Ontological relations and spatial reasoning in earth science ontologies

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Geological Society of America Special Papers Ontological relations and spatial reasoning in earth science ontologies Hassan A. Babaie Geological Society of America Special Papers 2011;482;13-27 doi: 10.1130/2011.2482(02)

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© 2011 Geological Society of America


The Geological Society of America Special Paper 482 2011

Ontological relations and spatial reasoning in earth science ontologies Hassan A. Babaie Department of Geosciences, Georgia State University, Atlanta, Georgia 30302-4105, USA

ABSTRACT Several types of fundamental ontological relations connect the endurant (continuant) and perdurant (occurrent) entities in every domain. These include: instantiation, parthood, location, and connection relations, and those that are derived from them, such as adjacency, overlap, containment, and coincidence. Some of these types of relations, and their subtypes, are formally defined in the context of the Web Ontology Language (OWL) for a variety of endurant geological examples, mostly from the Nankai Trough in southwest Japan and the San Andreas fault in California. Here, the foundational ontological relations are discussed to show their application in building useful earth science ontologies. These relations, defined as properties in OWL, are given in the context of the Resource Description Framework (RDF) triples and their relationship to relational databases. The role of properties in providing semantics, reasoning, and knowledge structuring and representation is discussed for various ontological relations. The semantics of classes are provided by the metaproperty and restrictions of the properties that use these classes as domain and range. Types of properties are described from different perspectives and for different purposes. Property subclassing, through OWL’s subproperty construct, is used to restrict properties. The formal definitions of the foundational taxonomic (isA), partonomic (partOf), location (locatedIn), containment (containedIn, componentOf), and topologic (overlap, adjacentTo) relations, at the class and instance levels, are given in first-order logic for continuant geological entities. Geologic examples for several other basic relations such as derivesFrom, transformationOf, and absorb are also given.

INTRODUCTION

ontologies (Kashyap et al., 2008; Donnelly et al., 2005; Smith, 2003, 2004) represent the reality by depicting the taxonomic, partonomic, and other types of hierarchical structure of objects and events in the real world (Lambe, 2007). Application of ontologies to support information management, knowledge discovery, and spatial and temporal reasoning has been progressively appreciated by earth scientists in recent years, as is apparent by

Depiction of the structure and dynamics of reality, to enable spatial and temporal reasoning of simple and composite components of Earth’s natural systems, requires construction of ontologies that are based on formalized representation of the static and dynamic entities and their complex relationships. Formal domain

Babaie, H.A., 2011, Ontological relations and spatial reasoning in earth science ontologies, in Sinha, A.K., Arctur, D., Jackson, I., and Gundersen, L., eds., Societal Challenges and Geoinformatics: Geological Society of America Special Paper 482, p. 13–27, doi:10.1130/2011.2482(02). For permission to copy, contact [email protected]. © 2011 The Geological Society of America. All rights reserved.

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a significant increase in the number of projects devoted to building top-level and domain ontologies (e.g., Raskin and Pan, 2005; Sinha, 2006). Construction of useful ontologies requires a thorough understanding of the formal meaning of the foundational ontological relations such as instantiation, parthood, location, and connection, and those that are derived from them, such as adjacency, overlap, containment, and coincidence (e.g., Randell et al., 1992; Cohn and Varzi, 2003; Donnelly, 2004b; Schulz et al., 2005, 2006). Despite the spatial and spatio-temporal nature of most components in Earth systems, and the fact that most natural objects are complex composite entities, the existing taxonomic structures in current earth science ontologies are mainly based on the primitive isA (i.e., subclass, instantiation, or subsumption) relation, and they under-represent the other types of ontological relation. The hierarchies in these ontologies generally do not include the mereological part-whole (i.e., partitive) and topological (e.g., connection) relations that are needed to depict composite, spatially and spatio-temporally related entities. Moreover, there seems to be confusion in the earth science community as to the difference between formal relations that hold at the universal level compared to those that exist among instances (individuals) in reality. In this paper, I focus on introducing the major fundamental relations (both Web Ontology Language [OWL] qualified names and user-defined) and the difference between those that hold between universal types and those that exist among instances in reality. The main objective of this paper is to introduce the formal ontological relations so that they can be used more consistently in designing better and more reliable earth science ontologies. Due to the scarcity of ontologies in the earth sciences, most of the material used in this paper is based on the work in artificial intelligence, medical informatics, and analytical philosophy, where significant progress has been done in ontological engineering. The formal relations in this paper are given in the first-order logic notation (Levesque and Lakemeyer, 2000; Brachman and Levesque, 2004), and the symbols used in this paper include: ∧ (and), ∨ (or), → (then), ¬ (not), ∃ (there exists), ⊆ (is a subclass of), ∩ (intersection of) and ∀ (for all, if any). All variables are given in Helvetica font, and those for types are given in the upper case letters, e.g., X, C, R. Variables representing particulars or instances of universal types are given in the lowercase letters, e.g., x, c, r. Universal types and particulars are also given in Helvetica font, with the first letter of every word written in the capital letters, e.g., Ocean, Rock, Fault, and the first letter of each word in compound names capitalized in the camel case, e.g., IgneousRock, AccretionaryPrism. The relation names start with a lowercase letter in Helvetica font, and the first letter of each subsequent word in composite names is in uppercase, i.e., camel case (e.g., partOf, isA, connectedTo). Concepts, i.e., term definitions, are given in italic font, e.g., mineral, water. Although the geological examples given to elucidate the semantics of each type of relation are varied, they mostly relate to the spatial relations in the Nankai Trough accretionary prism in southwest Japan (Tobin and Kinoshita, 2006) and the San Andreas fault in California.

PROPERTIES AND SEMANTIC LANGUAGES Semantic models (e.g., ontologies) have a graph-theoretical structure, consisting of nodes representing terms, and edges representing relations that link the nodes (Smith et al., 2005). The relata for the relations are classes (i.e., terms) that represent entities in reality. Ontology captures the formal relations and their universal relata, based on the real relationships and constraints between the instances in a specific domain of discourse, i.e., a discipline or field (e.g., Smith, 2003, 2004; Bains, 2006). This way, ontologies can be used to represent domain knowledge and theories, and support inquiries in the domain. Information models depict the reality by using classes to represent the universals and instances of these classes to represent the individuals. Knowledge in any domain is a collection of numerous true statements (propositions) (Carrara and Sacchi, 2006). For example, the statement, “Rock is an aggregate of one or more minerals,” is a piece of the petrology knowledge. The following are two other examples of tectonics and structural geology knowledge statements: “Temperature (T) increases with depth (z),” denoted by the geothermal gradient ∂T/∂z ≈ 30 °C/km, and “stress (σ) and strain (e) are linearly related (at low temperatures/pressures and shallow depth)” expressed by Hooke’s law of elasticity, σ = Ee, where E is the proportionality constant. These knowledge statements are composed of terms (e.g., stress, strain in the last statement) that represent universal classes that evoke domain concepts, and the relationships among them (e.g., the “linearly related to” relation between stress and strain). Knowledge statements are explicitly asserted in ontologies in the form of Resource Description Framework (RDF) triples, applying the semantic web languages such as OWL (Web Ontology Language) and its underlying RDF and RDF schema (RDFS) languages (Breitman et al., 2007; Allemang and Hendler, 2008; Antoniou and van Harmelen, 2008; Kashyap et al., 2008). Every statement in RDF is like a value in a single cell of a database table, which requires three components for its complete representation (Allemang and Hendler, 2008): a row identifier (subject, s), a column identifier (predicate, P), and the value in the cell (object, o) (Fig. 1). Subject is the thing (individual) for which we make the statement, the predicate is the property (relation) for the subject, and the object is the value for the property. We refer to the subject-predicateobject statement as an RDF triple. Figure 1 shows the concept of the RDF triple and its relationship to the relational database table structure, and Figure 2 gives an example for conversion of the relational database table into RDF triples. A knowledge base built based on a domain ontology is a large set of such RDF triples about the individuals in that domain. Because each row (i.e., record) in a relational database table has multiple columns (fields, attributes), several triples in an ontology often relate to one subject (Figs. 1B and 2B). In other words, a single subject in a knowledge base may relate to many objects through many predicates, and hence RDF triples. For example, a specific, individual sedimentary bed (subject) has specific values (objects) for its age, orientation, and composition predicates.


Ontological relations and spatial reasoning in earth science ontologies

Figure 1. The correspondence between the Resource Description Framework triple (subject-predicate-object) and a relational database table. (A) Each row (record) in a relational database table represents a subject (s). Each column (field or attribute) in the table is a predicate (property, Pi). The cell value at the intersection of a row and column is the object (Oi). (B) Each subject (S) corresponds with many objects (Oi) through many properties (Pi) (Allemang and Hendler, 2008).

