Extending DLR with Labelled Tuples, Projections, Functional Dependencies and Objectification

Extending DLR with Labelled Tuples, Projections, Functional Dependencies and Objectification

Alessandro Artale and Enrico Franconi

KRDB Research Centre, Free University of Bozen-Bolzano, Italy {artale,franconi}@inf.unibz.it

Abstract. We introduce an extension of the n-ary description logicDLRto deal with attribute-labelled tuples (generalising the positional notation), with arbitrary projections of relations (inclusion dependencies), generic functional dependen- cies and with global and local objectification (reifying relations or their projec- tions). We show how a simple syntactic condition on the appearance of projec- tions and functional dependencies in a knowledge base makes the language de- cidable without increasing the computational complexity of the basicDLRlan- guage.

1 Introduction

We introduce in this paper the language DLR^` which extends then-ary description logicsDLR[Calvaneseet al., 1998; Baaderet al., 2003] andDLR_ifd [Calvaneseet al., 2001] as follows:

– the semantics is based on attribute-labelled tuples: an element of a tuple is identi- fied by an attribute and not by its position in the tuple, e.g., the relationPerson has attributesfirstname,lastname,age,heightwith instance:

x firstname: Enrico, lastname: Franconi, age: 53, height:

1.90y;

– renaming of attributes is possible, e.g., to recover the positional semantics:

firstname,lastname,age,heightí1,2,3,4;

– it can express projections of relations, and therefore inclusion dependencies, e.g., Drfirstname,lastnamesStudentĎDrfirstname,lastnamesPerson;

– it can express multiple-attribute cardinalities, and therefore functional dependen- cies and multiple-attribute keys, e.g., the functional dependency fromfirstname, lastnametoageinPersoncan be written as:

Drfirstname,lastnamesPersonĎ

D^ď1rfirstname,lastnamespDrfirstname,lastname,agesPersonq;

– it can express global and local objectification (also known as reification): a tuple may be identified by a unique global identifier, or by an identifier which is unique only within the interpretation of a relation, e.g., to identify the name of a person we can writeNameĎÄ

Drfirstname,lastnamesPerson.

We show how a simple syntactic condition on the appearance of projections in the knowledge base makes the language decidable without increasing the computational

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

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

ϕ Ñ C1ĎC2 | R1ĎR2

ϑ Ñ U1 íU2

Fig. 1.Syntax ofDLR^`.

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

τpσU_i:CRq “ τpRq ifUiPτpRq

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

Fig. 2.The signature ofDLR^`relations.

complexity of the basic DLR language. We call DLR^˘ this fragment ofDLR^`. DLR^˘ is able to correctly express the UML fragment as introduced in [Berardi et al., 2005; Artaleet al., 2007] and the ORM fragment as introduced in [Franconi and Mosca, 2013].

2 Syntax of the Description Logic DLR

We first define the syntax of the language DLR^{`</sup>. AsignatureinDLR<sup>`} is a triple L“ pC,R,U, τqconsisting of a finite setCofconceptnames (denoted byCN), a finite setRofrelationnames (denoted byRN) disjoint fromC, and a finite setUofattributes (denoted byU), and arelation signaturefunctionτassociating a set of attributes to each relation name,τpRNq “ tU₁, . . . , U_nu ĎU withně2.

The syntax of conceptsC, relationsR, formulasϕ, and attribute renaming axiomsϑ is defined in Figure 1, whereqis a positive integer and2ďkăARITYpRq. We extend the signature functionτ to arbitrary relations as specified in Figure 2. We define the

ARITYof a relationRas the number of the attributes in its signature, namely|τpRq|.

ADLR^`TBoxT is a finite set of formulas, i.e.,concept inclusionaxioms of the formC1ĎC2andrelation inclusionaxioms of the formR1ĎR2.

A renaming schema induces an equivalence relationpí,Uqover the attributesU, pro- viding a partition ofU into equivalence classes each one representing the alternative ways to name attributes. We write rUs_< to denote the equivalence class of the at- tributeU w.r.t. the equivalence relationpí,Uq. We allow onlywell foundedrenaming schemas, namely schemas such that each equivalence classrUs<in the induced equiv- alence relation never contains two attributes from the same relation signature. In the following we use the shortcutU1. . . UníU₁¹. . . U_n¹ to group many renaming axioms, with the obvious meaning thatUiíU_i¹, for alli“1, . . . , n.

ADLR^`knowledge baseKB“ pT,<qis composed by a TBoxT and a renaming schema<.

