R2q “ τpR1q ifτpR1q “τpR2q

τpσUi:CRq “ τpRq ifUiPτpRq

τpD^ijqrU1, . . . , UksRq “ tU1, . . . , Uku iftU1, . . . , Uku ĂτpRq τpRq “ H otherwise

Fig. 2.The signature ofDLR^`relations.

as specified in Figure 2. Notice that, while global objectification (Å

R) can be applied to arbitrary relations, local ones (Ä

RN) can be applied just to relation names. We use the shortcutDrU1, . . . , UksRforD^ě1rU1, . . . , UksRforkě1.

ADLR^`TBoxT is a finite set ofconcept inclusionaxioms of the formC1ĎC2

andrelation inclusionaxioms of the formR₁ĎR₂. We useX₁”X₂as a shortcut for X₁ĎX₂andX₂ĎX₁. ADLR^{`</sup> ABoxAis a finite set ofconcept instanceaxioms of the formCNpoq,relation instanceaxioms of the formRNpU1:o1, . . . , Un:onq, and same/distinct individualaxioms of the formo1“o2ando1‰o2, withoiPO. Restricting ABox axioms to concept and relation names only does not affect the expressivity of DLR<sup>`}due to the availability of TBox axioms.

A set of renaming axioms forms arenaming schema, inducing an equivalence re- lationpÕ,Uqover the attributesU, providing a partition ofUinto equivalence classes each one representing alternative ways to name attributes. We writerUs<to denote the equivalence class of the attributeU w.r.t. the equivalence relationpÕ,Uq. We allow onlywell foundedrenaming schemas, i.e., there is no equivalence class containing two attributes from the same relation signature. We use the shortcutU1. . . UnÕU₁¹. . . U_n¹ to group many renaming axioms with the meaning thatUiÕU_i¹for alli“1, . . . , n.

The renaming schema reconciles the named attribute and the positional perspectives on relations. They are crucial when expressing both inclusion axioms and set operators ([, \, z) between relations, which make sense only overunion compatiblerelations.

Two relationsR₁, R₂ are union compatible if their signatures are equal up to the at- tribute renaming induced by the renaming schema <; that is,τpR1q “ tU1, . . . , Unu andτpR2q “ tV1, . . . , Vnuhave the same aritynandrUis<“ rVis<for each1ďiďn.

Notice that through the renaming schema relations can use just local attribute names that can then be renamed when composing relations. Also note that it is obviously pos- sible for the same attribute to appear in the signature of different relations.

ADLR^` knowledge base (KB)KB“ pT,A,<qis composed by a TBoxT, an ABoxA, and a renaming schema<.

Example 1. Consider the relation namesR1, R2 whereτpR1q “ tW1, W2, W3, W4u, τpR2q “ tV1, V2, V3, V4, V5u, and the renaming axiomW1W2W3ÕV3V4V5. The TBox Texaconsists of the axioms:

DrW₁, W₂sR₁ĎD^ď1rW₁, W₂sR₁ (1) DrV3, V4sR2ĎD^ď1rV3, V4spDrV3, V4, V5sR2q (2)

J^I“∆ K^I“ H p Cq^I“ J^IzC^I pC1[C2q^I“C1^IXC2^I

pC1\C2q^I“C1^IYC2^I

pD^ijqrUisRq^I“ tdP∆| ˇ

ˇttPR^I|trρpUiqs “duˇ ˇijqu pÅ

Rq^I“ tdP∆|d“ıptq ^tPR^Iu pÄ

RNq^I“ tdP∆|d“`RNptq ^tPRN^Iu pR1zR2q^I“R^I₁zR^I₂

pR1[R2q^I“R^I₁XR₂^I

pR1\R2q^I“ ttPR^I₁YR^I₂|ρpτpR1qq “ρpτpR2qqu pσU_i:CRq^I“ ttPR^I|trρpUiqs PC^Iu

pD^ijqrU1, . . . , UksRq^I“ txρpU1q:d1, . . . , ρpUkq:dky PT∆ptρpU1q, . . . , ρpUkquq | 1ďˇ

ˇttPR^I|trρpU1qs “d1, . . . , trρpUkqs “dkuˇ ˇijqu Fig. 3.Semantics ofDLR^`expressions.

DrW1, W2, W3sR1ĎDrV3, V4, V5sR2. (3) Intuitively, the axiom (1) expresses that W1, W2 form a multi-attribute key for R1; (2) introduces a functional dependency in the relation R₂ where the attribute V₅ is functionally dependent from attributesV₃, V₄, and (3) states an inclusion between two projections of the relation namesR1, R2based on the renaming schema axiom.

