An Ordering ofConvex Topological Relations

Matthew P. Dube Max J. Egenhofer

School of Computing and Information Science

max@spatial.maine.edumatthew.dube@umit.maine.edu

A World of Languages

with a truckload of expressions that

don’t exist in all languages... BogusławskiWierzbickaGoddard

чужбина чужбина buitenland ???

Lost in Translation...

Translation vs. Circumscription

•Translation: one word for another word

•Circumscription: one word to many potential words

.

.n..

Outline• Topological Spatial Relations

• Properties of Conceptual Neighborhood Graphs

• Connectivity

• Distance Hereditary

• Convexity

• Visual Example of Algorithm

• Mapping Language back to Theory

• Translation/Circumscription Example

• A Look to the Future

Topological Spatial Relations in

S2

Egenhofer

embraceentwined

overlap

attach

meetdisjoint equal covers containscoveredBy inside

A-Neighborhood

EgenhoferFreksa

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Al-Taha

Disjunctions of Relations

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

BennettCohn

Disjunctions of Relations

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Disjunctions of Relations

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Disjunctions of Relations

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Howorka

Disjunctions of Relations

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Ligozat

ConvexityConvex objects maintain all of their shortest paths

Concave objects lack at least one shortest pathArtigas SzwarcfiterDourado

Convex SubgraphsConnected

•Can all nodes be reached from all other nodes?

Distance Hereditary•Can all nodes be reached in regularly optimal

cost?

Multi-Path Preserving•Can all nodes be reached in as many regularly

optimal ways?

Adjacency Matrix

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Adjacency Matrix

d(A,B) + d(B,C) ≥ d(A,C)

Distance Matrix

Distance Matrix

33

d(A,B) + d(B,C) ≥ d(A,C)

Distance Matrix

d(A,B) + d(B,C) ≥ d(A,C)EgenhoferAl-Taha

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Connected Subgraphs

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Connected Subgraphs

0 1 0

1 0 1

0 1 0

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Distance Hereditary Subgraphs

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Howorka

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Distance Hereditary Subgraphs

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Multi-Path Preserving Subgraphs

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

Multi-Path Preserving Subgraphs

If a node v is on a shortest path

between nodes u and w, it must

satisfy:

d(u,v) + d(v,w) = d(u,w)

ifthen

Multi-Path Preserving Subgraphs

2 1 2

1 2 1

2 1 2

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Multi-Path Preserving Subgraphs

Convex Subgraphs of

the A-Neighborhood

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Connected

• All nodes can be reached from all other nodes

Distance Hereditary

• All nodes can be reached in regularly optimal cost

Multi-Path Preserving

• All nodes can be reached in as many regularly optimal ways

Convex Subgraphs of

the A-Neighborhood

Connected

• All nodes can be reached from all other nodes

Distance Hereditary

• All nodes can be reached in regularly optimal cost

Multi-Path Preserving

• All nodes can be reached in as many regularly optimal ways

Subset/Superset Ordering

Schilder

ContributionsAlgorithm for identifying subgraphs of arbitrary graphs that satisfy particular properties