The renaming schema reconciles the attribute and the positional perspectives on re- lations (see also the similar perspectives in relational databases [Abiteboulet al., 1995]).

They are crucial when expressing both inclusion axioms and operators ([, \, z) between relations, which make sense only over union compatible relations. Two re- lations R1, R2 are union compatible if their signatures are equal up to the attribute renaming induced by the renaming schema<, namely, τpR1q “ tU1, . . . , Unuand τpR2q “ tV1, . . . , Vnuhave the same aritynandrUis_< “ rVis_<for each1 ďi ďn.

Notice that, thanks to 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.

To show the expressive power of the language, let us consider the following example with tree relation namesR1, R2andR3with the following signature:

τpR1q “ tU1, U2, U3, U4, U5u τpR2q “ tV1, V2, V3, V4, V5u τpR₃q “ tW₁, W₂, W₃, W₄u

To state thattU1, U2uis themulti-attribute keyofR1we add the axiom:

DrU1, U2sR1ĎD^ď1rU1, U2sR1

whereDrU₁, . . . , U_ksRstands forD^ě1rU₁, . . . , U_ksR. To express that there is afunc- tional dependencyfrom the attributestV₃, V₄uto the attributetV₅uofR₂we add the axiom:

DrV3, V4sR2ĎD^ď1rV3, V4spDrV3, V4, V5sR2q (1) The following axioms express thatR2is a sub-relation ofR1and that a projection of R3is a sub-relation of a projection ofR1, together with the corresponding axioms for the renaming schema to explicitly specify the correspondences between the attributes of the two inclusion dependencies:

R₂ĎR₁

DrW1, W2, W3sR3ĎDrU3, U4, U5sR1

V1V2V3V4V5íU1U2U3U4U5

W₁W₂W₃íU₃U₄U₅

3 Semantics

The semantics makes use of the notion oflabelled tuplesover a domain set∆: aU- labelled tuple over ∆ is a function t: U Ñ ∆. ForU P U, we write trUs to refer to the domain element dP∆labelled by U, if the function tis defined forU – that is, if the attributeU is a label of the tuplet. Givend1, . . . , dn P ∆, the expression

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^I₁ XR^I₂

pR1\R2q^I “ ttPR^I₁ YR^I₂ |ρpτpR1qq “ρpτpR2qqu pσU_i:CRq^I “ ttPR^I |trρpUiqs PC^Iu

pD^ijqrU1, . . . , UksRq^I “ txρpU1q:d1, . . . , ρpUkq:dky PT∆ptρpU1q, . . . , ρpUkquq | ˇˇttPR^I |trρpU1qs “d1, . . . , trρpUkqs “dkuˇ

ˇijqu

Fig. 3.Semantics ofDLR^`expressions.

xU1:d1, . . . , Un:dnystands for theU-labelled tupletover∆(tuple, for short) such thattrUis “di, for1ď1ďn. We writetrU1, . . . , Uksto denote theprojectionof the tupletover the attributesU1, . . . , Uk, namely the functiontrestricted to be undefined for the labels not inU1, . . . , Uk. The set of allU-labelled tuples over∆is denoted by T∆pUq.

ADLR<sup>`</sup>interpretation,I “ p∆,¨<sup>I</sup>, ρ, ı, `RN₁, `RN<sub>2</sub>, . . .q, consists of a nonempty domain∆, aninterpretation function¨<sup>I</sup>, arenaming functionρ, aglobal objectification function ı, and a family oflocal objectification functions `RNi, one for each named relationRNiPR.

The renaming function ρ for attributes is a total function ρ:U ÑU represent- ing a canonical renaming for all attributes. We consider, as a shortcut, the notation ρptU₁, . . . , U_kuq “ tρpU₁q, . . . , ρpU_kqu.

The global objectification function is an injective function,ı:T_∆pUq Ñ∆, associating auniqueglobal identifier to each possible tuple.

The local objectification functions,`RN_i:T∆pUq Ñ∆, are distinct for each relation name in the signature, and as the global objectification function they are injective: they associate an identifier – which is unique only within the interpretation of a relation name – to each possible tuple.

The interpretation function¨^I assigns a set of domain elements to each concept name, CN^I Ď ∆, and a set of U-labelled tuples over ∆to each relation name conforming with its signature and the renaming function:

RN^I ĎT∆ptρpUq |U PτpRNquq.

The interpretation function¨^Iis unambiguously extended over concept and relation expressions as specified in the inductive definition of Fig. 3.

