Proceedings, Eleventh International Conference on Principles of Knowledge Representation and Reasoning (2008)
Formalising Temporal Constraints on Part-Whole Relations
Alessandro Artale1, Nicola Guarino2, and C. Maria Keet1 1Faculty of Computer Science, Free University of Bozen-Bolzano, Italy, {artale, keet}@inf.unibz.it 2National Research Council, Institute for Cognitive Sciences and Technologies, Trento, Italy, [email protected]
Abstract since we normally assume that each human has necessarily a specific brain, while not necessarily a specific heart (thanks Representing part-whole relations and effectively using to heart transplantation). In addition, if a human is also a them in domain ontologies and conceptual data mod- els poses multiple challenges. In this paper we face the boxer, he must necessarily have exactly his own two hands. issue of imposing temporal constraints on part-whole Standard discussions on parthood relationships distinguish relationships, introducing a way to account for essen- between essential (the brain) and mandatory (the heart) tial and immutable parts (and wholes) in addition to the parts of a whole (the human). In UML modeling, a fur- usual mandatory parts (and wholes). Our approach is ther case of parthood relationship is introduced: that of so- based on i) an explicit temporalization of the part-whole called immutable parts (Barbier et al. 2003; Guizzardi 2005; relation, which allows us to introduce a novel notion 2007). This is indeed the case of the boxer’s hands, which of status for part-whole relationships; ii) an explicit ac- must exist as long as a human is a boxer (assuming hand count of the ontological nature of the classes involved transplantation is forbidden for boxers), but are neither es- in a part-whole relationships, which distinguishes be- sential nor mandatory for the human’s life. tween rigid and anti-rigid classes. The main novelty in this paper is to resort to a temporal logic approach to capture the above mentioned notions. The formalization 1 1 proposed here is grounded on the temporal description Brain Human logic DLRUS and is based on previous successful ef- Heart 1 1 forts to formalize temporal conceptual models. 1 Introduction 0..2 Boxer Hand The proper account of parthood relations in knowledge rep- 1 2 resentation languages and formal conceptual data models has received much attention in the literature recently. For Figure 1: UML diagram with part-whole relations. instance, issues such as the transitivity of part-whole rela- tions and the specific behavior of functional parts can be Clearly, cardinality constraints alone are not enough to considered as relatively well clarified nowadays (Varzi 2006; distinguish between these interpretations: some kinds of Vieu 2006; Keet & Artale 2008). However, some subtle is- modal constraints have to be taken into account. A rep- sues concerning the various ways two classes may be re- resentation language with “full” modal operators would be lated by a formula containing modal constraints on parthood the obvious choice, but it would certainly present computa- relations are still a matter of discussion in the conceptual tional and semantic problems. For this reason, we adopt in modelling community. Moreover, a proper formalization for this paper a temporal logic approach, assuming that a tem- such modal constraints in terms of representation languages poral modality is enough to capture most practical cases, with well-understood computational properties, such as de- and reducing therefore the constraints concerning the so- scription logics, is still lacking. called “alethic modality”—pure possibility/necessity, inde- Consider, for instance, the UML class diagram shown pendently of time—to those expressible in terms of tempo- in Fig. 1, which constrains the part-whole relations in the ral modality only. In particular, we shall concentrate in this following way: every human has exactly one brain, one paper on what we may call the life cycle semantics of part- heart and at most two hands, while every boxer (a sub- whole relationships, focusing on the constraints concerning class of human) has exactly two hands. As discussed by the temporal existence of both their participants—the part Guizzardi (Guizzardi 2005; 2007), the intended interpreta- and the whole—and the part-whole relationship itself. The tion of these part-whole relationships is typically different, explicit recognition given to such distinct temporal behav- Copyright c 2008, Association for the Advancement of Artificial iors is one of the main contributions of this paper. Intelligence (www.aaai.org). All rights reserved. Currently, no commonly used conceptual data mod-
