This document summarizes a paper that proposes a new trajectory calculus (TC-6 and TC-10) for qualitative spatial reasoning about trajectories of moving objects. TC-6 defines 6 base relations between trajectories, while TC-10 defines 10 base relations by restricting trajectories to always have different start and end regions. Composition tables are provided to determine all possible relations between trajectories based on the relations between components. An Answer Set Programming implementation of the calculi is presented, along with optimizations to improve performance. A generalized encoding is also described that can be applied to other qualitative calculi.

1. A Trajectory Calculus for Qualitative
Spatial Reasoning Using ASP
George Baryannis, Ilias Tachmazidis, Sotiris Batsakis, Grigoris Antoniou
University of Huddersfield, UK
Mario Alviano
University of Calabria, Italy
Timos Sellis, Pei-Wei Tsai
Swinburne University of Technology, Australia
34th International Conference on Logic Programming, Oxford, UK

2. Trajectory-based Services
• Proliferation of location-aware devices in recent years, including
spatio-temporal information
– explicitly (GPS, RFID, GSM, etc.)
– implicitly (geo-tagged photos, addresses, etc.)
• Abundance of trajectory data, containing geographic information of
moving objects
– Billions of RFID tags generated daily
– Millions of sensors collecting measurements
– Figures expected to double every 2 years
• How can this data prove useful?
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
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Introduction Trajectory Calculi Implementation Evaluation Conclusion

3. Trajectory-based Services
• Traffic surveillance and control
– Detection of flocks or convoys
• Fleet management
– Identification of frequently followed routes
• Personalised services
– Carpooling
– Tracing objects or people
• Wildlife protection
– Clustering of animals
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

4. Comparing Trajectories
• Model moving objects as point entities moving on the Euclidean
plane across time
– Points can be continuous, or a discrete time approximation (e.g. GPS
readings taken every few seconds)
• Quantitative comparison
– Directly compare points within the trace
– Taking into account their (temporal) position in the trajectory
• Qualitative comparison
– Use natural language expressions to express relations between two
trajectories and reason about them
– Closer to human intuition, communication and decision making
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
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5. Qualitative Representation & Reasoning
• “High-level” approaches: focus only on modelling
spatial and/or temporal continuity
– Allen’s Interval Algebra
– Region Connection Calculus
– Double Cross Calculus
• “Low-level” approaches: detailed representation of
moving objects
– Qualitative Trajectory Calculus
– Noyon et al.’s Relative Trajectory Model
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
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6. Qualitative Trajectory Calculus
• QTC (and related literature) focuses on detailed
representation at the expense of efficient reasoning
– Assumes spatial and temporal continuity and takes into
account location, speed, and motion azimuth of MPOs
• Resulting into numerous relations, ranging from 27 to 81 for QTC
– No efficient implementations capable of reasoning
• Delafontaine et al. (2011) only yields the relation between two
MPOs and an ordering between successive relations
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
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7. Motivation
• How to achieve efficient qualitative reasoning?
• Focus on one feature at a time
– Martínez-Martín et al. (2012) use CSP formalisations that
reason only about one of distance, velocity or acceleration
of a moving object
• Simplify trajectory model
– Like earlier “high-level” approaches, suitable for
applications that view trajectories as complete paths
– Our work follows this direction
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

8. Proposed Simplifications
• Trajectories modelled as sequences of regions on a
partitioned map
– Given a map M, a partitioning R of M is defined as a set
of non-overlapping regions 𝑟𝑖, such that 𝑀 = 𝑟 𝑖∈𝑅 𝑟𝑖
• Trajectories are compared as a whole and not on the
basis of individual positions
• Individual features of moving objects such as velocity
and acceleration are not taken into account
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

9. Trajectory Calculus TC-6
• Simplest case: trajectories are arbitrary, but
consecutive regions within them must be different
• A trajectory is allowed to start and end at the same
region
– Given a partitioning R, a trajectory T is defined as a
sequence of regions 𝑡1, 𝑡2, … , 𝑡 𝑛 , 𝑛 ≥ 2, where 𝑡𝑖 ≠
𝑡𝑖+1, 1 ≤ 𝑖 < 𝑛
• Possible associations between two trajectories are
captured by 6 base relations, which are pairwise
disjoint and symmetric
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

