This section introduces a domain-neutral approach for representing and relating in- formation within a formal ontology model. The model serves as a common basis for reasoning about different types of information based on their shared semantics.
3.3.1
Information Dimensions
Newby et al. define an information space as “the set of concepts and relations among them held by an information system. [An] information space is produced by a set of known procedures, and is changed through intentional manipulation of its con- tent.” (New96). Here, we think of aninformation systemas termed by Newby, as the sensing capability of a collection of devices, or more generally the sensing capability of an environment such as a room, building, street, or city.
Information spaces, capturing all that can be sensed in an information system, can be subdivided into multiple dimensions of information, each with its own structural and relational characteristics. For example, within the information system of a house, one dimension of information is temperature, represented by a combined numeric value and scale; for example, 23°C, or 73.4°F, or a concept with meaning to a person likehot orcold. Another dimension with multiple representations is location, represented as a point or region within a 2D or 3D coordinate system, or with a symbolic name, like kitchen. Time (as an instant) also has multiple representations, for example 2:59pm or 14:59 in 12hr and 24hr clocks.
The set of relevant information dimensions varies with the information system being modelled. For example, a system tracking a mobile entity may be concerned with speed, acceleration, distance travelled, or distance to destination, while a personal health monitoring system may be concerned with heart rate, blood pressure, or respiration.
3.3.2
Abstracting Information
A small number of axiomatic definitions provide a formal basis for modelling dimen- sions of information. We begin with the concept ofunits—the set of smallest values that can bemeasuredin an information dimension. Units are disjoint and exhaustive so as to cover anything representable in the dimension, whether finite or infinite, continuous or discrete. Examples of units for different information dimensions are given in Table 3.1. In dimensions where multiple unit representations exist – for example, temperature
Dimension Features Units
Temperature Continuous, Finite Celsius (or Fahrenheit, Kelvin) Boolean Discrete, Finite TrueorFalse
Time Continuous, Infinite Millisecond (typically) Number of Interactions Discrete, Infinite Count (Integer)
Location Continuous, Infinite Cartesian coordinate system Table 3.1: Ground value measurements in different information dimensions.
values can be equally well expressed in Celsius, Fahrenheit, or Kelvin – we assume a primary measurement system, to which others are mapped.
While irreducible units of measurement can be used, the ability to measure units is a practical consideration that serves to simplify a model. For example, milliseconds typically serves as a good representation for temporal instants, despite being sub- divisible.
Building on unit measurements, each information dimension can have a set ofconcept values, or simplyconcepts– symbolic terms that are human-understandable or useful to the application at hand – to which unit values are mapped. For example, in the temperature dimension ranges of Celsius unit values might be mapped to the concepts cold,warm, andhot; in the location dimension, a region of a cartesian coordinate space may be mapped to a room, building, or city; and values in the speed dimension toslow orfast.
We formalise this mapping in Definition 1.
LetVgbe the unit value set of a dimension of information. A set of concept valuesVaare defined via a mapping functionµ∶Va→ P(Vg), whereP(Vg)is
the power set ofVg.
Definition 1(Mapping unit values to concepts).
3.3.3
Information Semantics
Using a set-theoretic approach, Definition 2 formalises how the semantics of concept values common to all information dimensions –equal,subsume,overlap,adjacent, and disjoint– are evaluated from their mappings from unit values. These relations are also illustrated in Figure 3.2.
Given two concepts in the same information dimension:vai,vaj ∈Va: • vai equalsvaj, denoted asvai =vaj, iff µ(vai) =µ(vaj);
• vaj subsumesvai, denoted asvai⊲vaj, iffµ(vai) ⊆µ(vaj).
• vai overlapsvaj, denoted asvai○○vaj, iffµ(via)∩µ(vaj) ≠ ∅,µ(vai) ⊈µ(vaj),
andµ(vaj) ⊈µ(vai).
• vai is adjacent to vaj, denoted as vai∥vaj, iff µ(vai) ∩µ(vaj) = ∅, and
∃k∈µ(vai) such thatk+ε ∈µ(vaj), where ε is the smallest unit value
increment.
• vai is disjoint withvaj, denoted asvai⊠vaj, iff µ(vai)∩µ(vaj) = ∅;
Definition 2(Primary semantic relations between concepts).
The semantics in Definition 2, allow us to define further semantics through their composition by logical OR (∨) or use of negation (¬), as listed in Table 3.2.
Each of the semantic relations is interpreted in the context of the information dimension it is applied to. Illustrating with temperature, as shown in Figure 3.3, we might define
Va i j Va Va i j Va Va i j Va V a i j Va Va i j Va
Equal Subsume Overlap Adjacent Disjoint
Va i Vj a Va i Vj a Va i Vj a Va i Vj a Va i Vj a
Figure 3.2: Relationships between concepts in an information dimension. Each circle represents the unit value set to which a concept maps.
