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1. Geoinformatics Mereotopology

2. Mereotopology • The formal theory for parthood and connection relations is called mereotopology • Mereotopology, built on mereology and some elements of topology, is about the contact of spatial entities whose boundaries are collocated • i.e., there is a point or area on their boundary interface at which the two objects touch • Mereotopology allows us to formulate ontological laws related to boundaries and interiors of wholes, to relations of contact and connectedness, and to concepts of surface, point, and neighborhood

3. Coincidence of Processes • Although traditionally used for reasoning about spatial relations among material objects and their region, mereotopology has been extended to deal with other types of coincident but non-overlapping entities, including qualities, processes, and holes • In the same sense, processes coincide the spatio-temporal regions that they occupy • For example, cataclasis coincides with the specific region in a shear zone over a given time interval • Processes can also coincide other processes, for example heating of a mass of rock coincides with (but is not part of) the thermal expansion of the rock • Holes, such as pore spaces in a rock, can become partially or completely coincident with the fluids (e.g., oil, water) that fill them

4. Why Mereotopology? • The axiomatics of mereotopology can significantly contribute to the building of effective, formal ontologies of the spatial and spatio-temporal entities in a domain • The formal ontological relations that may exist between entities in a domain include those of: identity, difference, parthood, overlap, inherence, dependence, participation, and location • The rules of inference and reasoning based on the axioms defining such relations will help build and query efficient knowledge bases

5. Mereology (Part-Whole) • Real objects, such as rock, molecule (e.g., Mg2SiO4), fault, fold, and river, are mostly, if not all, composites made of parts • These objects are called mereological complex, composite or compound object, or wholes • Mereology is the study of parts and wholes • The ontological parthood relation between two particular objects x and y is denoted by Pxy, or alternatively as P(x, y), which reads: x is a part of y, e.g.: part-of (axis, fold)part-of (seismogenicZone, plateBoundaryFault) Rock partOf Mineral

6. Endurants vs. Perdurants • The entities in any domain of discourse are of two types: continuant • perdurant • Continuant (endurant) entities endure or persist through time by being fully present at different times • They have spatial parts • Examples of continuants are aquifer, mineral, and river • Occurrent entities (perdurants) persist through time by having different temporal parts (phases) at different times • Examples of occurrents, that include processes and events, are deformation, flow, and diffusion

7. Spatial Parthood • Parthood for continuants depends on time, i.e., an entity (x) is a spatial part of a whole (y) during certain phases of whole’s lifespan • This modified version of the part-of relation for continuants is given by: part-of (x, y, t) partOf (SeismogenicZone, PlateBoundaryFaultZone, t) where t is the phase during the life of the plate-boundary fault when the fault zone is partly (i.e., locally) seismogenic due to the qualities (state of stress, strain rate, pressure, temperature) of the local spatial region of the fault zone

8. Concepts of Mereology • The concepts of the standard mereology include proper parthood (PP), improper parthood (P), overlap (O), disjointness (D), product, sum, difference, and complement • The proper parthood (PP) obtains between a part and a whole when the part is not the same thing as the whole itself, which is very common in natural systems • By definition PPxy = Pxyxy, which reads: x is a proper-part-of y, if x is part-of y, and x is any part-of y other than itself

9. Proper-part-of • The inverse of the proper-part-of (x, y) ishas-proper-part (y, x) which is denoted as PP-1yx • Two distinct objects cannot have the same proper parts, and a whole that has one proper part must have others • For example, a river delta is a proper-part-of a river, but is not the same thing as the river, i.e., PP (Delta, River) • The seismogenic zone of a subduction zone is a proper-part-of the PlateBoundaryFaultZone, but is not the same thing as the plateBoundaryFaultZone itself