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In the semantic web jargon, subjects are said to be instances of classes, predicates are properties (relations), and objects, which are either instances of other classes or are data types, provide values for the properties. There are two types of properties in OWL: data type and object. The data type property is a binary relation between a set of individuals (subjects) and a set of instances of a typed literal (e.g., XSD, XML Schema Definition data types) (Breitman et al., 2007). An object property is a binary relation between sets of individuals of two classes, i.e., the subject and objects of a RDF triple are both individuals. Properties are restricted by their domain and range. Domain relates a property (predicate) to a subject class of a triple that uses the property. This way, domain namespace imposes restriction on the type of the subject instances that use the property (Antoniou and van Harmelen, 2008); the domain is the set of values for which the property is defined, i.e., it is the collection of types that use the property. For example, the domain for the composition property in the triple: Mineral composition MineralGroup, is the Mineral class. Range is used to relate a property to a target, object class, or data type, and therefore it puts a restriction on the set of values a property can take. For example, the range for the composition property in the previous triple can be silicate, phosphate, sulfide, hydroxide, etc., which are all of type MineralGroup. Domain and range are constructs that give information on how a property may be used, and they should be declared with care. The direction of roles should be defined correctly when using the domain and range. Table 1 shows some statements in the structural geology domain (namespace prefix: struc), given in the N3 serialization format (Allemang and Hendler, 2008). In these statements, the Fold class has the foldDescription and tightness data type properties of string type, and a foldAxis object property for which the range is of the Line type. RELATIONS AND REASONING

Figure 2. A relational database table (A) converted into the Resource Description Framework triples (B) for the first row (record) of the table.

As conceptual units, relations constitute a major component of knowledge structuring, representation, and reasoning by providing semantics, i.e., meaning (Jouis, 2002; Lambe, 2007). Because semantic data are focused mainly on the relationship

TABLE 1. ABBREVIATED N3 SERIALIZATION OF THE STRUCTURAL GEOLOGY DOMAIN’S FOLD CLASS N3 stat ement Resource Description Framework triple struc:foldDescription rdf:type owl:DataTypeProperty. Fold tightness “open” Fold Description “Harmonic fold in bedding” struc:foldDescription rdfs:domain struc:Fold. Fold foldAxis Line1 struc:foldDescription rdfs:range XSD:string. struc:foldAxis rdf:type owl:ObjectProperty. struc:foldAxis rdfs:domain struc:Fold. struc:foldAxis rdfs:range struc:Line. struc:tightness rdf:type owl:DateTypeProperty. struc:tightness rdfs:domain struc:Fold. struc:tightness rdfs:range xsd:string.


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between individuals, ontologies are property-oriented in contrast to the object-oriented models (Sagaran et al., 2009). Properties in ontology languages such as OWL are first-class citizens, meaning that like classes, they can subsume subproperties (Antoniou and van Harmelen, 2008). Because properties are defined globally, they can be used anywhere by any class, i.e., unlike methods in an OO (object-oriented) language, a property in RDF does not belong to one class! This provides a great deal of flexibility in OWL and its sublanguages because there is no need to modify classes when we introduce new properties. Relations provide a framework to meaningfully connect categories of entities defined in classes that stand for our domain types. Properties stand for relations or constitute the arguments for relations. Classes are defined in ontologies based on properties or values of the properties, for example, we can define classes of minerals that are silicate, or classes of deformed rock that are mylonitic. Ontologies are often built in one domain and reused in another through importation. The reuse requires declaration of namespace for each ontology. The XML namespace standard, used by RDF, provides a global scope to domain classes and properties, allowing reuse of existing types into new ones without naming conflicts. The XML standard allows different communities of scientists (e.g., oceanography and atmospheric science) to independently develop their own ontologies and markup languages. There is often a need to integrate these autonomously developed vocabularies into other applications, and this is where the namespace becomes very useful. It is common for two domain vocabularies to contain classes or properties that have the same name but that are structured in different ways, and thus provide meaning differently. If the two vocabularies are shared by an application, there will be a name conflict, which would lead to an error during processing. The namespace prevents this kind of name collision by assigning the similarly named terms, which belong to different domains, to different uniform resource identifiers (URI) that reference these communities. Declaration of a namespace is done by the xmlns (XML namespace) attribute, which allows both an optional prefix and a URI to be chosen for the namespace. The prefix, which references the URI, qualifies each term within a vocabulary to a specific community, e.g., struc:Fold, ocean:Floor, which are qualified names for the structural geology and oceanography domains, respectively.

The fact that rocks have textures and textures have textual descriptions can be stated as: Rock texture Texture and Texture textureType XSD:string, respectively. Let’s assume that Rock and Texture are defined in the Petrology ontology (with the “petr” namespace prefix, which is a URIref), and then they are imported into another ontology. The imported Rock and Texture classes are referred to, in the ontology that is using the Petrology ontology, by the qualified names petr:Rock and petr:Texture, respectively. In practice, the petr prefix is a URI, which is a unique identifier for the Petrology ontology if it exists. Figure 3 shows the graphical and textual presentation of the texture property and its domain (Rock) and range (Texture) classes, and the textureType property and its domain (Texture) and range (XSD:string). Two instances (individuals) of the Rock and textureType are also shown. There are two general types of relation: static and dynamic (Jouis, 2002). While the static relations connect the spatial or static aspects of entities (i.e., no change in state), the dynamic relations deal with the temporal and spatio-temporal concepts (e.g., process, event) that involve change of state. Examples of the static relations are: IdahoBatholith locatedIn Idaho; Rock isA Solid. An example of the dynamic relation is: Faulting displace Rock. Relations are used to structure knowledge and provide semantics for reasoning through the following constructs: (1) hyponomy (Cruse, 2002), to specialize general spatial entities through the “isA” and “is a kind of” relations (e.g., NormalFault isA Fault), (2) troponymy (Fellbaum, 2002), to subclass verbs that represent the effects of processes (e.g., strain isA deform, slip isA move), and (3) meronymy (Pribbenow, 2002; Koslicki, 2008), to structure complex wholes by using the “part of” relation (e.g., Mineral partOf Rock). A statement such as “NormalFault isA Fault” or “strain isA deform” implies inclusion of meaning, that is, the meaning of the NormalFault type or the strain relation includes the meaning of the Fault type or deform relation, respectively. In other words, the hyperonym-hyponym pair (i.e., general-specialized pair, e.g., Fault and NormalFault, or strain and deform) are of the same semantic type. In a troponymy in which elongate and distort are subproperties of the strain relation, the subordinate (i.e., specialized) relations (elongate, distort) contain the meaning of the superordinate (general) relation (strain), but they add extra semantics to it, that is, whereas strain is change in something (volume, length, and

Figure 3. (A) Directed graph of two resource description framework (RDF) triples: Rock texture Texture and Texture textureType XSD:string. “andesite” and “porphyritic” are instances of the Rock and Texture classes. (B) The N3 serialization of the two RDF triples, defining the Rock class and the domain and range for the texture and textureType properties.


Ontological relations and spatial reasoning in earth science ontologies angle), the subordinate elongate or distort relation deals specifically with the change in length or shape, respectively (Fellbaum, 2002). The difference between the verbal relations elongate and distort is in the “manner” or “way”’ in which the processes that these relations represent occur. The elongate or distort relation, as in “Extension elongate Pebble” or “Shearing distort Fossil,” denotes the manner or way in which a pebble or fossil strains by elongation or change in angle, respectively. SUBSUMPTION OF PROPERTIES AND CLASSES Properties relate classes in the hierarchy. Importing and equating properties from different sources can be done with the use of the rdfs:subPropertyOf. When an ontology is imported into another one, we may need to make two properties equivalent. For example, assume that two groups of geologists (e.g., geochemistry and tectonics groups) have knowledge bases in which one domain uses the property study and the other one uses investigate for the scientific work done by its geologists. Let’s assume that the namespace prefixes for these two domains are geochem and tect, respectively. Assuming that the investigate and study verbal properties mean the same thing, we make them equivalent letting each property be the subproperty of the other (Allemang and Hendler, 2008), i.e., geochem:study rdfs:subPropertyOf tect:investigate and tect:investigate rdfs:subPropertyOf geochem:study. Or, we can use the owl:equivalentProperty for this purpose. As another example, if hydrogeology ontology calls flow for what structural geology ontology calls transport, we can use the rdfs:subPropertyOf as long as the domain and range of these various properties are of the same type. To state that all uses of the transport and move properties are the same, we assert: hydro:transport rdfs:subPropertyOf struc:move. If we have the triple: x transport y in the hydrogeology domain (hydro namespace prefix), we can infer x move y in the structural geology domain (struc namespace prefix), as long as x and y are of related types (e.g., water and ion). For example, since transport is a subproperty of the move property, the explicit assertion hydro:Water hydro:transport geochem:Ion infers that struc:Fluid struc:move geochem:Ion.