673 elling language, such as UML and EER (Extended Entity- relationship between the class that describes the whole and Relationship), is explicitly tailored to account for these dis- the one that describes the part—denoted in the following as tinctions, although efforts in this direction pointed out useful Whole and Part, respectively. Mandatory parts express a modeling patterns and clarified some expressivity require- generic dependence relationship in the sense that, although ments (Artale et al. 1996; Barbier et al. 2003; Guizzardi a part of a certain kind must be always present when the 2005; 2007; Keet 2006; Motschnig-Pitrik & Kaasboll 1999). whole exists, the particular part can be different at different In this paper we focus in particular on the formal clarifica- moments of time (namely the part can be replaced, as in the tion of the notions of mandatory, essential and immutable human’s heart example). On the other hand, essential parts parts when used in conceptual models. The formalization express a specific dependence relationship, that is, the whole proposed here is grounded on the temporal description logic must be always associated with the very same part—i.e., the DLRUS and is based on previous efforts to formalize tem- part must be the same along the entire lifetime of the whole. poral conceptual models. Namely, we rely on a previous If we assume that Whole is rigid (Guarino & Welty 2000; work to define the temporal EER model ERVT that can be Welty & Andersen 2005) (that is, if something is a Whole, fully captured with DLRUS (Artale et al. 2002; Artale, then it is a Whole forever), then the distinction between es- Franconi, & Mandreoli 2003; Artale, Parent, & Spaccapietra sential and mandatory parts—corresponding to the distinc- 2007). The choice of using DLRUS is motivated by its abil- tion between specific and generic dependence—is enough to ity to logically reconstruct and extend representational tools capture all the useful parthood behaviors. such as object-oriented and conceptual data models, frame- If Whole is not rigid, however, it is useful to introduce a based and web ontology languages (Berardi, Calvanese, & further notion, a kind of weaker version of ‘essential part’: De Giacomo 2005; Calvanese, Lenzerini, & Nardi 1999; the notion of immutable part. Immutable parts, like essential Horrocks, Patel-Schneider, & van Harmelen 2003). In par- parts, are bound to the instances of Whole by a specific de- ticular, we will show that DLRUS has the expressivity re- pendence relationship. However, this relationship does not quired to formalize the above mentioned distinctions, while hold necessarily through the whole life of such individuals, being able, at the same time, to capture two relevant notions but only as long as they are instances of the class Whole. like the rigidity properties of a class (Guarino & Welty 2000; Immutable parts are therefore only conditionally essential Welty & Andersen 2005) and the (newly introduced) notion for being a Whole, but not essential tout court. For this of status for relations. Concerning the use of automated rea- reason, immutable parts might also be called conditionally soning services to check quality properties of a conceptual essential parts. Consider again the class Boxer in Fig. 1, model we have to remember here that DLRUS is an unde- which we assume to be not rigid (indeed, anti-rigid). Every cidable language (Artale et al. 2002). A promising direction boxer must not only have exactly two hands, but he must is to resort to the a weaker but decidable temporal DL, TDL- have his own hands: if he does not, then he ceases to be a Lite (first results appeared in (Artale et al. 2007)), that tem- boxer, while still being a human. In this case, having two porally extends the computationally simpler DL-Lite with (specific) hands as parts is conditionally essential for be- both temporal roles and concepts. It will be part of our fur- ing a boxer. If, on the other hand, boxing regulations allow ther work to investigate the expressive power of TDL-Lite for substitution of hands (be it by hand transplantation or to capture temporal conceptual models. protheses), then having some hands is just mandatory for the In the remainder of this paper, we first discuss the no- boxer. In sum, what we need to account for is another kind tions of essential, immutable and mandatory parts exempli- of constraint for the parthood relationship that only holds for fied by means of the running example. We then analyse re- the time the Whole property holds. As we shall see, this will lated works on temporal part-whole relations and provide an be formalized by introducing a “guard” condition in the def- overview of the description logic DLRUS. We proceed by inition for specific dependence which will lead to the notion introducing and formalising the notion of status for relations, of conditionally essential parts (and, in analogy, condition- and then we show how axioms in DLRUS capture the dis- ally essential wholes). tinction between rigid and anti-rigid classes. The main sec- tion of the paper deals with the constraints on parts, wholes, Temporal part-whole relations: Related works and part-whole relations, proving that our formalism eas- Given that our focus is on temporal modality, let us briefly ily captures mandatory, essential, and immutable parts (and discuss related works on temporal part-whole relations. The wholes). We close with conclusions and future work. most straightforward, yet also limited, way to temporalize part-whole relations is to turn a part-of predicate from a Preliminaries on essential, immutable and binary into a ternary relation, such that we have p part of mandatory parts w at time t: part of(p, w, t). To the best of our knowl- edge, almost all current temporalizations of parthood take Considering the conceptual model of Fig. 1, we would like this approach (Bittner & Donnelly 2007; Masolo et al. 2003; to distinguish between the brain, which we assume to be an Smith et al. 2005)1, but do not go further to take advantage essential part of a human, the heart, which we assume to be of a temporal knowledge representation language. An ex- a mandatory part for humans, and the two hands, which we assume to be immutable parts of every boxer. 