10. TC-6 Base Relations
Relation Interpretation Illustration
Equal
(Eq)
T1 and T2 are equal (identical trajectories)
Alternative
(Alt)
T1 and T2 are alternative (different trajectories
for the same start and end regions)
Start
(S)
T1 and T2 start at the same region (but end at
different regions)
Finish
(F)
T1 and T2 end at the same region (but start at
different regions)
Intersect
(I)
T1 and T2 intersect (different start and end
regions but at least one common region)
Disjoint
(Dis)
T1 and T2 are disjoint (no common regions)
end
start
end
start
end
start end
start
start end
end
start end
start end
start
end
start

11. TC-6 Composition Table
• If we know the relation that holds between
trajectories T1 and T2 and the relation that
holds between trajectories T2 and T3 the
following table shows all possible relations
that may hold between trajectories T1 and T3
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Relations Eq Alt S F I Dis
Eq Eq Alt S F I Dis
Alt Alt Eq, Alt S F I, Dis I, Dis
S S S Eq, Alt, S I, Dis F, I, Dis F, I, Dis
F F F I, Dis Eq, Alt, F S, I, Dis S, I, Dis
I I I, Dis F, I, Dis S, I, Dis All Alt, S, F, I, Dis
Dis Dis I, Dis F, I, Dis S, I, Dis Alt, S, F, I, Dis All
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

12. TC-6 Composition Table
• If we know the relation that holds between
trajectories T1 and T2 and the relation that
holds between trajectories T2 and T3 the
following table shows all possible relations
that may hold between trajectories T1 and T3
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Relations Eq Alt S F I Dis
Eq Eq Alt S F I Dis
Alt Alt Eq, Alt S F I, Dis I, Dis
S S S Eq, Alt, S I, Dis F, I, Dis F, I, Dis
F F F I, Dis Eq, Alt, F S, I, Dis S, I, Dis
I I I, Dis F, I, Dis S, I, Dis All Alt, S, F, I, Dis
Dis Dis I, Dis F, I, Dis S, I, Dis Alt, S, F, I, Dis All
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
start end
start end
end start
start end
Introduction Trajectory Calculi Implementation Evaluation Conclusion

13. Trajectory Calculus TC-10
• We can further restrict trajectories to always have different
start and end regions
• A trajectory is still allowed to have cycles within itself
– Given a partitioning R, a trajectory T is defined as a sequence
of regions 𝑡1, 𝑡2, … , 𝑡 𝑛 , 𝑛 ≥ 2, where 𝑡1 ≠ 𝑡 𝑛 and
𝑡𝑖 ≠ 𝑡𝑖+1, 1 ≤ 𝑖 < 𝑛
• Possible relations between two trajectories are now
captured by 10 base relations, which are pairwise disjoint
– Relations are symmetric except two of the new relations, Ex and
Exi, which are each other’s inverse
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

14. TC-10 Base Relations
Relation Interpretation Illustration
Equal (Eq), Alternative (Alt), Start (S), Finish (F), Disjoint (Dis) and Intersect (I) are defined as in
TC-6, additionally ensuring that Intersect never happens at start or end regions
Reverse
(Rev)
T1 and T2 are the reverse of each other
Return
(Ret)
T1 and T2 are return trajectories (can be
used in order to return to the same region)
Extends
(Ex)
T1 extends T2 (the end region of T2 is the
start region of T1)
Extended By
(Exi)
T1 is extended by T2 (the end region of T1 is
the start region of T2)
end
start
start end
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T1 end/
T2 start
T1 start/
T2 end
T1 start/
T2 end
T1 end/
T2 start
T1 start/
T2 end
T1 end/
T2 start
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
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15. TC-10 Composition Table
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Relat
ions
Eq Rev Alt Ret S F Ex Exi I Dis
Eq Eq Rev Alt Ret S F Ex Exi I Dis
Rev Rev Eq Ret Alt Exi Ex F S I Dis
Alt Alt Ret Eq, Alt Rev, Ret S F Ex Exi I, Dis I, Dis
Ret Ret Alt Rev, Ret Eq, Alt Exi Ex F S I, Dis I, Dis
S S Ex S Ex Eq, Alt, S Exi, I, Dis Rev, Ret,
Ex
F, I, Dis F, Exi, I, Dis F, Exi, I, Dis
F F Exi F Exi Ex, I, Dis Eq, Alt, F S, I, Dis Rev, Ret,
Exi
S, Ex, I, Dis S, Ex, I, Dis
Ex Ex S Ex S F, I, Dis Rev, Ret,
Ex
Exi, I, Dis Eq, Alt, S F, Exi, I, Dis F, Exi, I, Dis
Exi Exi F Exi F Rev, Ret,
Exi
S, I, Dis Eq, Alt, F Ex, I, Dis S, Ex, I, Dis S, Ex, I, Dis
I I I I, Dis I, Dis F, Ex, I,
Dis
S, Exi, I,
Dis
S, Exi, I, Dis F, Ex, I,
Dis
All Alt, Ret, S, F, Ex,
Exi, I, Dis
Dis Dis Dis I, Dis I, Dis F, Ex, I,
Dis
S, Exi, I,
Dis
S, Exi, I, Dis F, Ex, I,
Dis
Alt, Ret, S, F,
Ex, Exi, I, Dis
All