Symbol Name Composition
≠ Not equal ¬ = ⊴ Subsumed by or equal to ⊲∨ = ⊳ Subsumes ¬⊴ ⊵ Subsumes or equal to ¬⊲ /○○ Not overlapping ¬○○ ∦ Not adjacent ¬∥ Not disjoint ¬⊠
four concepts:freezing,cold,warm, andhot. In this example,i) coldsubsumesfreezing; ii) freezingis disjoint withwarmand hot,iii) hot is adjacent tocold; and, iv) warm overlaps bothcoldandhot.
The process is similar for other dimensions:
• In a dimension with a boolean property, there are only two concepts, correspond- ing totrueandfalse, which are disjoint.
• In the time dimension, concepts such asafternoonorweekendare mapped to unit value sets containing time instances, which are comparable. For example, the conceptweekendsubsumes the conceptSaturday.
• In the location dimension, concepts take the form of symbolic locations, with their relationships evaluated by projecting their associated coordinate sets onto a coordinate system and comparing them (SYDN10). For example, the concept kitchen may be adjacent to the concept dining room. We develop a worked example using location in Section 3.7.
3.3.4
Relation properties
To this point we have defined an approach to inferring basic semantic relations between concepts in an information dimension, solely through a domain expert mapping concepts to sets of unit values. This allows concepts in an information dimension to be compared in a uniform way, regardless of the type information being modelled, be it speed, location, or membership of an organisation.
Thermometer
Name ___________________________ 120 °F ° C 50 40 30 20 10 0 – 10 – 20 – 30 110 100 90 80 70 60 50 40 30 20 10 0 – 10 – 20 Warm Hot Cold FreezingFigure 3.3: An illustration of the semantic relations between concepts in the temperature domain.
In this section we use these semantics to develop a general approach to reasoning over a modelled domain, uncovering the implicit knowledge in any single dimension. The starting point of this derivation process is the axiomatic properties of the five relationships introduced in Definition 2, presented in Definitions 3 – 7.
Equal relationships are reflexive, transitive, and symmetric. Definition 3(Properties of the equals relation).
Subsume relationships are reflexive, transitive, and antisymmetric. Definition 4(Properties of the subsume relation).
Overlap relationships are symmetric and irreflexive. Definition 5(Properties of the overlap relation).
Adjacent relationships are symmetric and irreflexive. Definition 6(Properties of the adjacent relation).
Disjoint relationships are symmetric and irreflexive. Definition 7(Properties of the disjoint relation).
In addition to the intrinsic properties of the semantic relations, further knowledge can be derived by considering the interactions between multiple relations:
• If two concepts are adjacent, they are necessarily disjoint (directly from Defini- tion 2).
• If two concepts are disjoint, then any pair of abstract values they subsume, one taken from each set, must also be disjoint (Lemma 1);
• If one concept is subsumed by two concepts, neither of which subsumes the other, then the latter two values overlap (Lemma 2).
• Thesubsume,overlap, anddisjointrelationships form an exhaustive and mutu- ally exclusive set of relationships between any two concepts; that is, any two concepts must have one and only one of these three relationships. If two of these relationships are known not to hold, the third must (Lemma 3).
∀vai,vaj∈Va, ifvai⊠vaj, then∀vai′⊴v a i and∀vaj′⊴v a j,vai′⊠v a j′. Proof. From Definition 2,
vai⊠vaj ⇒µ(vai)∩µ(vaj) = ∅(1), vai′⊴v a i ⇒µ(vai′) ⊆µ(v a i)(2), vaj′⊴v a j ⇒µ(vaj′) ⊆µ(v a j)(3), From (1), (2), (3)⇒µ(vai′)∩µ(v a j′) = ∅, and by Definition 2⇒v a i′⊠v a j′. Lemma 1(Subsumed concepts inherit disjointness).
Forvai,vaj,vak∈Va, ifµ(vak) ≠ ∅,vak⊴vai,vak⊴vaj,vai⋬vaj, andvaj⋬vai, thenvai○○vaj.
Proof. From Definition 2, vak⊴vai ⇒µ(vak) ⊆µ(vai)(1), vak⊴vaj ⇒µ(vak) ⊆µ(vaj)(2), From (1) and (2)⇒µ(vak) ⊆µ(vai)∩µ(vaj)(3). From (3) andµ(vak) ≠ ∅ ⇒µ(vai)∩µ(vaj) ≠ ∅(4). vai⋬vaj ⇒µ(vai) ⊈µ(vaj)(5), vaj⋬vai ⇒µ(vaj) ⊈µ(vai)(6),
From (4), (5), (6), and by Definition 2⇒vai○○vaj.