1We focus here on temporal parts of objects (i.e., endurants in The distinction between essential and mandatory parts can DOLCE). Neither parts of events nor parts of 4-dimensional enti- be explained in terms of a specific vs. generic dependence ties (Hawley 2004) are considered here.
674 ception is (Barbier et al. 2003), where life-time dependen- notes an n-ary relation whose i-th argument (i ≤ n), named cies are represented in UML class diagrams by introducing Ui, is of type C. If it is clear from the context, we omit n ≶k an oclUndefined observer function to “assert that all parts and write (Ui : C). The projection expression ∃ [Uj]R is of result do not exist before (@pre)” the creation of the a generalisation with cardinalities of the projection operator whole instance w. Barbier and colleagues also tried a rep- over argument Uj of relation R; the classical projection is ≥1 resentation of immutability, which, however, remained an ∃ [Uj]R. open problem due to the lack of a full implementation of The model-theoretic semantics of DLRUS assumes a temporal UML. In addition, (Barbier et al. 2003) discussed flow of time T = Tp,<, where Tp is a set of time points nine basic life-cycle cases, obtained by fixing the lifespan and < a binary precedence relation on Tp, assumed to be iso- of the Whole and varying the lifespan of the Part. Clearly, morphic to Z,<. The language of DLRUS is interpreted one can consider further cases corresponding to the opposite T in temporal. models over , which are triples of the form perspective, obtained by fixing the part’s lifespan, and as- I = T , ΔI , ·I(t), where ΔI is non-empty set of objects sessing its temporal relationships with the whole’s lifespan (the domain of I) and ·I(t) an interpretation function. Since (see also Fig. 5). This may sound as a simple inverse, but the domain, ΔI , is time independent, we assume here the we shall see in the following sections that these two views so called constant domain assumption with rigid designa- require distinct constrains. tor—i.e., an instance is always present in the interpretation Regarding the ternary temporal part-whole relation, we domain and it identifies the same instance at different points have, for instance, Bittner and Donnelly’s “temporal mere- in time. The interpretation function is such that, for every ology” (Bittner & Donnelly 2007), which was developed to t ∈T (a shortcut for t ∈Tp), every class C, and every n-ary deal with “portions of stuff”, i.e., to deal with amounts of I(t) I I(t) I n relation R,wehaveC ⊆ Δ and R ⊆ (Δ ) . The matter, such as gold, and mixtures, such as lemonade, vary- semantics of class and relation expressions is defined in the ing in time. lower part of Fig. 2, where (u, v)={w ∈T |u 675 ≶k C → |⊥|CN |¬C | C1 C2 | C1 C2 |∃ [Uj]R | + − + − 3 C | 3 C | 2 C | 2 C | ⊕ C | C | C1 U C2 | C1 S C2 R → n | RN |¬R | R1 R2 | R1 R2 | Ui/n : C | + − + − 3 R | 3 R | 2 R | 2 R | ⊕ R | R | R1 U R2 | R1 S R2 I(t) =ΔI ; ⊥I(t) = ∅; CN I(t) ⊆ I(t); (¬C)I(t) = I(t) \ CI(t); I(t) I(t) I(t) (C1 C2) = C1 ∩ C2 ; I(t) I(t) I(t) (C1 C2) = C1 ∪ C2 ; ≶k I(t) I(t) I(t) (∃ [Uj]R) = { d ∈ | {d1,...,dn∈R | dj = d} ≶ k}; I(t) I(t) I(v) I(w) (C1 U C2) = { d ∈ |∃v>t.(d ∈ C2 ∧∀w ∈ (t, v).d ∈ C1 )}; I(t) I(t) I(v) I(w) (C1 S C2) = { d ∈ |∃v Figure 2: Syntax and semantics of DLRUS. status modeling includes up to four different statuses: sched- Blanc mountain is part of the Alps mountain range, and uled, active, suspended, disabled, where each one entails the country Republic of Ireland is part of the European different constraints. Union. Only active classes can participate into an active Concerning status relations there are two options: (1) to relation. derive a relation’s status from the status of the classes par- • Suspended: to capture a temporarily inactive relation. ticipating in the relation, or (2) to explicitly define it on the For example, an instance of a CarEngine is removed from relation itself, where the latter, in turn, puts constraints on the instance of a Car it is part of, for purpose of mainte- the statuses of the classes. Since we are interested in mod- nance at the car mechanic. Note that at the moment of eling relations as first-class citizens, we choose to have a suspension, both the part and the whole must be active, means to explicitly model the status of a relation. There- but can upon suspension of the relation be either active or fore, as for classes, we have four