Ontology Matching
OM-2014
Proceedings of the ISWC Workshop
Introduction
Ontology matching1 is a key interoperability enabler for the semantic web, as well as a useful tactic in some classical data integration tasks dealing with the semantic heterogeneity problem. It takes the ontologies as input and determines as output an alignment, that is, a set of correspondences between the seman- tically related entities of those ontologies. These correspondences can be used for various tasks, such as ontology merging, data translation, query answering or navigation on the web of data. Thus, matching ontologies enables the knowl- edge and data expressed in the matched ontologies to interoperate.
The workshop has three goals:
• To bring together leaders from academia, industry and user institutions to assess how academic advances are addressing real-world requirements.
The workshop will strive to improve academic awareness of industrial and nal user needs, and therefore direct research towards those needs. Simul- taneously, the workshop will serve to inform industry and user represen- tatives about existing research eorts that may meet their requirements.
The workshop will also investigate how the ontology matching technology is going to evolve.
• To conduct an extensive and rigorous evaluation of ontology matching and instance matching (link discovery) approaches through the OAEI (Ontol- ogy Alignment Evaluation Initiative) 2014 campaign2. The particular fo- cus of this year's OAEI campaign is on real-world specic matching tasks as well as on evaluation of interactive matchers and matchers for query an- swering. Therefore, the ontology matching evaluation initiative itself will provide a solid ground for discussion of how well the current approaches are meeting business needs.
• To examine new uses, similarities and dierences from database schema matching, which has received decades of attention but is just beginning to transition to mainstream tools.
The program committee selected 5 submissions for oral presentation and 9 submissions for poster presentation. 14 matching system participated in this year's OAEI campaign. Further information about the Ontology Matching workshop can be found at: http://om2014.ontologymatching.org/.
Acknowledgments. We thank all members of the program committee, au- thors and local organizers for their eorts. We appreciate support from the Trentino as a Lab (TasLab)3 initiative of the European Network of the Living Labs4 at Informatica Trentina SpA5, the EU SEALS (Semantic Evaluation at Large Scale)6 project and the Semantic Valley7 initiative.
Pavel Shvaiko Jérôme Euzenat Ming Mao Juanzi Li
Ernesto Jiménez-Ruiz Axel Ngonga
October 2014
3http://www.taslab.eu
4
Organization
Organizing Committee
Pavel Shvaiko, Informatica Trentina SpA, Italy Jérôme Euzenat, INRIA & LIG, France Ming Mao, Electronic Arts, USA
Ernesto Jiménez-Ruiz, University of Oxford, UK Juanzi Li, Tsinghua University, China
Axel Ngonga, University of Leipzig, Germany
Program Committee
Alsayed Algergawy, Jena University, Germany Michele Barbera, Spazio Dati, Italy
Zohra Bellahsene, LRIMM, France
Chris Bizer, University of Mannheim, Germany
Olivier Bodenreider, National Library of Medicine, USA Michelle Cheatham, Write State University, USA Marco Combetto, Informatica Trentina, Italy Gianluca Correndo, University of Southampton, UK Isabel Cruz, The University of Illinois at Chicago, USA Jérôme David, INRIA & LIG, France
Stefan Dietze, L3S, Germany
Alo Ferrara, University of Milan, Italy Avigdor Gal, Technion, Israel
Fausto Giunchiglia, University of Trento, Italy Wei Hu, Nanjing University, China
Ryutaro Ichise, National Institute of Informatics, Japan
Antoine Isaac, Vrije Universiteit Amsterdam & Europeana, Netherlands Yannis Kalfoglou, Ricoh Europe plc, UK
Anastasios Kementsietsidis, IBM, USA
Patrick Lambrix, Linköpings Universitet, Sweden Nico Lavarini, Expert System, Italy
Tatiana Lesnikova, INRIA, France
Vincenzo Maltese, University of Trento, Italy Fiona McNeill, University of Edinburgh, UK
Christian Meilicke, University of Mannheim, Germany Andriy Nikolov, Open University, UK
Leo Obrst, The MITRE Corporation, USA
Heiko Paulheim, University of Mannheim, Germany Yefei Peng, Google, USA
Catia Pesquita, University of Lisbon, Portugal Alessandro Solimando, University of Genova, Italy Umberto Straccia, ISTI-C.N.R., Italy
Ond°ej Zamazal, Prague University of Economics, Czech Republic Cássia Trojahn, IRIT, France
Giovanni Tummarello, Fondazione Bruno Kessler - IRST, Italy
Lorenzino Vaccari, European Commission - Joint Research Center, Italy Ludger van Elst, DFKI, Germany
Shenghui Wang, Vrije Universiteit Amsterdam, Netherlands Songmao Zhang, Chinese Academy of Sciences, China
Table of Contents
PART 1 - Technical Papers
A categorical approach to ontology alignment
Mihai Codescu, Till Mossakowski, Oliver Kutz . . . 1 The properties of property alignment
Michelle Cheatham, Pascal Hitzler . . . .13 Completeness and optimality in ontology alignment debugging
Jan Noessner, Heiner Stuckenschmidt,
Christian Meilicke, Mathias Niepert . . . 25 Time-ecient execution of bounded Jaro-Winkler distances
Kevin Dreÿler, Axel-Cyrille Ngonga Ngomo . . . .37 A two-step blocking scheme learner for scalable link discovery
Mayank Kejriwal, Daniel P. Miranker . . . 49
PART 2 - OAEI Papers
Results of the Ontology Alignment Evaluation Initiative 2014 Zlatan Dragisic, Kai Eckert, Jérôme Euzenat, Daniel Faria, Alo Ferrara, Roger Granada, Valentina Ivanova,
Ernesto Jiménez-Ruiz, Andreas Oskar Kempf, Patrick Lambrix, Stefano Montanelli, Heiko Paulheim, Dominique Ritze,
Pavel Shvaiko, Alessandro Solimando, Cássia Trojahn,
Ond°ej Zamazal, Bernardo Cuenca Grau . . . .61 AgreementMakerLight results for OAEI 2014
