Consistency of Chordal RCC-8 Networks

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
Consistency of Chordal RCC-8 Networks
Outline

Introduction

PyRCC8
Michael Sioutis
-Path
Consistency
Department of Informatics and Telecommunications
Experimental
Results National and Kapodistrian University of Athens
Conclusions

Future Work November 8, 2012
Acknowledge

Bibliography

Michael Sioutis Consistency of Chordal RCC-8 Networks

Table of Contents

Consistency
of Chordal
RCC-8
Networks
1 Introduction
Michael
Sioutis

2 PyRCC8
Outline

Introduction

PyRCC8
3 -Path Consistency
-Path
Consistency
4 Experimental Results
Experimental
Results

Conclusions 5 Conclusions
Future Work

Acknowledge 6 Future Work
Bibliography


What is Qualitative Spatial Reasoning?

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline
Qualitative spatial reasoning is based on qualitative
Introduction

PyRCC8
abstractions of spatial aspects of the common-sense
-Path
background knowledge, on which our human perspective
Consistency on the physical reality is based
Experimental
Results

Conclusions

Future Work

Acknowledge

Bibliography


Reasons for Qualitative Spatial Reasoning

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline Two main reasons why non-precise, qualitative spatial
Introduction information may be useful:
PyRCC8
1 Only partial information may be available
-Path
Consistency 2 Spatial constraints are often most naturally stated in
Experimental qualitative terms
Results

Conclusions

Future Work

Acknowledge

Bibliography


Applications of Qualitative Spatial Reasoning

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
Qualitative spatial reasoning is an important subproblem
Outline
in many applications, such as:
Introduction

PyRCC8 Robotic navigation
-Path
High level vision
Consistency Geographical information systems (GIS)
Experimental Reasoning and querying with semantic geospatial query
Results
languages (e.g., stSPARQL, GeoSPARQL)
Conclusions

Future Work

Acknowledge

Bibliography


Region Connection Calculus

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline
The Region Connection Calculus (RCC) is a ﬁrst-order
Introduction
language for representation of and reasoning about
PyRCC8
topological relationships between extended spatial regions
-Path RCC abstractly describes regions, that are non-empty
Consistency

Experimental
regural subsets of some topological space which do not
Results have to be internally connected
Conclusions

Future Work

Acknowledge

Bibliography


The RCC-8 Calculus

Consistency
of Chordal
RCC-8
Networks

Michael
RCC-8 is a constraint language formed by the combination
Sioutis
of the following eight jointly exhaustive and pairwise
Outline disjoint base relations:
Introduction
disconnected (DC)
PyRCC8 externally connected (EC)
-Path
Consistency
equal (EQ)
partially overlapping (PO)
Experimental
Results tangential proper part (TPP)
Conclusions tangential proper part inverse (TPPi)
Future Work non-tangential proper part (NTPP)
Acknowledge non-tangential proper part inverse (NTPPi)
Bibliography


The Eight Basic Relations of the RCC-8 Calculus

Consistency
of Chordal
RCC-8
Networks

Michael Y Y
X Y X Y
Sioutis X X

X DC Y X EC Y X TPP Y X NTPP Y
Outline

Introduction
X X X
X Y
Y Y
PyRCC8 Y

-Path X PO Y X EQ Y X TPPi Y X NTPPi Y
Consistency

Experimental Figure: Two dimesional examples for the eight base relations of RCC-8
Results

Conclusions

Future Work From these basic relations, combinations can be built. For
Acknowledge
example, proper part (PP) is the union of TPP and NTPP.
Bibliography