different statuses for re- become suspended, too, but neither scheduled (see below lations, too—scheduled, active, suspended, disabled—each constraints on scheduled) nor disabled. illustrated with an example before we proceed to the formal characterization. • Disabled: to model expired relations that never again can • Scheduled: a relation is scheduled if its instantiation be used. For instance, to represent the donor of an organ is known but its membership will only become effective who has donated that organ and one wants to keep track some time later. Objects in its participating classes must of who donated what to whom: say, the heart p1 of donor be either scheduled, too, be active, or suspended. For in- w1 used to be a structural part of w1 but it will never be stance, a pillar for finishing the interior of the Sagrada again a part of it. The heart, p1, then may have become Familia in Barcelona is scheduled to become part of that participant in a new part-of relation with a new whole, w2 church, i.e., this part of relation between the pillar and where w1 = w2, but the original part-of between p1 and the church is scheduled. w1 remains disabled. Observe that participating objects can be member of the active, suspended or disabled class. • Active: the status of a relation is active if the particu- lar relation fully instantiates the type-level relation: the We assume that active relations involve only active classes part is currently part of the whole. For instance, the Mont and, by default, the name of a relation denotes already its ac- 676 TopR PROPOSITION 1 (Status Relations: Logical Implications). Given the set of axioms Σst and an n-ary relation, R U1 : C1 ... Un : Cn, the following logical implications hold: d (RACT) Active will possibly evolve into Suspended or Dis- - - Exists R Disabled R abled. . + Σst |= R 2 (R Suspended-R Disabled-R) d (RDISAB3) Disabled will never become active anymore. + Σst |= Disabled-R 2 ¬R - Suspended-R Scheduled R R (RDISAB4) Disabled classes can participate only in dis- abled relations. − Figure 3: Status relations. Σst |= Disabled-Ci 3 ∃[Ui]R ∃[Ui]Disabled-R (RDISAB5) Disabled relations involve active, suspended, or disabled classes. tive status—i.e., Active-R ≡ R. Disjointness and ISA con- Disabled-R Ui :(Ci Suspended-Ci Disabled-Ci), straints among the four status relations are analogous to the for all i =1,...,n. one for status classes and can be represented with an EER (RSCH3) Scheduled persists until active. diagram as illustrated in Fig. 3 (double arrows indicate cov- Σst |= Scheduled-R Scheduled-R U R ering and an encircled d means disjointness). In addition to (RSCH4) Scheduled cannot evolve directly to Disabled. hierarchical constraints, the following constraints hold (we Σ |= ⊕ ¬ present both the model-theoretic semantics and the corre- st Scheduled-R Disabled-R spondent DLRUS axioms considering, without loss of gen- (RSCH5) Scheduled relations do not involve disabled erality, binary relations only): classes. Scheduled-R Ui :¬Dibabled-Ci, for all i =1,...,n. (ACT) Active relations involve only active classes. o ,o ∈ I(t) → o ∈ CI(t) ∧ o ∈ CI(t) 1 2 R 1 1 2 2 Proof. See Appendix. R U1 :C1 U2 :C2 (REXISTS) Existence persists until Disabled. I(t) Lifespan and related notions. The lifespan of an ob- o1,o2∈Exists-R →∀t >t.(o1,o2∈ ject with respect to a class describes the set of temporal I(t) I(t) Exists-R ∨o1,o2∈Disabled-R ) instants where the object can be considered a member of Exists-R 2+(Exists-R Disabled-R) that class. We can distinguish between the following no- tions: EXISTENCESPANC ,LIFESPANC ,ACTIVESPANC , (RDISAB1) Disabled persists. I(t) BEGINC ,BIRTHC , and DEATHC depending on the status o1,o2∈Disabled-R →∀t >t.o1,o2∈ of the class, C, the object is member of. We briefly report Disabled-RI(t ) 2+ here their definition as presented in (Artale, Parent, & Spac- Disabled-R Disabled-R capietra 2007). (RDISAB2) Disabled was Active in the past. I(t) I(t) I(t) EXISTENCESPANC (o)={t ∈T |o ∈ Exists-C } o1,o2∈Disabled-R →∃t 677 Definition 2 (-R).Anon-rigid property φ is a prop- axioms hold globally, 2∗ means “at all moments”, and 3∗ erty that is not essential to some of its instances, i.e., means “at some moment”). ∃xφ(x) ∧¬ φ(x). (RIGID) Rigid Class Definition 3 (∼R).Ananti-rigid property φ is a C 2∗C property that is not essential to all its instances, i.e., ∀xφ(x) →¬ φ(x) (A-RIGID) Anti-Rigid Class . C 3∗¬C In addition, we have the constraint that for each anti-rigid Category Endurant, Abstract entity class CA we must have a rigid class CR that subsumes Non-sortal Attribution Blue, Spherical CA (Guarino & Welty 2000; Guizzardi 2005): Formal role Recipient (A-SUB-R) Each Anti-Rigid Class is a subclass of a Rigid Class Role Material role CA CR Property Anti-rigid Student, Food Phased sortal Caterpillar, Chrysalis, Butterfly (for Papilionoidae) Finally, due to the constant domain assumption in Non-rigid Mixin DLRUS, we can capture the notion of existential rigidity Sortal Type by introducing an exist class, E, collecting, at each point in Cat, Chair Rigid Quasi-type time, the objects that actually exist. Herbivore (EXIST-RIGID) Existentially Rigid Class 2∗( → ) Figure 4: Taxonomy of properties with several examples C E C (Source: based on (Guarino & Welty 2000)). Given the above formalization for anti-rigid classes, then for each object, o, member of an anti-rigid class, CA,wehave Practically, for conceptual data modeling and ontology that: development the anti-rigid properties are highly useful to ACTIVESPANC (o) ⊂ ACTIVESPANC (o) represent entity types such as boxer and, more generally, A R those object types for which object migration may apply. On the other hand, for rigid and existentially rigid classes For example, a human, h1, may be recorded in a university the notions of EXISTENCESPANC (o),LIFESPANC (o) and database as an instance of PostDoc during some time and be ACTIVESPANC (o) all denote the full set of time points, T , promoted to Professor at a later time. A common example or the set ACTIVESPANE(o), respectively. for bio-ontologies is the transformation from caterpillar to We now have sufficient ingredients to formally character- butterfly where the same entity changes membership from ize the different kinds of dependence between a whole (part) one class to another. and its part (whole). Furthermore, as pointed out in (Welty & Andersen 2005), in the literature different kinds of rigidity have been intro- Representing mandatory, immutable, and duced. Here we also consider the so called existential rigid- essential parts and wholes ity that limits rigidity to the actual existence of the instance (captured by the exist predicate “E(x)”). The distinction between essential, immutable and manda- tory parts (Guizzardi 2007) can be formalized in terms of Definition 4 (Existential Rigidity).Anexistentially two criteria: i) the nature of the dependence relationship be- rigid property φ is a property that is essential to all tween the class that describes the whole and the one that its instances as long as they exist, i.e., ∀xφ(x) → describes the part (which can be either specific or generic), (E(x) → φ(x)). and ii) the strength of the membership between the whole To capture the distinction between immutable and essen- and the class that describes it (which, depending whether tial parts and wholes, we need to capture the distinction the class is rigid or not, will result in an unconditional or between rigid and anti-rigid classes. Indeed, while essen- conditional dependence). We thus distinguish between the tial parts are properties of a whole that cannot change with- following different cases. out destroying the whole, the fact that a part-whole relation 1. Generic Dependence – Mandatory Part. The whole must is immutable is conditional to the whole belonging contin- have a part at each instant of its lifetime. Thus, the pres- gently to a particular class in a given interval of time. Thus, ence of the part is mandatory, but it can be replaced over while essential parts are properties of a whole as being mem- time (e.g., the human heart example). ber of a rigid class, immutable parts describe properties of a whole as being member of an anti-rigid class. Furthermore, 2. Unconditional Specific Dependence – Essential Part. since for those instances that belong to the same non-rigid The part is mandatory, but it cannot be replaced without class at all points in time it is more appropriate to speak of destroying the whole (e.g., the human brain example). essential parts, we will formally speak of immutable parts 3. Conditional Specific Dependence – Immutable Part (also only in the stricter case of an anti-rigid class. called conditionally essential part). The part is manda- Starting from Definition 1 and Definition 3, we can en- tory and cannot be replaced, but only as long as the whole force a class C to be either rigid or anti-rigid with the follow- belongs to the class that describes it (e.g., the boxer’s hand ing DLRUS axioms, respectively (remember that: DLRUS example). 678 In a symmetric way we can define the notions of manda- A. whole's lifetime time B. part's lifetime time tory, immutable, and essential wholes. Furthermore, we say p1 w1 that a part is exclusive if it can be part of at most one whole p2 w2 (similarly for exclusive wholes). p3 w3 In the following we provide a formalization using an ax- p4 w4 iomatization expressed with the temporal description logic DLRUS of such mandatory, immutable, essential and ex- Figure 5: Lifelines of immutable parts (p1 ...p4) w.r.t. the clusive parts and wholes. lifeline of the whole (A)—and vice versa (B). Mandatory and exclusive parts and wholes : : Let PartWhole part P whole W be a generic part- ACTIVESPANP(op)=T and op is (an active) part of ow whole relation between a whole, W, and a part, P. The fol- at each instant t ∈T. Thus, essential parts are global prop- DLR lowing US axioms give a formalization of mandatory erties of rigid wholes that can be formalized in DLRUS with and exclusive parts and wholes: the following axioms: (MANP) Mandatory Part ∃[ ] (RIGID) Rigid Whole W whole PartWhole W 2∗W (MANW) Mandatory Whole ∃[ ] (ESSP) Essential Part P part PartWhole W ∃[whole]2∗PartWhole (EXLP) Exclusive Part ≤1 While essential