Lemma 2(Concepts sharing a subsumed concept overlap).
Forvai,vaj∈Va,
(i) if¬(vai○○vaj),vaj⋬via, andvai⋬vaj, thenvai⊠vaj; (ii) if¬(vai ⊠vaj),vaj⋬via, andvai⋬vaj, thenvai○○vaj; (iii) if¬(vai○○vaj),¬(vai⊠vaj), andvaj⋬vai, thenvai⊲vaj;
(iv) if¬(vai○○vaj),¬(vai⊠vaj), andvai ⋬vaj, thenvaj⊲vai.
Definitions 3 – 7 and Lemmata 1 – 3 provide a sound basis for deriving further know- ledge from that which is explicitly provided. The main benefit of this approach is that time can be saved in the knowledge specification process. A domain expert need only define the mapping from concepts to the unit value set for an information domain, or specify the primary relationships between concepts directly, in order for these derivation rules to be applicable.
A worked example demonstrating this process is given in Section 3.7.
3.4
Context model
Dourish (Dou04) describes context as a “slippery notion”; it is highly application specific, giving rise to definitions that are too restrictive to obtain consensus or too broad to be meaningful. Many have attempted to formalise this notion of contextas it applies to computer systems. For example, Schilit et al. (SAW94) state:
“[three] important aspects of context are: where you are, who you are with, and what resources are nearby ... [context] includes lighting, noise level, network connectivity, communication costs, communication bandwidth, and even the social situation; e.g., whether you are with your manager or with a co-worker”.
while Dey et al. (DA00) define context more generally as:
“any information that can be used to characterize the situation of an entity. An entity is a person, place, or object that is considered relevant to the interaction between a user and an application, including the user and the application themselves”.
more broadly again, Yao et al. (YHGY06) and Loke (Lok04b) describe context as:
“any information acquired from a system or an environment”.
This final form most closely resembles the definition of an information space given by Newby et al. introduced in Section 3.3.1.
We define a context model that expands upon the concept model present above by relating conceptsacross dimensions of information. Thus,Alices’s living roommay be hot,her kettleswitchedon,her carmay bestationary, and3people may be inher house.
Figure 3.4 illustrates these examples, and other relationships between dimensions of context. Concepts sharing an information domain are denoted by shared colour. Note that often the same knowledge can be expressed in multiple ways. For example, we can equivalently state thatAlice’s housecontainsAlice’s living room, orAlice’s living roomiscontained byAlice’s house.
Alice's Kettle Alice's House Bob Mary Alice's Kitchen Hot Alice's Living Room Stationary Alice On Alice's Car Alice's Garage 3 containsPeople hasTemperature locatedIn locatedIn containedBy containsRoom containsRoom containsObject hasState locatedIn hasSpeedState locatedIn containedBy containsPerson
Figure 3.4: An illustration of relationships between information dimensions.
3.4.1
Context statements
Given two information domains, acontext statementrepresents a relation between a pair of concepts, mapping to the cartesian square or product set of the information domains. We represent a context statement as a triple,[:s, :p,:o], where:s and:o, termed the subjectandobjectrespectively, are concepts in two information dimensions, and:p, termed the predicate, represents the named relation that exists between the two concepts. This naming scheme and notation is similar to RDF and its markup languageNotation 3(BLC11), a mapping we explore further in Section 3.7.
We say that a (context) statement holds when the pair of concepts from two informa- tion domains are present in the product space of a predicate. For example, from the illustration given in Figure 3.4, the (subject, object) pair (mary,alice’sKitchen) is present in the product space of predicatelocatedInand therefore the statement[:mary, :locatedIn,:alice’sKitchen]holds. Definition 8 formally defines a context statement.
LetSa andOa be two information domains and let predicate pbe a relation mapping to their cartesian productSa×Oa, the set of ordered pairs of concepts
(s,o)wheres∈Saando∈Oa.
We represent a context statement as[:s, :p,:o], wheres∈Sa ando∈Oa. We say the pair(s,o)entails the statement[:s, :p,:o]underp.
Definition 8(A context statement).
Further to this, Definition 9 defines four general types of predicate that support the inference of additional statements beyond the statement directly entailed by the presence of a given pair of concepts. Given asubject-associativepredicate, if the subject of a statement is subsumed by any concepts, the set of corresponding statements with the subject substituted for each of these subsuming concepts also holds. Given anobject- associativepredicate, if the object of a statement is subsumed by any concepts, the set of corresponding statements with the object substituted for each of these subsuming concepts also holds. Given anall-associativepredicate, if the subject or object are both subsumed by concepts, the set of corresponding statements with all combinations of the subject and object being substituted for their subsuming concepts also holds. Finally, given anot-associativepredicate, no additional statements are directly inferred. It is the responsibility of the model designer to define which, if any, of these types apply to a particular predicate and, further, to restrict the domains describing the subject, object or both where the predicate type applies. For example, the predicatelocatedInmay be defined to be object-associative on the information domain describing locations and their containment relationships. This allows the statement[:alice, :locatedIn,:alice’s house]to be inferred from the statement[:alice, :locatedIn,:alice’sLounge]. Similarly, the same predicate could be defined to be subject-associative on the information domain describing residency relationships of people in the house. For example, to support inference of the statement[:houseResident, :locatedIn,:alice’sLounge]. IflocatedInis defined as all-associative on these domains, the statement[:houseResident, :locatedIn, :alice’sHouse]can be inferred.