16. Implementation (1): COI7
• TC-6 and TC-10 implemented as ASP programs
– Each answer set corresponds to an assignment of relations to a set of
trajectories that conforms to the composition table
• Initial encoding: COI7 in Brenton et al. (2016)
1. Possible relations represented as a Choice rule 𝑟1 𝑋, 𝑍 ; … ; 𝑟𝑛 𝑋, 𝑍 =
1 ← 𝑡𝑟𝑎𝑗 𝑋 , 𝑡𝑟𝑎𝑗 𝑍 , 𝑋! = 𝑍
2. Inverse relations modelled using One predicate (e.g. ex(X,Y) and ex(Y,X))
3. Composition table entries are encoded as:
1. Integrity constraints when entry contains > 7 relations: ← 𝑟𝑖 𝑋, 𝑍 , 𝑟𝑎 𝑋, 𝑌 , 𝑟𝑏 𝑌, 𝑍
for all 𝑟𝑖 that are not in the entry
2. Disjunctive rules when entry contains ≤ 7 relations: 𝑟1 𝑋, 𝑍 | … |𝑟𝑛 𝑋, 𝑍 ← 𝑟𝑎 𝑋, 𝑌 ,
𝑟𝑏 𝑌, 𝑍
4. An additional integrity constraint per pair of relations, to express that they
are pairwise disjoint
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

17. Implementation (2): CTSA
• Use the specific characteristics of TC-6 and TC-10 to improve on COI7, resulting in
new encoding named CTSA
1. Use Two predicates per inverse pair
– No benefit from using one as there is only one such pair: ex and exi
2. Antisymmetric optimisation of choice rule (𝑋 < 𝑍)
– Reduces possible pairs by half
– Only possible when using two predicates per pair, needs two extra rules for ex and exi
3. Remove integrity constraints for pairwise disjointness
– The choice rule suffices
4. Simplified Integrity constraints for all composition table entries
– A single constraint per entry: ← 𝑛𝑜𝑡 𝑟𝑖 𝑋, 𝑍 , … , 𝑛𝑜𝑡 𝑟𝑛 𝑋, 𝑍 , 𝑟𝑎 𝑋, 𝑌 , 𝑟𝑏 𝑌, 𝑍
– Disjunctive rules are avoided, since they may lead to ASP programs with head cycles
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

18. Implementation (3): CTSA2 & GEN
• Ensure that no grounding is performed for trajectory pairs when the relation is
already known – encoding CTSA2
– Predicate fact 𝑟𝑖, 𝑋, 𝑍 to express that the relation between trajectories 𝑋 and 𝑍 is known to
be 𝑟𝑖
– Append #count 𝑅: fact 𝑅, 𝑋, 𝑍 = 0 to the choice rule body to ensure rule applies only
when relation is unknown
• Generalised encoding (GEN), applicable to any qualitative calculus with a
composition table
– Domain: 𝑒𝑙𝑒𝑚𝑒𝑛𝑡(𝑋)
– Base relations: 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(<rel_name>)
– Composition table entries: 𝑡𝑎𝑏𝑙𝑒 <row_relation>, <column_relation>,<valid_relation>
– Generalised integrity constraint ← 𝑡𝑟𝑢𝑒 𝑋, 𝑅1, 𝑌 , 𝑡𝑟𝑢𝑒 𝑌, 𝑅2, 𝑍 , 𝑛𝑜𝑡 𝑡𝑟𝑢𝑒 𝑋, 𝑅 𝑜𝑢𝑡, 𝑌 ∶
𝑡𝑎𝑏𝑙𝑒 𝑅1, 𝑅2, 𝑅 𝑜𝑢𝑡
– Choice rule: 𝑡𝑟𝑢𝑒 𝑋, 𝑅, 𝑍 ∶ 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑅) = 1 ← 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑋 , 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑍 , 𝑋! = 𝑍
– Equality relation: 𝑡𝑟𝑢𝑒 𝑋, 𝑒𝑞, 𝑋 ← 𝑒𝑙𝑒𝑚𝑒𝑛𝑡(𝑋)
– Input: 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑋, 𝑅, 𝑌
– Input integrity constraint: ← 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑋, _, 𝑌 , 𝑛𝑜𝑡 𝑡𝑟𝑢𝑒 𝑋, 𝑅, 𝑌 ∶ 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑋, 𝑅, 𝑌 18
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