•connected

•distance hereditary

•multi-path preserving

•convexity

Contributions

•104 subgraphs of the A-neighborhood identified as convex

•Partial ordering based on subset/superset relationship

Natural Spatial Language

Communication is often spoken or written

Spatial information is important

Often comes through the realm of prepositional phrases

“Fairly complete” list of 90 prepositions

LandauJackendoff

aboutaboveacrossafter

against

alongalongsideamidstamongst

around

atatop

behindbelowbeneathbesidebetweenbetwixtbeyondby

down

from

ininsideinto

nearnearby

offononto

opposite

outoutside

overpast

throughthroughoutto

towardunderunderneath

upupon

via

withwithinwithout

far from

in back ofin betweenin front ofin line with

on top of

to the left ofto the right ofto the side of

afterward

apart

awaybackbackward

downstairsdownward

east

forward

here

inward

left

homeward

north

outward

rightsideways

souththere

together

upstairsupward

westago

as

because of

before

despite

during

for

like

of

since

until

English Spatial Prepositions

aboutacrossagainstalongalongsideamidstamongst

aroundat

besideby

ininsidenearoutoutsidethroughthroughout

withwithin

far from

together

in back ofin betweenin front of

on top of

to the left ofto the right of

afterward

apart

away

backbackward

downstairsdownwardeastforward

here

inward

left

homeward

north

outward

rightsidewayssouththere

to the side of

upstairsupward

west

Topologically Driven Other Dimensionsaboveafter

atop

behindbelowbeneathbetweenbetwixtbeyonddown

from

into

nearby

offononto

opposite

over

past

totoward

underunderneathupupon

via

without

in line with

English Spatial Prepositions

aboutacrossagainstalongalongsideamidstamongst

aroundat

besideby

ininsidenearoutoutsidethroughthroughout

withwithin

far from

to the side ofapart

awaytogether

Topologically Driven

English Spatial Prepositions

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Along/Alongside

EgenhoferMarkShariff

Along/Alongside

Along/Alongside

embrace

entwined

overlap

attach

meet

disjoint

equal

covers

contains

coveredBy

inside

Along/Alongside

embrace

entwined

attach

equal

covers

contains

overlap

meet

disjoint

coveredBy

inside

Along/Alongside

C47

Along represents a convex subgraph

embrace

entwined

attach

equal

covers

contains

Topologically DrivenSpatial Prepositions

Prepositions Disjunctions of Topological Spatial Relations Convex Relation

about equal, coveredBy, inside C30

across overlap, coveredBy C19

against meet, attach C15

along/alongside disjoint, meet, overlap, coveredBy, inside C47

around disjoint, meet C13

at equal, coveredBy, inside C30

beside disjoint, meet, attach C29

by disjoint, meet, attach C29

far from/apart disjoint C2

in/inside/within coveredBy, inside C18

near disjoint, meet, attach C29

out/outside disjoint, meet, attach C29

through overlap C9

throughout equal, covers, contains C31

togetheroverlap, equal, coveredBy, inside, covers, contains, entwined,

embraceC91

to the side of disjoint, meet, attach C29

withoverlap, equal, coveredBy, inside, covers, contains, entwined,

embraceC91

English Spatial Prepositions

aboutacrossagainstalong

alongsideamidstamongst

aroundat

besideby

ininsidenear

outoutsidethroughthroughout

with

within

far from

to the side of

apart

away

together

aboutacrossagainstalong

alongsideamidstamongst

aroundat

besideby

ininsidenear

outoutsidethroughthroughout

with

within

far from

to the side of

apart

away

together

English Spatial Prepositions

English Spatial Preposition Ordering

aboutacross

against

alongalongside

amidstamongst

around

at

besideby

ininside

nearout

outside

through

throughout

with

within

far from

to the side of

apartaway

together

universal

nothing

How Can This Be Used?

Language Circumscription

The phrase “goes to” has been shown to be a problematic translation from English to Spanish and its related dialects

How can it be conveyed without drawing or showing it to the other person?

EgenhoferMark

чужбина

Bulgarian word

Foreigner’s land

Examples:

Norway (disjoint)

Greece (meet)

Not in Bulgaria (attach)

чужбина → “Outside”

aboutacross

against

alongalongside

amidstamongst

around

at

besideby

ininside

nearout

outside

through

throughout

with

within

far from

to the side of

apartaway

together

universal

nothing

Contributions

25 topologically driven English spatial prepositions

Partial ordering based on the ordering of convex relations

Mathematically-assisted translation and circumscription between languages

Conclusions•104 convex subgraphs identified

within the A-neighborhood of S2 region-region relations

•All topologically driven English terms from Landau and Jackendoff parsed to these subgraphs

•Translation and circumscription have the potential for mathematical analysis

A Look to the Future

•A mathematical strategy to circumscript between languages

•Convex subgraphs in other neighborhoods and for various sets of relations

•The impacts of convexity on relation composition = ?

Acknowledgments• Brian Lopez-Cornier

• University of Maine Upward Bound Math-Science Student

• DOE P047M080002

• Sensor Science, Engineering, and Informatics IGERT

• DGE 0504494 (PI: Kate Beard)

• NSF IIS 1016740 (PI: Max Egenhofer)

Lopez-Cornier

Beard

Questions?