wholes can be captured as: P ∃ [part]PartWhole (EXLW) Exclusive Whole (RIGID) Rigid Part ≤1 2∗ W ∃ [whole]PartWhole P P The human’s heart example can thus be represented by in- (ESSW) Essential Whole ∃[ ]2∗ troducing the part-whole relation HumanHeartPW as a sub- P part PartWhole relation of the generic PartWhole relation: By substituting the above rigidity axioms with the ex- HumanHeartPW PartWhole istential rigidity axiom we can capture essential parts and HumanHeartPW part : Heart whole : Human wholes just along the existence lifespan of an object, i.e., it must be that ACTIVESPANW(ow)=ACTIVESPANP(op)= and then constraining every human to participate at least ACTIVESPANE(ow), where E is the exist class. once in the HumanHeartPW relation by adding the follow- The human’s brain example can be represented by in- ing (MANP) axiom: troducing the part-whole relation HumanBrainPW as a sub- ∃[ ] Human whole HumanHeartPW relation of the generic PartWhole relation: DLR Given the semantics of US axioms, a generic heart HumanBrainPW PartWhole must be always associated to a human, i.e., the heart is possi- HumanBrainPW part : Brain whole : Human bly a different heart at different times but each human must have a heart along its lifetime. While this example speci- and then adding both the (RIGID) and (ESSP) axioms: fies a mandatory part (heart) for a rigid class (human), in the Human 2∗Human general case mandatory parts can be specified regardless of Human ∃[whole]2∗HumanBrainPW the meta-properties concerning rigidity of the whole. This asserts that the very same brain is a part of the very Finally, if we want to stress the exclusiveness of the heart, same human in all possible time instants. Obviously, exclu- i.e., that a given heart can be part of at most one human, then siveness axioms can be added to further constrain essential we add the (EXLP) axiom: ≤1 parts and wholes. Heart ∃ [part]HumanHeartPW One can also conceive examples of exclusive essen- Note that to represent mandatory and exclusive parts tial parts or wholes for artifacts, such as that each (wholes) we do not need to use the temporal operators of DisposableLighter has as an essential part at most DLRUS. one Firestone, or that each USBStick has exactly one MemoryCard as essential part. Essential parts and wholes Essential parts express a specific dependence between the Immutable parts and wholes whole and the correspondent part as opposed to the generic As we said above, immutable parts are a form of condition- dependence of a mandatory part. Furthermore, the very ally essential parts—i.e., the part is mandatory and cannot same part (i.e., the specific part) must actively exist during be replaced, but only as long as the whole belongs to the the entire existence of the whole, unconditionally whether class that describes it. Thus, given an anti-rigid class, WA, to- the whole belongs to different classes during its exis- gether with its rigid super-class, WR, and an individual whole, o tence. Given an instance, w, of the rigid class, W, then, ow, that is member of WA, with ACTIVESPANWA (ow) ⊂ op is an essential part of ow if ACTIVESPANW(ow)= ACTIVESPANWR (ow), then, op is an immutable part of ow if 679 ACTIVESPANWA (ow) ⊆ ACTIVESPANP(op) and op is (an ac- 3. p3 if (MANP), (SUSW), (DISW), (SCHPW), (SCHP) tive) part of ow at each instant t ∈ ACTIVESPANWA (ow). The hold. p1,...,p4 temporal relations of Fig. 5-A illustrate exactly 4. p1 if (MANP), (SUSW), (DISW), (DISP), (SCHPW), all the possible different temporal relationships between the (SCHP) hold. activespan of the given whole (top line of the figure) and the activespan of its immutable part. Proof. See Appendix. A similar reading can be associated to Fig. 5-B, where A similar result can be proved for immutable wholes. In w1,...,w4 now denote all the possible different temporal analogy with the (SUSW) axiom, we need to introduce an relations between a fixed part and its immutable whole. Note axiom that governs the case in which the part is suspended: that the temporal relation p2 (w2) can be regarded as the most general notion of immutable part (whole). (SUSP) If part-whole is suspended then the part is sus- To formalize the notion of immutable parts (wholes) in pended. DLRUS, we need to resort to the notion of status for both Suspended-PartWhole part : Suspended-P classes and relations as illustrated previously. In addition, THEOREM 3 (Immutable Wholes). Let PR be a rigid class the whole (part) must belong to an anti-rigid class2. ∗ ∗ (i.e., PR 2 PR), P be an anti-rigid class (i.e., P 3 ¬P) Let us show now that DLRUS has enough expressivity to s.t. P PR, and PartWhole part : P whole : W be a account for the various cases of immutable parts and wholes generic part-whole relation satisfying Σst. Then, for each introduced above. To see that, we shall first introduce a set part, op, of type P there exists an immutable whole, ow,of of useful DLRUS axioms expressing some basic constraints type W that is temporally related to op with the relation: and then we shall prove that each of the possibilities de- scribed by Fig. 5-A (Fig. 5-B) can be expressed by suitable 1. w2 if (MANW), (SUSP), (DISP) hold. combination of these axioms. The DLRUS axioms are as 2. w4 if (MANW), (SUSP), (DISP), (DISW) hold. follows: 3. w3 if (MANW), (SUSP), (DISP), (SCHPW), (SCHW) (SUSW) If part-whole is suspended then the whole is sus- hold. pended. 