Daniel Faria, Catarina Martins, Amruta Nanavaty, Aynaz Taheri, Catia Pesquita, Emanuel Santos,
Isabel F. Cruz, Francisco M. Couto . . . .105 AOT / AOTL results for OAEI 2014
Abderrahmane Khiat, Moussa Benaissa . . . .113 InsMT / InsMTL results for OAEI 2014 instance matching
Abderrahmane Khiat, Moussa Benaissa . . . .120 LogMap family results for OAEI 2014
Ernesto Jiménez-Ruiz, Bernardo Cuenca Grau, Weiguo Xia, Alessandro Solimando, Xi Chen, Valerie Cross, Yuan Gong,
Shuo Zhang, Anu Chennai-Thiagarajan . . . .126 Alignment evaluation of MaasMatch for the OAEI 2014 campaign
Frederik C. Schadd, Nico Roos . . . 135 OMReasoner: combination of multi-matchers for ontology matching:
results for OAEI 2014
Guohua Shen, Yinling Liu, Fei Wang,
Jia Si, Zi Wang, Zhiqiu Huang, Dazhou Kang . . . 142 RiMOM-IM results for OAEI 2014
Chao Shao, Linmei Hu, Juanzi Li . . . 149 RSDL workbench results for OAEI 2014
Simon Schwichtenberg, Christian Gerth, Gregor Engels . . . 155 XMap++: results for OAEI 2014
Warith Eddine Djeddi, Mohamed Tarek Khadir . . . 163
PART 3 - Posters
Evaluation of string normalisation modules for string-based biomedical vocabularies alignment with AnAGram
Anique van Berne, Veronique Malaisé . . . 170 Building reference alignments for compound matching
of multiple ontologies using OBO cross-products Catia Pesquita, Michelle Cheatham, Daniel Faria,
Joana Barros, Emanuel Santos, Francisco M. Couto . . . 172 A term-based approach for matching multilingual thesauri
Mauro Dragoni, Andi Rexha, Matteo Casu, Alessio Bosca . . . 174 The importance of cross-lingual information
for matching Wikipedia with the Cyc ontology
Aleksander Smywinski-Pohl, Krzysztof Wróbel . . . 176 Constructing a class hierarchy with properties
by rening and aligning Japanese wikipedia ontology and Japanese WordNet
Takeshi Morita, Susumu Tamagawa, Takahira Yamaguchi . . . 178 Partitioning-based ontology matching approaches:
a comparative analysis
Alsayed Algergawy, Friederike Klan, Birgitta Konig-Ries . . . .180 Towards a cluster-based approach for user participation
in ontology maching
Vinicius Lopes, Fernanda Baião, Kate Revoredo . . . 182 One query at a time: incremental, collective ontology matching
Thomas Kowark, Hasso Plattner . . . 184 Enabling semantic search for EO products:
an ontology matching approach
Maria Karpathiotaki, Konstantina Dogani, Manolis Koubarakis . . . 186
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❡①✐(5✐♥❣ (❡♠❛♥5✐❝ ❢♦3♠❛❧✐(♠( ✭❞✐(53✐❜✉5❡❞ ✜3(5✲♦3❞❡3 ❧♦❣✐❝( ✭❉❋❖▲✮ ❬✶✵❪✱ ❞✐(53✐❜✉5❡❞
❞❡(❝3✐♣5✐♦♥ ❧♦❣✐❝( ✭❉❉▲✮ ❬✷❪ ❛♥❞ ❝♦♥5❡①5✉❛❧✐(❡❞ ♦♥5♦❧♦❣✐❡( ✭❈✲❖❲▲✮ ❬✸❪✮ ❛♥❞ ❧❛5❡3
❝♦33❡❝5❡❞ 5♦ ❛ 3❡❧❛5✐♦♥❛❧ (❡♠❛♥5✐❝( ✐♥ ❬✷✷❪✳ T❛❝❦❛❣❡✲❜❛(❡❞ ❞❡(❝3✐♣5✐♦♥ ❧♦❣✐❝( ✭T❉▲✮ ❬✶❪
❛❧(♦ ❢❛❧❧ ✐♥ 5❤✐( (❡♠❛♥5✐❝ ❝❛5❡❣♦3②✳ ▼♦3❡♦✈❡3✱ ❬✷✸❪ ❞✐(❝✉((❡( 5❤❡ ✐♠♣❧✐❝❛5✐♦♥( ♦❢ 5❤❡(❡
♣♦((✐❜❧❡ ✐♥5❡3♣3❡5❛5✐♦♥( ♦❢ ❛❧✐❣♥♠❡♥5( ✇✐5❤ 3❡(♣❡❝5 5♦ 3❡❛(♦♥✐♥❣ ❛♥❞ ❝♦♠♣♦(✐5✐♦♥ ♦❢
❛❧✐❣♥♠❡♥5(✳
❆ ♠❛❥♦% ♣%♦❜❧❡♠ ✇✐,❤ ,❤❡.❡ ❛♣♣%♦❛❝❤❡. ✐. ,❤❡✐% ❞✐✈❡%.✐,②✳ ❚❤❡%❡ ❡①✐., .♦♠❡ ❛,✲
,❡♠♣,. ❢♦% ✉♥✐✜❝❛,✐♦♥✱ ✇❤✐❝❤ ❤♦✇❡✈❡% %❡♠❛✐♥ ✉♥.❛,✐.❢❛❝,♦%②✿ ,❤❡%❡ ✐. ♥♦ ❝♦♠♠♦♥
.②♥,❛①✱ ♥♦ ❝♦♠♠♦♥ .❡♠❛♥,✐❝ ❢%❛♠❡✇♦%❦✱ ❛♥❞ ♥♦ ❝♦♠♠♦♥ ,♦♦❧ .✉♣♣♦%,✳ ■♥ ,❤✐. ✇♦%❦✱
✇❡ .❤♦✇ ❤♦✇ ❝❛,❡❣♦%② ,❤❡♦%② ❝❛♥ ♣%♦✈✐❞❡ .✉❝❤ ❛ ✉♥✐❢②✐♥❣ ❢%❛♠❡✇♦%❦ ❛, ✈❛%✐♦✉. ❧❡✈❡❧.✱
✐♠♣%♦✈✐♥❣ ♣%❡✈✐♦✉. %❡❧❛,❡❞ ✇♦%❦ ❬✷✹✱ ✶✺✱ ✷✷✱ ✶✶❪ ✇❤✐❝❤ ❞✐❞ ♥♦, .♣❡❧❧ ♦✉, ❞❡,❛✐❧.✱ ❛♥❞
❞✐❞ ♥♦, ♠❛❦❡ ,❤❡ .,❡♣ ❢%♦♠ ❛❜.,%❛❝, ❞❡.❝%✐♣,✐♦♥ ❛♥❞ ❝❛.❡ .,✉❞✐❡. ,♦ ❧❛♥❣✉❛❣❡ ❞❡.✐❣♥
❛♥❞ ✐♠♣❧❡♠❡♥,❛,✐♦♥✳
✷ ●❡♥❡$❛❧ ❛♣♣$♦❛❝❤
❚❤❡ ❣❡♥❡%❛❧ %❡♣%❡.❡♥,❛,✐♦♥ ❛♥❞ %❡❛.♦♥✐♥❣ ❢%❛♠❡✇♦%❦ ,❤❛, ✇❡ ♣%♦♣♦.❡ ✐♥❝❧✉❞❡.✿ ✶✮
❛ ❞❡❝❧❛%❛,✐✈❡ ❧❛♥❣✉❛❣❡ ,♦ .♣❡❝✐❢② ♥❡,✇♦%❦. ♦❢ ♦♥,♦❧♦❣✐❡. ❛♥❞ ❛❧✐❣♥♠❡♥,.✱ ✇✐,❤ ✐♥❞❡✲
♣❡♥❞❡♥, ❝♦♥,%♦❧ ♦✈❡% .♣❡❝✐❢②✐♥❣ ❧♦❝❛❧ ♦♥,♦❧♦❣✐❡. ❛♥❞ ❝♦♠♣❧❡① ❛❧✐❣♥♠❡♥, %❡❧❛,✐♦♥.✱ ✷✮
,❤❡ ♣♦..✐❜✐❧✐,② ,♦ ❛❧✐❣♥ ❤❡,❡%♦❣❡♥❡♦✉. ♦♥,♦❧♦❣✐❡.✱ ❛♥❞ ✸✮ ✐♥ ♣%✐♥❝✐♣❧❡✱ ,❤❡ ♣♦..✐❜✐❧✐,② ,♦ ❝♦♠❜✐♥❡ ❞✐✛❡%❡♥, ❛❧✐❣♥♠❡♥, ♣❛%❛❞✐❣♠. ✭.✐♠♣❧❡✴✐♥,❡❣%❛,❡❞✴❝♦♥,❡①,✉❛❧✐.❡❞✮ ✇✐,❤✐♥
♦♥❡ ♥❡,✇♦%❦✳
❚❤%♦✉❣❤ ❝❛,❡❣♦%② ,❤❡♦%②✱ ✇❡ ♦❜,❛✐♥ ❛ ✉♥✐❢②✐♥❣ ❢%❛♠❡✇♦%❦ ❛, ✈❛%✐♦✉. ❧❡✈❡❧.✿
❡♠❛♥%✐❝ ❧❡✈❡❧ ❲❡ ❣✐✈❡ ❛ ✉♥✐❢♦%♠ .❡♠❛♥,✐❝. ❢♦% ❞✐.,%✐❜✉,❡❞ ♥❡,✇♦%❦. ♦❢ ❛❧✐❣♥❡❞
♦♥,♦❧♦❣✐❡.✱ ✉.✐♥❣ ,❤❡ ♣♦✇❡%❢✉❧ ♥♦,✐♦♥ ♦❢ ❝♦❧✐♠✐%✱ ✇❤✐❧❡ %❡✢❡❝,✐♥❣ ♣%♦♣❡%❧② ,❤❡
.❡♠❛♥,✐❝ ✈❛%✐❛,✐♦♥ ♣♦✐♥,. ✐♥❞✐❝❛,❡❞ ❛❜♦✈❡✳
✭♠❡%❛✮ ❧❛♥❣✉❛❣❡ ❧❡✈❡❧ ❲❡ ♣%♦✈✐❞❡ ❛ ✉♥✐❢♦%♠ ♥♦,❛,✐♦♥ ✭❜❛.❡❞ ♦♥ ,❤❡ ❞✐.,%✐❜✉,❡❞
♦♥,♦❧♦❣② ❧❛♥❣✉❛❣❡ ❉❖▲✮ ❢♦% ❞✐.,%✐❜✉,❡❞ ♥❡,✇♦%❦. ♦❢ ❛❧✐❣♥❡❞ ♦♥,♦❧♦❣✐❡.✱ .♣❛♥♥✐♥❣
,❤❡ ❞✐✛❡%❡♥, ♣♦..✐❜❧❡ .❡♠❛♥,✐❝ ❝❤♦✐❝❡.✳
.❡❛ ♦♥✐♥❣ ❧❡✈❡❧ ❯.✐♥❣ ,❤❡ ♥♦,✐♦♥ ♦❢ ❝♦❧✐♠✐,✱ ✇❡ ❝❛♥ ♣%♦✈✐❞❡ %❡❛.♦♥✐♥❣ ♠❡,❤♦❞. ❢♦%
❞✐.,%✐❜✉,❡❞ ♥❡,✇♦%❦. ♦❢ ❛❧✐❣♥❡❞ ♦♥,♦❧♦❣✐❡.✱ ❛❣❛✐♥ ❛❝%♦.. ❛❧❧ .❡♠❛♥,✐❝ ❝❤♦✐❝❡.✳✶
%♦♦❧ ❧❡✈❡❧ ❚❤❡ ,♦♦❧ ♦♥"♦❤✉❜✳♦'❣ ♣%♦✈✐❞❡. ❛♥ ✐♠♣❧❡♠❡♥,❛,✐♦♥ ♦❢ ❛♥❛❧②.✐. ❛♥❞ %❡❛✲
.♦♥✐♥❣ ❢♦% ❞✐.,%✐❜✉,❡❞ ♥❡,✇♦%❦. ♦❢ ❛❧✐❣♥❡❞ ♦♥,♦❧♦❣✐❡.✱ ❛❣❛✐♥ ✉.✐♥❣ ,❤❡ ♣♦✇❡%❢✉❧
❛❜.,%❛❝,✐♦♥. ♣%♦✈✐❞❡❞ ❜② ❝❛,❡❣♦%② ,❤❡♦%②✳
❧♦❣✐❝ ❧❡✈❡❧ ❖✉% .❡♠❛♥,✐❝. ✐. ❣✐✈❡♥ ❢♦% ,❤❡ ♦♥,♦❧♦❣② ❧❛♥❣✉❛❣❡ ❖❲▲✱ ❜✉, ❞✉❡ ,♦ ,❤❡
❛❜.,%❛❝,✐♦♥ ♣♦✇❡% ♦❢ ,❤❡ ❢%❛♠❡✇♦%❦✱ ✐, ❡❛.✐❧② ❝❛%%✐❡. ♦✈❡% ,♦ ♦,❤❡% ❧♦❣✐❝. ✉.❡❞ ✐♥
♦♥,♦❧♦❣② ❡♥❣✐♥❡❡%✐♥❣✱ ❧✐❦❡ ❘❉❋❙✱ ✜%.,✲♦%❞❡% ❧♦❣✐❝ ♦% ❋✲❧♦❣✐❝✳
❚❤✐. .❤♦✇. ,❤❛, ❝❛,❡❣♦%② ,❤❡♦%② ✐. ♥♦, ♦♥❧② ❛ ♣♦✇❡%❢✉❧ ❛❜.,%❛❝,✐♦♥ ❛, ,❤❡ .❡♠❛♥,✐❝
❧❡✈❡❧✱ ❜✉, ❝❛♥ ♣%♦♣❡%❧② ❣✉✐❞❡ ❧❛♥❣✉❛❣❡ ❞❡.✐❣♥ ❛♥❞ ,♦♦❧ ✐♠♣❧❡♠❡♥,❛,✐♦♥. ❛♥❞ ,❤✉.
♣%♦✈✐❞❡ ✉.❡❢✉❧ ❛❜.,%❛❝,✐♦♥ ❜❛%%✐❡%. ❢%♦♠ ❛ .♦❢,✇❛%❡ ❡♥❣✐♥❡❡%✐♥❣ ♣♦✐♥, ♦❢ ✈✐❡✇✳
❚❤❡ ❞✐.,%✐❜✉,❡❞ ♦♥,♦❧♦❣② ❧❛♥❣✉❛❣❡ DOL ✐. ❛ ♠❡,❛❧❛♥❣✉❛❣❡ ✐♥ ,❤❡ .❡♥.❡ ,❤❛, ✐,
❡♥❛❜❧❡. ,❤❡ %❡✉.❡ ♦❢ ❡①✐.,✐♥❣ ♦♥,♦❧♦❣✐❡. ❛. ❜✉✐❧❞✐♥❣ ❜❧♦❝❦. ❢♦% ♥❡✇ ♦♥,♦❧♦❣✐❡. ✉.✐♥❣ ❛
✈❛%✐❡,② ♦❢ .,%✉❝,✉%✐♥❣ ,❡❝❤♥✐T✉❡.✱ ❛. ✇❡❧❧ ❛. ,❤❡ .♣❡❝✐✜❝❛,✐♦♥ ♦❢ %❡❧❛,✐♦♥.❤✐♣. ❜❡,✇❡❡♥
♦♥,♦❧♦❣✐❡.✳ ❖♥❡ ✐♠♣♦%,❛♥, ❢❡❛,✉%❡ ♦❢ DOL ✐. ,❤❡ ❛❜✐❧✐,② ,♦ ❝♦♠❜✐♥❡ ♦♥,♦❧♦❣✐❡. ,❤❛,
❛%❡ ✇%✐,,❡♥ ✐♥ ❞✐✛❡%❡♥, ❧❛♥❣✉❛❣❡. ✇✐,❤♦✉, ❝❤❛♥❣✐♥❣ ,❤❡✐% .❡♠❛♥,✐❝.✳ ❆ ❢♦%♠❛❧ .♣❡❝✐✜✲
❝❛,✐♦♥ ♦❢ ,❤❡ ❧❛♥❣✉❛❣❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✼❪✳ ❍♦✇❡✈❡% ♥♦,❡ ,❤❛, .②♥,❛① ❛♥❞ .❡♠❛♥,✐❝.