10. x < y • The relation x < y, which means x is a proper part of y, is irreflexive, asymmetric, and transitive • For example, a cutoff bank is a proper part of a river • The Mg ions are proper parts of the olivine molecule (Mg2SiO4) • Irreflexivity states that nothing is a proper part of itself, i.e.,(x < x), or alternatively as PPxx • Asymmetry asserts that if an object is a proper part of a second object, then the second object cannot be a proper part of the first object (x < y) (y , x), or alternatively: PPxyPPyx, orPPxyPxyPyx • For example, if the Earth is a proper part of the Solar System, the Solar System cannot be a proper part of the Earth

11. PP is antisymmetry, transitive • Notice that if x is part-of y and y is part-of x, then x and y are the same thing, i.e., they are identical, i.e., PxyPyx x=y (antisymmetry) • The transitivity means that if an object is a proper part of a second object, and the second object is a proper part of a third object, then the first object is a proper part of the third object (x < y  y < z)  (x < z) • Notice that an alternative way of writing the transitive axiom for proper parts is: PP (x, y)  PP (y, z)  PP (x, z) or PPxyPPyzPPxz • If a xenolith is a proper part of an intrusion, and the intrusion is a proper part of a pluton, then the xenolith is a proper part of the pluton

12. x  y • The proper or improper parthood, denoted with the  symbol, holds when an object is either a proper part of a second object or identical to it (i.e., x  y) • The x  y relation is reflexive, non-symmetric, and transitive • Any object is an improper part of itself (x  x) • Non-symmetry: if an object is a proper or improper part of another object, then there are some cases in which the second is also proper or improper part of the first, and in other cases the second is not also a proper or improper part of the first. This is given by the axiom: (x)(y) (x  y  y  x)  (x)(y) (x  y y  x) • Transitivity: if an object is a proper or improper part of a second object, and the second object is a proper or improper part of a third object, then the first object is a proper or improper part of the third object (x  y  y  z)  (x  z)

13. Transitivity • The three ontological axioms: • Everything is part of itself (reflexivity) • Two distinct things cannot be part of each other (antisymmetry) • Any part of a part of a thing is itself part of that thing (transitive)

14. Transitivity • The transitivity axiom is especially useful for faults because of their fractal geometry • In this case, a bend or step (x), which is a part-of a fault segment (y), which is itself a part-of a larger fault (z), is also part-of the large fault at time t, i.e., PxyPyzPxz at t part-of (FaultStep, FaultSegment)  part-of (FaultSegment, Fault)  part-of (FaultStep, Fault) • If a fluid inclusion (x) is part-of a quartz crystal (y) in a vein (z), it (i.e., x) is also a part-of the vein (z) at time t: part-of (FluidInclusion, Quartz)  part-of (Quartz, Vein)  part-of (FluidInclusion, Vein)

15. Has-part • The has-part, also denoted as: part-of -1, is the inverse of the part-of relation, and may be written as: Pxy P-1yx , or alternatively as: part-of (x, y)  has-part (y, x) has-part (Vein, FluidInclusion)has-part (AccretionaryPrism, ThrustSheet)has-part (Formation, Member)has-part (SubductionComplex, UnderplatedSediment) • Of course, an accretionary prism may not have any underplated sediment, and therefore, this partitive relation should be refined, or defined more strictly

16. Mineral isA Meronomy vs. hyponomy (Part-of vs. subclassOf) Silicate • Classes whose individuals are part-of individuals of another class should be modeled with mereology role (part-of), not specialization (is-a) • The specialization should only be used if every instance of the subclass is also an instance of the superclass(class A is a subclass of B if every A is a B) • FaultSegment is-a Fault is correct, i.e., every segment of a fault is itself a fault • Silicate is-a Mineral, i.e., every individual silicate is also a mineral Rock partOf Mineral

17. Which one to use? Rock partOf Phenocryst • We cannot say that the phenocrysts in a porphyritic igneous rock are the same thing as the igneous rock itself • These grains are actually part of the rock • We can verify (with the instance test) if a relation is a subsumption by asking if every instance of the subclass is also an instance of the superclass • If it is, we use the is-a relation, otherwise, we may use the part-of relation • Notice that in some cases, grains in a rock, in addition to be part of the rock, are themselves rocks • For example, gravels in a conglomerate, in addition to be part of the conglomerate, may be rock or mineral, among other types. These relations should be captured in the ontology.

