Integrated Uncertainty in Knowledge Modelling and Decision Making - IUKM 2011

Lecture Notes in Artificial Intelligence Subseries of Lecture Notes in Computer Science LNAI Series Editors Randy Goebel University of Alberta, Edmonton, Canada Yuzuru Tanaka Hokkaido University, Sapporo, Japan Wolfgang Wahlster DFKI and Saarland University, Saarbrücken, Germany

LNAI Founding Series Editor Joerg Siekmann DFKI and Saarland University, Saarbrücken, Germany

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Yongchuan Tang Van-Nam Huynh Jonathan Lawry (Eds.)

Integrated Uncertainty in Knowledge Modelling and Decision Making International Symposium, IUKM 2011 Hangzhou, China, October 28-30, 2011 Proceedings

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Series Editors Randy Goebel, University of Alberta, Edmonton, Canada Jörg Siekmann, University of Saarland, Saarbrücken, Germany Wolfgang Wahlster, DFKI and University of Saarland, Saarbrücken, Germany Volume Editors Yongchuan Tang Zhejiang University, College of Computer Science Hangzhou, 310027, P.R. China E-mail: [email protected] Van-Nam Huynh Japan Advanced Institute of Science and Technology School of Knowledge Science 1-1 Asahidai, Nomi City, Ishikawa, 923-1292, Japan E-mail: [email protected] Jonathan Lawry University of Bristol Department of Engineering Mathematics Bristol, BS8 1TR, UK E-mail: [email protected]

ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-24917-4 e-ISBN 978-3-642-24918-1 DOI 10.1007/978-3-642-24918-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011939089 CR Subject Classification (1998): I.2.6, I.2, H.2.8, H.3-5, F.1, F.2.2, J.1 LNCS Sublibrary: SL 7 – Artificial Intelligence

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Preface

This volume contains papers presented at the 2011 International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2011), which was held at Zhejiang University, Hangzhou, China, during October 28–30, 2011. The principal aim of IUKM 2011 was to provide a forum in which researchers could exchange ideas and results on both theoretical and applied research relating to all aspects of uncertainty management and their applications. The organizers received 55 papers. Each paper was peer reviewed by two members of the Program Committee. Finally, 21 papers were chosen for presentation at IUKM 2011 and publication in the proceedings. The keynote and invited talks presented at the symposium are also included in this volume. As a follow-up of the symposium, a special issue of the International Journal of Approximate Reasoning is anticipated to include a small number of extended papers selected from the symposium as well as other relevant contributions received in response to subsequent open calls. These journal submissions will go through a fresh round of reviews in accordance with the journal’s guidelines. The IUKM 2011 symposium was partially supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61075046, Zhejiang Natural Science Foundation under Grant No. Y1090003, JSPS Grant-in-Aid for Scientific Research [KAKENHI(B) No. 22300074], and SCOPE 102305001 of the Ministry of Internal Affairs and Communications (MIC), Japan. We are very thankful to the College of Computer Science and Technology of Zhejiang University for providing crucial support through out the organization of IUKM 2011. We would like to express our appreciation to the members of the Program Committee for their support and cooperation in this publication. Last, but not the least, we wish to thank all the authors and participants for their contributions and fruitful discussions that made this symposium a success. October 2011

Yongchuan Tang Van-Nam Huynh Jonathan Lawry

Organization

IUKM 2011 was co-organized by the College of Computer Science and Technology, Zhejiang University, and the School of Knowledge Science, Japan Advanced Institute of Science and Technology.

General Co-chairs Yueting Zhuang Yoshiteru Nakamori

Zhejiang University, China Japan Advanced Institute of Science and Technology, Japan

Program Co-chairs Yongchuan Tang Van-Nam Huynh Jonathan Lawry

Zhejiang University, China Japan Advanced Institute of Science and Technology, Japan University of Bristol, UK

Program Committee Byeong Seok Ahn Bernard De Baets Yaxin Bi Bernadette Bouchon-Meunier Tru H. Cao Fabio Cuzzolin Van Hung Dang Thierry Denoeux Gary Gang Feng Lluis Godo Yongyong He Enrique Herrera-Viedma Kaoru Hirota Tu Bao Ho

Chung-Ang University, Korea Ghent University, Belgium University of Ulster at Jordanstown, UK University Pierre and Marie Curie, France Ho Chi Minh City University of Technology, Vietnam Oxford Brookes University, UK Hanoi National University, Vietnam; UNU-IIST, Macau University of Technology of Compi`egne, France Hong Kong City University, Hong Kong Artificial Intelligence Research Institute, CSIC, Spain Tsinghua University, China University of Granada, Spain Tokyo Institute of Technology, Japan Japan Advanced Institute of Science and Technology, Japan