The semantics ofDLR^`is based onlabelled tuplesover a domain∆: aU-labelled tuple over∆(ortuplefor short) is a functiont:UÑ∆. ForUPU, we writetrUsto refer to the domain elementdP∆labelled byU, if the functiontis defined forU— that is, if the attributeU is a label of the tuplet. Givend₁, . . . , d_nP∆, the expression xU1:d1, . . . , Un:dny stands for the tuplet such that trUis “di, for1ď1ďn. The projectionof the tupletover the attributesU1, . . . , Uk (i.e., the functiontrestricted to be undefined for the labels not inU1, . . . , Uk) is denoted bytrU1, . . . , Uks. The relation signature functionτcan be applied also to labelled tuples to obtain the set of labels on which the tuple is defined.T∆pUqdenotes the set of allU-labelled tuples over∆.

ADLR<sup>`</sup> interpretation is a tuple I“ p∆,¨<sup>I</sup>, ρ, ı, Lqconsisting of a nonempty domain∆, aninterpretation function¨<sup>I</sup>, arenaming functionρ, aglobal objectification functionı, and a familyLcontaining onelocal objectification function`RNi for each named relationRNiPR.

The renaming function ρ is a total function ρ:UÑU representing a canonical renaming for all attributes. We use ρptU1, . . . , Ukuq to denote tρpU1q, . . . , ρpUkqu.

The global objectification function is an injective function, ı:T∆pUq Ñ∆, associat- ing auniqueglobal identifier to each tuple. The local objectification functions,`RN_i: T_∆pUq Ñ∆, are associated to each relation name in the signature, and as the global objectification function they are injective: they associate an identifier—which is guar- anteed to be unique only within the interpretation of a relation name—to each tuple.

The interpretation function¨^I assigns a domain element to each individualo^IP∆, a set of domain elements to each concept nameCN^IĎ∆, and a set ofU-labelled tuples over∆to each relation name conforming with its signature and to the renaming func- tion RN^IĎT_∆ptρpUq |U PτpRNquq. Note that the unique name assumption is not enforced. The interpretation function¨^Iis unambiguously extended over concept and relation expressions as specified in Figure 3.

The interpretationI satisfies the concept inclusion axiomC₁ĎC₂ ifC₁^IĎC₂^I, and the relation inclusion axiom R1ĎR2 if R^I₁ ĎR^I₂. It satisfies the concept in- stance axiomCNpoqifo^IPCN^I, the relation instance axiomRNpU1:o1, . . . , Un:onq ifxρpU₁q:o^I₁, . . . , ρpU_nq: o^I_ny PRN^I, and the axiomso₁“o₂ando₁‰o₂ifo^I₁“o^I₂, ando^I₁‰o^I₂, respectively.I satisfies a renaming schema<if for everyU, V PU, (i) ρpUq P rUs<, and (ii) ifVP rUs<, thenρpUq “ρpVq.Iis amodelof the KBpT,A,<q if it satisfies the axioms in the TBoxT, in the ABoxA, and the renaming schema<.

KB satisfiabilityrefers to the problem of deciding the existence of a model of a given KB;concept satisfiability(resp.relation satisfiability) is the problem of deciding whether there is a model of the KB with a non-empty interpretation of a given concept (resp. relation). A KBentails(orlogically implies) an axiom if all models of the KB are also models of the axiom. For instance, the TBox in Example 1 entails that axiom (2) is redundant sinceV₃, V₄are a key forR₂:Texa|ù DrV₃, V₄sR₂ĎD^ď1rV₃, V₄sR₂.

3 Expressiveness of DLR

DLR^`is an expressive description logic able to assert relevant constraints in the con- text of relational databases, such asinclusion dependencies(inclusion axioms among ar- bitrary projections of relations),equijoins,functional dependencyaxioms,keyaxioms, external uniquenessaxioms,identificationaxioms, andpath functional dependencies.

Anequijoinamong two relations with disjoint signatures is the set of all combi- nations of tuples in the relations that are equal on their selected attribute names. Let R1, R2 be relations with τpR1q “ tU, U1, . . . , Un₁u and τpR2q “ tV, V1, . . . , Vn₂u;

their equijoin over U and V is the relationR“R1

’

U“V

R2 that uses the signature τpRq “τpR₁q YτpR₂qztVu, and is expressed by theDLR^`axioms:

DrU, U₁, . . . , U_n1sR”σ_{U:pDrUsR1[DrVsR2q}R₁ DrV, V₁, . . . , V_n2sR”σ_{V:pDrUsR1[DrVsR2q}R₂ UÕV .