18. Rock Examples to clarify the difference: isA IgneousRock IgneousRock Rock SedimentaryRock Rock IgneousRockSedimentaryRocki.e., IgneousRockowl.disjointWithSedimentaryRock Conglomerate SedimentaryRock Conglomerate has.Graini.e., conglomerates have grains Grain partOf.Rock i.e., grain could be part of any kind of rock Grain  Mineral i.e., grain can be a mineral Grain  Rock i.e., grain can be a rock

19. Examples … • Mineral  Rock  • (i.e., rock and mineral are disjoint) • PhenocrystpartOf.IgneousRock • (i.e., some ignoues rocks have phenocryst) • Matrix partOf.SedimentaryRock • (i.e., some sedimentary rocks have matrix) • GroundMasspartOf.IgneousRock • (i.e., some igneous rocks have groundmass) • Rock (SedimentaryRock) • SedimentaryRock (limestone) (an instance) • Conglomerate (BathtiyariConglomerate) (an instance) • Grain (limestone) (an instance)

20. Entailments • Does this ontology entail that limestone is a rock or is part of a rock • SedimentaryRock (limestone) • By being a sedimentary rock, the instance of limestone is a rock • SedimentaryRock (limestone) • Rock (SedimentaryRock) • However, a grain can also be a mineral, but we have specified that limestone is a sedimentary rock • The ontology also entails that limestone, by being a grain, can also be part of a conglomerate which is a sedimentary rock, which is a rock

21. Equivalence vs. Subsumption • Confusing class equivalence (owl : equivalentClass) with subsumption (rdfs : subClassOf) is a common error • Equivalence is used when: • Two or more classes were defined in different domains, which satisfy the same necessary and sufficient conditions of a class definition • A class can be defined by restricting an existing classes in the same ontology (e.g., mylonite is a fault rock deformed through crystal plastic mechanisms) Mylonite deformed.CrystalPlasticMechanism • Classes defined in different natural languages are the same

22. Example • For example, the LithicUnit class in the Rock ontology may be equivalent to the StratigraphicUnit class of the Stratigraphy ontology by having the same necessary and sufficient conditions. • If so, they can be declared as equivalent rock : LithicUnitstrat: StratigraphicUnit Mylonite FaultRockdeformed.CrystalPlasticMechanism fre : Fenêtre eng : Window  Per : پنجره

23. Overlap and Disjointness • Objects can overlap each other if they have a proper or improper part in common • Although this may imply only spatial overlap, it applies to other cases that are not spatial (e.g., processes) • The fact that x overlaps y is denoted as Oxy or xy, and can hold: • (i) if x and y share a proper part • (ii) if x and y are identical • (iii) if x is a proper part of y • (iv) if y is a proper part of x

24. Overlap Relation • The mereological binary overlap relation obtains when there is a region z such that z is part-of both x and y Oxy = z (PzxPzy) O(x,y) = z (P(z,x)  P(z,y)) • i.e., two objects overlap if they share a common part • Shoulder is a common part for arm and chest • Two segments of a strike-slip fault may overlap (by fault steps or bends), and the step or bend region (i.e., jog) between the two segments is part of both segments that define the step

25. Overlap … • Overlap is reflexive and symmetric, but not transitive • Reflexive: every object overlaps itself Oxx or xx • Symmetry: if an object overlaps a second object, the second object overlaps the first Oxy Oyx or (xy) (yx) • In general, if a first object overlaps a second, and the second overlaps a third object, it does not always follow that the first also overlaps the third