♦❢DOL❛❧✐❣♥♠❡♥,. ✐. ✐♥,%♦❞✉❝❡❞ ✐♥ ,❤✐. ♣❛♣❡% ❢♦% ,❤❡ ✜%., ,✐♠❡✳
✶ ❲❡ ❞♦ ♥♦% ❝❧❛✐♠ ❤❡,❡ %❤❛% %❤❡ ,❡❛-♦♥✐♥❣ ♠❡%❤♦❞- ✇❡ ♣,♦✈✐❞❡ ♦✉%♣❡,❢♦,♠ ♠♦,❡ -♣❡❝✐❛❧✐-❡❞
❛❧✐❣♥♠❡♥% ,❡❛-♦♥✐♥❣ ♠❡%❤♦❞-✱ -❛② ❢♦, ❉❉▲✱ ♦, ❛❧✐❣♥♠❡♥% ❞❡❜✉❣❣✐♥❣✿ ♦✉, ♠❛✐♥ ❝♦♥%,✐❜✉✲
%✐♦♥ ✐- %❤❡ ♣,♦✈✐-✐♦♥ ♦❢ ❛ ✉♥✐❢②✐♥❣ ❢,❛♠❡✇♦,❦ %❤❛% ✇♦,❦- -✐♠✉❧%❛♥❡♦✉-❧② ❛% %❤❡ ✈❛,✐♦✉-
❚❤❡ ❣❡♥❡%❛❧ ♣✐❝+✉%❡ ✐- +❤❡♥ ❛- ❢♦❧❧♦✇-✿ ❡①✐-+✐♥❣ ♦♥+♦❧♦❣✐❡- ❝❛♥ ❜❡ ✐♥+❡❣%❛+❡❞ ❛-✲✐-
✐♥+♦ +❤❡DOL❢%❛♠❡✇♦%❦✳ ❲✐+❤ ♦✉% ♥❡✇ ❡①+❡♥❞❡❞DOL-②♥+❛①✱ ✇❡ ❝❛♥ -♣❡❝✐❢② ❞✐✛❡%❡♥+
❦✐♥❞- ♦❢ ❛❧✐♥❣♠❡♥+-✳ ❋%♦♠ -✉❝❤ ❛♥ ❛❧✐❣♥♠❡♥+✱ ✇❡ ❝♦♥-+%✉❝+ ❛ ❣%❛♣❤ ♦❢ ♦♥+♦❧♦❣✐❡- ❛♥❞
♠♦%♣❤✐-♠- ❜❡+✇❡❡♥ +❤❡♠✖✐♥ ❛ ✇❛② ❞❡♣❡♥❞✐♥❣ ♦♥ +❤❡ ❝❤♦-❡♥ ❛❧✐❣♥♠❡♥+ ❢%❛♠❡✇♦%❦✳
❙♦♠❡+✐♠❡-✱ +❤✐- -+❡♣ ❛❧-♦ ✐♥✈♦❧✈❡- +%❛♥-❢♦%♠❛+✐♦♥- ♦♥ +❤❡ ♦♥+♦❧♦❣✐❡-✱ -✉❝❤ ❛- %❡❧❛✲
+✐✈✐-❛+✐♦♥ ♦❢ +❤❡ ✭❣❧♦❜❛❧✮ ❞♦♠❛✐♥ ✉-✐♥❣ ♣%❡❞✐❝❛+❡-✳ ❆ ♥❡+✇♦%❦ ♦❢ ❛❧✐❣♥♠❡♥+- ❝❛♥ +❤❡♥
❜❡ ❝♦♠❜✐♥❡❞ +♦ ❛♥ ✐♥+❡❣%❛+❡❞ ❛❧✐❣♥♠❡♥+ ♦♥+♦❧♦❣② ✈✐❛ ❛ -♦✲❝❛❧❧❡❞ ❝♦❧✐♠✐+✳ ❘❡❛-♦♥✐♥❣
✐♥ ❛ ♥❡+✇♦%❦ ♦❢ ❛❧✐❣♥❡❞ ♦♥+♦❧♦❣✐❡- ✐- +❤❡♥ +❤❡ -❛♠❡ ❛- %❡❛-♦♥✐♥❣ ✐♥ +❤❡ ❝♦♠❜✐♥❡❞
♦♥+♦❧♦❣②✳ ❚❤✉-✱ ✐♥ ♦%❞❡% +♦ ✐♠♣❧❡♠❡♥+ ❛ %❡❛-♦♥❡%✱ ✐+ ✐- ✐♥ ♣%✐♥❝✐♣❧❡ -✉✣❝✐❡♥+ +♦ ❞❡✜♥❡
+❤❡ %❡❧❛+✐✈✐-❛+✐♦♥ ♣%♦❝❡❞✉%❡ ❢♦% +❤❡ ❧♦❝❛❧ ❧♦❣✐❝- ❛♥❞ +❤❡ ❛❧✐❣♥♠❡♥+ +%❛♥-❢♦%♠❛+✐♦♥ ❢♦%
❡❛❝❤ ❦✐♥❞- ♦❢ -❡♠❛♥+✐❝-✳
✸ ◆❡#✇♦&❦( ♦❢ ♦♥#♦❧♦❣✐❡( ❛♥❞ #❤❡✐& (❡♠❛♥#✐❝(
■♥ +❤✐- -❡❝+✐♦♥ ✇❡ %❡❝❛❧❧ ♥❡+✇♦%❦- ♦❢ ♦♥+♦❧♦❣✐❡- ❛♥❞ +❤❡✐% -❡♠❛♥+✐❝- ✐♥+%♦❞✉❝❡❞ ✐♥ ❬✷✸✱
✽❪✳ ◆❡+✇♦%❦- ♦❢ ♦♥+♦❧♦❣✐❡- ✭❤❡%❡ ❞❡♥♦+❡❞ ◆❡❖✮ ❬✽❪✱ ❝❛❧❧❡❞ ❞✐-+%✐❜✉+❡❞ -②-+❡♠- ✐♥ ❬✷✸❪✱
❝♦♥-✐-+ ♦❢ ❛ ❢❛♠✐❧② (Oi)i∈I ♦❢ ♦♥+♦❧♦❣✐❡- ♦✈❡% ❛ -❡+ ♦❢ ✐♥❞❡①❡- I ✐♥+❡%❝♦♥♥❡❝+❡❞ ❜② ❛ -❡+ ♦❢ ❛❧✐❣♥♠❡♥+- (Aij)i,j∈I ❜❡+✇❡❡♥ +❤❡♠✳ ❆❧✐❣♥♠❡♥+- ❛%❡ -❡+- ♦❢ ❝♦""❡$♣♦♥❞❡♥❝❡$
❜❡+✇❡❡♥ +❤❡ +❛%❣❡+ ♦♥+♦❧♦❣②O1❛♥❞ -♦✉%❝❡ ♦♥+♦❧♦❣②O2♦❢ +❤❡ ❛❧✐❣♥♠❡♥+✳ ❈♦%%❡-♣♦♥✲
❞❡♥❝❡- ❛%❡ +%✐♣❧❡- (e1, e2, R)✇❤❡%❡ e1 ❛♥❞ e2 ❛%❡ ❡♥+✐+✐❡- ❜✉✐❧+ ✇✐+❤ +❤❡ ❤❡❧♣ ♦❢ ❛♥
❡♥+✐+② ❧❛♥❣✉❛❣❡ ♦✈❡% O1 ❛♥❞ O2✱ %❡-♣❡❝+✐✈❡❧②✱ ❛♥❞ R ✐- ❛ %❡❧❛+✐♦♥ ❜❡+✇❡❡♥ ❡♥+✐+✐❡-
❢%♦♠ ❛ -❡+ ♦❢ %❡❧❛+✐♦♥- R✳
❆ -❡♠❛♥+✐❝- ♦❢ ♥❡+✇♦%❦- ♦❢ ♦♥+♦❧♦❣✐❡- ✐- ❣✐✈❡♥ ✐♥ +❡%♠- ♦❢ ❧♦❝❛❧ ✐♥+❡%♣%❡+❛+✐♦♥ ♦❢
+❤❡ ♦♥+♦❧♦❣✐❡- ❛♥❞ ❛❧✐❣♥♠❡♥+- ✐+ ❝♦♥-✐-+- ♦❢✳ ❚♦ ❜❡ ❛❜❧❡ +♦ ❣✐✈❡ -✉❝❤ ❛ -❡♠❛♥+✐❝-✱ ♦♥❡
♥❡❡❞- +♦ ❣✐✈❡ ❛♥ ✐♥+❡%♣%❡+❛+✐♦♥ ♦❢ +❤❡ %❡❧❛+✐♦♥- ❜❡+✇❡❡♥ ❡♥+✐+✐❡- +❤❛+ ❛%❡ ❡①♣%❡--❡❞ ✐♥
+❤❡ ❝♦%%❡-♣♦♥❞❡♥❝❡-✳ ■♥ +❤❡ ❢♦❧❧♦✇✐♥❣ +❤%❡❡ -✉❜-❡❝+✐♦♥- ❧❡+ S={(Oi)i∈I,(Aij)i,j∈I}
❜❡ ❛ ◆❡❖ ♦✈❡% ❛ -❡+ ♦❢ ✐♥❞❡①❡-I✳
O1
m1
((
O2
m2
!!
. . . On
mn
wwD
❙✐♠♣❧❡ &❡♠❛♥)✐❝& ■♥ +❤❡ -✐♠♣❧❡ -❡✲
♠❛♥+✐❝-✱ +❤❡ ❛--✉♠♣+✐♦♥ ✐- +❤❛+ ❛❧❧ ♦♥✲
+♦❧♦❣✐❡- ❛%❡ ✐♥+❡%♣%❡+❡❞ ♦✈❡% +❤❡ -❛♠❡
❞♦♠❛✐♥ ✭♦% ✉♥✐✈❡%-❡ ♦❢ ✐♥+❡%♣%❡+❛+✐♦♥✮
D✳ ❚❤❡ %❡❧❛+✐♦♥- ✐♥ R ❛%❡ ✐♥+❡%♣%❡+❡❞
❛- %❡❧❛+✐♦♥- ♦✈❡%D✱ ❛♥❞ ✇❡ ❞❡♥♦+❡ +❤❡ ✐♥+❡%♣%❡+❛+✐♦♥ ♦❢ R∈R❜② RD✳
■❢ O1✱ O2 ❛%❡ +✇♦ ♦♥+♦❧♦❣✐❡- ❛♥❞ c = (e1, e2, R) ✐- ❛ ❝♦%%❡-♣♦♥❞❡♥❝❡ ❜❡+✇❡❡♥
O1 ❛♥❞ O2✱ ✇❡ -❛② +❤❛+ c ✐- -❛+✐-✜❡❞ ❜② ✐♥+❡%♣%❡+❛+✐♦♥- m1✱ m2 ♦❢ O1✱ O2 ✐✛
m1(e1)RDm2(e2)✳ ❚❤✐- ✐- ✇%✐++❡♥m1, m2|=Sc✳ ❆ ♠♦❞❡❧ ♦❢ ❛♥ ❛❧✐❣♥♠❡♥+A❜❡+✇❡❡♥
♦♥+♦❧♦❣✐❡-O1❛♥❞O2✐- +❤❡♥ ❛ ♣❛✐%m1✱m2♦❢ ✐♥+❡%♣%❡+❛+✐♦♥- ♦❢O1✱O2-✉❝❤ +❤❛+ ❢♦%
❛❧❧c∈A✱m1, m2|=S c✳ ❲❡ ❞❡♥♦+❡ +❤✐- ❜②m1, m2|=SA✳ ❆♥ ✐♥+❡%♣%❡+❛+✐♦♥ ♦❢S ✐- ❛
❢❛♠✐❧②(mi)i∈I ♦❢ ♠♦❞❡❧-mi ♦❢Oi✳ ❆ -✐♠♣❧❡ ✐♥+❡%♣%❡+❛+✐♦♥ ♦❢S ✐- ❛♥ ✐♥+❡%♣%❡+❛+✐♦♥
(mi)i∈I ♦❢S♦✈❡% +❤❡ -❛♠❡ ❞♦♠❛✐♥ D✳
❉❡✜♥✐)✐♦♥ ✶✳ ❬✷✸❪ ❆ $✐♠♣❧❡ ♠♦❞❡❧ ♦❢ ❛ S ✐$ ❛ $✐♠♣❧❡ ✐♥2❡"♣"❡2❛2✐♦♥ (mi)i∈I ♦❢ S
$✉❝❤ 2❤❛2 ❢♦" ❡❛❝❤i, j ∈I✱mi, mj|=S Aij✳ ❚❤✐$ ✐$ ✇"✐22❡♥(mi)i∈I |=SS✳ ❲❡ ❞❡♥♦2❡
❜②M odsim(S)2❤❡ ❝❧❛$$ ♦❢ ❛❧❧ $✐♠♣❧❡ ♠♦❞❡❧$ ♦❢ S✳
O1
m
O2
m2
. . . On
mn
D1
γ1
((
D2
γ2
!!