4. w1 if (MANW), (SUSP), (DISP), (DISW), (SCHPW), Suspended-PartWhole whole : Suspended-W (SCHW) hold. (DISP) If part-whole is disabled then the part is disabled. Proof. Similar to Theorem 2. Disabled-PartWhole part : Disabled-P Observe that our formalization of immutability allows for (DISW) If part-whole is disabled then the whole is dis- suspension of parts, wholes and their part-whole relations, abled. : thanks to the introduction of status for both classes and re- Disabled-PartWhole whole Disabled-W lations. Since status is not considered in the literature for (SCHPW) The part-whole was scheduled sometime in the immutability of part-whole relations (Barbier et al. 2003; past. Guizzardi 2005; 2007), we can enforce immutability to hold PartWhole 3−Scheduled-PartWhole just for continuously active classes. To achieve this, it is enough to substitute the (SUSP) and (SUSW) axioms with (SCHP) If part-whole is scheduled then the part is sched- uled. the following axiom that forbids suspension of the part- Scheduled-PartWhole part : Scheduled-P whole relation: (SCHW) If part-whole is scheduled then the whole is sched- (CONPW) Continuous active part-whole relation. ⊥ uled. Suspended-PartWhole Scheduled-PartWhole whole : Scheduled-W Using the (CONPW) axiom, the immutable parts and The following Theorem shows the DLRUS axioms wholes are always active while participating in the part- needed to represent the various forms of immutable parts whole relation and cannot be suspended. Furthermore, when as illustrated in Fig. 5-A. the stricter case is allowed (i.e. either p1 or w1 holds), then they are either both member of their respective Scheduled THEOREM 2 (Immutable Parts). Let WR be a rigid class class, or both Active, or both member of their respective 2∗ 3∗¬ (i.e., WR WR), W be an anti-rigid class (i.e., W W) Disabled classes. Hence, a change of status from one of the : : s.t. W WR, and PartWhole part P whole W be a two objects implies an instantaneous change of the other in Σ generic part-whole relation satisfying st. Then, for each the same type of status class. o o whole, w, of type W there exists an immutable part, p,of The boxer’s hands example can be represented with the o type P that is temporally related to w with the relation: following axioms—enforcing e.g. case p2. First, we intro- 1. p2 if (MANP), (SUSW), (DISW) hold. duce the part-whole relation HumanHandPW as a sub-relation 2. p4 if (MANP), (SUSW), (DISW), (DISP) hold. of the generic PartWhole relation: 2 HumanHandPW PartWhole To capture immutable parts for the more general case of non- HumanHandPW part : Hand whole : Human rigid classes, our approach forces the introduction of an anti-rigid sub-class typing the instances having a limited lifetime for which then we assert that boxer is anti-rigid and it is a sub-class of we can speak of immutable parts. the rigid class human: 680 ∗ Boxer 3 ¬Boxer mous reviewers for their insightful comments. Boxer Human and finally we add the (MANP), (SUSW), (DISW) axioms References rephrased in terms of the HuamnHandPW part-whole relation: =2 Artale, A.; Franconi, E.; Guarino, N.; and Pazzi, L. Boxer ∃ [whole]HumanHandPW 1996. Part-whole relations in object-centered systems: An Suspended-HumanHandPW whole : Suspended-Boxer overview. DKE 20(3):347–383. Disabled-HumanHandPW whole : Disabled-Boxer Artale, A.; Franconi, E.; Wolter, F.; and Zakharyaschev, The use of the (SUSW) axiom instead of (CONPW) cap- M. 2002. A temporal description logic for reasoning about tures the scenario where it is permitted for a boxer to be conceptual schemas and queries. In Flesca, S.; Greco, S.; temporarily member of the Suspended-Boxer class during Leone, N.; and Ianni, G., eds., Proc. of JELIA-02, volume which the part-whole relation is suspended as well (e.g., the 2424 of LNAI, 98–110. Springer. boxer has a broken hand that is healing). To be compatible Artale, A.; Kontchakov, R.; Lutz, C.; Wolter, F.; and Za- with extant literature on immutability, we can enforce a con- kharyaschev, M. 2007. Temporalising tractable description tinuously active boxer by replacing the (SUSW) axiom with logics. In Inter. Symposium on Temporal Representation the (CONPW) axiom: and Reasoning (TIME07). IEEE Computer Society. Suspended-HumanHandPW ⊥ Artale, A.; Franconi, E.; and Mandreoli, F. 2003. De- Thinking of examples of immutable parts for artifacts, scription logics for modelling dynamic information. In we may consider, e.g., an EcoFarm (anti-rigid class) as a Chomicki, J.; van der Meyden, R.; and Saake, G., eds., sub-class of RealEstate (rigid class), where an area of Logics for Emerging Applications of Databases, LNCS. Farmland is an immutable part of the EcoFarm: if, say, the Springer-Verlag. ecofarm’s farmland is confiscated, then the residual real es- tate becomes member of either the Suspended-EcoFarm or Artale, A.; Parent, C.; and Spaccapietra, S. 2007. Evolving Disabled-EcoFarm class (while continuing to be an active objects in temporal information systems. AMAI 50(1-2):5– member of RealEstate). 