26. Other Mereological Relations • There are other derived mereological relations that include underlap, over-crossing, and undercrossing • Underlap, denoted by Uxy, is a relation of two objects x and y, when there is a larger region z, that includes both x and y, Uxy = z (PxzPyz) • Underlap is used in layered mereotopology where regions and objects of the same kind are taken to lie in the same layer • An object x over-crosses y, if x overlaps y but is not part of y, i.e., OXxy = Oxy Pxy • Object x under-crosses y if x underlaps y but y is not part of x, i.e., UXxy = UxyPyx z x y y x z y x

27. Achille C. Varzi, PARTS, WHOLES, AND PART-WHOLE RELATIONS: THE PROSPECTS OF MEREOTOPOLOGY, Data and Knowledge Engineering 20 (1996), 259–286. The four basic patterns of mereological relationship. The leftmost pattern in turn corresponds to two distinct situations (validating or falsifying the clauses in parenthesis) depending on whether or not there is a larger z including both x and y. y x x x y x y y

28. Relation of Oxy and PPxy y x • The overlap and proper-part-of mereological relations are related • If x is a proper-part-of y, then x overlaps y, i.e., PPxy Oxy • If a pseudotachylite body is part of a seismogenic segment of the plate-boundary fault zone, then the fault rock is also part of the larger plate-boundary fault zone PP (Pseudotachylite, SeismogenicZone)  O (Pseudotachylite, SeismogenicZone) • If x overlaps y, and y is part-of z, then x overlaps z, i.e., Oxy PyzOxz z y x

29. Example • If a cataclasite (x) overlaps a bend (y) of a fault (z) (i.e., y is a part of z), then the cataclasite overlaps the fault for which the bend is a part of • In other words: O (Cataclasite, Bend)  P (Bend, Fault)  O (Catalasite, Fault)

30. discrete-from • Object x is discrete-from object y (i.e., Dxy) if x does not overlap y, i.e., Dxy = Oxy • Thus, two objects that do not share parts are said to be discrete • For example, the forearc and the plate-boundary fault zone in a subduction zone are discrete objects y x

31. Disjointness • If two objects do not overlap, they are said to be disjoint • This means that they do not share any proper or improper parts • Disjointness is symmetric, but neither reflexive nor transitive • Disjointness is denoted with the ʅ symbol • Symmetry: (x ʅ y)  (y ʅ x), which means that if x is disjoint with y, then y is disjoint with x • Non-reflexivity: (x ʅ x), which means that no object can be disjoint from itself • Non-transitivity: [(x ʅ y & y ʅ z)  (x ʅ z)], which states that if x is disjoint with y, and y is disjoint with z, it does not in general mean that x is disjoint with z

32. located-in is related to part-of • Any entity (x) that exists at time t, can be mapped to a spatial region by the r(x, t) function • The located-in relation can then be given in terms of this function as located-in (x, y, t) = part-of (r(x, t), r(y, t), t), which reads: object x is located-in object y (a whole) at time t, if the region of x at time t is a part of region y at t • This means that parts of geological entities are located-in their corresponding wholes located-in (SeismogenicZone, PlateBoundaryFaultZone, t), i.e., the seismogenic zone is located-in the plate-boundary fault zone, because it is part of the fault zone

33. Containment • For cases where an object is not a part of a whole, and the relation is between a material (e.g., water, mineral) and immaterial (e.g., hole, pore) objects, we use the containment contained-in (x, y, t) relation, defined as: contained-in (x, y, t) = located-in (x, y, t) part-of (x, y, t) • This asserts that: x is contained-in y at time t if x is located-in y at t, and x is not part-of y at t • This relation can be used for the common case of the existence of an entity, or a portion of a homogeneous, composite entity, in a container entity (e.g., pore, interstitial space) as a non-part • contained-in (Cement, Porosity, t) • contained-in (Water, Fracture, t) • contained-in (Contaminant, IntergranularSpace, t)