If class (type, set) C is a subclass of A and a subclass of B (i.e., C rdfs:subClassOf A, and C rdfs:subClassOf B), then C is in the intersection of sets A and B (i.e., C ⊆ A ∩ B). In this case, if individual x is in C, then x is also in both A and B. For example, pyroclastic rocks (e.g., tuff) have the properties of both volcanic and clastic sedimentary rocks. In a default namespace (where no prefix precedes the colon in the qualified name), this fact is asserted as :PyroclasticRock rdfs:subClassOf:VolcanicRock, and :PyroclasticRock rdfs:subClassOf:DepositionalRock. If we now assert that :HuckleberryTuff rdf:type:PyroclasticRock, we infer (i.e., reason through the inference rules) that the Huckleberry tuff (in Wyoming) is both depositional (kind of sedimentary) and volcanic, i.e., we derive the following two inferred statements: :HuckleberryTuff rdf:type:VolcanicRock, and :HuckleberryTuff rdf:type:DepositionalRock. Notice that the inference is unidirec-

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tional, i.e., pyroclastic rock is both volcanic and depositional, but every depositional sedimentary or volcanic rock is not pyroclastic! Each class defines the essential and accidental properties (Colomb, 2007). The essential properties are necessary for the individuals to have if they are members of the class. Individuals may or may not have the accidental properties. For example, one essential property of the silicate family of minerals is to have a composition made of silicon, oxygen, and certain metals. If a mineral does not have Si or O (essential properties), then it does not belong to the Silicate class. Thus, essential values for all instances of a class must necessarily be the same, and they must always be present. However, being glassy, blue, smoky, purple, or milky for quartz (a silicate) is accidental. Notice that what is essential for a subclass may be accidental for its superclass. For example, in the IgneousRock isA Rock taxonomy, formation from magma, which is essential for a member of the IgneousRock subclass, is accidental for a member of the Rock superclass. An essential whole is a complex individual with essential unifying properties (relating all the essential parts) that are required by the whole. For example, the Fold essential whole must have limb and hinge line as essential parts. If a property is both essential and sufficient to identify the class, it is called rigid. For example, physical, optical, and chemical properties of minerals are rigid. All instances of a given mineral (rigid class) have the same set of values drawn from its rigid properties. Properties represent the binary relations (predicates) between resources or individuals, referenced by the subjects and objects in the RDF triples (Hitzeler et al., 2009). Properties are defined in RDF as instances of the rdf:Property class (e.g., solidify rdf:type rdf:Property). The meaning of the recrystallize property, relating metamorphism and rocks (Metamorphism recrystallize Rock), is the set of all individuals that are recrystallized. As sets, properties exhibit a similarity to classes rather than to individuals. Properties are subclassed in OWL by applying the owl:subPropertyOf construct (e.g., grainBoundaryMigrate owl:subPropertyOf recrystallize). The rdfs:subPropertyOf provides a mechanism to extend properties, from more general to more specific, i.e., allow hierarchy of properties. “P is said to be a subproperty of P′ if every instance of P is also an instance of P′” (Colomb, 2007), or, stated differently, “P is a subproperty of P′ if P′(x,y) whenever P(x,y)” (Fig. 4). In general, P rdfs:subPropertyOf P′ means: if x P y, then x P′ y, i.e., if x and y are related by P, then they are also related by P′. To clarify this, think of the shear and extend properties, which are more specific types of the displace property (i.e., shear rdfs:subPropertyOf displace and extend rdfs:subPropertyOf displace) (Fig. 4). In this case, if Fault shear Grain, then Fault displace Grain. The brittlyDeform property is more specific than deform, and rotate is more specific than deform, i.e., rotate rdfs:subPropertyOf deform. This means that if a fault rotates a fold, it deforms it (if Fault rotate Fold, then Fault deform Fold). The crystallize property is a subproperty of solidify (crystallize owl:subPropertyOf solidify), and strain, rotate, and translate are subproperties of the deform property (e.g., strain owl:subPropertyOf


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Figure 5. A property inheriting from two other properties, where x and y are instances of classes A and B, which are related by property P. The relation x P y implies the two top x R y and x S y relations (properties).

Figure 4. Subclassing of properties, where x and y are instances of the A and B classes, respectively. (A) P is a subproperty of P′ if x p y infers x p′ y, i.e., the relation of two instances (resources) x and y by the subproperty infers the relation by the superproperty.

deform). So crystallize owl:subPropertyOf solidify means that if a magma x crystallizes into a mineral y (i.e., if x P y), then x also solidifies into mineral y (i.e., implies x P′ y). Also, grainBoundaryMigrate owl:subPropertyOf recrystallize means that grain boundary migration during metamorphism brings recrystallization. Notice that subsumption is transitive, i.e., material that is strained is also deformed; material that is crystallized is also solidified. The converse is not necessarily true, that is, something that solidifies does not necessarily do it through crystallization. There are cases where a property (P) inherits meaning from two (R and S) (or more) other properties. This can be done by defining property P to be the logical intersection of R and S (i.e., P ⊆ R ∩ S), and two resources x and y are related by property P. In other words, if x P y, then x R y and x S y (Fig. 5). For example, recrystallization (defined in the structural geology ontology with the struc namespace) implies both crystal plastic deformation and strain softening. These relations are asserted as follows: struc:recrystallize rdfs:subPropertyOf struc: crystalPlasticallyDeform and struc:recrystallize rdfs:subPropertyOf struc:strainSoften. This is a unidirectional inference, i.e., when recrystallization occurs, it strain softens the rock, but all strain softenings are not achieved via recrystallization (for example, it can occur via recovery).

Optional properties can subsume mandatory subproperties, but mandatory properties can only have mandatory subproperties (Colomb, 2007). Instances of the domain classes for an optional property are not required to participate in it. For example, ductilelyDeform may be optional for a Rock, but it can subsume mylonitize, which can subsume recrystallize. If the mylonitize property is mandatory for a crystal-plastically deformed fault rock, then its recrystallize and recover subproperties must be mandatory too. Mandatory properties are expressed with the existential quantifier (∃). For example, Folding (a process) must involve either a planar object (e.g., bedding, foliation) or a linear object (flute cast, lineation). So, if there is an instance of Folding, there also exists an instance of a Planar or Linear class through the deform property. For example, if the Folding and Bedding classes are defined in the structural geology ontology (namespace prefix: struc), then the domain and range for the deform property are given by: struc:deform rdfs:domain struc:Folding, and struc:deform rdfs:range struc:Bedding. A class can “supply” or “carry” a property. It supplies the property if the property only holds for the class and not its superclass, i.e., it is defined in the subclass. It carries the property if it is inherited from the superclass (Guarino and Welty, 2002; Colomb, 2007). Properties can carry essentiality (+E), unity (+U), identity (+I), and rigidity (+R). These metaproperties are defined in the Ontoclean method (Guarino and Welty, 2002). It should be noted that the Ontoclean method is only one of several possible ontology evaluation methods. Essentiality means that all instances must have the same value for their property. Unity relates to the parts that are needed to make the whole. It is concerned about how parts of a complex object are put together to make the whole object. Identity refers to the properties that are needed to identify instances of a class. Rigidity is provided by the necessary and sufficient properties that identify the class of an individual. If any of these metaproperties is annotated with a negative prefix (read: “not”), e.g., –E, –U, –I, and –R, it means that the metaproperty does not necessarily hold (may hold by accident) for all instances of the class for which the property is defined (Guarino and Welty, 2002). Thus, –E means that property is not essential, although it could be accidental. Same is true for –U, –I,