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Organization

Kaizhu Huang Wei Huang Eyke Hullermeier Mitsuru Ikeda Masahiro Inuiguchi Gabriele Kern-Isberner Etienne E. Kerre Hiroaki Kikuchi Laszlo Koczy Vladik Kreinovich Ming Li Churn-Jung Liau Jun Liu Weiru Liu Trevor Martin Radko Mesiar Sadaaki Miyamoto Tetsuya Murai Hung T. Nguyen Witold Pedrycz Zengchang Qin Jordi Recasens Jonathan Rossiter Andrzej Skowron Noboru Takagi Vicenc Torra Milan Vlach Junzo Watada Ronald Yager

Chinese Academy of Sciences, China Huazhong University of Science and Technology, China Philipps-Universit¨ at Marburg, Germany Japan Advanced Institute of Science and Technology, Japan Osaka University, Japan University of Dortmund, Germany University of Ghent, Belgium Tokai University, Japan Budapest University of Technology and Economics, Hungary University of Texas at El Paso, USA Nanjing University, China Academia Sinica, Taiwan University of Ulster at Jordanstown, UK Queens University Belfast, UK University of Bristol, UK Slovak University of Technology Bratislava, Slovakia University of Tsukuba, Japan University of Hokkaido, Japan New Mexico State University, USA University of Alberta, Canada Beihang University, China UPC Barcelona, Spain University of Bristol, UK Warsaw University, Poland Toyama Prefectural University, Japan Artificial Intelligence Research Institute, CSIC, Spain Charles University, Czech Republic Waseda University, Japan Machine Intelligence Institute, USA

Sponsoring Institutions Zhejiang University Japan Advanced Institute of Science and Technology The University of Bristol Springer Elsevier

Table of Contents

Keynote and Invited Talks Talking with Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toyoaki Nishida

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Introduction to the ER Rule for Evidence Combination . . . . . . . . . . . . . . . Jian-Bo Yang and Dong-Ling Xu

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Theories and Approaches to Treat Incomparability . . . . . . . . . . . . . . . . . . . Yang Xu and Jun Liu

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Knowledge Science – Modeling the Knowledge Creation Process . . . . . . . Yoshiteru Nakamori

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Two Classes of Algorithms for Data Clustering . . . . . . . . . . . . . . . . . . . . . . Sadaaki Miyamoto

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Fusing Conceptual Graphs and Fuzzy Logic: Towards the Structure and Expressiveness of Natural Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tru H. Cao

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Contributed Papers A MMORPG Decision-Making Model Based on Persuasive Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helio C. Silva Neto, Leonardo F.B.S. Carvalho, F´ abio Paragua¸cu, and Roberta V.V. Lopes A Computing with Words Based Approach to Multicriteria Energy Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hong-Bin Yan, Tieju Ma, Yoshiteru Nakamori, and Van-Nam Huynh Bipolar Semantic Cells: An Interval Model for Linguistic Labels . . . . . . . . Yongchuan Tang and Jonathan Lawry A Fuzzy Rule-Based Classification System Using Interval Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Min Tang, Xia Chen, Weidong Hu, and Wenxian Yu

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Rough Approximations in General Approximation Spaces . . . . . . . . . . . . . Keyun Qin, Zheng Pei, and Yang Xu

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Multi-agents and Non-classical Logic Systems . . . . . . . . . . . . . . . . . . . . . . . . Chenfang Zhao and Zheng Pei

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An Information Processing Model for Emotional Agents Based on the OCC Model and the Mood Congruent Effect . . . . . . . . . . . . . . . . . . . . . . . . Chao Ma, Guanghong Gong, and Yaofei Ma

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On Distributive Equations of Implications and Contrapositive Symmetry Equations of Implications Based on a Continuous t -Norm . . . Feng Qin and Meihua Lu

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A Novel Cultural Algorithm Based on Differential Evolution for Hybrid Flow Shop Scheduling Problems with Fuzzy Processing Time . . . . . . . . . . Qun Niu, Tingting Zeng, and Zhuo Zhou

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An Over-Relaxed (A, η, m)-Proximal Point Algorithm for System of Nonlinear Fuzzy-Set Valued Operator Equation Frameworks and Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heng-you Lan, Xiao Wang, Tingjian Xiong, and Yumin Xiang

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Reliability-Based Route Optimization of a Transportation Network with Random Arc Capacities and Time Threshold . . . . . . . . . . . . . . . . . . . Tao Zhang, Bo Guo, and Yuejin Tan

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Modeling Multilocation Transshipment with Application of Stochastic Programming Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jingxian Chen and Jianxin Lu

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Identifying a Non-normal Evolving Stochastic Process Based upon the Genetic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kangrong Tan, Meifen Chu, and Shozo Tokinaga

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Clustering Based Bagging Algorithm on Imbalanced Data Sets . . . . . . . . . Xiao-Yan Sun, Hua-Xiang Zhang, and Zhi-Chao Wang

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Agglomerative Hierarchical Clustering Using Asymmetric Similarity Based on a Bag Model and Application to Information on the Web . . . . . Satoshi Takumi and Sadaaki Miyamoto