. . . Dn
γn
wwU
■♥"❡❣%❛"❡❞ ❙❡♠❛♥"✐❝, ❆♥♦#❤❡& ♣♦(✲
(✐❜✐❧✐#② ✐( #♦ ❝♦♥(✐❞❡& #❤❛# #❤❡ ❞♦♠❛✐♥
♦❢ ✐♥#❡&♣&❡#❛#✐♦♥ ♦❢ #❤❡ ♦♥#♦❧♦❣✐❡( ♦❢ ❛
◆❡❖ ✐( ♥♦# ❝♦♥(#&❛✐♥❡❞✱ ❛♥❞ ❛ ❣❧♦❜❛❧
❞♦♠❛✐♥ ♦❢ ✐♥#❡&♣&❡#❛#✐♦♥ U ❡①✐(#(✱ #♦✲
❣❡#❤❡& ✇✐#❤ ❛ ❢❛♠✐❧② ♦❢ ❡!✉❛❧✐&✐♥❣ ❢✉♥❝✲
,✐♦♥& γi :Di →U✱ ✇❤❡&❡ Di ✐( #❤❡ ❞♦✲
♠❛✐♥ ♦❢ Oi✱ ❢♦& ❡❛❝❤ i ∈ I✳ ❆ &❡❧❛#✐♦♥ R ✐♥ R ✐( ✐♥#❡&♣&❡#❡❞ ❛( ❛ &❡❧❛#✐♦♥ RU ♦♥
#❤❡ ❣❧♦❜❛❧ ❞♦♠❛✐♥✳ ❙❛#✐(❢❛❝#✐♦♥ ♦❢ ❛ ❝♦&&❡(♣♦♥❞❡♥❝❡ c = (e1, e2, R) ❜② #✇♦ ♠♦❞❡❧(
m1 ♦❢ O1 ❛♥❞ m2 ♦❢ O2 ♠❡❛♥( #❤❛# γi(mi(e1))RUγj(mj(e2))✳ ❲❡ ❞❡♥♦#❡ #❤✐( ❜② m1, m2 |=Iγ1,γ2 c❛♥❞ ❜② m1, m2 |=Iγ1,γ2 A ✇❡ ❞❡♥♦#❡ #❤❛# m1, m2 |=Iγ1,γ2 c ❢♦& ❡❛❝❤
c∈A✳
❆♥ ✐♥#❡❣&❛#❡❞ ✐♥#❡&♣&❡#❛#✐♦♥ ♦❢ S ✐( #❤❡♥ {(mi)i∈I,(γi)i∈I} ✇❤❡&❡ (mi)i∈I ✐( ❛♥
✐♥#❡&♣&❡#❛#✐♦♥ ♦❢S ❛♥❞γi:Di→U ✐( ❛ ❢✉♥❝#✐♦♥ #♦ ❛ ❝♦♠♠♦♥ ❣❧♦❜❛❧ ❞♦♠❛✐♥ U ❢♦&
❡❛❝❤i∈I✳ ❲❡ ❤❡&❡ ❛((✉♠❡ #❤❛# #❤❡γi❛&❡ ✐♥❝❧✉(✐♦♥(✳✷
❉❡✜♥✐"✐♦♥ ✷✳ ❬✷✸❪ ❆♥ ✐♥,❡❣3❛,❡❞ ✐♥,❡3♣3❡,❛,✐♦♥ {(mi),(γi)} ♦❢ S ✐& ❛♥ ✐♥,❡❣3❛,❡❞
♠♦❞❡❧ ♦❢S✐✛ ❢♦3 ❡❛❝❤i, j∈I✱mi, mj|=Iγi,γj Aij✳ ❲❡ ❞❡♥♦,❡ ❜② M odint(S),❤❡ ❝❧❛&&
♦❢ ❛❧❧ ✐♥,❡❣3❛,❡❞ ♠♦❞❡❧& ♦❢ ❛ ◆❡❖S✳
O1
m
O2
m2
. . . On
mn
D1
r1,2
//
r1,3
;;D2
r2,3
//. . . Dn
rn,1
ii
❈♦♥"❡①"✉❛❧✐,❡❞ ❙❡♠❛♥"✐❝, ❚❤❡ ❢✉♥❝✲
#✐♦♥❛❧ ♥♦#✐♦♥ ♦❢ ❝♦♥#❡①#✉❛❧✐(❡❞ (❡♠❛♥✲
#✐❝( ✐♥ ❬✷✸❪ ✐( ♥♦# ✈❡&② ✉(❡❢✉❧ ❛♥❞ ❤❛(
❜❡❡♥ &❡♣❧❛❝❡❞ ❜② ❛ ♠♦&❡ ✢❡①✐❜❧❡ &❡❧❛✲
#✐♦♥❛❧ ♥♦#✐♦♥ (✉❜(❡B✉❡♥#❧② ❬✽❪✱ ❝❧♦(❡❧② &❡✲
❧❛#❡❞ #♦ #❤❡ (❡♠❛♥#✐❝( ♦❢ ❉❉▲( ❬✷❪ ❛♥❞
E✲❝♦♥♥❡❝#✐♦♥( ❬✶✹❪✳
❚❤❡ ✐❞❡❛ ✐( #♦ &❡❧❛#❡ #❤❡ ❞♦♠❛✐♥( ♦❢ #❤❡ ♦♥#♦❧♦❣✐❡( ❜② ❛ ❢❛♠✐❧② ♦❢ &❡❧❛#✐♦♥( r= (rij)i,j∈I✳ ❚❤❡ &❡❧❛#✐♦♥(R ✐♥R ❛&❡ ✐♥#❡&♣&❡#❡❞ ✐♥ ❡❛❝❤ ❞♦♠❛✐♥ ♦❢ #❤❡ ♦♥#♦❧♦❣✐❡( ✐♥
#❤❡ ◆❡❖✳ ❙❛#✐(❢❛❝#✐♦♥ ♦❢ ❛ ❝♦&&❡(♣♦♥❞❡♥❝❡c= (e1, e2, R)❜② #✇♦ ♠♦❞❡❧(m1♦❢O1❛♥❞
m2♦❢O2♠❡❛♥( #❤❛#mi(e1)Rirji(mj(e2))✱ ✇❤❡&❡Ri✐( #❤❡ ✐♥#❡&♣&❡#❛#✐♦♥ ♦❢R✐♥Di✳
❲❡ ❞❡♥♦#❡ ✐# ❜②m1, m2|=Cr c✱ ❛♥❞ ❡①#❡♥❞ #❤✐( #♦ ❛❧✐❣♥♠❡♥#(✱ ❞❡♥♦#❡❞ m1, m2|=Cr A
✐❢ ❛❧❧ ❝♦&&❡(♣♦♥❞❡♥❝❡( ♦❢ #❤❡ ❛❧✐❣♥♠❡♥# ❛&❡ (❛#✐(✜❡❞ ❜② m1, m2 ✇✳&✳#✳r✳
❆ ❝♦♥#❡①#✉❛❧✐(❡❞ ✐♥#❡&♣&❡#❛#✐♦♥ ♦❢ S✐( ❛ ♣❛✐& {(mi)i∈I,(rij)i,j∈I} ✇❤❡&❡(mi)i∈I
✐( ❛♥ ✐♥#❡&♣&❡#❛#✐♦♥ ♦❢ S ❛♥❞(rij)i,j∈I ✐( ❛ ❢❛♠✐❧② ♦❢ ❞♦♠❛✐♥ &❡❧❛#✐♦♥( (✉❝❤ #❤❛# rij
&❡❧❛#❡( #❤❡ ❞♦♠❛✐♥ ♦❢ mi #♦ #❤❡ ❞♦♠❛✐♥ ♦❢ mj ❛♥❞ rii ✐( #❤❡ ✐❞❡♥#✐#② ✭❞✐❛❣♦♥❛❧✮
&❡❧❛#✐♦♥✳ ❋✉&#❤❡& ❛((✉♠♣#✐♦♥( ❛❜♦✉# ❞♦♠❛✐♥ &❡❧❛#✐♦♥( ❝❛♥ ❜❡ ❛❞❞❡❞✱ #❤✉( &❡(#&✐❝#✐♥❣
♠♦&❡ #❤❡ ❝❧❛(( ♦❢ ✐♥#❡&♣&❡#❛#✐♦♥( ♦❢ ❛ ◆❡❖✳
❉❡✜♥✐"✐♦♥ ✸✳ ❆ ❝♦♥,❡①,✉❛❧✐&❡❞ ♠♦❞❡❧ ♦❢ ,❤❡ ◆❡❖S✐& ❛ ❝♦♥,❡①,✉❛❧✐&❡❞ ✐♥,❡3♣3❡,❛,✐♦♥
((mi)i∈I,(rij)i,j∈I) ♦❢ S &✉❝❤ ,❤❛, ❢♦3 ❡❛❝❤ i, j ∈I✱ mi, mj |=Cr Aij✳ ❲❡ ❞❡♥♦,❡ ❜② M odcon(S),❤❡ ❝❧❛&& ♦❢ ❛❧❧ ❝♦♥,❡①,✉❛❧✐&❡❞ ♠♦❞❡❧& ♦❢ ❛ ◆❡❖ S✳
✷ ❚❤❡ #❤❡♦%② ❛❧)♦ ✇♦%❦) ❢♦% ✐♥❥❡❝#✐♦♥) ✇✐#❤♦✉# ♠✉❝❤ ❝❤❛♥❣❡✳ ❆%❜✐#%❛%②✱ ✐✳❡✳ ♣♦))✐❜❧② ♥♦♥✲
✐♥❥❡❝#✐✈❡ ♠❛♣)✱ ❛%❡ ❝♦♥❝❡♣#✉❛❧❧② ♥♦# ♥❡❝❡))❛%②✿ ❛ ❧♦❝❛❧ ♠♦❞❡❧ ❝❛♥ ❜❡ =✉♦#✐❡♥#❡❞ ❜② #❤❡
❦❡%♥❡❧ ♦❢ ❛ ♥♦♥✲✐♥❥❡❝#✐✈❡ )✉❝❤ ♠❛♣✱ ❛♥❞ #❤❡♥ ❜❡ %❡♣❧❛❝❡❞ ❜② #❤❡ =✉♦#✐❡♥#✱ ❧❡❛❞✐♥❣ #♦ ❛♥
✹ DOL ❆❧✐❣♥♠❡♥()
■♥ "❤✐% %❡❝"✐♦♥ ✇❡ %"❛+" ❜② ✐♥"+♦❞✉❝✐♥❣ "❤❡DOL❝♦♥❝❡♣"% ♥❡❝❡%%❛+② ❢♦+ ❣✐✈✐♥❣ %❡♠❛♥✲
"✐❝% ♦❢ ❛❧✐❣♥♠❡♥"%✳ ❲❡ "❤❡♥ ✐♥"+♦❞✉❝❡ "❤❡ %②♥"❛① ♦❢ ❛❧✐❣♥♠❡♥"% ✐♥ DOL❛♥❞ ✐❧❧✉%"+❛"❡
