Territorios de la métrica

A valuable extension to DLs are concrete domains, based on the framework proposed in [BH91].9 In contrast to individuals from the general-purpose (or abstract) domain∆I we have considered thus far, values of concrete domains are “eternal” (mathematical) abstractions. Notable examples include numerical, temporal, and spatial concrete do- mains, the latter of which is not considered in the following. Characteristic to values of concrete domains is that their identity is determined by some procedure usually involv- ing their structure. For instance, the strings 24.10.2010 and 2010/10/24 both identify

9_{Combinations of “standard” DLs with concrete domains are commonly denoted by appending(D)} to the name of the DL (e.g.,S HOI N (D),S ROI Q(D)).

Table 3.2: Additional Constructs inS ROI Qand their Semantics

DL OWL

Ex. Syntax Semantics

∃S.Self {x| (x, x) ∈SI} SelfRestriction S U UI =∆I×_∆I _{topObjectProperty} R S1◦ · · · ◦SnvR S₁I◦ · · · ◦SIn ⊆RI propertyChain ≥nS.C {x| ]{y | (x, y) ∈SI and y∈CI} ≥n} minCardinalityQ Q with onClass ≤nS.C {x| ]{y | (x, y) ∈SI and y∈CI} ≤n} maxCardinalityQ with onClass Role assertions

Sym(R) (x, y) ∈RI implies(y, x) ∈RI SymmetricProperty Asy(S) (x, y) ∈SI implies(y, x) 6∈SI AsymmetricProperty

Ref(R) DiagI ⊆ RI ReflexiveProperty

Irr(S) SI∩DiagI = ∅ IrreflexiveProperty

Dis(S1, S2) S1I∩SI2 = ∅ disjointObject-

Properties ◦is overloaded to denote the standard composition of binary relations.

]N denotes the cardinality of the set N. DiagI denotes the set{(x, x) | x∈∆I_}.

the same date – October 24th2010 – after normalization from the (locale-specific) lexical space into the value space, which is done by a procedure based on the structure.

Since concrete domains, in their unrestricted form, have severe consequences on decidability and computational complexity of reasoning, more recent works then in- troduced practicable notions of datatype maps and data ranges [HS01, MH08]. In short, datatype maps are formalizations of (i) a set of available datatypes, (ii) their value space and lexical space, and (iii) facets and facet expressions. The latter are expressions over a datatype, that further restrict their range of values. Datatypes are usually thought to be unary. Nevertheless, the definitions introduced in the following can be extended to n-ary datatypes as in [PH03] (e.g., date can be considered a 3-ary datatype consisting of the components year, month, and day). Finally, data ranges are basically expressions over the elements in a datatype map.

Definition 3.8(Datatype Map). A datatype map is a 4-tupleD = (VD, VLS, VF,·D), where • VD is a set of datatypes d,

• VLS is a function assigning a set of lexical forms VLS(d) to each d ∈ VD where VLS(d) is called the lexical space of d,

• VF is a function assigning a set of facets VF(d)to each d ∈VD, and

• ·D is a function assigning a datatype interpretation dD to each datatype d ∈ VD called the value space, a facet interpretation fD ⊆ dD to each facet f ∈ VF(d), and a data value vD ∈ dD to each lexical form (constant) v∈ VLS(d).

A facet expression for a datatype d ∈ VD is a formula ϕ built using propositional connectives over the elements from VF(d) ∪ {>d,⊥d}. The function ·D is extended to facet expressions by setting, for f(i) ∈ VF(d), (>d)D = dD, (⊥d)D = ∅, (¬f)D = dD\fD, (f1∧ f2)D =

f₁D∩ f₂D, and(f1∨ f2)D = f1D∪ f2D.

Observe that the symbols >_d and ⊥_d denote built-in universal and empty facets, respectively, specific to a datatype d ∈VD.

Example 3.1

Imagine a datatype mapDwith VD = {string, real}, where stringDshall be the set of all strings over some alphabetΣ (i.e., stringD =Σ∗) and realDthe set of real numbers. The set VLS(string) would then contain all lexical string forms and VLS(real) shall contain decimal representations of real numbers.10 Finally, the set VF(real) might contain the facet int, interpreted as the set of integers, and facets of the form<q,>q,≤q, and≥q for decimal numbers q. The facet expression int ∧ >10 ∧ <20 would then represent the integers 11, . . . , 19.