An interpretationI satisfies a concept inclusion axiomC₁ ĎC₂ ifC₁^I Ď C₂^I, it satisfies a relation inclusion axiomR₁ ĎR₂ifR^I₁ ĎR^I₂, and it satisfies a renaming schema<if the renaming functionρrenames the attributes in a consistent way with respect to<, namely if

@U.ρpUq P rUs<^ @V P rUs_<.ρpUq “ρpVq.

An interpretation is amodelfor a knowledge basepT,<qif it satisfies all the formu- las in the TBoxT and it satisfies the renaming schema<. We defineKB satisfiabilityas the problem of deciding the existence of a model of a given knowledge base,concept satisfiability(resp.relation satisfiability) as the problem of deciding whether there is a model of the knowledge base that assigns a non-empty extension to a given concept (resp. relation), andentailmentas the problem to check whether a given knowledge base logically implies a formula, that is, whenever all the models of the knowledge base are also models of the formula.

For example, from the knowledge baseKBintroduced in the previous Section the fol- lowing logical implication holds:

KB|ù DrV1, V2sR2ĎD^ď1rV1, V2sR2

i.e., the attributesV₁, V₂are a key for the relationR₂.

Proposition 1. The problems of KB satisfiability, concept and relation satisfiability, and entailment are mutually reducible inDLR^`.

DLR^`can express complex inclusion and functional dependencies, for which it is well known that reasoning is undecidable [Mitchell, 1983; Chandra and Vardi, 1985].

DLR^` also includes theDLRextensionDLR_ifd together with unary functional de- pendencies [Calvaneseet al., 2001], which also has been proved to be undecidable.

4 The DLR

fragment of DLR

Given aDLR^`knowledge basepT,<q, we define theprojection signatureas the set T including the signaturesτpRNqof the relationsRN PR, the singletons associated with each attribute nameU PU, and the relation signatures as they appear explicitly in projection constructs in the relation inclusion axioms of the knowledge base, together with their implicit occurrences due to the renaming schema:

1. τpRNq PT if RN PR;

2. tUu PT if U PU;

3. tU₁, . . . , U_ku PT if D^ijqrV₁, . . . , V_ksRPT and tU_i, V_iu Ď rU_is< for1ďiďk.

We callprojection signature graphthe directed acyclic graphpĄ,Tqwith the at- tribute singletonstUubeing the sinks. TheDLR^˘fragment ofDLR^`allows only for knowledge bases with a projection signature graph being amultitree, namely the set of nodes reachable from any node of the projection signature graph should form a tree.

!U1 V₁,^U_V²

2,^UV3³ W₁,^UV44

W₂,^UV5⁵ W₃

) !_U3

V₃ W₁,^UV44

W₂,^UV55 W₃,W₄

)

U₁ V₁,^U_V2²(

!_U5

V₅ W₃

)

!_U3

V3 W₁,^UV44

W₂,^UV5⁵ W₃

)

tW₄u

U1 V1

( _U2

V2

( !_U3

V₃ W₁,^UV4⁴

W₂

)

!U₃ V3 W1

) !U₄

V4 W2

)

Fig. 4.The projection signature graph of the example.

Given a relation nameRN, the subgraph of the projection signature graph dominated byRNis a tree where the leaves are all the attributes inτpRNqand the root isτpRNq.

We callTtU₁,...,U_kuthe tree formed by the nodes in the projection signature graph dom- inated by the set of attributestU₁, . . . , U_ku. Given two relation signatures (i.e., two sets of attributes)τ₁, τ₂ ĎU, byPATH_Tpτ₁, τ₂qwe denote the path inpĄ,Tqbetweenτ₁ andτ2, if it exists. Note thatPATH_Tpτ1, τ2q “ Hboth when a path does not exist and whenτ1 Ďτ2, andPATH_T is functional inDLR^˘due to the multitree restriction on projection signatures. The notation CHILD_Tpτ1, τ2qmeans thatτ2 is a child ofτ1 in pĄ,Tq.

In addition to the above multitree condition, theDLR^˘fragment ofDLR^`allows for knowledge bases with projection constructs D^ijqrU1, . . . , UksR (resp. D^ijqrUsR) with a cardinalityq ą 1 only if the length of the pathPATH_TptU₁, . . . , U_ku, τpRqq (resp.PATH_TptUu, τpRqq) is 1. This allows to map cardinalities inDLR^˘into cardi- nalities inALCQI.

Figure 4 shows that the projection signature graph of the knowledge base introduced in Section 2 is indeed a multitree. Note that in the figure we have collapsed equivalent attributes in a unique equivalence class, according to the renaming schema.

