Motivation Diagrams Conclusion

Formal Multi-Lane Traffic Diagrams

Sven Linker

Carl von Ossietzky Universitat Oldenburg

07.12.2012

1/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

2/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Motivation

Scenario: Automated traffic on highways

Defined maneuovres for cars [HESV91]

Changing lanesKeeping distance. . .

Ensure safety of traffic by formal methods

E.g., no collisions

Highly dynamical, complex systems

Formal machinery also very complex, hard to understand

Use suitable abstraction of dynamics

3/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstraction

E

Cars use full width of lane

Physical size + length for braking

Distinguish space used by cars:

Reservation must not be violated by other cars, needed foremergency brake, etc.

Claim indicator, whenever a car wants to change a lane(setting turn signals)

4/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstraction

E

Cars use full width of lane

Physical size + length for braking

Distinguish space used by cars:

Reservation must not be violated by other cars, needed foremergency brake, etc.

Claim indicator, whenever a car wants to change a lane(setting turn signals)

4/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstraction

E

E

Reservation

Claim

Cars use full width of lane

Physical size + length for braking

Distinguish space used by cars:

Reservation must not be violated by other cars, needed foremergency brake, etc.

Claim indicator, whenever a car wants to change a lane(setting turn signals)

4/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Diagrams with mathematical semantics

Intuitive way to reason about space

Logical reasoning with diagrams possible [AB96]

Different diagrammatic languages established

Formal Euler/Venn Diagrams

Conceptual Graphs CAT : ∗ on MAT : ∗

Spider Diagrams. . .

Spatial properties seldom captured by diagrams

Queries for spatial databases [ES99]Time-space diagrams [KRL+08]

t

x

5/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Traffic Diagrams

ego

a c(2, 5)

(2, 5)

Layer

Lane separation

Claim/Reservation

Distance Arrow

Auxillary lines

Central Car

6/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstract Syntax

Mathematical structure capturing elements/relations

Transformation rules

Suitable formalisms:

Meta-modelsGraph grammars

Here: Graph grammars

Spatial constraintsApplication conditions of rulesTransformation rules also within grammar

Non-terminal edges replaced by graphs

Graphs without NT ⇒ syntactically correct diagrams

7/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstract Syntax: Example

start

layer

lane

sep

lane

cars free cars

({a}, ∅) ({c}, ∅)

free cars free

({ego}, ∅)

at

ci

at in

s t s t s t

l

r l

r

s

t

at

n i s t s t s t

l r

ego

a c(2, 5)

(2, 5)

8/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstract Syntax: Example

start

layer lane

sep

lane

cars free cars

({a}, ∅) ({c}, ∅)

free cars free

({ego}, ∅)

at

ci at i

n

s t s t s t

l

r l

r

s

t

at

n i

s t s t s t

l r

ego

a c(2, 5)

(2, 5)

8/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstract Syntax: Example

start

layer lane

sep

lane

cars free cars

({a}, ∅) ({c}, ∅)

free cars free

({ego}, ∅)

at

ci at i

ns t s t s t

l

r l

r

s

t

at

n i s t s t s t

l r

ego

a c

(2, 5)

(2, 5)

8/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstract Syntax: Example

start

layer lane

sep

lane

cars free cars

({a}, ∅) ({c}, ∅)

free cars free

({ego}, ∅)sarrow

constr

(2, 5)

at

ci at i

ns t s t s t

l

r l

r

s

t

at

n i s t s t s t

l r

s

tty

cis

ego

a c(2, 5)

(2, 5)

8/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Abstract Syntax: Example

start

layer lane

sep

lane

cars free cars

({a}, ∅) ({c}, ∅)

free cars free

({ego}, ∅)sarrow

constr

(2, 5)

sarrow

constr (2, 5)

at

ci at i

ns t s t s t

l

r l

r

s

t

at

n i s t s t s t

l r

s

tty

cis

s

t

tycis

ego

a c(2, 5)

(2, 5)

8/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Conclusion

Diagrams expressing traffic situations

Abstract syntax by graph grammar

Not Shown

Real-time aspects

Semantics

Compatible with existing logic (same semantic model)

Future Work

Expressivity

Calculus

Decidability

9/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Conclusion

Diagrams expressing traffic situations

Abstract syntax by graph grammar

Not Shown

Real-time aspects

Semantics

Compatible with existing logic (same semantic model)

Future Work

Expressivity

Calculus

Decidability

9/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Conclusion

Diagrams expressing traffic situations

Abstract syntax by graph grammar

Not Shown

Real-time aspects

Semantics

Compatible with existing logic (same semantic model)

Future Work

Expressivity

Calculus

Decidability

9/9 Sven Linker Traffic Diagrams

Motivation Diagrams Conclusion

Gerard Allwein and Jon Barwise, editors.Logical reasoning with diagrams.Oxford University Press, Inc., New York, NY, USA, 1996.

Martin Erwig and Markus Schneider.Visual Specification of Spatio-Temporal Developments.In VL ’99, page 187. IEEE Computer Society, 1999.

Ann Hsu, Farokh Eskafi, Sonia Sachs, and Pravin Varaiya.The design of platoon maneuver protocols for ivhs.Technical Report UCB-ITS-PRR-91-6, University of California,Berkeley, April 1991.

Peter Koonce, Lee Rodegerdts, Kevin Lee, Shaun Quayle,Scott Beaird, Cade Braud, Jim Bonneson, Phil Tarnoff, andTom Urbanik.Traffic signal timing manual.Technical report, U.S. Department of Transportation, FederalHighway Administration, 2008.

9/9 Sven Linker Traffic Diagrams