The RCC-8 Composition Table

Consistency
of Chordal DC EC PO TPP NTPP TPPi NTPPi EQ
RCC-8 DC,EC DC,EC DC,EC DC,EC
Networks DC * PO,TPP PO,TPP PO,TPP PO,TPP DC DC DC
NTPP NTPP NTPP NTPP
Michael
DC,EC DC,EC DC,EC EC,PO PO
Sioutis
EC PO,TPPi PO,TPP PO,TPP TPP TPP DC,EC DC EC
NTPPi TPPi,EQ NTPP NTPP NTPP
Outline DC,EC DC,EC PO PO DC,EC DC,EC
PO PO,TPPi PO,TPPi * TPP TPP PO,TPPi PO,TPPi PO
Introduction NTPPi NTPPi NTPP NTPP NTPPi NTPPi
DC,EC TPP DC,EC DC,EC
PyRCC8
TPP DC DC,EC PO,TPP NTPP NTPP PO,TPP PO,TPPi TPP
-Path NTPP TPPi,EQ NTPPi
Consistency DC,EC DC,EC
NTPP DC DC PO,TPP NTPP NTPP PO,TPP * NTPP
Experimental NTPP NTPP
Results DC,EC EC,PO PO PO,EQ PO TPPi
TPPi PO,TPPi TPPi TPPi TPP TPP NTPPi NTPPi TPPi
Conclusions NTPPi NTPPi NTPPi TPPi NTPP
DC,EC PO PO PO PO,TPP
Future Work
NTPPi PO TPPi TPPi TPPi NTPP NTPPi NTPPi NTPPi
Acknowledge TPPi NTPPi NTPPi NTPPi NTPPi
NTPPi TPPi,EQ
Bibliography
EQ DC EC PO TPP NTPP TPPi NTPPi EQ


The RSAT Reasoning Problem

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis RSAT in the RCC-8 framework is the reasoning problem of
Outline
deciding whether there is a spatial conﬁguration where the
Introduction relations between the regions can be described by a spatial
PyRCC8 formula Θ
-Path
Consistency
RSAT is NP-Complete!
Experimental
ˆ
However, tractable subsets S of RCC-8 exist, such as H8 ,
Results
C8 , Q8 [5], for which the consistency problem can be
Conclusions
decided in polynomial time
Future Work

Acknowledge

Bibliography


Path Concistency

Consistency
of Chordal
RCC-8
Networks
Approximates consistency and realizes forward checking in
Michael
Sioutis a backtracking algorithm
Outline
Checks the consistency of triples of relations and
Introduction
eliminates relations that are impossible though iteravely
PyRCC8 performing the operation
-Path
Consistency Rij ← Rij ∩ Rik Rkj
Experimental
Results until a ﬁxed point R is reached
Conclusions If Rij = ∅ for a pair (i, j) then R is inconsistent, otherwise
Future Work R is path-consistent.
Acknowledge
Computing R is done in O(n3 )
Bibliography


About PyRCC8..

Consistency
of Chordal
RCC-8
Networks
PyRCC81 is an eﬃcient qualitative spatial reasoner written
Michael
Sioutis in pure Python. It employs PyPy2 , a fast, compliant
implementation of the Python 2 language
Outline

Introduction

PyRCC8

-Path
Consistency

Experimental
Results PyRCC8 oﬀers a path consistency algorithm for solving
Conclusions tractable RCC-8 networks and a backtracking-based
Future Work algorithm for general networks
Acknowledge

Bibliography

1
http://pypi.python.org/pypi/PyRCC8
2
http://pypy.org/

Comparing PC Implementations of Different
Reasoners
Consistency
of Chordal
RCC-8
Networks We compare the PC implementation of PyRCC8 to the PC
Michael implementations of the following qualitative spatial
Sioutis
reasoners:
Outline
Renz’s solver3
Introduction
GQR4
PyRCC8
Pellet Spatial5
-Path
Consistency
The admingeo6 dataset (11761 regions / 77910 relations)
Experimental
Results was used which was properly translated to fit the input
Conclusions format of the different PC implementations
Future Work

Acknowledge
3
Bibliography http://users.rsise.anu.edu.au/%7Ejrenz/software/rcc8-csp-solving.tar.gz
4
http://sfbtr8.informatik.uni-freiburg.de/R4LogoSpace/Tools/gqr.html
5
http://clarkparsia.com/pellet/spatial/
6
http://data.ordnancesurvey.co.uk/ontology/admingeo/

Evaluation with a Large Dataset

Consistency
of Chordal
RCC-8
Networks Performance of four QSRs for the admingeo dataset
105
Michael
Sioutis
104
Outline 103
Introduction
102
CPU time (sec)

PyRCC8

-Path 101
Consistency

Experimental
100
Results
10-1 Renz
Conclusions
PyRCC8
Future Work 10 -2
GQR
Pellet-Spatial
Acknowledge
10-3 0 10000 20000 30000 40000 50000 60000 70000
Bibliography Number of relations