Ontological relations and spatial reasoning in earth science ontologies and –R, which read “not unity,” “not identity,” and “not rigidity,” respectively. A metaproperty annotated with ~ (read “anti”), e.g., ~E (anti-essential), ~U (anti-unity), ~I (anti-identity), or ~R (antirigid), means that it necessarily does not hold for any instance of the class. An anti-identity and anti-unity property cannot be used as a basis of an identity or unifying relation for any instance, respectively. An anti-essential property can be updated (i.e., the value may change) in all instances (Colomb, 2007). The +R and +E properties must be mandatory, whereas –R, ~R, –E, and ~E can be optional. Subclasses cannot weaken the strength of the metaproperty for the superclass, i.e., a subclass cannot have an ~E, ~I, or ~U if the superclass has the property +E, +I, and +U. However, the opposite is possible, i.e., if a superclass has a property with metaproperties ~E, ~I, or ~U, the subclass can have that property with +E, +I, and +U, respectively (Colomb, 2007). RELATIONS THAT HOLD BETWEEN FUNDAMENTAL TYPES OF ENTITIES The entities in a domain (e.g., subduction zone, fold-andthrust belt) fall into two broad, disjoint (i.e., nonoverlapping) categories (e.g., Smith and Grenon, 2004; Bittner et al., 2004): (1) continuants (endurants), and (2) occurrents (perdurants). The continuants include material and immaterial substances, parts (both fiat and bona fide parts; Smith 2001), boundaries, aggregates, qualities, roles, function, and spatial regions (Smith and Grenon, 2004). The continuant objects, such as fault, lake, accretionary prism, rock, and porosity, exist in their entirety (i.e., as a mereological whole) at any temporal slice (i.e., at a time instant, ti) of their four-dimensional (spatio-temporal) life history. Despite the continuous change in the object (attribute) and relational properties (e.g., partitive and connection relations) of the continuants, these entities maintain their identity through time and space, as long as they exist. For example, the continuous qualitative changes in the type, thickness, and spatial location of sediments in the Kumano forearc basin, and its underlying accretionary prism, do not change the identity of these components of the Nankai Trough (just like the change in the color of your hair does not change you). While continuants represent the static part of reality, the occurrents correspond to the dynamics of the world. The occurrents include events that signify the instantaneous beginning and end of state change in objects (e.g., rock, fault) through homogeneous processes that bring qualitative change to the continuants. For example, the accretionary prism (a continuant) grows through processes of offscraping, underplating, and sedimentation, which modify the structure of the prism over time. In this paper, I focus on the formal spatial relations and do not cover the temporal and spatio-temporal relations, which require discussions of occurrents, such as processes and events, that are not in the scope of the paper. These relations can be found in Babaie (2011). Formal, in this case, means that the relations, which are defined in first-order logic, apply to any domain of

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reality, for example, to subduction zone, strike-slip fault, experimental rock deformation, or atmospheric science. Notice that the formal relations, such as partOf, are not necessarily part of the OWL language. The formal relations (e.g., partOf, locatedIn) may hold between: (1) continuant objects, e.g., SeismogenicZone partOf PlateBoundaryFaultZone; SplayFault locatedIn AccretionaryPrism; UnderplatedSediment externallyConnectedTo AccretionaryPrism, (2) occurrent objects, e.g., Comminution partOf Cataclasis; Fracturing partOf Faulting, or (3) between objects of the two disjoint types, e.g., Folding involves Discontinuity; Ions participate-in Mineralization. Notice that the relations between two continuants

are defined at the time instants that the two objects are related to each other, i.e., xi partOf yi, or xi properPartOf yi, for the time index i. The related objects can have different granularities, for example, Microfracture partOf DamageZone; TwinBanding locatedIn Mineral. A damage zone, a fractal entity, can exist over a large range of scale, microscopic to regional, compared to twin banding, which is microscopic. The universal term entity refers to objects, processes, events, functions, times, and places (Smith and Rosse, 2004). Entities are represented in information systems by classes (universals, types, kinds) and their instances (individuals, particulars, tokens), which exist in a specific region of space and time. Examples of entities include Ocean (a continuant type) and its instances, e.g., IndianOcean and PacificOcean, and Subduction (an occurrent type). Smith (2004) and Klein and Smith (2006) defined concept to refer to the meaning of a consensual, general term that has been agreed upon by domain experts and has been used in making scientific assertions. For example, the San Andreas fault in California is an individual of the Fault type (class). The string: “San Andreas fault” is a term (symbol) that refers to the actual, individual San Andreas fault, and it evokes the abstract fault concept, which has a welldefined meaning to the domain experts. The concept fault means “a planar discontinuity in rock along which there has been some displacement.” The concept ocean refers to the universal Ocean type, the largest body of water on Earth. The fault and ocean concepts refer to (i.e., stand for) the universal types Fault and Ocean that have instances (particulars, individuals) in reality, such as the San Andreas Fault and Pacific Ocean. Concepts do not exist; they are used to represent the universal types that can be instantiated in reality. Ontologies are not about concepts; they are models of individual entities that instantiate the universals in space and time, e.g., SanAndreasFault, NankaiTroughAccretionaryPrism. These two examples are instances of the Fault and AccretionaryPrism type, respectively. Universal ontological types are represented in information models as artifacts, such as classes in UML (Unified Modeling Language) diagrams, entities in entity relationship diagrams, elements in XML schema, and tables in databases. Thus, UML classes represent ontological types that have instances in reality, and they are given specific terms that refer to our concepts. It is imperative that we not think of ontologies as hierarchies


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of concepts, but of types and instances and the ways in which they are related. For example, the rock and water concepts are not related in dictionaries. In reality, however, rock can contain water in its pores, i.e., Pore partOf Rock and Pore contains Water (inverse of Water containedIn Pore). Notice that the types Rock, Water, and Pore are related at the universal level. At the instance level in reality, however, there are some real rocks (i.e., instances of the Rock type) that do not have pores and therefore do not contain water. Ontologies are depictions of both the universal types and the real relations that may exist among instances of these types, based on domain theories and knowledge. METAPROPERTIES Complex entities can be partitioned either through decomposition into subclasses (taxonomy) or into parts (partonomy) (Tversky, 1990). Class inclusion through taxonomy is based on similarity. Whereas the meronomic relations are between concepts, allowing no inheritance by the subclass, the taxonomic relations are within concepts, making it possible for a class to inherit properties from its superclass (Pribbenow, 2002). In contrast to the downward inheritance in a taxonomy, a partonomy may allow an upward inheritance, whereby a whole inherits from its parts. For example, an ultramafic rock inherits its dark color and high density from its mineral parts. A fold inherits its shape from the constituent layers; a molecule inherits its composition from its elements. Relations can be unary, binary, ternary, or n-ary (e.g., Smith, 2003; Smith et al., 2005). There are three general types of relations (Smith et al., 2005) that hold between: (1) classes, i.e., , e.g., isA; (2) a class and it instances, i.e., , e.g., instanceOf, and (3) instances, i.e., , e.g., partOf. In the following, some of the formal, primitive, foundational relations that obtain between different classes are described. Ontological relations are those that obtain between instances in reality independent of the ways in which we gain knowledge about them or represent them (Smith et al., 2005). Formal means that the relations are general and domainneutral, and primitive means that they are atomic and other relations can be derived from that. The metaproperties, which define the properties for properties, are defined next (e.g., Breitman et al., 2007). If the property that relates two classes is the same in both directions, we declare the property to be symmetric. P is symmetric if and only if, for any x and y, P(x,y) if and only if P(y,x). For example: “twinsWith” is a symmetric property (if x twinsWith y, then y twinsWith x). Symmetric properties must be declared as such (P rdf:type owl:SymmetricProperty). The inference for a symmetric property is as follows: P owl:inverseOf P. That is, the symmetric property is an inverse property. Property R is said to be inverse of property P if for any x, y, P(x,y) if and only if R(y,x). Many properties in one direction have an inverse property in the opposite direction but are named differently. For exam-

ple, the first property in each of the following pairs reverses the direction of the second property: analyzes and analyzedBy, investigates and investigatedBy, hasSample and sampleOf, wrote and writtenBy, and locatedIn and locationOf. The partOf property is an inverse property. This means that if fold has limb as part, then limb is part of fold (struc:Fold hasPart struc:limb; partOf owl:inverseOf struc:hasPart.). In mathematics, a relation P is said to be transitive, for any x, y, and z, if and only if P(x,y) and P(y,z), then P(x,z). This is represented by the owl:transitiveProperty and may be declared as follows: P rdf:type owl:TransitiveProperty. The inference for this property is as follows: If x P y and y P z, then x P z. For example, C partOf B, B partOf A, then C partOf A. For example, by being fractal, faults have segments that have smaller fault segments, which have even smaller segments, which are themselves faults, such that struc:FaultSegment struc:partOf struc:FaultSegment; struc:partOf rdf:type owl:TransitiveProperty; struc:FaultSegment rdfs:subClassOf struc:Fault. The locatedIn property is also transitive: geo:locatedIn rdf:type owl:TransitiveProperty. For example, if tect:SanAndreasFault geo:locatedIn geo:California, and geo:California geo:locatedIn geo:United States, then, geo:SanAndreasFault geo:locatedIn geo:United States. Property P is functional, for any x, y, and z, if P(x,y) and P(x,z), then y = z. It is the one for which there exists only one value. This is in analogy with mathematical function (y = 3x), which for any single input value (e.g., x = 2) returns one unique value (y = 6). For example, the location of a sample given by its longitude and latitude is a functional property. The owl:FunctionalProperty and owl:InverseFunctionalProperty allow merging data for a same individual from different sources. The owl:FunctionalProperty can only take one value for any individual, allowing sameness to be inferred. The inference rule for this construct is as follows: if P rdf:type owl:FunctionalProperty, X P A and X P B, then A owl:sameAs B. Property P is inverse functional, for any x, y, and z, if P(y,x) and P(z,x), then y = z. The owl:InverseFunctionalProperty, which is the inverse of the owl:FunctionalProperty, is very useful for merging data from different sources. This property is equivalent to the key in relational databases, such as social security number and driver’s license number. The inference rule of this construct is as follows: if P rdf:type owl:InverseFunctionalProperty, and A P X, B P X, then A owl:sameAs B. For example, assume that the location of any measurement is uniquely identified by the longitude and latitude (defined by the location class), i.e., no two samples can be taken at the same exact spot, given :Station134 :locatedAt :LocationA and :Station346 :locatedAt :LocationA, we infer that :Station134 owl:sameAs :Station346. For a one-to-one relationship, we use a combination of the owl:FunctionalProperty and owl:inverseFunctionalProperty. For example, we can define sample ID to be unique. :sampleId rdfs:domain :Sample :sampleId rdfs:range xsd:Integer :sampleId rdf:type owl:FunctionalProperty :sampleId rdf:type owl:Inverse FunctionalProperty. So, any two samples with the same ID must