✇✐"❤ "❤❡ ❤❡❧♣ ♦❢ ❛♥ ❡①❛♠♣❧❡ ✐♥✈♦❧✈✐♥❣ OWL ♦♥"♦❧♦❣✐❡% ❤♦✇ "❤❡ %❡♠❛♥"✐❝% ♦❢ ❛❧✐❣♥✲
♠❡♥"% ❝❛♥ ❜❡ ❣✐✈❡♥ ✉%✐♥❣ ❞✐❛❣+❛♠% ❛♥❞ ❝♦❧✐♠✐"%✳ ❲❡ "❤❡♥ ♣+❡%❡♥" "❤❡ ♠❛✐♥ +❡%✉❧"
♦❢ "❤❡ ♣❛♣❡+✱ %❤♦✇✐♥❣ ❤♦✇ "❤❡ ❝❛"❡❣♦+✐❝❛❧ %❡♠❛♥"✐❝% ♦❢ DOL❛❧✐❣♥♠❡♥"% ❝❛♣"✉+❡% "❤❡
"❤+❡❡ %❡♠❛♥"✐❝% ♦❢ ♥❡"✇♦+❦% ♦❢ ♦♥"♦❧♦❣✐❡%✳
✹✳✶ DOL ❉✐❛❣'❛♠) ❛♥❞ ❈♦♠❜✐♥❛/✐♦♥)
❚❤❡ %②♥"❛① ❢♦+ %♣❡❝✐❢②✐♥❣ ❞✐❛❣+❛♠% ✐♥DOL✐%
❣!❛♣❤ ❉ ❂ D1, . . . , Dm, O1, . . . , On, M1, . . . , Mp, A1, . . . , Ak
✇❤❡+❡Di❛+❡ ✭%✉❜✲✮❞✐❛❣+❛♠%✱Oi❛+❡ ♦♥"♦❧♦❣✐❡%✱Mi❛+❡ ♠♦+♣❤✐%♠% ❛♥❞Ai❛+❡ ❛❧✐❣♥✲
♠❡♥"%✳ ❚❤❡ ✉%❡+ %♣❡❝✐✜❡% ❛ ❞✐❛❣+❛♠D❢♦+♠❡❞ ✇✐"❤ "❤❡ %✉❜❣+❛♣❤% ❣✐✈❡♥ ❜② ❞✐❛❣+❛♠%
Di✱ ❡①"❡♥❞❡❞ ✇✐"❤ ♦♥"♦❧♦❣✐❡%Oi ❛♥❞ "❤❡ ♠♦+♣❤✐%♠%Mi ❛♥❞ "❤❡ %✉❜❞✐❛❣+❛♠% ♦❢ "❤❡
❛❧✐❣♥♠❡♥"%Ai
DOL❛❧%♦ ♣+♦✈✐❞❡% ♠❡❛♥% ❢♦+ ❝♦♠❜✐♥✐♥❣ ❛ ❞✐❛❣+❛♠ ♦❢ ♦♥"♦❧♦❣✐❡% ✐♥"♦ ❛ ♥❡✇ ♦♥✲
"♦❧♦❣②✱ %✉❝❤ "❤❛" "❤❡ %②♠❜♦❧% +❡❧❛"❡❞ ✐♥ "❤❡ ❞✐❛❣+❛♠ ❛+❡ ✐❞❡♥"✐✜❡❞✳ ❚❤❡ %②♥"❛① ♦❢
❝♦♠❜✐♥❛"✐♦♥% ✐% ♦♥"♦❧♦❣② ❖ ❂ ❝♦♠❜✐♥❡ ❉✱ ✇❤❡+❡D ✐% ❛ ❞✐❛❣+❛♠✱ ♥❛♠❡❞ ♦+ %♣❡❝✐✲
✜❡❞ ❛% ❛❜♦✈❡✳ ❚❤❡ %❡♠❛♥"✐❝% ♦❢ ❛ ❝♦♠❜✐♥❛"✐♦♥O✐% "❤❡ ❝❧❛%% ♦❢ ♠♦❞❡❧% ♦❢ "❤❡ ❝♦❧✐♠✐"
♦♥"♦❧♦❣② ♦❢ "❤❡ ❞✐❛❣+❛♠ %♣❡❝✐✜❡❞ ✐♥ "❤❡ ❝♦♠❜✐♥❛"✐♦♥✳ ❯♥❞❡+ +❛"❤❡+ ♠✐❧❞ "❡❝❤♥✐❝❛❧
❛%%✉♠♣"✐♦♥%✱ "❤✐% ♠♦❞❡❧ ❝❧❛%% ❝❛♣"✉+❡% ❡①❛❝"❧② "❤❡ ♠♦❞❡❧% ♦❢ "❤❡ ❞✐❛❣+❛♠✳
✹✳✷ ❙②♥/❛① ♦❢ DOL ❆❧✐❣♥♠❡♥/)
DOL+❡♣+❡%❡♥"% "❤❡ ❣❡♥❡+❛❧ ❛❧✐❣♥♠❡♥" ❢♦+♠❛" ✐♥ ❛ %✐♠✐❧❛+ ✇❛② "♦ "❤❡ ❆❧✐❣♥♠❡♥" ❆B■
❬✼❪ ❛% ❢♦❧❧♦✇%✿
❛❧✐❣♥♠❡♥, ❆ ✿ O1 ,♦ O2 ❂ s11 REL1 s12✱ . . .✱ sn1 RELn sn2
❛11✉♠✐♥❣ ❉❖▼❆■◆
❡♥❞
✇❤❡+❡ O1 ❛♥❞ O2 ❛+❡ "❤❡ ♦♥"♦❧♦❣✐❡% "♦ ❜❡ ❛❧✐❣♥❡❞✱ si1 ❛♥❞ si2 ❛+❡ O1 ❛♥❞ +❡%♣❡❝✲
"✐✈❡❧②O2 %②♠❜♦❧%✱ ❢♦+ i= 1, . . . , n✱si1 RELi si2✐% ❛ ❝♦""❡$♣♦♥❞❡♥❝❡ ✇❤✐❝❤ ✐❞❡♥"✐✜❡%
❛ +❡❧❛"✐♦♥ ❜❡"✇❡❡♥ "❤❡ ♦♥"♦❧♦❣② %②♠❜♦❧%✱ ✉%✐♥❣ ♦♥❡ ♦❢ "❤❡ %②♠❜♦❧% >✭%✉❜%✉♠❡%✮✱<
✭✐% %✉❜%✉♠❡❞✮✱ = ✭❡G✉✐✈❛❧❡♥"✮✱ % ✭✐♥❝♦♠♣❛"✐❜❧❡✮✱ ∈✭✐♥%"❛♥❝❡✮ ♦+ ∋ ✭❤❛% ✐♥%"❛♥❝❡✮
❛♥❞ ❉❖▼❆■◆ +❡❝♦+❞% ✇❤❡"❤❡+ %✐♥❣❧❡✱ ✐♥"❡❣+❛"❡❞ ♦+ ❝♦♥"❡①"✉❛❧✐%❡❞ %❡♠❛♥"✐❝% ✐% ✉%❡❞✱
✉%✐♥❣ "❤❡ ❝♦♥%"❛♥" ❙✐♥❣❧❡❉♦♠❛✐♥✱ ●❧♦❜❛❧❉♦♠❛✐♥ ❛♥❞ ❈♦♥"❡①"✉❛❧✐9❡❞❉♦♠❛✐♥ +❡✲
%♣❡❝"✐✈❡❧②✳
❇❡❢♦+❡ %"❛+"✐♥❣ "♦ ❛♥❛❧②%❡ "❤❡ "❤+❡❡ %❡♠❛♥"✐❝% ❢♦+ ◆❡❖# ✐♥ ♦✉+ %❡""✐♥❣✱ ✇❡ ❝❛♥
✜+%" ❞❡✜♥❡ "❤❡ ❞✐❛❣+❛♠ ♦❢ ❛ ◆❡❖ ✐♥ "❡+♠% ♦❢ "❤❡ ❞✐❛❣+❛♠% ♦❢ ✐"% ♣❛+"%✳
❉❡✜♥✐/✐♦♥ ✹✳ ❚❤❡ ❞✐❛❣"❛♠ ♦❢ ❛ ◆❡❖S={(Oi)i∈I,(Aij)i,j∈I}✐$ ♦❜0❛✐♥❡❞ ❜② ♣✉00✐♥❣
0♦❣❡0❤❡" 0❤❡ ❞✐❛❣"❛♠$ ♦❢ ❛❧❧ ❛❧✐❣♥♠❡♥0$ Aij ✐0 ❝♦♥$✐$0$ ♦❢✳
=
=
= ∈
⊑ ¬
: = =
∈
<¬
A
OWL
A
O1 O2
B
O1' O2'
Bridge
■!" ❝♦♥"!✐!✉❡♥!" ❛*❡ ♦❜!❛✐♥❡❞ ❛" ❢♦❧❧♦✇"✳ ❚❤❡ ♦♥!♦❧♦❣✐❡"O1′ ❛♥❞O′2❝♦❧❧❡❝!✱ *❡"♣❡❝!✐✈❡❧②✱
❛❧❧ !❤❡ "②♠❜♦❧"s1 ❛♥❞ s2 !❤❛! ❛♣♣❡❛* ✐♥ ❛ ❝♦**❡"♣♦♥❞❡♥❝❡ s1RELs2 ✐♥A✱ ❛♥❞ ❤❛✈❡
♥♦ "❡♥!❡♥❝❡"✳ ❚❤❡ ♠♦*♣❤✐"♠" ιi ❢*♦♠ O′i !♦ Oi✱ ✇❤❡*❡ i= 1,2✱ ❛*❡ ✐♥❝❧✉"✐♦♥"✳ ❚❤❡
♦♥!♦❧♦❣②B ✐" ❝♦♥"!*✉❝!❡❞ ❜② !✉*♥✐♥❣ !❤❡ ❝♦**❡"♣♦♥❞❡♥❝❡" ♦❢ !❤❡ ❛❧✐❣♥♠❡♥! ✐♥!♦OWL
❛①✐♦♠"✳ ❚❤❡ ♠♦*♣❤✐"♠" σ1 ❛♥❞σ2 ♠❛♣ !❤❡ "②♠❜♦❧" ♦❝❝✉**✐♥❣ ✐♥ ❝♦**❡"♣♦♥❞❡♥❝❡" !♦
!❤❡✐* ❝♦✉♥!❡*♣❛*! ✐♥ B✳ ❚❤❡ ❛❧✐❣♥♠❡♥! ✐" ✐❧❧✲❢♦*♠❡❞ ✇❤❡♥ ✐! ❝♦♥!❛✐♥" ❛♥ ❡;✉✐✈❛❧❡♥❝❡
❜❡!✇❡❡♥ "②♠❜♦❧" ♦❢ ❞✐✛❡*❡♥! ❦✐♥❞"✱ ♦* ✐❢ B ❢❛✐❧" !♦ ❜❡ ❛ ✇❡❧❧✲❢♦*♠❡❞ ♦♥!♦❧♦❣②✳
❊①❛♠♣❧❡ ✷✳ ❲❡ "#❛%# ❜② ❛❞❞✐♥❣ #❤❡ ❛""✉♠♣#✐♦♥ #❤❛# ✇❡ ❤❛✈❡ ❛ "❤❛%❡❞ ❞♦♠❛✐♥ ❢♦%
#❤❡ ♦♥#♦❧♦❣✐❡" ✐♥ #❤❡ ❛❧✐❣♥♠❡♥# ♦❢ ❊①✳ ✶✿
❛❧✐❣♥♠❡♥' ❆:❙ '♦ ❚=. . .