Consistency

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

To explore the search space for the general case of RCC-8
Outline

Introduction
networks in order to solve an instance Θ of RSAT, some
PyRCC8
sort of backtracking must be used
-Path We implemented two backtracking algorithms:
Consistency

Experimental 1 A strictly recursive one
Results 2 An equivalent iterative one which resembles recursion
Conclusions

Future Work

Acknowledge

Bibliography


Comparing PyRCC8 to Other Reasoners

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
We compare PyRCC8 to the following qualitative spatial
Outline reasoners:
Introduction
Renz’s solver7
PyRCC8
GQR8
-Path
Consistency
Diﬀerent size n of instances from A(n, d = 9.5, l = 4.0)
Experimental
Results were used
Conclusions

Future Work

Acknowledge

Bibliography

7
http://users.rsise.anu.edu.au/%7Ejrenz/software/rcc8-csp-solving.tar.gz
8
http://sfbtr8.informatik.uni-freiburg.de/R4LogoSpace/Tools/gqr.html

Comparison Diagram

Consistency
of Chordal
RCC-8
Networks Performance of three QSRs for A(n,d=9.5,l=4.0)
102
Michael Renz
Sioutis PyRCC8
GQR
Outline 101
Introduction
CPU time (sec)

PyRCC8

-Path 100
Consistency

Experimental
Results
10-1
Conclusions

Future Work

Acknowledge
10-2
100 200 300 400 500 600 700 800 900
Bibliography Number of nodes


-Path Concistency

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline
Up till now, all aproaches in qualitative spatial reasoning
Introduction
enforce path consistency on a complete spatial network
PyRCC8 We propose enforcing path consistency on a chordal
-Path spatial network [2] as Chmeiss and Condotta have done for
Consistency

Experimental
temporal networks [3], and we call this type of local
Results consistency as -path consistency for clarity
Conclusions

Future Work

Acknowledge

Bibliography


Chordal Graph

Consistency
of Chordal
RCC-8
A graph is chordal if each of its cycles of four or more
Networks nodes has a chord, which is an edge joining two nodes
Michael
Sioutis
that are not adjacent in the cycle
An example of a chordal graph is shown below:
Outline

Introduction

PyRCC8

-Path
Consistency

Experimental
Results

Conclusions

Future Work

Acknowledge

Bibliography


Triangulation

Consistency
of Chordal
RCC-8
Networks

Michael Triangulation of a given graph is done by eliminating the
Sioutis
vertices one by one and connecting all vertices in the
Outline
neighbourhood of each eliminated vertex with fill edges
Introduction

PyRCC8
Determining a minimum triangulation is an NP-hard
-Path
problem
Consistency
Use of several heuristics for sub-optimal solutions (e.g.
Experimental
Results minimum degree, minimum fill)
Conclusions
Chordality checking can be done efficiently in O(|V | + |E |)
Future Work
time, for a graph G = (V , E ) (e.g., with MCS, LexBFS)
Acknowledge

Bibliography


Preliminaries

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
Let G = (V , E ) be an undirected chordal graph. There
Outline exists a tree T , called a clique tree of G , whose vertex set
Introduction
is the set of maximal cliques of G
PyRCC8

-Path
Let C be a constraint network from a given CSP. Then,
Consistency VC refers to the set of variables of C
Experimental
Results If V is any set of variables, CV will be the constraint
Conclusions network C that involves variables of V
Future Work

Acknowledge

Bibliography


Patchwork Property in RCC-8 Networks

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
Definition

Outline
We will say that a CSP has the patchwork property if for any
Introduction
finite satisfiable constraint networks C and C of the CSP such
PyRCC8
that CVC ∩VC = C VC ∩VC , the constraint network C ∪ C is
-Path satisfiable [4].
Consistency

Experimental
Results
Proposition
Conclusions ˆ
The three CSPs for path consistent H8 , C8 , and Q8 networks,
Future Work respectively, all have patchwork [4].
Acknowledge

Bibliography


Proposition

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
Proposition
Outline

Introduction Let C be an RCC-8 constraint network with relations from
PyRCC8 ˆ
H8 , C8 , and Q8 on its edges. Let G be the chordal graph that
-Path results from triangulating the associated constraint graph of C ,
Consistency

Experimental
and T a clique tree of G . C is consistent if all the networks
Results corresponding to the nodes of T are path consistent.
Conclusions