DLR^˘ restrictsDLR^` only in the way multiple projections of relations appear in the knowledge base. It is easy to see that DLRis included in DLR^˘, since the projection signature graph of anyDLRknowledge base has maximum depth equal to 1.DLR_ifd [Calvaneseet al., 2001] together with (unary) functional dependencies is also included inDLR^˘, with the proviso that projections of relations in the knowledge base form a multitree projection signature graph. Since (unary) functional dependencies are expressed via the inclusions of projections of relations (see, e.g., the functional dependency (1) in the previous example), 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. Also note that DLR^˘is able to correctly express the UML fragment as introduced in [Berardiet al., 2005; Artaleet al., 2007] and the ORM fragment as introduced in [Franconi and Mosca, 2013].

AR1, AR2 AR3

AtU1,U2u

R1 , AtU1,U2u R2

AtU5u R1 , AtU5u

R2 , AtU5u R3 AtU3,U4,U5u

R1 , AtU3,U4,U5u

R2 , AtU3,U4,U5u

R3 AtW4u

R3

AtU1u R1 , AtU1u

R2 AtU2u R1 , AtU2u

R2 AtU3,U4u

R1 , AtU3,U4u

R2 , AtU3,U4u R3

AtU3u R1 , AtU3u

R2 , AtU3u

R3 AtU4u

R1 , AtU4u R2 , AtU4u

R3 QtU1,U2u QtU3,U4,U5u

QtU3,U4,U5u QtW4u

QtU1u QtU2u QtU3,U4u QtU5u

QtU3u QtU4u

Fig. 5.TheALCQIsignature generated by the example.

5 Mapping DLR

to ALCQI

We show that reasoning in DLR^˘ is EXPTIME-complete by providing a mapping fromDLR^˘knowledge bases toALCQIknowledge bases; the reverse mapping from ALCQIknowledge bases toDLRknowledge bases is well known. The proof is based on the fact that reasoning withALCQIknowledge bases is EXPTIME-complete [Baader et al., 2003]. We adapt and extend the mapping presented forDLRin [Calvaneseet al., 1998].

In the following we use the shortcutpS1˝. . .˝Snq^´forS_n^´˝. . .˝S₁^´, the shortcut D^ij1S1˝. . .˝Sn.C forD^ij1S1.. . ..D^ij1Sn.C and the shortcut@S1˝. . .˝Sn.Cfor

@S1.. . ..@Sn.C. Note that these shortcuts for the role chain constructor “˝” are not correct in general, but they are correct in the context of the specificALCQIknowledge bases used in this paper.

LetKB “ pT,<qbe aDLR^˘ knowledge base. We first rewrite the knowledge base as follows: for each equivalence classrUs< a singlecanonicalrepresentative of the class is chosen, and theKB is consistently rewritten by substituting each attribute with its canonical representative. After this rewriting, the renaming schema does not play any role in the mapping.

The mapping function¨^: maps each concept nameCN in theDLR^˘ knowledge base to anALCQI concept nameCN, each relation nameRN in theDLR^˘ knowl- edge base to anALCQI concept nameARN (its global reification), and each attribute nameU in theDLR^˘knowledge base to anALCQIrole name, as detailed below.

For each relation nameRN the mapping introduces a concept nameA^l_RN and a role name QRN (to capture the local reification), and a concept nameA^τ_RNⁱ for each pro- jected signatureτiin the projection signature graph dominated byτpRNq,τiPTτpRNq

(to capture global reifications of the projections ofRN). Note thatA^τpRN_RN ^qcoincides withA_RN. Furthermore, the mapping introduces a role name Q_τi for each projected signatureτi in the projection signature,τi P T, such that there existsτj P T with

CHILD_Tpτj, τiq, i.e., we exclude the case whereτi is one of the roots of the multitree induced by the projection signature.

p Cq^: “ C^: pC1[C2q^: “ C₁^:[C₂^: pC1\C2q^: “ C₁^:\C₂^: pD^ijqrUisRq^: “ D^ijq`

PATH_TpτpRq,tUiuq^:˘´

.R^: pÅ

Rq^: “ R^: pÄ

RNq^: “ A^l_RN pR1zR2q^: “ R^:₁[ R^:₂ pR1[R2q^: “ R^:₁[R^:₂ pR1\R2q^: “ R^:₁\R^:₂

pσU_i:CRq^: “ R^:[ @PATH_TpτpRq,tUiuq^:.C^: pD^ijqrU1, . . . , UksRq^: “ D^ijq`

PATH_TpτpRq,tU1, . . . , Ukuq^:˘´

.R^:

Fig. 6.The mapping for concept and relation expressions.