However it would be incorrect to apply a subject-associative locatedIn relation to familial relationships (for example, incorrectly inferring the statement[:mary’sFamily, :locatedIn,:alice’sKitchen]). Similarly, thehasTemperaturerelation is not subject- associative on the information domain describing locations and their containment relationships; that is,[:alice’sKitchen, :hasTemperature,:hot]does not support the inference of the statement[:alice’sHouse, :hasTemperature,:hot].
Following on from Definition 8, let ps,po,pa, and pnbe subject-associative, object-associative, all-associative, and not-associative predicates respectively, such that:
• Ifs⊴s′,[:s, :ps,:o]entails context statement[:s’, :ps,:o],∀s′∈Sa. • Ifo⊴o′,[:s, :po,:o]entails context statement[:s, :po,:o’],∀o′∈Oa. • Ifs⊴s′,o⊴o′,[:s, :pa,:o]entails context statement[:s’, :pa,:o’],∀s′∈
Sa,∀o′∈Oa.
pndescribes a predicate where no entailment rule applies. Definition 9(Predicate types).
3.4.2
Relationships between context statements
We build upon the relations we have defined to this point to study the semantic rela- tions between context statements in the same manner as concepts. Returning to the example in Figure 3.3 we can see that the statement[:edinburgh, :hasTemperature, :freezing]is subsumed by the statement[:edinburgh, :hasTemperature,:cold], and is disjoint with the statements[:edinburgh, :hasTemperature,:warm]and[:edinburgh, :hasTemperature,:hot]. Similarly, the statement[:dublin, :hasTemperature,:hot]is disjoint with the statement[:dublin, :hasTemperature,:freezing], is adjacent to the statement[:dublin, :hasTemperature,:cold], and overlaps with the statement[:dublin, :hasTemperature,:warm]. Definition 10 provides these definitions formally.
These relationships can also be inferred directly from the relationships in their corres- ponding dimensions of information, details of which are relegated to Appendix A. As with concepts, this provides an approach for deriving information from that which has been explicitly profiled or sensed from the environment. We demonstrate the utility of this approach further via a worked example in Section 3.7.
Given two context statements [si,p,oi] and [sj,p,oj] that share the same relationp: Sa×Oa,
• [si,p,oi] equals [sj,p,oj], denoted [si,p,oi] = [sj,p,oj], iff ∀(s,o) ∈ Sa×Oawhere(s,o)entails[si,p,oi],[sj,p,oj]also holds; and∀(s,o) ∈ Sa×Oawhere(s,o)entails[sj,p,oj],[si,p,oi]also holds;
• [sj,p,oj]subsumes[si,p,oi], denoted[si,p,oi] ⊲ [sj,p,oj], iff∀(s,o) ∈ Sa×Oawhere(s,o)entails[si,p,oi],[sj,p,oj]also holds; and∃(s,o) ∈ Sa×Oawhere(s,o)entails[s
j,p,oj]such that[si,p,oi]does not also hold;
• [si,p,oi] overlaps with [sj,p,oj], denoted [si,p,oi] ○○ [sj,p,oj], iff
∃(s,o),(s′,o′),(s′′,o′′) ∈ Sa×Oa, (s,o) such that both [s
i,p,oi] and
[sj,p,oj]hold,(s′,o′)such that[si,p,oi]holds but[sj,p,oj]does not, and(s′′,o′′)such that[sj,p,oj]holds but[si,p,oi]does not;
• [si,p,oi] is adjacent to [sj,p,oj], denoted [si,p,oi] ∥ [sj,p,oj], iff
∃(s,o) and (s,o′) or (s′,o) ∈Sa×Oa, s∥s′, o∥o′ where (s,o) entails
[si,p,oi]but not[sj,p,oj]while either(s′,o)or(s,o′)entails[sj,p,oj] but not[si,p,oi];
• [si,p,oi] is disjoint with [sj,p,oj], denoted [si,p,oi] ⊠ [sj,p,oj], iff
/∃ (s,o) ∈Sa×Oawhere(s,o)entails both[s
i,p,oi]and[sj,p,oj]. Definition 10(Primary semantic relations between context statements).