be the same sample!


Ontological relations and spatial reasoning in earth science ontologies Notice that not every functional property can also be an inverse functional property. For example, mineral composition can be functional only, because every mineral has a unique composition, but many individuals can share that same composition (if they belong to the same class) (compare with the hasFather property). Some properties can only be inverse functional, but not functional. For example, a single-author publication of an author (or description of an outcrop of a thin section by one person) may be inverse functional, because it only belongs to one person. The person can have several such publications or descriptions. INSTANTIATION RELATION Individuals (instances) are related to the universals (types) with the primitive isA relation, e.g., SanAndreasFault isA Fault. The isA relation, as a mechanism for subtyping in reality, provides for the specialization of a class in an information model. In scientific investigations, we deal with the individuals, not the universals. These instances can be of the continuant or occurrent types (Klein and Smith, 2006). As scientists, we study individuals such as the SanAndreasFault, or a specific specimen of a rock or water sample from a particular river (i.e., an instance of the River type). The instances stand in different relations to each other in the real world. At the class level, we may have: Mylonite isA Rock; Pore properPartOf Rock; Recrystallization potentialPartOf Deformation, Basin contains Water, and AccretonaryPrism tangentiallyConnectedTo subductingPlate. Notice that in reality a particular deformation (e.g., brittle deformation), somewhere in space and time, may not include recrystallization. The assertion Mylonite isA Rock implies that all instances of the type Mylonite are also instances of the type Rock. However, notice that only some instances of the type Rock are also instances of the Mylonite type. The assertion ForearcBasin potentiallyAdjacentTo AccretionaryPrism implies that all instances of the type ForearcBasin may be adjacent to some instance of the type AccretionaryPrism. Two conditions are needed for a type to be a subtype of another type (Smith, 2004): C isA D, if: C and D are universals, and for all times t, if anything instantiates universal C at t, then that same thing must also instantiate the universal D at t. A universal is anything that can be instantiated by an individual (particular). For example, if contraction in a subduction complex leads to the creation of an instance of a SplayFault in the accretionary prism, it also forms an instance of the Fault super type at the same time. Instantiation is represented, at the instance level, by the instanceOf relation, for example, c instanceOf C at t, which means that the continuant, particular c instantiates universal C at time t (Smith et al., 2005), e.g., BereaSandstone instanceOf SedimentaryRock. The binary instanceOf relation is written as Inst (c, C), or Cct, where the particular c is an instanceOf the universal C. Every universal has a particular (i.e., ∃c Inst [c, C]), and every particular is an instance of a universal (i.e., ∃C Inst [c, C]) (Donnelly et al., 2005). The symbol ∃ is the existential quantifier, which

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reads: there exists some (at least one). The instance relation can be used to define subsumption at the universal level: isA (C, D) = ∀c (Inst[c, C] → Inst[c, D]), which says that C is subsumed by D (i.e., C isA D), if every instance (c) of C is also an instance of D. The ∀ symbol is the universal quantifier, which reads: for all, if any. For example, isA (Pseudotachylyte, FaultRock), or isA (Silicate, Mineral). These can also be written as: Pseudotachylyte isA FaultRock, and Silicate isA Mineral. The universal-level assertion: C isA D means: for all c, t, if c instanceOf C at t, then c instanceOf D at t (Smith et al., 2005). For example, ThrustFault isA Fault, which is true only if instantiation of a structure of type ThrustFault, say in an accretionary prism, leads to the instantiation of the structure of type Fault at the same time in the prism. Thus, in the examples: FaultSegment isA Fault and Mylonite isA FaultRock, the instantiation of the FaultSegment or Mylonite leads to simultaneous instantiation of the Fault or FaultRock, respectively. Notice that the isA relation does not hold between the concepts (i.e., meaning of the terms); it holds between universals with actual instances in reality. When we assert that: ThrustFault isA Fault, we are not saying that the meaning of the term thrust fault is the same as the meaning of the term fault. The meanings of these two terms are of course different. The assertion means that the universal ThrustFault isA type of the universal Fault type. Thus, the isA relation is used here to mean subtyping between universals and between their corresponding instances. The isA relation represents a necessary, but not a sufficient, condition for an entity. For example, the universal type Mylonite is necessarily a FaultRock, but it has other properties that the more general FaultRock type does not have. A ThrustFault is necessarily a Fault, but a thrust has some properties that are unique to itself, that a general type of Fault may lack. In other words, it is not sufficient to say that a ThrustFault isA Fault or Mylonite isA Rock. Even though a human being is necessarily a mammal, it is not sufficient to say that a human being isA mammal (dogs are also mammals). There is a difference between dogs and humans even though both are necessarily mammals. The difference is represented by additional and unique object and relational properties of the subclasses in the ontology model. For example, an instance of the type Mylonite may have foliation, lineation, and zones of recrystallization that an instance of Rock may not have. PARTONOMIC RELATION Although most ontologies apply the isA relation for class hierarchies, the mereological (partOf, part-whole, partitive) relation is probably of equal value for building the hierarchical structure of ontologies (Pribbenow, 2002; Schulz and Hahn, 2005; Koslicki, 2008). Notice that, although OWL uses the owl:subclassOf property to construct the isA relation, it does not have a qualified name to construct the partOf or hasPart relation. Many entities of interest to Earth scientists are composite, i.e., aggregates made of several parts, which have complex spatial or spatio-temporal structures. The following is a discussion of


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the formal semantics of the non-OWL constructs such as partOf and hasPart properties. Composite entities can be separated into parts, which maybe spatial objects, temporal entities (events), or spatio-temporal entities such as processes. For example, a subduction complex above the plate-boundary fault is a composite whole made of several parts that include the accretionary prism, forearc basin, plate-boundary fault zone, and inner and outer slope basins. A strike-slip fault is a composite of many segments, steps, and bends. A ductile deformation (process) along a fault may contain several spatio-temporal parts, which may include subprocesses of recrystallization, recovery, or cataclastic flow. Each of these parts may have its own parts, e.g., the prism part includes the offscraped and underplated thrust sheets of sediment; a segment is made of many other segments, steps, and bends (because fault is fractal). The offscraped thrust sheets are bounded thrust faults that are members (i.e., parts) of a collection of faults (a whole). The plate-boundary fault zone, at the base of the prism, may have the following parts: seismogenic zone, mylonite zone, cataclastic zone, aseismic zone, and boundaries that include the décollement. Notice that even though the seismogenic zone may be part of some plate-boundary fault zone in a subduction zone, not all such fault zones have a seismogenic zone. Facts like this need to be included in the ontology. The recrystallization may include dynamic recrystallization by grain boundary migration or rotation, or static recrystallization. The partitive partOf relation, like the isA relation, only holds between universals or instances, not between concepts or classes in information models. The formal definitions for the partOf relation at the instance level is given by (Smith et al., 2005): c partOf d at t, which means that the particular, continuant c is an instancelevel partOf the particular, continuant d at time t. The equivalent universal-level assertion: C partOf D means that for all c, t, if c instance of C at t, then there is some d such that d instanceOf D at t, and c partOf d at t. Notice the all-some structure in these definitions (given in italic font) (Smith et al., 2005). For example, Xenolith partOf IgneousRock means that all xenoliths, if any exists at any time anywhere, should be part of some igneous rock. This reflects the domainal fact that only some igneous rocks have xenolith, and that xenoliths do not make any sense if they are not part of an igneous rock. Smith (2004) defined the partitive partOf relation as a combination of the partFor and hasPart relations, which are defined as follows. The universal assertion X partFor Y provides information mainly about X, and it asserts that if x (an instance of X) exists at time t, then y (an instance of Y) also exists at t, and that x exists only as a partOf y (i.e., at instance level). For example, UnderplatedSediment partFor AccretionaryPrism, means that if there is an instance of underplated sediment at time t, then it is a part of an instance of an accretionary prism that exists at the same time. This means that underplated sediment does not exist (or mean anything) if there is no prism. It does not deny the fact that accretionary prisms may have no underplated sediment. The assertion Y hasPart X, on the other hand, provides information about Y, and asserts that if y (an instance of Y) exists at time t, then x (an instance