❛))✉♠✐♥❣ ❙✐♥❣❧❡❉♦♠❛✐♥
❚❤❡ ❞✐❛❣%❛♠ ♦❢ ❆ ✐" #❤❡♥
S B T
S′
σ1
>>
ι1
__
T′
ι2
>>
σ2
``
✇❤❡%❡S′ ❝♦♥"✐"#" ♦❢ #❤❡ ❝♦♥❝❡♣#" @❡*"♦♥ ❛♥❞ ❈❤✐❧❞ ❛♥❞ #❤❡ ✐♥❞✐✈✐❞✉❛❧ ❛❧❡① ❛♥❞ T′
❝♦♥"✐"#" ♦❢ #❤❡ ❝♦♥❝❡♣#" ❍✉♠❛♥❇❡✐♥❣✱ ❊♠♣❧♦②❡❡ ❛♥❞ ▼❛❧❡✱ ι1 ❛♥❞ ι2 ❛%❡ ✐♥❝❧✉"✐♦♥"
❛♥❞σ1❛♥❞σ2♠❛♣✱ %❡"♣❡❝#✐✈❡❧②✱ @❡*"♦♥ ❛♥❞ ❍✉♠❛♥❇❡✐♥❣ #♦ @❡*"♦♥❴❍✉♠❛♥❇❡✐♥❣
❛♥❞ ❛❧❧ ♦#❤❡% ❝♦♥❝❡♣#" ❛♥❞✴♦% ✐♥❞✐✈✐❞✉❛❧" ✐❞❡♥#✐❝❛❧❧②✳
❚❤❡ ❜%✐❞❣❡ ♦♥#♦❧♦❣②B ✐"✿
♦♥'♦❧♦❣② ❇=❈❧❛))✿ ❡"#♦♥❴❍✉♠❛♥❇❡✐♥❣
❈❧❛))✿ ❊♠♣❧♦②❡❡
❈❧❛))✿ ▼❛❧❡
❈❧❛))✿ ❈❤✐❧❞ ❙✉❜❈❧❛))❖❢✿¬❊♠♣❧♦②❡❡
■♥❞✐✈✐❞✉❛❧✿ ❛❧❡① ❚②♣❡)✿ ▼❛❧❡
❚❤❡ ❝♦❧✐♠✐# ♦♥#♦❧♦❣② ♦❢ #❤❡ ❞✐❛❣%❛♠ ♦❢ ❆ ✐"✿
♦♥'♦❧♦❣② ❈=❈❧❛))✿ ❡"#♦♥❴❍✉♠❛♥❇❡✐♥❣
❈❧❛))✿ ❊♠♣❧♦②❡❡
❈❧❛))✿ ▼❛❧❡ ❙✉❜❈❧❛))❖❢✿ ❡"#♦♥❴❍✉♠❛♥❇❡✐♥❣
❈❧❛))✿ ❈❤✐❧❞ ❙✉❜❈❧❛))❖❢✿¬❊♠♣❧♦②❡❡
■♥❞✐✈✐❞✉❛❧✿ ❛❧❡① ❚②♣❡)✿ ▼❛❧❡, ❡"#♦♥❴❍✉♠❛♥❇❡✐♥❣
✹✳✹ ■♥$❡❣'❛$❡❞ ❙❡♠❛♥$✐❝.
❈❛♣#✉%✐♥❣ ✐♥#❡❣%❛#❡❞ "❡♠❛♥#✐❝" ✐♥ DOL✉"✐♥❣ ❢❛♠✐❧✐❡" ♦❢ ♠♦❞❡❧" ❝♦♠♣❛#✐❜❧❡ ✇✐#❤ ❛
❞✐❛❣%❛♠ ✐" ♠♦%❡ ❞✐✣❝✉❧#✱ ❛" ❝♦♠♣❛#✐❜✐❧✐#② ✇✐#❤ #❤❡ ❞✐❛❣%❛♠ ✐♠♣❧✐❡" ✉♥✐@✉❡♥❡"" ♦❢
#❤❡ ❞♦♠❛✐♥✳ ❚♦ %❡♠❡❞② #❤✐"✱ ✇❡ ✉"❡ %❡❧❛#✐✈✐"❛#✐♦♥ ♦❢ ❛♥ ♦♥#♦❧♦❣② ✇❤❡%❡ #❤❡ ✉♥✐✈❡%"❛❧
❝♦♥❝❡♣# ❜❡❝♦♠❡" ❛ ♥❡✇ ❝♦♥❝❡♣# ❛♥❞ #❤✉" ❝❛♥ ❜❡ ✐♥#❡%♣%❡#❡❞ ❛" ❛ "✉❜"❡# ♦❢ #❤❡ %❡❧❛✲
#✐✈✐"❡❞ ❞♦♠❛✐♥✳ ❘❡❧❛#✐✈✐"❛#✐♦♥" ❤❛✈❡ ♣%❡✈✐♦✉"❧② ❜❡❡♥ ✉"❡❞ ✐♥ ❞❡✜♥✐♥❣ ❈♦♠♠♦♥ ▲♦❣✐❝
♠♦❞✉❧❡" ❬✶✾❪ ♦% ✐♥ #❤❡ %❡✲❡♥❝♦❞✐♥❣ ♦❢ ❉❉▲ ✐♥#♦ ❖❲▲ ❬✻❪✳
❉❡✜♥✐$✐♦♥ ✻✳ ▲❡!O❜❡ ❛♥OWL♦♥!♦❧♦❣②✳ ❲❡ ❞❡✜♥❡ !❤❡ %❡❧❛#✐✈✐"❛#✐♦♥ ♦❢O✱ ❞❡♥♦!❡❞
O˜✱ ❛" ❢♦❧❧♦✇"✳ ❚❤❡ ❝♦♥❝❡♣!" ♦❢ O˜ ❛*❡ !❤❡ ❝♦♥❝❡♣!" ♦❢ O!♦❣❡!❤❡* ✇✐!❤ ❛ ♥❡✇ ❝♦♥❝❡♣!✱
❞❡♥♦!❡❞⊤O✳ ❚❤❡ *♦❧❡" ❛♥❞ ✐♥❞✐✈✐❞✉❛❧" ♦❢O˜❛*❡ !❤❡ "❛♠❡ ❛" ✐♥O✳O˜ ❝♦♥!❛✐♥" ❛①✐♦♠"
"!❛!✐♥❣ !❤❛!
✕ ❡❛❝❤ ❝♦♥❝❡♣' C ♦❢O✐* *✉❜*✉♠❡❞ ❜②⊤O✱
✕ ❡❛❝❤ ✐♥❞✐✈✐❞✉❛❧ i♦❢O✐* ❛♥ ✐♥*'❛♥❝❡ ♦❢ ⊤O✱
✕ ❡❛❝❤ 3♦❧❡ r❤❛* ✐'* ❞♦♠❛✐♥ ❛♥❞ 3❛♥❣❡✱ ✐❢ ♣3❡*❡♥'✱ ✐♥'❡3*❡❝'❡❞ ✇✐'❤ ⊤O✱ ♦'❤❡3✇✐*❡
'❤❡② ❛3❡⊤O✳
❛♥❞ '❤❡ ❛①✐♦♠* ♦❢O ✇❤❡3❡ '❤❡ ❢♦❧❧♦✇✐♥❣ 3❡♣❧❛❝❡♠❡♥' ♦❢ ❝♦♥❝❡♣'* ✐* ♠❛❞❡✿
✕ ❡❛❝❤ ♦❝❝✉3❡♥❝❡ ♦❢ ⊤✐* 3❡♣❧❛❝❡❞ ❜② ⊤O✱ ❛♥❞
✕ ❡❛❝❤ ❝♦♥❝❡♣' ¬C ✐* 3❡♣❧❛❝❡❞ ❜② ⊤O⊓ ¬C
✕ ❡❛❝❤ ❝♦♥❝❡♣' ∀R.C ✐* 3❡♣❧❛❝❡❞ ❜②⊤O⊓ ∀R.C✳
❊①❛♠♣❧❡ ✸✳ ❲❡ ❛❞❞ $❤❡ ❛&&✉♠♣$✐♦♥ $❤❛$ ✇❡ ❤❛✈❡ ❛ ❣❧♦❜❛❧ ❞♦♠❛✐♥ ✇❤❡2❡ $❤❡ ❞♦♠❛✐♥&
♦❢ $❤❡ ♦♥$♦❧♦❣✐❡& ✐♥ ♦✉2 ❛❧✐❣♥♠❡♥$ ❛2❡ ✐♥❝❧✉❞❡❞✿
❛❧✐❣♥♠❡♥' ❆:❙ '♦ ❚=. . .
❛))✉♠✐♥❣ ●❧♦❜❛❧❉♦♠❛✐♥
❚❤❡ ❞✐❛❣2❛♠ ♦❢ ❆ ✐& $❤❡♥
S˜ B˜ T˜
S′
σ1
??
ι1
__
T′
ι2
??