Afunctional dependencyaxiom pR:U1. . . UjÑUq(also calledinternal uniqueness axiom [Halpin and Morgan, 2008]) states that the values of the attributes U₁. . . U_j uniquely determine the value of the attribute U in the relation R. Formally, the in- terpretation I satisfies this functional dependency axiom if, for all tuples s, tPR^I, srU₁s “trU₁s, . . . , srU_js “trU_jsimplysrUs “trUs. Functional dependencies can be expressed inDLR^`, assuming thattU1, . . . , Uj, Uu ĎτpRq, with the axiom:

DrU1, . . . , UjsRĎD^ď1rU1, . . . , UjspDrU1, . . . , Uj, UsRq.

A special case of functional dependencies arekeyaxiomspR:U1. . . UjÑRq, which state that the values of the key attributesU1. . . Uj of a relation R uniquely identify tuples inR. A key axiom can be expressed inDLR^`, assuming thattU1. . . UjuĎτpRq, with the axiom:

DrU1, . . . , UjsRĎD^ď1rU1, . . . , UjsR .

Theexternal uniquenessaxiomprU¹sR1Ó. . .Ó rU^hsRhqstates that the joinR of the relations R₁, . . . , R_h via the attributesU¹, . . . , U^h has the joined attribute function- ally dependent on all the others [Halpin and Morgan, 2008]. This can be expressed in DLR^`with the axioms:

R”R1

’

U¹“U²

¨ ¨ ¨

’

U^h´1“U^h

Rh

R:U₁¹, . . . , U_n¹

1, . . . , U₁^h, . . . , U_n^h

hÑU¹

whereτpR_iq “ tUⁱ, U₁ⁱ, . . . , U_nⁱ

iu,1ďiďh, andRa new relation name with signature τpRq “ tU¹, U₁¹, . . . , U_n¹₁, . . . , U₁^h, . . . , U_n^h_hu.

Identificationaxioms as defined inDLR_ifd [Calvaneseet al., 2001] (an extension ofDLRwith functional dependencies and identification axioms) are a variant of ex- ternal uniqueness axioms, constraining only the elements of a conceptC; they can be expressed inDLR^`with the axiom:

rU¹sσ_U1:CR₁Ó. . .Ó rU^hsσ_Uh:CR_h.

Path functional dependencies—as defined in the DL familyCF D[Toman and Weddell, 2009]—can be expressed inDLR^{`</sup>as identification axioms involving joined sequences of functional binary relations.DLR<sup>`}also capturestree-based identification constraint (tid)introduced in [Calvaneseet al., 2014] to capture fds inDL-LiteRDFS,tid.

The rich set of constructors inDLR^` allows us to extend the known mappings in description logics of popular conceptual data models. The EER mapping as introduced in [Artale et al., 2007] can be extended to deal with multi-attribute keys (by using identification axioms) and named roles in relations; the ORM mapping as introduced in [Franconiet al., 2012; Sportelli and Franconi, 2016] can be extended to deal with arbitrary subset and exclusive relation constructs (by using inclusions among global objectifications of projections of relations), arbitrary internal and external uniqueness constraints, arbitrary frequency constraints (by using projections), local objectification, named roles in relations, and fact type readings (by using renaming axioms); the UML mapping as introduced in [Berardiet al., 2005] can be fixed to deal properly with asso- ciation classes (by using local objectification) and named roles in associations.

4 The DLR

fragment of DLR

Since aDLR^{`</sup>KB can express inclusions and functional dependencies, the reasoning is undecidable [Chandra and Vardi, 1985]. In this section we presentDLR<sup>˘</sup>, a decidable syntactic fragment ofDLR<sup>`}limiting the co-occurence of relation projections in a KB.

Given aDLR^` knowledge basepT,A,<q, theprojection signatureis the setT containing the signaturesτpRNqof the relationsRNPR, the singleton sets associated with each attribute name UPU, and the relation signatures that appear explicitly in projection constructs in some axiom fromT, together with their implicit occurrences due to the renaming schema. Formally,T is the smallest set where (i)τpRNqPT for all RNPR; (ii)tUu PT for allUPU; and (iii)tU1, . . . , Uku PT for allD^ijqrV1, . . . , VksR appearing as sub-formulas inT andtUi, Viu Ď rUis< for1ďiďk.