of X) exists at the same time as a partOf y (i.e., at the instance level). This means that y does not exist unless it has an instance of X as part. For example, NankaiTroughAccretionaryPrism hasPart AccretedSediment, which means that the accretionary prism cannot exist at time t (e.g., today) if it does not have accreted sediment as a part. Thus, X partOf Y holds, by combining the two assertions, if and only if for any instance x of X existing at time t, there is some simultaneously existing instance y of Y, such that x is an instance-level part of y, and y has x as part. For example, SubductingPlate partOf SubductionZone, which means that if the PhilippinePlate is an instance of the SubductingPlate and the NankaiTrough is an instance of the SubductionZone at the present time, then the subducting Philippine plate can only exist today as a part of the present-day Nankai Trough subduction zone. The relationship between parts and the whole that composes or aggregates the parts is the subject of the formal theory of part-whole structure called mereology (e.g., Simon, 1987; Varzi, 2003). The mereological partOf relation (denoted as P) and its inverse, hasPart (denoted as P-1), that obtain between two individuals (a part and its whole) constitute the fundamental relations for composite entities (e.g., Varzi, 1996; Casati and Varzi, 1999; Pontow, 2004; Schulz and Hahn, 2005; Koslicki, 2008). The relations may hold in the following cases (e.g., Schulz et al., 2006): between material and nonmaterial objects, parthood over time, parthood and spatial location, and parthood between occurrents (Schulz et al., 2005, 2006). For example, Mineral partOf Rock (or Rock hasPart Mineral) signifies that an instance of the class Mineral (which represents individual minerals in reality in a domain model) is a part of an instance of the class Rock at a specific instant of time, t. In UML the isA and partOf relations are represented by subclassing and composition (filled black diamond) or aggregation (open diamond), respectively. The partitive partOf relation is needed, along with other semantic relations such as attribution (e.g., thickness of the forearc sediments), class-subclass (e.g., ThrustFault isA Fault), spatial inclusion (e.g., NankaiTrough locatedIn Japan), and connection relation (e.g., ForearcBasin externallyConnectedTo Prism), for a better representation of reality (e.g., Donnelly et al., 2005). Partitive relations in ontologies, designed to portray the reality in natural systems, hold between universals and then are applied to individuals using constraints based on domain knowledge. We need to make a distinction between parthood at the universal level, i.e., PartOf (A, B) versus that at the instance or individual level, i.e., partOf (x, y) (Schulz et al., 2006). For example, compare PartOf (PlateBoundaryFault, SubductionZone) versus partOf (NankaiTroughPlateBoundaryFault, NankaiSubductionZone). The class-level PartOf (A, B) means that the universal A is part of universal B if every instance of B has some instance of A as part, and every instance of A is part of some instance of B. Schulz et al. (2006) introduced the ternary relation Inst, which relates an individual to a universal at time t, i.e., Inst (x, A, t). The formal definition of the class-level PartOf (A, B) is then given based on the Inst (x, A, t) as follows: ∀x,t Inst (x, A, t) ∃ y


Ontological relations and spatial reasoning in earth science ontologies Inst (y, B, t) ∧ p (x, y, t), which reads: for all x and t, x is an instanceOf A at time t, if there exists a y, where y is an instanceOf B at t, and x is a partOf y at t. For example, Inst (SanAndreasFault, StrikeSlipFault, t) means that the San Andreas fault is a strike-slip

fault at a given time; it may become a thrust fault at another geological time if the stress field changes due to plate reconfiguration. The definition of the universal-level HasPart (A, B) relation (which is an inverse of the universal PartOf [A, B] relation) is as follows: ∀y,t Inst (y, B, t) ∃ x Inst (x, A, t) ∧ p (x, y, t). The classlevel PartOf (A, B) can be interpreted in the following different ways (Schulz et al., 2006), which need to be clarified using constraints such as cardinality in the ontology model. It means that all instances of A are partOf some instances of B, for example: PartOf (underplatedSediment, AccretionaryPrism), i.e., there are some prisms in which there is no underplated sediment as part. It means that all instances of B have some instances of A as part, for example: PartOf (Bed, Formation). It means that all instances of A are partOf some instances of B and all instances of B have some instance of A as part. For example, PartOf (OuterSlopeBasin, AccretionaryPrism). It implies that there is at least one instance of A which is partOf some instance of B, for example, PartOf (SplayFault, AccretionaryPrism). Notice, however, that not all splay faults are part of accretionary prisms, and not all prisms have splay faults. The relationship between parts and the whole may be functional, structural, spatial, or based on whether or not parts are separable from the whole, or are homeomerous (Lambrix, 2000). An example of a functional partOf relation is that between cement/matrix (parts) and the sediment (a whole); the function of cement is to hold the grain together. The function of pores or open fractures (parts) in a rock (a whole) is to store or transmit fluids, respectively. Parts are separable if the whole survives after the parts are separated from it, for example, when dissolved ions (parts) are removed from water (the whole). Parts are homeomerous if they are the same kind of thing as their whole, e.g., calcite or quartz crystal (part) in a monomineralic marble or quartz quartzite (whole), respectively. Specimens taken from a sample of a core of granite are homeomerous with the sample, and with the core itself. Gerstl and Pribbenow (1996) divided a composite whole into three end-member types: heterogeneous complex, uniform collection, and homogeneous mass. The mereological partOf relation is further extended (i.e., specialized) by the following relations (Lambrix, 2000). The componentOf relation implies the existence of a complex, heterogeneous structure in the whole, and functional, structural, spatial, temporal, and other relations among the components (Gerstl and Pribbenow, 1996), and between components and the whole. The parts (i.e., components) are separable and nonhomeomerous, and they have spatial and temporal relations to the whole. Examples: UnderplatedSediment componentOf AccretionaryPrism; ForearcBasin componentOf SubductionComplex; PlateBoundaryFaultZone componentOf SubductionComplex. Both prism and the subduction complex, in these cases, are heterogeneous, complex wholes, with several

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internal components that have spatial and temporal relations to each other. The memberOf (elementOf) relation relates all members (parts) to a compositionally uniform collection (a whole), in the same way, without any functional or subclassing implication. The parts in this case have membership based on spatial proximity, not type (i.e., are not subclasses of the whole), and they are separable and nonhomeomerous. Examples: RiserCore memberOf CoreCollection of the boreholes in the NanTroSEIZE project in the Nankai Trough. For a splay fault propagating from the plate-boundary fault: SplayFault memberOf SplayFaultCollection. FractureSet memberOf FractureSystem, for the sets of fractures forming in shear zones. SiteNT3-01A memberOf NanTroSEIZESiteCollection. The portionOf relation relates a homeomerous, separable, and nonfunctional part to a homogeneous whole. For example, WorkingCoreSplit portionOf Core represents the portion of the core, other than the archive core split, that is set aside for different analyses. Other examples include: SpotCoreSample portionOf RoundCoreSample; DrillingMudSample portionOf DrillingMud. The nonfunctional, separable, and nonhomeomerous stuffOf relation holds between a whole and the material that it is partly or wholly made of. Examples: Rock stuffOf Core, in addition to other stuff, such as air, water, and mud; Mineral stuffOf Vein; IgneousRock stuffOf Dike. There are several criteria that can be used to identify genuine parthood of entities (Smith et al., 2005). The main criterion is sortality, which means that parts need to be of the right type to enable instantiation of the whole. For example, a drilling pipe stuck in a bore hole is not part of rock; monitoring equipment in a borehole is not part of the borehole, they are containedIn it. Metal and plastic cannot be part of rock. A question arises as to whether a piece of meteorite embedded in a clastic sedimentary rock (e.g., mudstone) is part of the rock, even though meteorite and mudstone are not of the same type. In this case, the meteorite is a clast, and clasts are parts of the clastic rock. It is possible for the provenance to help in this case, where a meteorite that is extraterrestrial in origin does not fit the terrestrial origin of other clasts in the mudstone. The function of a part may be essential to the functioning of the whole. For example, the cement of a conglomerate is a part of the conglomerate because it is holding the clasts together. If there is no cement, the conglomerate would be another entity (loose gravel aggregate). A part may be a structural element of the whole. For example, one or more limbs of a fold are needed for a fold to exist. In most cases, the life cycle of the part and the whole correspond to each other. For example, quartz, feldspar, and mica are parts of a granitic rock. However, the feldspar may be altered into clay at a time when other components (i.e., mica and quartz) are still parts of the granite. There may be exceptions to the life cycle rule if temporary parthood occurs. For example, the seismogenic zone in the plate-boundary fault zone under the accretionary prism may migrate into the prism and become part of the prism at a later time.