σ2
__
✇❤❡2❡ S′ ❝♦♥&✐&$& ♦❢ $❤❡ ❝♦♥❝❡♣$& T hingS✱ ;❡3*♦♥ ❛♥❞ ❈❤✐❧❞ ❛♥❞ $❤❡ ✐♥❞✐✈✐❞✉❛❧
❛❧❡① ❛♥❞T′ ❝♦♥&✐&$& ♦❢ $❤❡ ❝♦♥❝❡♣$& T hingT✱ ❍✉♠❛♥❇❡✐♥❣✱ ❊♠♣❧♦②❡❡ ❛♥❞ ▼❛❧❡✱ ι1
❛♥❞ ι2 ❛2❡ ✐♥❝❧✉&✐♦♥& ❛♥❞ σ1 ❛♥❞ σ2 ♠❛♣ ;❡3*♦♥ ❛♥❞ 2❡&♣❡❝$✐✈❡❧② ❍✉♠❛♥❇❡✐♥❣ $♦
;❡3*♦♥❴❍✉♠❛♥❇❡✐♥❣ ❛♥❞ ❛❧❧ ♦$❤❡2 ❝♦♥❝❡♣$& ❛♥❞✴♦2 ✐♥❞✐✈✐❞✉❛❧& ✐❞❡♥$✐❝❛❧❧②✳
❚❤❡ 2❡❧❛$✐✈✐&❛$✐♦♥&S˜ ❛♥❞T˜♦❢ $❤❡ ♦♥$♦❧♦❣✐❡& ❙ ❛♥❞ ❚ ❛2❡
♦♥'♦❧♦❣②S˜=❈❧❛))✿T hingS
❈❧❛))✿ ❡"#♦♥ ❙✉❜❈❧❛))❖❢✿T hingS
■♥❞✐✈✐❞✉❛❧✿ ❛❧❡① ❚②♣❡)✿ ❡"#♦♥✱T hingS
❈❧❛))✿ ❈❤✐❧❞ ❙✉❜❈❧❛))❖❢✿T hingS
♦♥'♦❧♦❣②T˜=❈❧❛))✿T hingT
❈❧❛))✿ ❍✉♠❛♥❇❡✐♥❣ ❙✉❜❈❧❛))❖❢✿T hingT
❈❧❛))✿ ▼❛❧❡ ❙✉❜❈❧❛))❖❢✿ ❍✉♠❛♥❇❡✐♥❣✱T hingT
❈❧❛))✿ ❊♠♣❧♦②❡❡ ❙✉❜❈❧❛))❖❢✿T hingT
❚❤❡ 2❡❧❛$✐✈✐&❡❞ ❜2✐❞❣❡ ♦♥$♦❧♦❣② ♦❢ ❛♥ ❛❧✐❣♥♠❡♥$ ✐& ❜✉✐❧$ ❜② 2❡❧❛$✐✈✐&✐♥❣ $❤❡ ❛①✐♦♠&
$❤❛$ 2❡&✉❧$ ❢2♦♠ $2❛♥&❧❛$✐♥❣ $❤❡ ❝♦22❡&♣♦♥❞❡♥❝❡& ♦❢ ❆ $♦ ❖❲▲ &❡♥$❡♥❝❡&✳ ❙✐♥❝❡ ✇❡
♠❛❞❡ $❤❡ ❛&&✉♠♣$✐♦♥ $❤❛$ ❡?✉❛❧✐&✐♥❣ ❢✉♥❝$✐♦♥& ❛2❡ ❛❧❧ ✐♥❝❧✉&✐♦♥&✱ $❤❡2❡ ✐& ♥♦ ♥❡❡❞ $♦
✐♥$2♦❞✉❝❡ ❡①♣❧✐❝✐$ &②♠❜♦❧& ❢♦2 $❤❡♠ ✐♥ $❤❡ ❜2✐❞❣❡ ♦♥$♦❧♦❣②✳ ■♥ ♦✉2 ❝❛&❡✱ $❤❡ ❜2✐❞❣❡
♦♥$♦❧♦❣② ♦❢ ❆ ✐&
♦♥'♦❧♦❣②B˜ =❈❧❛))✿T hingS❈❧❛))✿T hingT
❈❧❛))✿ ❡"#♦♥❴❍✉♠❛♥❇❡✐♥❣ ❙✉❜❈❧❛))❖❢✿T hingS✱T hingT
❈❧❛))✿ ▼❛❧❡ ❈❧❛))✿ ❊♠♣❧♦②❡❡
❈❧❛))✿ ❈❤✐❧❞ ❙✉❜❈❧❛))❖❢✿T hingT ❛♥❞¬❊♠♣❧♦②❡❡
❚❤❡ ❝♦❧✐♠✐( ♦♥(♦❧♦❣② ♦❢ (❤❡ -❡❧❛(✐✈✐0❡❞ ❞✐❛❣-❛♠ ♦❢ (❤❡ ❛❧✐❣♥♠❡♥( ✐♥ ❊①✳ ✶ ✐0✿
♦♥"♦❧♦❣② ❈=❈❧❛((✿ ❚❤✐♥❣❙
❈❧❛((✿ ❚❤✐♥❣❚
❈❧❛((✿ &❡()♦♥❴❍✉♠❛♥❇❡✐♥❣ ❙✉❜❈❧❛((❖❢✿ ❚❤✐♥❣❙✱ ❚❤✐♥❣❈
❈❧❛((✿ ▼❛❧❡ ❙✉❜❈❧❛((❖❢✿ &❡()♦♥❴❍✉♠❛♥❇❡✐♥❣
❈❧❛((✿ ❊♠♣❧♦②❡❡ ❙✉❜❈❧❛((❖❢✿ ❚❤✐♥❣❚
❈❧❛((✿ ❈❤✐❧❞ ❙✉❜❈❧❛((❖❢✿ ❚❤✐♥❣❙
❈❧❛((✿ ❈❤✐❧❞ ❙✉❜❈❧❛((❖❢✿ ❚❤✐♥❣❚ ❛♥❞¬❊♠♣❧♦②❡❡
■♥❞✐✈✐❞✉❛❧✿ ❛❧❡① ❚②♣❡(✿ ▼❛❧❡✱ &❡()♦♥❴❍✉♠❛♥❇❡✐♥❣
✹✳✺ ❈♦♥&❡①&✉❛❧✐-❡❞ ❙❡♠❛♥&✐❝-
❍❡-❡ ✇❡ ♥❡❡❞ (♦ ✐♥(-♦❞✉❝❡ ❡①♣❧✐❝✐(❧② (❤❡ -❡❧❛(✐♦♥0 ❜❡(✇❡❡♥ (❤❡ ❞♦♠❛✐♥0 ✐♥ (❤❡ ❧❛♥✲
❣✉❛❣❡ ♦❢ (❤❡ ❜-✐❞❣❡ ♦♥(♦❧♦❣②✳ ❚❤❡ ❞✐❛❣-❛♠ ♦❢ (❤❡ ❛❧✐❣♥♠❡♥( ❤❛0 (❤✉0 (❤❡ 0❛♠❡ 0❤❛♣❡
❛0 ✐♥ ❉❡❢✳ ✺✱ ❜✉( ♥♦✇ (❤❡ ❜-✐❞❣❡ ♦♥(♦❧♦❣② ✐0 ❝♦♠♣✉(❡❞ ❞✐✛❡-❡♥(❧② ❛♥❞✱ ❛0 ✐♥ (❤❡ ♣-❡✲
✈✐♦✉0 0❡❝(✐♦♥✱ (❤❡ ♦♥(♦❧♦❣✐❡0 ❛-❡ -❡❧❛(✐✈✐0❡❞✳ ❲❡ ❞❡♥♦(❡ (❤❡ ❜-✐❞❣❡ ♦♥(♦❧♦❣② ❜② B❛♥❞
❞❡✜♥❡ ✐( (♦ ♠♦❞✐❢②B ❛0 ❢♦❧❧♦✇0✿
✕ rji✐0 ❛❞❞❡❞ (♦B ❛0 ❛ -♦❧❡ ✇✐(❤ ❞♦♠❛✐♥⊤T ❛♥❞ -❛♥❣❡⊤S
✕ (❤❡ ❝♦--❡0♣♦♥❞❡♥❝❡0 ❛-❡ (-❛♥0❧❛(❡❞ (♦ ❛①✐♦♠0 ✐♥✈♦❧✈✐♥❣ (❤❡0❡ -♦❧❡0✿
• Ci=Cj ❜❡❝♦♠❡0Ci≡ ∃rji•Cj
• ai=aj ❜❡❝♦♠❡0ai rjiaj
• ai∈Cj ❜❡❝♦♠❡0ai∈ ∃rji•Cj
• Ci< Cj ❜❡❝♦♠❡0Ci⊑ ∃rji•Cj
• Ci%Cj ❜❡❝♦♠❡0Ci⊓ ∃rji•Cj=∅
✕ (❤❡ ♣-♦♣❡-(✐❡0 ♦❢ (❤❡ rji❛-❡ ❛❞❞❡❞ ❛0 ❛①✐♦♠0 ✐♥ B✳
❍❡-❡ ✇❡ ❛00✉♠❡ (❤❛( (❤❡ ❛❧✐❣♥♠❡♥( Aij ❝♦♥(❛✐♥0 ♥♦ ❝♦--❡0♣♦♥❞❡♥❝❡ (ri, rj, R)✱
✇❤❡-❡ri❛♥❞rj❛-❡ -♦❧❡0✳ ❍❛✈✐♥❣ 0✉❝❤ ❝♦--❡0♣♦♥❞❡♥❝❡0 ❧❡❛❞0 (♦ 0❡♥(❡♥❝❡0 (❤❛( ❝❛♥♥♦(
❜❡ ❡①♣-❡00❡❞ ✐♥OWL✳
❊①❛♠♣❧❡ ✹✳ ❲❡ ❛❞❞ (❤❡ ❛00✉♠♣(✐♦♥ (❤❛( ✇❡ ❤❛✈❡ ❞✐✛❡-❡♥( ❞♦♠❛✐♥0 ❢♦- (❤❡ ♦♥(♦❧♦❣✐❡0✱
✇❤✐❝❤ ❛-❡ -❡❧❛(❡❞ ❜② ❞♦♠❛✐♥ -❡❧❛(✐♦♥0✿
❛❧✐❣♥♠❡♥" ❆:❙ "♦ ❚=. . .
❛((✉♠✐♥❣ ❈♦♥"❡①"✉❛❧✐(❡❞❉♦♠❛✐♥
❚❤❡ ❞✐❛❣-❛♠ ♦❢ ❆ ✐0 (❤❡♥
S˜ B T˜
S′
σ1
??
ι1
__
T′
ι2
??