34. Containment … • The containment relation is transitive: contained-in (x, y, t)  contained-in (y, z, t)  contained-in (x, z, t) • If drilling mud is in the pore space when the core is retrieved at time t, and pore is in the core, then the mud is in the core at the time (t) of the retrieval contained-in (DrillingMud, Pore, t)  contained-in (Pore, Core, t)  contained-in (DrillingMud, Core, t)

35. Located-in and contained-in • If an object is located-in another object, it is also contained-in that object • However, the reverse is not true, i.e., anything which is contained-in another object is not located-in it unless it is a part-of it • Drilling mud, contained-in a core, is not part of the core, but offscraped sediments of an accretionary prism are both located-in and contained-in the prism • A contaminant in water is not a necessary part of the water molecule, therefore we say: contained-in (contaminant, water, t) • The following is true at all times for the case of oxygen in the water molecule: located-in (Oxygen, WaterMolecule)

36. Connection Relations • Mereotopology includes one primitive binary relation for connection (or contact), denoted by C, and several derived relations • The connection relation brings topology into mereotopology • The primitive, bidirectional connection relation, Cxy or C(x, y), reads: x is connected to y, or x is in contact with y • The connection Cxy also implies that y is connected to x, and that the distance between x and y is zero • Disconnection, DCxy is then defined as: DCxy = Cxy

37. Axioms of Connection Relation … • Reflexivity: everything is connected to itself: Cxx • Symmetry: if an object is connected to another object, the second object is connected to the first: CxyCyx • In contrast to parthood, connection may not be transitive • e.g., the forearc basin is connected to the accretionary prism, and prism is connected to the subducting plate, but the forearc basin is not connected to the subducting plate • Other derived relations are as follows: two distinct things cannot have the same connections; everything is connected with its mereological complement

38. Enclosure Relation y • The enclosure relation, Exy, which means x is enclosed-in y, is related to the connection relation: Exy = z (CzxCzy) • The enclosure relation is: • Reflexive (Exx), i.e., everything is enclosed-in itself • Transitive (ExyEyzExz), which means that if x is enclosed-in y and y is enclosed-in z, then x is enclosed-in z • Everything is connected to anything to which its parts are connected, i.e., PxyExy (monotonicity) • In other words, if x is part of y, whatever is connected to x is connected to y, i.e., x is topologically enclosed-in y z x

39. Parthood and Connection • This implies that mereological overlap is a form of connection, i.e., Oxy Cxy, but any two objectsconnected to each other do not have to overlap • In other words, there is connection without sharing parts, which is the external connection (EC) discussed below • Thus, the slope basin which may be part of an accretionary prism is connected to the prism • However, the subducting plate, which despite being connected to the prism, is not part of the prism, i.e., does not overlap it • Parthood can be written in terms of the primitive connection relation: Pxy = z (CzxCzy), i.e., x is part-of y if there is a region z which is connected to x and y

40. x y External Connection • External connection, EC, is defined as: ECxy = CxyOxy, i.e., x externally-connected-to y, if x connected-to y, but does not overlap it (i.e., x not part-of y) • EC is symmetric, i.e., if x externally-connected-to y, then y externally-connected-to x EC (ForearcBasin, AccretionaryPrism) EC (PlateBoundaryFaultZone, AccretionaryPrism) Means that the forearc basin and accretionary prism, or plate-boundary-fault zone and accretionary prism, are mutually connected to each other

41. EC … • External connection is neither transitive nor reflexive • Irreflexive: no entity can be externally-connected-to itself (ECxx) • Non-transitive: • Even though a subducting plate is externally-connected-to the plate-boundary fault zone at the base of the prism: EC (SubductingPlate, PlateBoundaryFaultZone) It is not externally connected to the prism despite the external connection between the fault zone and the prism: EC (PlatebloundaryFaultZone, AccretionaryPrism) EC(subductingPlate, AccretionaryPrism)