The mapping¨^:applies also to a path. Letτ, τ¹ P T be two generic sets of attributes such that the functionPATH_Tpτ, τ¹q “ τ, τ1, . . . , τn, τ¹, then, a path is mapped as fol- lows:

PATH_Tpτ, τ¹q^:“Q_τ1˝. . .˝Q_τn˝Q_τ1.

Intuitively, the mapping reifies each node in the projection signature graph: the tar- getALCQI signature of the example of the previous section is partially presented in Fig. 5, together with the projection signature graph. Each node is labelled with the corresponding (global) reification concept (A^τ_R^j

i), for each relation nameRiand each projected signatureτjin the projection signature graph dominated byτpRiq, while the edges are labelled by the roles (Qτ_i) needed for the reification.

The mapping¨^:is extended to concept and relation expressions as in Figure 6, with the proviso that wheneverPATH_Tpτ1, τ2qreturns an empty path then the translation for the corresponding expression becomes the bottom concept. Note that inDLR^˘the car- dinalities on a path are restricted to the caseq“1whenever a path is of length greater than1, so we still remain within theALCQI description logic when the mapping ap- plies to cardinalities. So, if we need to express a cardinality constraintD^ijqrU_isR,] with qą1, thenU_ishould not be mentioned in any other projection of the relationRin such a way that|PATH_TpτpRq,tU_iuq| “1.

In order to explain the need for the path function in the mapping, notice that a rela- tion is reified according to the decomposition dictated by projection signature graph it dominates. Thus, to access an attributeU_jof a relationR_iit is necessary to follow the path through the projections that use that attribute. This path is a role chain from the signature of the relation (the root) to the attribute as returned by thePATH_TpτpR_iq, U_iq function. For example, considering Fig. 5, in order to access the attributeU₄of the re- lationR₃in the expressionpσ_U4:CR₃q, the pathPATH_TpτpR₃q,tU₄uq^:is equal to the role chainQ_tU3,U4,U5u˝Q_tU3,U4u˝Q_tU4u, so thatpσU₄:CR3q^: “ AR₃[@Q_tU3,U4,U5u˝ Q_tU3,U4u˝Q_tU4u.C.

Similar considerations can be done when mapping cardinalities over relation projec- tions.

The mappingγpKBqof aDLR^˘knowledge baseKBwith a signaturepC,R,U, τq is defined as the followingALCQITBox:

γpKBq “ γdsj Y ď

RNPR

γrelpRNq Y ď

RNPR

γlobjpRNq Y

ď

C₁ĎC₂PKB

C₁^:ĎC₂^: Y ď

R₁ĎR₂PKB

R^:₁ĎR₂^:

where

γdsj“ A^τ_RNⁱ

1 Ď A^τ_RN^j

2 |RN1, RN2PR, τi, τj PT,|τi| ě2,|τj| ě2, τi ‰τj

(

γrelpRNq “ ď

τ_iPTτpRNq

ď

CHILD_Tpτi,τjq

A^τ_RNⁱ ĎDQτ_j.A^τ_RN^j , D^ě2Qτ_j.JĎK(

γ_lobjpRNq “ tARN ĎDQRN.A^l_RN, D^ě2QRN.JĎK, A^l_RN ĎDQ^´_RN.ARN, D^ě2Q^´_RN.JĎKu.

Intuitively, γ_dsj ensures that relations with different signatures are disjoint, thus, e.g., enforcing the union compatibility. The axioms inγ_relintroduce classical reification axioms for each relation and its relevant projections. The axioms inγ_lobjmake sure that each local objectification differs form the global one.

Clearly, the size ofγpKBqis polynomial in the size ofKB(under the same coding of the numerical parameters), and thus we are able to state the main result of this paper (see [Artale and Franconi, 2016] for a complete set of proofs of the Theorem).

Theorem 2. A DLR^˘ knowledge baseKB is satisfiable iff theALCQI knowledge baseγpKBqis satisfiable.

As a direct consequence of the above theorem and the fact thatDLRis a sublan- guage ofDLR^˘, we have that

Corollary 3. Reasoning inDLR^˘is anEXPTIME-complete problem.

6 Acknowledgements

We thank Alessandro Mosca for working with us on all the preliminary work necessary to understand how to get these technical results.

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