Future Work

Acknowledge

Bibliography


Example

Consistency
of Chordal
RCC-8
Networks
v1 v2 v1 v2
Michael
Sioutis
(a)
Outline
v5 (b)
v5
Introduction

PyRCC8
v3 v4 v3 v4

-Path
v1 v2 v2 v2
Consistency {v1, v2, v3}
Experimental {v2 , v3 }
Results (c)
v5 {v2, v3, v4}
Conclusions
{v2 , v4 }

Future Work v3 v3 v4 {v2, v4, v5}
v4
Acknowledge

Bibliography


PyRCC8

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline PyRCC8 is a chordal reasoner which was developed by
Introduction extending PyRCC8
PyRCC8
Similarly to PyRCC8, PyRCC8 oﬀers a -path
-Path
Consistency consistency algorithm for solving tractable RCC-8 networks
Experimental and a backtracking-based algorithm for general networks
Results

Conclusions

Future Work

Acknowledge

Bibliography


-Path Consistency Algorithm

Consistency
of Chordal
RCC-8 -Path-Consistency(C, G)
Networks Input: A constraint network C and its chordal graph G
Michael Output: True or False
Sioutis
1: Q ← {(i, j) | (i, j) ∈ E } // Initialize the queue
2: while Q is not empty do
Outline 3: select and delete an (i, j) from Q
4: for each k such that (i, k), (k, j) ∈ E do
Introduction 5: t ← Cik ∩ (Cij Cjk )
6: if t = Cik then
PyRCC8
7: if t = ∅ then
-Path 8: return False
Consistency 9: Cik ← t
10: Cki ← ˘ t
Experimental 11: Q ← Q ∪ {(i, k)}
Results 12: t ← Ckj ∩ (Cki Cij )
13: if t = Ckj then
Conclusions 14: if t = ∅ then
15: return False
Future Work 16: Ckj ← t
Acknowledge 17: Cjk ← ˘ t
18: Q ← Q ∪ {(k, j)}
Bibliography 19:return True


Complexity Analysis

Consistency
of Chordal
RCC-8
Networks

Michael Let δ denote the maximum degree of a vertex of G
Sioutis
For each arc (i, j) selected at line 3, we have at most δ
Outline
vertices of G corresponding to index k such that vi , vj , vk
Introduction
forms a triangle
PyRCC8

-Path
Additionaly, there exist |E | arcs in the network and one
Consistency can remove at most |B|9 values from any relation that
Experimental
Results
corresponds to an arc
Conclusions It results that the time complexity of -path consistency
Future Work is O(δ · |E | · |B|)
Acknowledge

Bibliography

9
B refers to the set of base relations of RCC-8

Recursive -Consistency Algorithm

Consistency
of Chordal
RCC-8
Networks

Michael -Consistency(C, G)
Sioutis Input: A constraint network C and its chordal graph G
Output: A reﬁned constraint network C’ if C is satisﬁable or None
Outline
1: if not -Path-Consistency(C, G) then
Introduction 2: return None
3: if no constraint can be split then
PyRCC8 4: return C
5: else
-Path 6: choose unprocessed constraint xi Rxj ; split R into S1 , ..., Sk ∈ S: S1 ∪ ... ∪ Sk = R
Consistency 7: Values ← {Sl | 1 ≤ l ≤ k}
8: for V in Values do
Experimental 9: replace xi Rxj with xi Vxj in C
Results 10: result = -Consistency(C, G)
11: if result = None then
Conclusions
12: return result
Future Work 13: return None

Acknowledge

Bibliography


Iterative -Consistency Algorithm

Consistency
of Chordal -Consistency(C, G)
RCC-8 Input: A constraint network C, A chordal graph G
Networks Output: A reﬁned constraint network C’ if C is satisﬁable or None

Michael 1: Stack ← {} // Initialize stack
Sioutis 2: if not -Path-Consistency(C, G) then
3: return None
4: while 1 do
Outline
5: if no constraint can be split then
Introduction 6: return C
7: else
PyRCC8 8: choose unprocessed constraint xi Rxj ; split R into S1 , ..., Sk ∈ S: S1 ∪ ... ∪ Sk = R
9: Values ← {Sl | 1 ≤ l ≤ k}
-Path 10: while 1 do
Consistency 11: if not Values then
12: while Stack do
Experimental 13: C, Values = Stack.pop()
Results 14: if Values then
15: break
Conclusions
16: else
Future Work 17: return None
18: V = Values.pop()
Acknowledge 19: replace xi Rxj with xi Vxj in C
20: if -Path-Consistency(C, G) then
Bibliography 21: break
22: Stack.push(C, Values)
23:raise RuntimeError, Can’t happen