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The relata of the part-whole relations, in addition to objects, can be activities and processes. For example, the phaseOf or stageOf relation holds between different (spatio-temporal) parts of activities or processes. The phaseOf and stageOf relations are functional but not separable or homeomerous. Activity in this case involves human or machine agents, and it may be sampling, examination, simulation, or drilling. Examples: Stage3 stageOf NanTroSEIZEDrilling; PaleomagneticMeasurement phaseOf NonDestructivePhysicalPropertyMeasurement; Microscopic Measurement phaseOf OnBoardAnalysis. All nonabstract objects occupy spatial and temporal regions. The placeOf relation holds

between an area (a whole) and the places that are located in it. Examples: Japan placeOf Nankai Trough. LOCATION AND CONTAINMENT RELATIONS The locatedIn relation (inverse of the placeOf relation) holds between a continuant and a spatial region (r) at time t (i.e., c locatedIn r at t), i.e., it depends on a function (Galton, 2000, 2004; Donnelly, 2004a), which assigns a region r(c, t) that any continuant instance c exactly occupies at time t. Thus, c locatedIn d at t means that r(c, t) partOf r(d, t) at t (Smith et al., 2005), which means c is locatedIn d if c’s region is part of d’s region (Donnelly, 2004a). Example: Vein locatedIn fracture at time t, Mylonite locatedIn ShearZone at time t. In all cases, scientists measure instances of the continuants at the present time, and therefore the present time is implied in the assertions (i.e., t is dropped). However, the present time becomes part of the past as new measurements are done in the “present times,” i.e., time instants of the future. Many of the locatedIn relations actually represent instantaneous parthood, for example, the SeismogenicZone partOf PlateBoundaryFault only applies at a specific time, and it may not be true at other times because the zone may migrate with time. Symbolically, the location relation is given as: locatedIn (c, d) = P(r[c], r[d]), which reads: c is locatedIn d if region c is partOf region d. Thus, if x is partOf y, then x is locatedIn y. Examples: if Mylonite locatedIn ShearZone at time t, then the region of Mylonite is partOf the region of the ShearZone at time t. If VolcaniclasticSediment is locatedIn ForearcBasin at t, then the region of VolcaniclasticSediment is partOf the region of the Forearc Basin at t. At the universal level, the assertion C locatedIn D means for all c, t, if c instanceOf C at time t, then there is some d such that d instanceOf D at t, and c locatedIn d at t. For example, at the present time: SanAndreasFault partiallyLocatedIn California, or YellowstoneHotSpot partiallyLocatedIn Wyoming. We can derive the following transitive relations from the locatedIn relation (Donnelly, 2004c): if c is locatedIn d and d is locatedIn z, then c is locatedIn z. Example: Mg2+ ion locatedIn water sample, which occupies a fracture, is located in the fracture. If c is part of d and d is locatedIn z, then c is locatedIn z. Example: A fracture that is part of a thrust sheet located in the accretionary prism is located in the prism. If c is locatedIn d and d is partOf z, then c is locatedIn z. Example: If volcaniclastic sediment locatedIn

the Kumano forearc basin sequence, and the forearc is partOf the Nankai Trough, then the volcaniclastic sediment is partOf the trough. If two objects coincide partially or wholly without being part of one another, then we use the coincidence relation, which is another kind of location relation. In other words, a continuant may exist in a spatial region occupied by another continuant of which it is not a part. In this case, the first entity may coincideWith but is not locatedIn the larger entity, which means that it is not part of the second entity. Examples are: water in an open fracture, and ions or contaminants in the pores of sediment or an aquifer. In all of these cases, there is no parthood relation, just spatial coincidence. It is not necessary for a fracture or pore to have water or a contaminant. Object x is said to overlap object y if x and y share a common part z (Pontow, 2004). In other words, Oxy = ∃z (Pzx ∧ Pzy). Examples: NankaiAccretionaryPrism overlap Nankai Trough. Object x is discreteFrom object y if x does not overlap y, i.e., Dxy = ¬Oxy. Example: SubductingPlate discreteFrom ForearcBasin; SlopeBasin discreteFrom ForearcBasin. In the partial coincidence (PCoin [x, y]) case, the regions of the two objects, x and y, overlap without a part-whole relation (i.e., ¬Oxy). Thus, PCoin (x, y) = Or(x)r(y). Partial coincidence is reflexive, PCoin (x, x), which means that any object partially coincides with itself, and it is symmetric, PCoin (x, y) = PCoin (y, x), which means that if x partially coincides with y, then y partially coincides with x. For example, ForearcBasin partiallyCoincidesWith AccretionaryPrism, which means that even though the spatial regions of both of these spatial entities overlap, the forearc basin is not part of the prism. Partial coincidence is more common than the total (whole) coincidence. The coincidesWith and locatedIn relations are related: locatedIn (x, y) → PCoin (x, y), i.e., if x is locatedIn y, then x coincides with y. Other relations include the containedIn, which obtains between a material continuant and a site, i.e., empty space that can contain the object. For example, ZeoliteCrystal containedIn Vug; Water containedIn Fracture; Contaminant containedIn Pore. Again, like the partOf relation, the containedIn relation holds at a certain time index, t. Some composite spatial entities are derived from other entities through the Boolean-like operations of sum, product, and complement, which are equivalent to the set theory’s union, intersection, and complement, respectively (Galton, 2004). For example, a fracture system in the plate-boundary fault zone is a sum of several variably oriented fracture sets of different types (e.g., Riedel shear, Y-shear, and P-shear fractures). This is true for onedimensional fracture traces and two-dimensional fracture planes. An example in three dimensions is the accretionary prism, which is the sum of all thrust sheets of accreted and underplated sediments and rocks. The derivesFrom relation obtains between two nonidentical individuals. The assertion c derivesFrom c1 means that continuant object c is derived from the continuant object c1 at some time t, and that c1 does not exist anymore. At the universal level, we


Ontological relations and spatial reasoning in earth science ontologies can have Gneiss derivesFrom Granite. There are three types of the derivesFrom relations: (1) Continuation, where an instance of a continuant succeeds another at time t. (2) Fusion, where two or more continuants fuse into a new continuant (earlier continuants cease to exist; examples include water and ions crystallizing into a new mineral at t; two magmas mixing at instant t; P- and Y-slip surfaces merging and becoming a new slip surface). Unification (Smith and Brogaard, 2003), which is a close relative of the fusion relation, in which two or more continuants join, but continue to exist, in a new complex continuant (for example, pebbles of many older rocks unifying in a conglomerate; porphyroclasts of older minerals unifying in a mylonite). (3) Fission, where a part of a continuant breaks into two or more other continuants at time t, which will exist on their own. Example: metamorphism of a rock leading to the formation of a new mineral at the expense of several older minerals (parts) that existed in the rock. The adjacentTo (Donnelly et al., 2005) relation is a proximity (topology) relation that holds between two disjoint continuants. The transformationOf relation represents change in the continuants over time. It obtains between an instance of a class C at time t, which used to be an instance of another disjoint class C1 at an earlier time t1. Examples: QuartzofeldspathicMylonite transformationOf Granite; Soil transformationOf Rock at t. The absorb relation obtains when a continuant continues to exist but absorbs another continuant, which ceases to exist on its own, for example, Mineral absorb Water, which means a mineral may absorb a water molecule, H2O as hydroxyl ion, OH–1, or an ion into its crystalline structure. Rector et al. (2006) introduced the notion of collectives and granular parts, which relate closely to the formalization of entities such as rocks and sediment that are made of many parts. Such collective wholes (e.g., rock, sediment) have “emergent” properties that do not exist in the individual parts (minerals, grains), for example, the emergent property of the silicate chain structure when the silicon-oxygen tetrahedra connect to each other to make a whole molecule. The emergent property of the silicate mineral (a collective) is not the same as the individual atoms (silicon, oxygen). A collective of grains in a sedimentary rock has emergent properties such as porosity, hydraulic conductivity, texture, and fabric that do not make sense for individual grains. Collectives are themselves a part of larger collectives, for example, minerals, which are collectives of atoms, are part of larger rock collectives, which are themselves part of even larger collectives organized in different packages such as lithostratigraphic rock units, members, formations, groups, and sequences. The emergent properties of a lithostratigraphic rock unit may include such things as its anisotropy and homogeneity in all sorts of physical properties. Rector et al. (2006) distinguished two types of subrelation under the parthood relation related to collectivity: granular parthood and determinate parthood. The collectives are aggregates of grains that play the same role in the whole, and they do not depend on the number of grains in the collective. For example, the relation of grains in a layer of sediment is a granular part-