42. y z x Interior-part-of (IP) • The interior-part-of (IP) of an object is that part which does not share any part with the boundary of that object, i.e., it is the part which is neither tangential nor the boundary of the object • The interior part is a kind of parthood, i.e., IPxy = Pxy, or IPxy = Pxyz (CzxOzy) interior-part-of (DeformedRock, ThrustSheet) = P (DeformedRock, ThrustSheet) z ( C (z, DeformedRock)  O (z, ThrustSheet) ) • Implies that the deformed interior part of offscraped thrust packets, between two thrusts in an accretionary prism, are part of the thrust sheet

43. Relations derived from interior-part-of • IPxyPyzIPxz, means that if x is the interior-part-of y, and y is part-of z, then x is the interior-part-of z • The interior of the thrust packets are parts of the accretionary prism which contains the thrust packets as parts • PxyIPyzIPxz, i.e., if x is part-of y, and y is the interior-part-of z, then x is the interior-part-of z • If a pseudotachylite zone is part-of a shear zone, and the shear zone is the interior-part-of the prism, then the shear zone is also the interior-part-of the prism • IPxyIPxzIPx (y  z), i.e., if x is the interior-parts-of both y and z, then x is the interior-part-of the intersection of y and z • A fluid inclusion (x) in the interior of a vein (y), filling a fracture in the interior of a cataclastic shear zone (z), is also in the interior of the shear zone z y x

44. y x z Internal Overlap and proper-part • The internal overlap, IO, is related to the notion of the internal part: IOxy = z (IPzxIPzy), and is the case when there is a region z which is an internal-part-of both x and y • The internal-proper-part: IPPxy = IPxyIpyx applies when x is an internal-part-of y but y is not an internal-part-of x

45. Internal underlap • Another useful relation is the internal underlap: IUxy = z (IPxzIPyz) • which applies when two objects are the internal-part-of a single region (layer) • This notion is very important in the so-called layered mereotopology which places related objects on the same layer • All objects that are part of a layer underlap each other • Tangential underlap: TUxy = Uxy, IUxy, is a special case of underlap when the objects underlap but not internally

46. Tangential part and proper-part x • Entities are tangential-to other entities when they touch or cross the exterior boundaries of otherentities • Tangential-part is that part of a whole which is not an internal part: TPxy = PxyIPxy • Sediments in a forearc basin are the tangential-part-of the basin; the other part of the basin is the water which is in contact with it • Tangential-proper-part: TPPxy = PPxyz (ECzxECzy) • x is a tangential-proper-part-of y if it is a proper-part-of y, and there is a region z where z is externally-connected-to x and y tangential-proper-part-of (SeismogenicZone, PlateBoundaryFaultZone) y

47. Non-tangential-proper-part • Non-tangential-proper-part: NTPPxy = PPxyTPPxy • x is non-tangential-proper-part-of y if x is a proper part of y, but is not the tangential proper-part-of y • nontangential-proper-part-of (CoverSequence, ForearcBasin) • Tangential overlap: TOxy = Oxy IOxy • x overlaps y, but the overlap is not internal y x

48. x y Product • Other relations between objects include product, sum, difference, universe, and complement, which are used to define singular terms • The binary product of two objects (x * y) is the object which is part of both x and y • This means that any common part of both x and y is a part of it • The product is the mereological analogue to the set-theoretic intersection; the difference is that two disjoint sets can have an intersection (null-set), but null-object does not exist in mereology

49. x y Sum and difference • The binary sum of x and y is denoted by (x + y) • Sum is the mereological analogue to the set-theoretic union • Any collection of objects, even if they are dissimilar can arbitrarily put together to make a sum, representing an existent and unique object • The difference of two objects (x – y) is the largest object in x which has no part in common with y • It only exists if x is not part of y. If x and y overlap, and x is not part of y, then the difference is a proper part of x. • The sum of all objects is called the universe (U). The complement of x, is then defined as (U-x), which denotes the object constituting the remainder of the universe outside of x • A mereological atom (unlike atom in physics), is an object that has no proper part, i.e., it is indivisible