Comparing PyRCC8 to PyRCC8

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline We compare PyRCC8 to PyRCC8, a complete graph
Introduction dedicated reasoner, using the following data:
PyRCC8
Random instances composed from the set of all RCC-8
-Path
Consistency
relations
Experimental
The admingeo10 dataset
Results

Conclusions

Future Work

Acknowledge

Bibliography

10

Experimenting with Random Instances

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis We generated instances from A(100, d, l = 4.0), for d
varying from 3 to 15 with a step of 0.5. For each series,
Outline

Introduction
300 networks were generated using Renz’s network
PyRCC8
generator11
-Path We used the Horn relations set as our split set, and the
Consistency
dynamic/local constraint scheme with a weighted queue
Experimental
Results conﬁguration, since it proved to be the best combination
Conclusions for both reasoners, conﬁrming the results in [5]
Future Work

Acknowledge

Bibliography

11
http://users.rsise.anu.edu.au/%7Ejrenz/software/rcc8-csp-solving.tar.gz

Comparison Diagram on CPU time

Consistency
of Chordal
RCC-8
Networks Performance of function Consistency for A(100,d,l=4.0)
0.70
Michael PyRCC8
Sioutis
0.65
PyRCC8
Outline
0.60
Introduction
CPU time (sec)

PyRCC8 0.55
-Path
Consistency 0.50
Experimental
Results 0.45
Conclusions
0.40
Future Work

Acknowledge 0.35 4 6 8 10 12 14
Bibliography Average degree of network for non trivial constraints


Comparison Diagram on # of Revised Arcs

Consistency
of Chordal
RCC-8
3000
Michael PyRCC8
Sioutis PyRCC8
2500
Outline

Introduction 2000
# of revised arcs

PyRCC8

-Path 1500
Consistency

Experimental 1000
Results

Conclusions
500
Future Work

Acknowledge 0 4 6 8 10 12 14


Comparison Diagram on # of Checked Constraints

Consistency
of Chordal
RCC-8
300000
Michael PyRCC8
Sioutis PyRCC8
250000
Outline
# of consistency checks

Introduction 200000
PyRCC8

-Path 150000
Consistency

Experimental 100000
Results

Conclusions
50000
Future Work

Acknowledge 0 4 6 8 10 12 14


Results Summary

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline
PyRCC8 PyRCC8 %
Introduction CPU time 0.524s 0.509s 2.80%
PyRCC8 revised arcs 1300.681 801.204 38.40%
-Path checked constraints 105751.173 74864.985 29.21%
Consistency

Experimental Table: Comparison based on the average of diﬀerent parameters
Results

Conclusions

Future Work

Acknowledge

Bibliography


Experimenting with the Admingeo Dataset

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis
The admingeo12 dataset consists of 11761 regions and
Outline 77910 base relations, thus being an extremely large and
Introduction sparse network, making itself a good candidate for stress
PyRCC8 testing different path consistency implementations
-Path
Consistency We used a simple queue configuration, since the weighted
Experimental variants made no difference on this dataset other than
Results
using much more memory
Conclusions

Future Work

Acknowledge

Bibliography

12

Comparison Diagram on CPU Time

Consistency
of Chordal
RCC-8
Networks Performance of function PC for the admingeo dataset
105
Michael PyRCC8
Sioutis PyRCC8
104
Outline

Introduction 103
CPU time (sec)

PyRCC8

-Path 102
Consistency

Experimental 101
Results

Conclusions
100
Future Work

Acknowledge
10-1 0 10000 20000 30000 40000 50000 60000 70000


Comparison Diagram on # of Revised Arcs

Consistency
of Chordal
RCC-8
108
Michael PyRCC8
Sioutis
107
PyRCC8
Outline
106
Introduction
# of revised arcs

PyRCC8 105
-Path
Consistency 104
Experimental
Results 103
Conclusions

Future Work
102

Acknowledge
101 0 10000 20000 30000 40000 50000 60000 70000


Comparison Diagram on # of Checked Constraints

Consistency
of Chordal
RCC-8
1012
Michael PyRCC8
Sioutis 1011 PyRCC8
1010
Outline
109
# of consistency checks