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hood. In this case, there is an indeterminate number of grains in the layer, and removal of one part (i.e., grain) does not necessarily diminish the whole (sedimentary layer). Compare this with the relation of a collective of several tectonostratigraphic units of different ages separated by unconformities or faults (e.g., in an accretionary prism). In this case, which represents the determinate parthood, the removal of any of the parts (e.g., thrust sheet), which are limited in number, will necessarily diminish the whole, and the integrity of the collective will be lost. DISCUSSION AND SUMMARY Continuant entities that exist in any domain of the natural world stand in a variety of ontological relations to each other, including instantiation, parthood (P), overlap (O), location (LocIn), containment, and adjacency. These types of binary relations, denoted by R (Smith and Rosse, 2004; Donnelly et al., 2005), can hold between universals or particulars of any earth science discipline. We use consensual, agreed-upon terms to define the universals in our domain, and we represent them as classes in information systems. The relations that exist among the instances of the universal may not exist among the concepts (terms) that represent them. While only one instance of a material, universal type can occupy a unique spatial region in a specific temporal region, many instances of a given universal type can synchronously exist in different spatial regions. The spatial regions of these simultaneously existing instances of the same universal type may or may not overlap. Smith and Rosse (2004) and Donnelly et al. (2005) introduced a refinement of each of the binary relations (R) of parthood (P), overlap (O), instantiation, and location (LocIn). In the following, R can stand for P, O, LocIn, or other types of binary relations. Donnelly et al. (2005) defined R1(A, B), R2 (A, B), and R12 (A, B) relations among universal types A and B, depending on whether the restriction is put on the first argument (A) or the second (B). These R-structures are defined next (notice the allsome structure in the three cases). R1(A, B) = ∀x {Inst (x, A) → ∃y (Inst[y, B] ∧ Rxy)}, i.e., A is related to B (e.g., by instantiation, parthood, location) if all instances of A are related to some instances of B. This means that each A stands in R relation (e.g., proper parthood) to some B. Notice that the emphasis is on A, i.e., something is true about A; hence subscript 1 in R1 to emphasize the first argument. For example the assertion: PP1(ForearcBasin, SubductionComplex) means that each forearc basin is a properPartOf some subduction zone. This does not say that each subduction zone must have a forearc basin as proper part. The assertion: O1(SeismogenicZone, PlateBoundaryFaultZone) means that every seismogenic zone overlaps (i.e., shares a common part with) some plate-boundary fault zone under the prism. This assertion does not say that each plate-boundary fault zone must overlap a seismogenic zone. O1(SplayFault, Décollement) means that every splay fault overlaps some décollement, but not the other way around. LocIn1(Vein, Fracture) asserts that every vein is located in some fracture. It does not mean that every fracture


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has vein in it. PCoin1(AccretionaryPrism, SubductingPlate) asserts that every accretionary prism partially coincides with some subducting plate, but every subducting plate does not have to partially coincide with an accretionary prism. Notice that we can assert PP1(AccretedSediment, AccretionaryPrism), but not PP1(UnderplatedSediment, AccretionaryPrism). We can assert PCoin1(Trench, SubductingPlate) but not PCoin1(SubductingPlate, Trench). B2(A, B) = ∀y {Inst (y, B) → ∃x (Inst[x, A] ∧ Rxy)}, i.e., the relation between A and B stands if all instances of B are related to some instance of A. Here, restriction is on the second argument (B), i.e., it says that for each B, there is some A that stands in R relation to it, for example, PP2(Sediment, ForearcBasin), which states that each forearc basin has some sediment as a proper part. Notice that the statement does not say that each instance of sediment is a proper part of a forearc basin. The assertion: O2(AccretionaryPrism, SlopeBasin) states that every slope basin overlaps some accretionary prism. However, it does not assert that every accretionary prism overlaps some slope basin. LocIn2(Sediment, ForearcBasin) states that every forearc basin contains sediment. However, not all sediments are located in forearc basins. PCoin2(SubductionZone, UnderplatedSediment) states that every underplated sediment partially coincides with some subduction zone. It does not say every subduction zone partially coincides with underplated sediment. R12(A, B) = R1(A, B) ∧ R2(A, B) conjuncts the above two cases and states that each instance of A stands in R relation to some B, and each instance of B stands in R relation to some instance of A. In this case, the restriction is on all instances of both A and B. For example, PP12(SubductingPlate, SubductionZone), which says that each subducting plate is a proper part of a subduction zone, and each subduction zone has a subducting plate as a proper part. O12(SubductingSediment, PlateBoundaryFault) asserts that every subducting sediment overlaps some plate-boundary fault, and every plate-boundary fault overlaps some subducting sediment. LocIn12(SubductionZone, Ocean) states that every subduction zone is located in some ocean, and every ocean has some subduction zone. PCoin12(TransformFault, MidOceanRidge) asserts every transform fault partially coincides with some mid-ocean ridge, and every mid-ocean ridge partially coincides with some transform fault. These R structures, defined by Donnelly et al. (2005), provide a powerful means for spatial reasoning. It can be concluded that a complete and comprehensive representation of knowledge in a specific earth science domain requires the application of these R structures. The axioms of the ontologies in these domains need to differentiate among R1, R2, and R12 structures to enable effective spatial reasoning. REFERENCES CITED Allemang, D., and Hendler, J., 2008, Semantic Web for the Working Ontologist: Effective Modeling in RDFS and OWL: Amsterdam, Morgan Kaufmann Publishers, 330 p.

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Simon, P., 1987, Parts: A Study in Ontology: Oxford, UK, Clarendon Press, 390 p. Sinha, K., ed., 2006, Geoinformatics: Data to Knowledge: Geological Society of America Special Paper 397, 288 p. Smith, B., 2001, Fiat objects: Topoi, v. 20, no. 2, p. 131–148, doi:10.1023/A: 1017948522031. Smith, B., 2003, Ontology, in Floridi, L., ed., Blackwell Guide to the Philosophy of Computing and Information: Oxford, UK, Blackwell, p. 155–166. Smith, B., 2004, Beyond concepts: Ontology as reality representation, in Varzi, A., and Vieu, L., eds., Formal Ontology and Information Systems: Amsterdam, IOS Press, p. 73–84. Smith, B., and Brogaard, B., 2003, Sixteen days: The Journal of Medicine and Philosophy, v. 28, p. 45–78, doi:10.1076/jmep.28.1.45.14172. Smith, B., and Grenon, P., 2004, The cornucopia of formal-ontological relations: Dialectica, v. 58, p. 279–296, doi:10.1111/j.1746-8361.2004 .tb00305.x. Smith, B., and Rosse, C., 2004, The role of foundational relations in the alignment of biomedical ontologies, in Fiechi, M., et al., eds., Proceedings MEDINFO 2004: Amsterdam, IOS Press, p. 444–448. Smith, B., Ceuster, W., Kellages, B., Kohler, J., Kumar, A., Lomax, J., Nungall, C., Neuhaus, F., Rector, A.L., and Rosse, C., 2005, Relations in biomedical ontologies: Genome Biology, v. 6, p. R46, doi:10.1186/gb-2005-6 -5-r46. Tobin, H.J., and Kinoshita, M., 2006, NanTroSEIZE: The IODP Nankai Trough Seismogenic Zone Experiment: Scientific Drilling, v. 2, p. 23–27. Tversky, B., 1990, Where partonomies and taxonomies meet, in Tsohatzidis, S.L., ed., Meanings and Prototypes: Studies in Linguistics and Categorization: New York, Routledge, p. 334–344. Varzi, A.C., 1996, Parts, wholes, and part-whole relations: The prospect of mereotopology: Data and Knowledge Engineering, v. 20, p. 259–286, doi:10.1016/S0169-023X(96)00017-1. Varzi, A.C., 2003, Mereology, in Zalta, E.N., ed., Stanford Encyclopedia of Philosophy: CSLI (Center for the Study of Language and Information) Internet publication: http://plato.stanford.edu/contents.html (accessed May 2011). MANUSCRIPT ACCEPTED BY THE SOCIETY 17 FEBRUARY 2011

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