38. Finally, observe that exclusiveness and immutability are Barbier, F.; Henderson-Sellers, B.; Le Parc-Lacayrelle, A.; orthogonal notions in our modeling framework. The for- and Bruel, J.-M. 2003. Formalization of the whole-part re- malization of immutable parts and wholes can be easily ex- lationship in the Unified Modelling Language. IEEE Trans. tended by adding to the axiomatization of Theorems 2-3 ei- on SE 29(5):459–470. ther (EXLP) or (EXLW), depending whether we want to cap- Berardi, D.; Calvanese, D.; and De Giacomo, G. 2005. ture exclusive immutable parts or wholes. Reasoning on UML class diagrams. AI Journal 168(1– 2):70–118. Conclusions Bittner, T., and Donnelly, M. 2007. A temporal mereology In this paper we proposed a formalization of mandatory, im- for distinguishing between integral objects and portions of mutable, and essential parts and wholes adopting a temporal stuff. In Proc. of AAAI’07, 287–292. Vancouver, Canada. logic approach. For each different parthood property we pre- sented its formalization with the aim to clarify their meaning Calvanese, D., and De Giacomo, G. 2003. Expressive de- when used in a conceptual modelling language. The tempo- scription logics. In Baader, F.; Calvanese, D.; McGuinness, D.; Nardi, D.; and Patel-Schneider, P., eds., The DL Hand- ral description logic DLRUS has been adopted thanks to its already established ability to give a logical reconstruction of book. Cambridge University Press. 178–218. (temporal) conceptual modelling languages. In addition to Calvanese, D.; Lenzerini, M.; and Nardi, D. 1999. Unify- represent the different interpretations of a part-whole rela- ing class-based representation formalisms. JAIR 11:199– tion, we introduced the novel notion of status for relations 240. that temporally constrains the evolution of a relation. The Chomicki, J., and Toman, D. 1998. Temporal logic in in- notion of status relations has been showed crucial to capture formation systems. In Chomicki, J., and Saake, G., eds., DLR immutable parts and has been formalized in US. Logics for Databases and Information Systems. Kluwer. Concerning the computational properties, we are aware chapter 1. of the undecidability of DLRUS. A promising direction is to resort to the a weaker but decidable temporal DL, TDL- Etzion, O.; Gal, A.; and Segev, A. 1998. Extended update Lite (Artale et al. 2007) that temporally extends DL-Lite functionality in temporal databases. In Etzion, O.; Jajodia, with both temporal roles and concepts. We are currently S.; and Sripada, S., eds., Temporal Databases - Research working on both extending the framework of TDL-Lite with and Practice, LNCS. Springer-Verlag. 56–95. sub-roles and showing the ability of TDL-Lite to represent Gabbay, D.; Kurucz, A.; Wolter, F.; and Zakharyaschev, temporal conceptual models. M. 2003. Many-dimensional modal logics: theory and applications. Studies in Logic. Elsevier. Acknowledgement Guarino, N., and Welty, C. 2000. A formal ontology of We would like to thank Giancarlo Guizzardi for the exten- properties. In Dieng, R., ed., Proc. of EKAW ’00, LNCS. sive discussions we had with him. We also thank the anony- Springer Verlag. 681 Guarino, N., and Welty, C. 2004. An overview of On- (RDISAB4) Disabled classes can participate only in dis- toClean. In Stab, S., and Studer, R., eds., Handbook on abled relations. − ontologies. Springer Verlag. 151–159. Σst |= Disabled-Ci 3 ∃[Ui]R ∃[Ui]Disabled-R Guizzardi, G. 2005. Ontological Foundations for Struc- (RDISAB5) Disabled relations involve active, suspended, tural Conceptual Models. Phd thesis, University of Twente, or disabled classes. The Netherlands. Telematica Instituut Fundamental Re- Disabled-R Ui:(Ci Suspended-Ci search Series No. 15. Disabled-Ci), for all i =1,...,n. Guizzardi, G. 2007. Modal aspects of object types and part- (RSCH3) Scheduled persists until active. whole relations and the de re/de dicto distinction. In Proc. Σst |= Scheduled-R Scheduled-R U R of CAISE’07, volume 4495 of LNCS. Springer-Verlag. (RSCH4) Scheduled cannot evolve directly to Disabled. Hawley, K. 2004. Temporal parts. In Zalta, E., ed., The Σst |= Scheduled-R ⊕ ¬Disabled-R Stanford Encyclopedia of Philosophy. Winter’04 edition. (RSCH5) Scheduled relations do not involve disabled http://plato.stanford.edu/archives/win2004/entries/temporal- classes. parts/. Scheduled-R Ui :¬Dibabled-Ci, for all i =1,...,n. Hodgkinson, I. M.; Wolter, F.; and Zakharyaschev, M. Proof. We prove here just (RDISAB4). The others cases 1999. Decidable fragments of first-order temporal logics. are similar to what was already proved in (Artale, Parent, Annals of pure and applied logic 106:85–134. & Spaccapietra 2007). Horrocks, I.; Patel-Schneider, P.; and van Harmelen, F. I(t) (RDISAB4). Let oi ∈ Disabled-Ci and r = 2003. From SHIQ and RDF to OWL: The making of a web I(t) o1,...,oi,...,on∈R for some t 682 since by assumption active relations involve only active I(t) classes, op ∈ P , for all t s.t. t0 683