Introduction

PyRCC8 108
-Path 107
Consistency
106
Experimental
Results 105
Conclusions 104
Future Work 103
Acknowledge
102 0 10000 20000 30000 40000 50000 60000 70000


Results Summary

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline
PyRCC8 PyRCC8 %
Introduction CPU time 1825.129s 289.203s 84.15%
PyRCC8 revised arcs 4834133.78 373080.28 92.28%
-Path checked constraints 3.606e + 10 1.181e + 09 96.72%
Consistency

Experimental Table: Comparison based on the average of diﬀerent parameters
Results

Conclusions

Future Work

Acknowledge

Bibliography


Test Machine

Consistency
of Chordal
RCC-8
Networks

Michael All experiments were carried out on a computer with an
Sioutis
Intel Xeon 4 Core X3220 processor with a CPU frequency
Outline of 2.40 GHz, 8 GB RAM, and the Debian Lenny x86 64 OS
Introduction
Renz’s solver and GQR were compiled with gcc/g++ 4.4.3
PyRCC8

-Path PelletSpatial was run with OpenJDK 6 build 19, which
Consistency
implements Java SE 6
Experimental
Results PyRCC8 was run with PyPy 1.8, which implements
Conclusions Python 2.7.2
Future Work
Only one of the CPU cores was used for the experiments
Acknowledge

Bibliography


Conclusions

Consistency
of Chordal
RCC-8 We made the case for a new generation of RCC-8
Networks
reasoners implemented in Python, and making use of
Michael
Sioutis advanced Python environments, such as PyPy, utilizing
trace-based JIT compilation techniques
Outline

Introduction We introduced -path consistency for RCC-8 networks
PyRCC8 We showed that -path consistency is suﬃcient to decide
-Path
Consistency
the consistency problem for the maximal tractable subsets
ˆ
H8 , C8 , and Q8 of RCC-8
Experimental
Results
We implemented a chordal graph dedicated reasoner for
Conclusions

Future Work
RCC-8 networks
Acknowledge We showed expirimentally that -path consistency can
Bibliography oﬀer a great advantage over full path consistency on
sparse graphs


Main Points

Consistency
of Chordal
RCC-8
Networks

Michael Explore self learning heuristics regarding variable and value
Sioutis
selection
Outline Create module to generate spatial CSPs
Introduction
Transform PyRCC8 into a generic qualitative reasoner
PyRCC8

-Path Use other methods of triangulation and compare the
Consistency
behavior of partial path consistency for these diﬀerent
Experimental
Results methods
Conclusions Perform experiments with other possible real datasets,
Future Work
such as GADM13
Acknowledge

Bibliography

13
http://gadm.geovocab.org/

Acknowledge

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline This work was funded by the FP7 project TELEIOS
Introduction (257662)
PyRCC8
I would also like to thank my colleagues, and Katia
-Path
Consistency Papakonstantinopoulou especially, for their help, interest,
Experimental and advice
Results

Conclusions

Future Work

Acknowledge

Bibliography


References

Consistency
of Chordal
RCC-8 [Van Beek and Manchak]
Networks
The design and experimental analysis of algorithms for temporal reasoning
Michael JAIR, vol. 4, pages 1–18, 1996
Sioutis
[Bliek and Sam-Haroud]
Outline Path Consistency on Triangulated Constraint Graphs
Introduction In IJCAI, 1999
PyRCC8 [Chmeiss and Condotta]
-Path
Consistency of Triangulated Temporal Qualitative Constraint Networks
Consistency In ICTAI, 2011
Experimental [Huang]
Results
Compactness and Its Implications for Qualitative Spatial and Temporal
Conclusions Reasoning
Future Work In KR, 2012
Acknowledge [Renz and Nebel]
Bibliography
Eﬃcient Methods for Qualitative Spatial Reasoning
JAIR, vol. 15, pages 289–318, 2001


The End

Consistency
of Chordal
RCC-8
Networks

Michael
Sioutis

Outline

Introduction

PyRCC8
Any Questions?
-Path
Consistency

Experimental
Results

Conclusions

Future Work

Acknowledge

Bibliography


Consistency of Chordal RCC-8 Networks

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