σ2
__
✇❤❡-❡ (❤❡ ❝♦♥0(✐(✉❡♥(0 ♦❢ (❤❡ ❞✐❛❣-❛♠✱ ❡①❝❡♣(B✱ ❛-❡ ❛0 ❞❡✜♥❡❞ ✐♥ ❊①✳ ✸✳ ❚❤❡ ❜-✐❞❣❡
♦♥(♦❧♦❣② ♦❢ ❆ ♥♦✇ ❜❡❝♦♠❡0✿
♦♥"♦❧♦❣②B =❈❧❛((✿ ❚❤✐♥❣❙
❈❧❛((✿ ❚❤✐♥❣❚
❖❜❥❡❝";<♦♣❡<②✿rT S ❉♦♠❛✐♥✿ ❚❤✐♥❣❚ ❘❛♥❣❡✿ ❚❤✐♥❣❙
❈❧❛((✿ &❡()♦♥ ❊?✉✐✈❛❧❡♥"❚♦✿ rT S (♦♠❡ ❍✉♠❛♥❇❡✐♥❣
❈❧❛((✿ ❊♠♣❧♦②❡❡
❈❧❛((✿ ▼❛❧❡
❈❧❛((✿ ❈❤✐❧❞ ❙✉❜❈❧❛((❖❢✿rT S (♦♠❡¬❊♠♣❧♦②❡❡
■♥❞✐✈✐❞✉❛❧✿ ❛❧❡① ❚②♣❡(✿rT S (♦♠❡ ▼❛❧❡
❚❤❡ ❝♦❧✐♠✐( ♦♥(♦❧♦❣② ♦❢ (❤✐- ❞✐❛❣0❛♠ ✐-✿
♦♥"♦❧♦❣② ❈=❈❧❛((✿ ❚❤✐♥❣❙
❈❧❛((✿ ❚❤✐♥❣❚
❖❜❥❡❝"/0♦♣❡0②✿rT S ❉♦♠❛✐♥✿ ❚❤✐♥❣❚ ❘❛♥❣❡✿ ❚❤✐♥❣❙
❈❧❛((✿ &❡()♦♥ ❊7✉✐✈❛❧❡♥"❚♦✿ rT S (♦♠❡ ❍✉♠❛♥❇❡✐♥❣
❈❧❛((✿ ▼❛❧❡ ❙✉❜❈❧❛((❖❢✿ &❡()♦♥❴❍✉♠❛♥❇❡✐♥❣
❈❧❛((✿ ❊♠♣❧♦②❡❡
❈❧❛((✿ ❈❤✐❧❞ ❙✉❜❈❧❛((❖❢✿rT S (♦♠❡¬❊♠♣❧♦②❡❡
■♥❞✐✈✐❞✉❛❧✿ ❛❧❡① ❚②♣❡(✿rT S (♦♠❡ ▼❛❧❡✱ &❡()♦♥
✹✳✻ ❚❤❡ &❤'❡❡ (❡♠❛♥&✐❝( ✐♥ ❉❖▲
■♥ (❤✐- -❡❝(✐♦♥ ❧❡( S = ((Oi)i∈I,(Aij)i,j∈I) ❜❡ ❛ ♥❡(✇♦0❦ ♦❢ OWL ♦♥(♦❧♦❣✐❡-✳ ❲❡
❞❡♥♦(❡ C(S)(❤❡ ❝♦❧✐♠✐( ♦♥(♦❧♦❣② ♦❢ (❤❡ ❞✐❛❣0❛♠ ❛--♦❝✐❛(❡❞ (♦ S✱ 0❡❣❛0❞❧❡-- ✐❢ (❤❡
❛--✉♠♣(✐♦♥ ❛❜♦✉( (❤❡ ❛❧✐❣♥♠❡♥(- ✐♥ S ✐- (❤❛( (❤❡② ✉-❡ -✐♥❣❧❡✱ ✐♥(❡❣0❛(❡❞ ♦0 ❝♦♥(❡①✲
(✉❛❧✐-❡❞ -❡♠❛♥(✐❝-✳ ❚❤❡ ♠♦❞❡❧ ❝❧❛-- ♦❢ C(S)✐- ❞❡♥♦(❡❞JC(S)K✳
❚❤❡♦'❡♠ ✶✳ ✶✳ ■❢ $❤❡ ❛❧✐❣♥♠❡♥$- ♦❢S ✉-❡ ❙✐♥❣❧❡❉♦♠❛✐♥ ❛♥❞ $❤❡ ❞✐❛❣1❛♠ ♦❢ S ✐-
❝♦♥♥❡❝$❡❞✱ $❤❡♥JC(S)K✐- ✐♥ ❜✐❥❡❝$✐♦♥ ✇✐$❤ M odsim(S)✳
✷✳ ■❢ $❤❡ ❛❧✐❣♥♠❡♥$- ♦❢ S ✉-❡ ●❧♦❜❛❧❉♦♠❛✐♥✱ $❤❡♥ JC(S)K ✐- ✐♥ ❜✐❥❡❝$✐♦♥ ✇✐$❤ $❤❡
❝❧❛--M odint(S)♦❢ ✐♥$❡❣1❛$❡❞ ♠♦❞❡❧-((mi),(γi)) ♦❢S ✇❤❡1❡γi ❛1❡ ✐♥❝❧✉-✐♦♥-✳
✸✳ ■❢ $❤❡ ❛❧✐❣♥♠❡♥$- ♦❢ S ✉-❡ ❈♦♥-❡①-✉❛❧✐0❡❞❉♦♠❛✐♥✱ $❤❡♥ JC(S)K✐- ✐♥ ❜✐❥❡❝$✐♦♥
✇✐$❤M odcon(S)✳
❉❖▲ ✐- -✉♣♣♦0(❡❞ ❜② ❖♥(♦❤✉❜ ✭❤--♣0✿✴✴♦♥-♦❤✉❜✳♦7❣✮✱ ❛ ❲❡❜✲❜❛-❡❞ 0❡♣♦-✐(♦0②
❡♥❣✐♥❡ ❢♦0 ♠❛♥❛❣✐♥❣ ❞✐-(0✐❜✉(❡❞ ❤❡(❡0♦❣❡♥♦✉- ♦♥(♦❧♦❣✐❡-✳ ❚❤❡ ❜❛❝❦✲❡♥❞ ♦❢ ❖♥(♦❤✉❜
✐- (❤❡ ❍❡(❡0♦❣❡♥❡♦✉- ❚♦♦❧ ❙❡( ❍❊❚❙ ❬✶✽❪ ✇❤✐❝❤ ✐- ✉-❡❞ ❢♦0 ♣❛0-✐♥❣✱ -(❛(✐❝ ❛♥❛❧②-✐-
❛♥❞ ♣0♦♦❢ ♠❛♥❛❣❡♠❡♥( ♦❢ ♦♥(♦❧♦❣✐❡-✳ ❍❊❚❙ -✉♣♣♦0(- ❛❧✐❣♥♠❡♥(- ❛♥❞ ❝♦♠❜✐♥❛(✐♦♥-✿
✐( ❣❡♥❡0❛(❡- (❤❡ ❞✐❛❣0❛♠ ♦❢ ❛♥ ❛❧✐❣♥♠❡♥( ❛❝❝♦0❞✐♥❣ (♦ (❤❡ ❛--✉♠♣(✐♦♥ ♦♥ (❤❡ ❞♦♠❛✐♥
❛♥❞ ❝❛♥ ❝♦♠♣✉(❡ ❝♦❧✐♠✐(- ♦❢ ❖❲▲ ♦♥(♦❧♦❣✐❡- ❛✉(♦♠❛(✐❝❛❧❧②✳
✺ ❈♦♥❝❧✉'✐♦♥' ❛♥❞ ❋✉,✉-❡ ❲♦-❦
❖✉0 (❤❡♦0❡(✐❝❛❧ ❝♦♥(0✐❜✉(✐♦♥- (♦ (❤❡ ❢♦✉♥❞❛(✐♦♥- ♦❢ ♦♥(♦❧♦❣② ❛❧✐❣♥♠❡♥( ❛♥❞ ❝♦♠❜✐✲
♥❛(✐♦♥ ❤❛✈❡ ❛ ♣♦(❡♥(✐❛❧❧② ❧❛0❣❡ ✐♠♣❛❝( ♦♥ ❢✉(✉0❡ ❛❧✐❣♥♠❡♥( ♣0❛❝(✐❝❡- ❛♥❞ 0❡❛-♦♥✐♥❣✳
❘❡❣❛0❞❧❡-- ♦❢ (❤❡ -❡♠❛♥(✐❝ ♣❛0❛❞✐❣♠ ❡♠♣❧♦②❡❞✱ ❵0❡❛-♦♥✐♥❣✬ ✇✐(❤ ❛❧✐❣♥♠❡♥(- ✐♥✈♦❧✈❡-
❛( ❧❡❛-( (❤0❡❡ ❧❡✈❡❧-✿ ✭✶✮ (❤❡ ✜♥❞✐♥❣✴❞✐-❝♦✈❡0② ♦❢ ❛❧✐❣♥♠❡♥(- ✭♦❢(❡♥ ❜❛-❡❞ ❤❡❛✈✐❧② ♦♥
-(❛(✐-(✐❝❛❧ ♠❡(❤♦❞-✮✱ ✭✷✮ (❤❡ ❝♦♥-(0✉❝(✐♦♥ ♦❢ (❤❡ ❛❧✐❣♥❡❞ ♦♥(♦❧♦❣② ✭(❤❡ ❵❝♦❧✐♠✐(✬✮✱ ❛♥❞
✭✸✮ 0❡❛-♦♥✐♥❣ ♦✈❡0 (❤❡ ❛❧✐❣♥❡❞ 0❡-✉❧(✱ 0❡-♣❡❝(✐✈❡❧② ❞❡❜✉❣❣✐♥❣ ❛♥❞ 0❡♣❛✐0✱ ❝❧♦-✐♥❣ (❤❡
❧♦♦♣ (♦ ✭✶✮✳ ❖✉0 ❝♦♥(0✐❜✉(✐♦♥- ✐♥ (❤✐- ♣❛♣❡0 ❛❞❞0❡-- ❧❡✈❡❧- ✭✷✮ ❛♥❞ ✭✸✮✳
❘❡❣❛0❞✐♥❣ ✭✷✮✱ ♣❧❛(❢♦0♠- -✉❝❤ ❛- ❇✐♦♣♦0(❛❧ ✭✇✐(❤ ❤✉♥❞0❡❞ (❤♦✉-❛♥❞- ♦❢ ♠❛♣✲
♣✐♥❣-✮ ✐❧❧✉-(0❛(❡ (❤❛( ♠❛♣♣✐♥❣- ❜❡(✇❡❡♥ ♦♥(♦❧♦❣✐❡-✱ ♦♥(♦❧♦❣② ♠♦❞✉❧❡-✱ ❛♥❞ (❤❡ ❝♦♥✲
❝❡♣(- ❛♥❞ ❞❡✜♥✐(✐♦♥- ❧✐✈✐♥❣ ✐♥ (❤❡♠✱ ❛0❡ ♦❢ ❣0❡❛( ✐♠♣♦0(❛♥❝❡ (♦ -✉♣♣♦0( 0❡✲✉-❡✳ ❚❤❡
✐♠♣♦0(❛♥❝❡ ♦❢ ❛❧✐❣♥♠❡♥( ❤❛- ❛❧-♦ ❜❡❡♥ ✇❡❧❧ ❞❡♠♦♥-(0❛(❡❞ ❢♦0 ❢♦✉♥❞❛(✐♦♥❛❧ ♦♥(♦❧♦✲
❣✐❡- ✐♥ (❤❡ 0❡♣♦-✐(♦0② ❘❖▼❯▲❯❙ ❬✶✸❪✳ ■♥ (❤❡ ❝❛-❡ ♦❢ ❇✐♦♣♦0(❛❧✱ (❤❡ ❉❖▲ ❧❛♥❣✉❛❣❡
❛❧❧♦✇- (♦ ❞❡❝❧❛0❛(✐✈❡❧② ♠❛♥❛❣❡ -❡(- ♦❢ ❛❧✐❣♥♠❡♥(-✱ ❛♥❞ (♦ ❣✐✈❡ ♣0❡❝✐-❡ -❡♠❛♥(✐❝-✳ ■♥
(❤❡ ❝❛-❡ ♦❢ ❘❖▼❯▲❯❙✱ ✐( ❛❧❧♦✇- (♦ ❛❧✐❣♥ ♦♥(♦❧♦❣✐❡- -✉❝❤ ❛- ❉♦❧❝❡ ♦0 ❇❋❖ ❡①♣0❡--❡❞