The syntax and semantics of the description logicsS HOI N (D) and S ROI Q(D)

will be introduced next. In essence, they are obtained by extending the “core” descrip- tion logic with a datatype system. Prior to that, the notion of data ranges is yet to be introduced.

Definition 3.9(Data Ranges). Let D = (VD, VLS, VF,·D)be a datatype map and letVLS = S

d∈VDVLS(d) be the union of all lexical forms over all datatypes in D. The set of data ranges

forDis the smallest set that contains (1) >D (universal data range),

(2) d (datatype range),

(3) d[ϕ](facet data range over a datatype),

(4) {v1, . . . , vn}(enumeration), (5) dr (negated data range),

for d∈ VD, ϕ a facet expression for d, vi ∈VLS, and dr a data range.

Definition 3.10(Syntax ofDL+D). LetDLbe a description logic defined over the vocabulary

(VC, VOP, VI). Let VDP be a countable set of concrete role names11 pair-wise disjoint from VC, VOP, VI. LetD be a datatype map. Let T(i) ∈ VDP be a concrete role, a ∈ VI an individual, v ∈ V_LS a lexical form, dr a data range forD, and n a non-negative integer.

The logicDL+D, obtained by extendingDLwithD, extends the concepts ofDLwith concepts of the form∃T.dr,∀T.dr,≥nT.dr, and≤nT.dr. The set of TBox axioms is extended by inclusion axioms of the form T1 v T2 and disjointness axioms of the form Dis(T1, T2). The set of ABox assertions is extended by assertions of the form T(a, v).

10_{Of course, one could also represent real numbers in binary or hexadecimal form.} 11_{We use the subscript DP to indicate that they are called data properties in OWL.}

Table 3.3: Model-Theoretic Semantics of DL + D data ranges, concepts, axioms, and assertions Data Ranges (>D)D =∆D (d[ϕ])D = ϕD ({v1, . . . , vn})D = {vD₁, . . . , vDn } dr D =∆D \ drD Concepts Axioms (∃T.dr)I = {x | ∃y.(x, y) ∈TI and y∈drD} Dis(T1, T2) ⇒T₁I∩T₂I = ∅ (∀T.dr)I = {x | ∀y.(x, y) ∈TI implies y∈drD} T1v T2⇒T₁I ⊆T2I (≥nT.dr)I = {x | ]{y| (x, y) ∈TI and y∈ drD} ≥n} Assertions (≤nT.dr)I = {x | ]{y| (x, y) ∈TI and y∈ drD} ≤n} T(a, v) ⇒ (aI, vD) ∈TI ]N denotes the cardinality of a set N.

In order to define formal semantics for the constructs of data ranges, the datatype domain∆D needs to be introduced, which is done byDefinition 3.11. The function·D is then extended for the constructs as shown inTable 3.3.

Definition 3.11 (Semantics of DL + D). An interpretation for DL + D is a triple I = (∆I,∆D,·I), where∆I and∆Dare nonempty disjoint sets such that dD ⊆∆Dfor each d∈ VD. The interpretation function·I is derived fromDL and extended to assign to each concrete role T ∈ VDP the interpretation TI ⊆ ∆I ×∆D. Furthermore, ·I and ·D are extended to data ranges, complex concepts, axioms, and assertions as shown inTable 3.3.

An interpretationI is a model of aDL+Dknowledge base K, writtenI |= K, if it satisfies all axioms of the TBoxT and all assertions of the ABoxA.

In [MH08] it is shown that, without losing generality, the assumption can be made that datatypes d(i) ∈ VD are pairwise disjoint; that is, if d1, d2 ∈ VD and d1 6= d2 then d₁D∩dD₂ = ∅. This allows for a modular treatment of different datatypes for reasoning

(i.e., handling a datatype d does not necessitate considering other supported datatypes VD\d). Furthermore, it has been pointed out in [MH08] that K can be interpreted by considering only those datatypes from the datatype map that are explicitly mentioned inK. This is the reason why it is not necessary to define the consequence relation|=in

Definition 3.11 w.r.t. the entire datatype map (i.e., including those datatypes not men- tioned inK) provided that∆Dis the set that contains at least the interpretations for each d ∈ VD, which is the case. Note, however, that the consequences of K might change under the extension of a datatype map with a new datatype (i.e., if∆Dis enlarged).