19. Evaluation: TC-6 Experiment 1
• 1000 trajectories, mean length of 282
regions, generated from T-Drive dataset
• clingo 5.2.0 run on Intel® Xeon® X3430
@2.4GHz, 16GB RAM
• Find a consistent configuration when
only one relation per trajectory is
known
• CTSA and CTSA2 show significantly
improved results over COI7
– Subtle differences between CTSA and
CTSA2 since input is relatively small (one
relation per distinct pair of trajectories)
• GEN trades off performance for
universal applicability
– Encoding can be used as a starting point
on which calculus-specific improvements
can be applied
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

20. Evaluation: TC-6 Experiment 2
• Find a consistent configuration
when trajectories are fixed to
50 and the number of known
relations increases
• Improvement of CTSA2 over
CTSA more pronounced up to
knowing 25 out of 50 relations
• GEN grounding time constant
due to input taking part only in
a single integrity constraint
• Program size increases similarly
across all encodings
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

21. Evaluation: TC-10 Experiment 1
• Find a consistent configuration
when only one relation per
trajectory is known
• Similarly to TC-6, CTSA and
CTSA2 show significantly
improved results over COI7
– Differences between CTSA and
CTSA2 even less visible due to
higher complexity
• GEN consumes less memory
than COI7, allowing reasoning
for up to 90 trajectories
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

22. Evaluation: TC-10 Experiment 2
• Find a consistent configuration
when trajectories are fixed to 50
and the number of known
relations increases
• Results mirror those of TC-6: an
increase of relations from 6 to 10
does not affect behaviour when
more relations are known
– Improvement of CTSA2 over CTSA
again peaks when knowing 25 out
of 50 relations
– Limited variations in performance
of GEN
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

23. Conclusion
• Simplified representation of trajectories as sequences of non-
overlapping regions on partitioned maps
• TC-6 allows arbitrary trajectories, TC-10 requires that they do not
end where they start
• ASP encodings scale up to 250 trajectories for TC-6 and 150
trajectories for TC-10
• GEN encoding applicable to any quantitative calculus that includes a
composition table in its definition
• TC-6 and TC-10 especially useful for applications that rely on large
trajectory databases: Origin/Destination analysis, discovery of
transportation demand and monitoring of stream of population
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

24. Future Work
• Explore a more restrictive calculus where no cycles are
allowed at all
– Potentially more useful for applications focusing on point-to-
point transportation of people or goods
• Extend calculi with a temporal dimension using Allen’s
interval algebra
• Determine whether knowledge gained through CTSA and
CTSA2 can be carried over to ASP implementations of other
qualitative calculi
• Investigate the trade-off between performance and
applicability of GEN 24
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
Introduction Trajectory Calculi Implementation Evaluation Conclusion

25. 25
George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK
g.bargiannis@hud.ac.uk
Questions?

26. References
• BRENTON, C., FABER, W., AND BATSAKIS, S. 2016. Answer set programming for qualitative
spatiotemporal reasoning: Methods and experiments. In ICLP (Technical Communications),
M. Carro, A. King, N. Saeedloei, and M. D. Vos, Eds. OASICS, vol. 52. Schloss Dagstuhl -
Leibniz-Zentrum fuer Informatik, 4:1–4:15.
• DELAFONTAINE, M., COHN, A. G., AND DE WEGHE, N. V. 2011. Implementing a qualitative
calculus to analyse moving point objects. Expert Syst. Appl. 38, 5, 5187–5196.
• MARTÍNEZ-MARTÍN, E., ESCRIG, M. T., AND DEL POBIL, A. P. 2012. A general qualitative
spatiotemporal model based on intervals. J. UCS 18, 10, 1343–1378.
• NOYON, V., CLARAMUNT, C., AND DEVOGELE, T. 2007. A relative representation of
trajectories in geogaphical spaces. GeoInformatica 11, 4, 479–496.
• YUAN, J., ZHENG, Y., XIE, X., AND SUN, G. 2013. T-drive: Enhancing driving directions with taxi
drivers’ intelligence. IEEE Trans. Knowl. Data Eng. 25, 1, 220–232.
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George Baryannis  A Trajectory Calculus for QSR with ASP  ICLP 2018  Oxford, UK