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