Theprojection signature graphis the directed acyclic graphpĄ,Tqwhose sinks are the attribute singletonstUu. Given a set of attributesτ“ tU1, . . . , Uku ĎU, thepro- jection signature graph dominated byτ, denoted as Tτ, is the sub-graph ofpĄ,Tq containing all the nodes reachable from τ. Given two sets of attributes τ₁, τ₂ĎU,

PATH_Tpτ₁, τ₂q denotes the set of paths in pĄ,Tq between τ₁ and τ₂. Notice that

PATH_Tpτ1, τ2q “ Hboth when a path does not exist and whenτ1Ďτ2. The notation

CHILD_Tpτ1, τ2qmeans thatτ2is a child ofτ1inpĄ,Tq. We now introduceDLR^˘. Definition 1. ADLR^˘knowledge baseis aDLR^`knowledge base that satisfies the following syntactic conditions:

1. the projection signature graphpĄ,Tqis a multitree: i.e., for every nodeτPT, the graphTτis a tree; and

2. for every projection constructD^ijqrU1, . . . , UksRappearing inT, ifqą1then the length of the pathPATH_TpτpRq,tU1, . . . , Ukuqis 1.

tW1, W2, W3, W4u tV1, V2, V3, V4, V5u

tW4u

"

W1, W2, W3

V3, V4, V5

*

tV1u tV2u

"

W1, W2

V3, V4

* "

W3

V5

*

"

W1

V3

* "

W2

V4

*

Fig. 4.The projection signature graph of Example 1.

The conditions inDLR^˘ restrictDLR^`in the way that multiple projections of rela- tions may appear in a KB: intuitively, there can not be projections of a relation sharing a common attribute. Moreover, observe that inDLR^˘ PATH_T is necessarily functional, due to the multitree restriction. Figure 4 shows that the projection signature graph of the knowledge base from Example 1 is indeed a multitree. Note that in the figure we have collapsed equivalent attributes in a unique equivalence class, according to the renaming schema. Furthermore, since all its projection constructs haveq“1, this knowledge base belongs toDLR^˘.

DLR is included in DLR^˘, since the projection signature graph of any DLR knowledge base is always a degenerate multitree with maximum depth equal to 1.

Not all the database constraints as introduced in Section 3 can be directly expressed in DLR^˘. While functional dependency and key axioms can be expressed directly inDLR^˘, equijoins, external uniqueness axioms, and identification axioms introduce projections of a relation which share common attributes, thus violating the multitree restriction. However, inDLR^˘it is still possible to reason over both external unique- ness and identification axioms by encoding them into a set of saturated ABoxes (as originally proposed in [Calvaneseet al., 2001]) and check whether there is a saturation that satisfies the constraints. Therefore, we can conclude thatDLR_ifd extended with unary functional dependencies is included inDLR^˘, provided that projections of rela- tions in the knowledge base form a multitree projection signature graph. Since (unary) functional dependencies are expressed via the inclusions of projections of relations, by constraining the projection signature graph to be a multitree, the possibility to build combinations of functional dependencies as the ones in [Calvaneseet al., 2001] leading to undecidability is ruled out. Concerning the ability ofDLR^˘to capture conceptual data models, only the mapping of ORM schemas is affected by theDLR^˘restrictions:

the projections involved in the ORM schema should satisfy theDLR^˘ multitree re- striction.

The main result of this work is that reasoning inDLR^˘is an EXPTIME-complete problem. The lower bound comes by observing thatDLRis a sublanguage ofDLR^˘, the upper bound is proved by providing a mapping fromDLR^˘KBs toALCQI KBs.

5 Implementation

We have implemented the framework discussed in this paper. DLRtoOWL is a Java library fully implementingDLR^˘ reasoning services. The library is based on the tool ANTLR4 to parse serialised input, and on OWLAPI4 for the OWL2 encoding. The

system includes JFact, the Java version of the popular Fact++ reasoner. DLRtoOWL provides a JavaDLRAPI package to allow developers to create, manipulate, serialise, and reason withDLR^˘knowledge bases in their Java-based application, extending in a compatible way the standard OWL API with the DLR^˘ TELL and ASK services.

During the development of this new library we strongly focused on performance. Since the OWL encoding is only possible if we have already built theALCQI projection signature multitree, in principle the program should perform two parsing rounds: one to create the multitree and the other one to generate the OWL mapping. We faced this issue using dynamic programming: during the first (and only) parsing round we store in a data structure each axiom that we want to translate in OWL and, after building the multitree, by the dynamic programming technique we build on-the-fly a Java class which generates the required axioms.

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