mereotopology. the formal theory for parthood and connection relations is called mereotopology...
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Mereotopology
MereotopologyThe 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 collocatedi.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
Coincidence of ProcessesAlthough 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
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
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
Mineral
partOf
Endurants vs. PerdurantsThe entities in any domain of discourse are of two
types: continuantperdurant
Continuant (endurant) entities endure or persist through time by being fully present at different timesThey have spatial partsExamples of continuants are aquifer, mineral, and
river
Occurrent entities (perdurants) persist through time by having different temporal parts (phases) at different timesExamples of occurrents, that include processes and
events, are deformation, flow, and diffusion
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
Concepts of MereologyThe 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 xy, 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
Proper-part-ofThe inverse of the proper-part-of (x, y) is
has-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
x < yThe 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 riverThe 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, or PPxy 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
PP is antisymmetry, transitiveNotice 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
x yThe 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 transitiveAny 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)
TransitivityThe 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)
TransitivityThe 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)
Has-partThe 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
Meronomy vs. hyponomy (Part-of vs. subclassOf)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
Mineral
partOf
Mineral
Silicate
isA
Which one to use?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.
Rock
Phenocryst
partOf
Examples to clarify the difference:IgneousRock RockSedimentaryRock Rock IgneousRock SedimentaryRock
i.e., IgneousRock owl.disjointWith SedimentaryRock
Conglomerate SedimentaryRockConglomerate has.Grain
i.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 mineralGrain Rock i.e., grain can be a rock
Rock
IgneousRock
isA
Examples …Mineral Rock
(i.e., rock and mineral are disjoint)
Phenocryst partOf.IgneousRock(i.e., some ignoues rocks have phenocryst as part)
Matrix partOf.SedimentaryRock(i.e., some sedimentary rocks have matrix as part)
GroundMass partOf.IgneousRock(i.e., some igneous rocks have groundmass as part)
Rock (SedimentaryRock)SedimentaryRock (limestone) (an instance)Conglomerate (BathtiyariConglomerate)(an instance)Grain (limestone) (an instance)
EntailmentsDoes this ontology entail that limestone is a rock
or is part of a rockSedimentaryRock (limestone)
By being a sedimentary rock, the instance of limestone is a rockSedimentaryRock (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
Equivalence vs. SubsumptionConfusing class equivalence (owl : equivalentClass)
with subsumption (rdfs : subClassOf) is a common error
Equivalence is used when: 1.Two or more classes were defined in different
domains, which satisfy the same necessary and sufficient conditions of a class definition
2.A class can be defined by restricting an existing class in the same ontology (e.g., mylonite is a fault rock deformed through crystal plastic mechanisms)
Mylonite deformed.CrystalPlasticMechanism
3.Classes defined in different natural languages are the same
ExampleFor 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
Overlap and DisjointnessObjects 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 xy, 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
Overlap RelationThe 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
Overlap …Overlap is reflexive and symmetric, but not
transitive
Reflexive: every object overlaps itself Oxx or xx
Symmetry: if an object overlaps a second object, the second object overlaps the first Oxy Oyx or (xy) (yx)
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
z
Other Mereological RelationsThere 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
zx y
yx
x y
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.
x y x y yx yx
Oxy Oyx
(Uxy) (Uyx)
OXxy OXyx
(UXxy) (Uxyx)
POxy POyx
(PUxy) (PUyx)
Oxy Oyx
Uxy Uyx
Pxy OXyx
PPxy
UXxy
Oxy Oyx
Uxy Uyx
OXxy Pyx
PPyx
UXyx
Oxy Oyx
Uxy Uyx
Pxy Pyx
x=y y=x
Achille C. Varzi, PARTS, WHOLES, AND PART-WHOLE RELATIONS: THE PROSPECTS OF MEREOTOPOLOGY, Data and Knowledge Engineering 20 (1996), 259–286.
z
Relation of Oxy and PPxyThe 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
y x
y x
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)
discrete-fromObject 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
DisjointnessIf 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
located-in is related to part-ofAny 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
ContainmentFor 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)
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)
Located-in and contained-inIf an object is located-in another object, it is also
contained-in that objectHowever, 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)
Connection RelationsMereotopology 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
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 transitivee.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
Enclosure RelationThe 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
y
x z
Parthood and ConnectionThis 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 prismHowever, 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
External ConnectionExternal 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
x y
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)
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
y
xz
Relations derived from interior-part-ofIPxy 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
Internal Overlap and proper-partThe 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
y
x z
Internal underlapAnother 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
Tangential part and proper-partEntities 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)
x y
Non-tangential-proper-partNon-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 IOxyx overlaps y, but the overlap is not internal
y x
ProductOther 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
x y
Sum and differenceThe 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
x y