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Applying Agglomerative Fuzzy K-Means to Reduce the Cost of Telephone Marketing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ming-Jia Hsu, Ping-Yu Hsu, and Bayarmaa Dashnyam

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Kansei Information Transfer Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yoshiteru Nakamori

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Combining Binary Classifiers with Imprecise Probabilities . . . . . . . . . . . . . S´ebastien Destercke and Benjamin Quost

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Applying Hierarchical Information with Learning Approach for Activity Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hoai-Viet To, Hoai-Bac Le, and Mitsuru Ikeda

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Querying in Spaces of Music Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wladyslaw Homenda and Mariusz Rybnik

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Automatic Music Transcription: An Experiment with Simple Tools . . . . . Wladyslaw Homenda

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Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Talking with Uncertainty Toyoaki Nishida Graduate School of Informatics, Kyoto University, Kyoto, Japan [email protected]

Abstract. We are living in the world with uncertainty. Our artificial partners should be able to not only reason about uncertainty but also communicate it in an empathic fashion. Combination of an immersive WOZ environment and the learning by mimicking framework allows for the development of robotic agent’s communication competence in a data-driven fashion. Conversational decision making permits an artificial partner to interactively help people make decision under uncertainty. A cognitive architecture needs to be considered for provoking empathy in social decision making. Keywords: Communicating Uncertainty, Conversational Informatics, Conversational Artifacts, Immersive WOZ Environment, Conversational Decision Making, Empathic Agent.

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Living with Uncertainty

One of the ultimate goals of artificial intelligence research is to build an artificial partner [1] with whom we live and work together to solve confronting problems. An artificial partner may be contrasted with an intelligent tool. The former maintains its own goal and belief to behave autonomously to help people, while the latter helps only when the owner uses it properly. The advantage of an artificial partner is that, if successfully implemented, it will solely depend on the natural communication skill of the user, not requiring any special user training for using it. Surely, our artificial partner should be able to reason about uncertainty and make decisions, as the world we live in is filled with uncertainty and little is certain. In addition, in order to be dependable and empathic, our artificial partner should be able to communicate uncertainty with us so as to share thought and decision about it. Although plenty of research has been conducted on reasoning about uncertainty [2], little has been done on the communication side; we need to address how to build a common ground regarding uncertainty, how to maintain and update a shared notion of uncertainty, how to carve plans for joint activities under uncertainty, etc. How can we bestow a qualified competence of communicating uncertainty on artificial partners? It appears that we need to address multiple aspects. First, we need to incorporate into an artificial partner an ability of communicating with a repertoire of verbal and nonverbal communication expression types for people to communicate uncertainty with each other. It might highly depend on the culture underlying the discourse of conversation. Second, we need to shed light on high-level decisionmaking support process, to enable an artificial partner to interactively help people Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 1–6, 2011. © Springer-Verlag Berlin Heidelberg 2011

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make decision under uncertainty. As we will discuss later, a key appears to be facilitating dialectic conversation to incrementally formulate a joint intention. Third, we need to think about a cognitive architecture that may provoke empathy in managing and communicating uncertainty, as social decision making needs to be supported by empathy among the participants.

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Building Conversational Artifacts

The most popular communicative activity is face-to-face conversation. In order for an artificial partner to participate in conversation to communicate with people, it should be able to not only express and recognize a rich vocabulary for representing uncertainty but also follow and control the conversation flow. Nonverbal communication is deemed useful for the speaker, when s/he wants to directly refer to the real world, illustrate some semanic features [3], or exhibit her/his attitude towards a subject as a social signal [4]. Meanwhile, nonverbal communication allows the participant to control or even negotiate the discourse flow of conversation, e.g., by gazing the partner to monitor the current state or by averting eye gaze in order not to be interrupted after turn is obtained, as reported by [5]. Our group addressed how much we can measure social signals from nonverbal cues. Huang et al [6] introduced two heuristics to drive the utterance policy in multiple-user interaction: Interaction atmosphere of the participants (AT) and the participant who tends to lead the conversation at a specific time point (CLP). AT and CLP are estimated from the activities of the participants’ face movement and acoustic information. Ohmoto et al [7] introduced the notion of I-degree, the degree of involvement in interaction. It is assumed that the I-degree naturally manifests as physiological indices such as skin conductance response and respiration as reflecting a person’s mental state. A method is proposed for estimating the I-degree of participants from visual cues, by exploiting measured correlation between the visual and physiological cues. I-degree is used to detect the atmosphere of conversation in multi-user interaction. How can we turn these insights into the design of artificial partners? A baseline approach is “behavior from observation”, i.e., to observe exactly how people behave in face-to-face conversation and create a quantitative model for reproducing the behavior. Recently, numerous works have been done with this approach, e.g., [8, 9]. This approach leverages recent advances in sensing technology, which allows for capturing nonverbal communication behaviors that people exhibit consciously or unconsciously during conversation to build a quantitative model for conveying propositions encompassing uncertainty in varying situations or controlling the discourse of conversation. Unfortunately, this approach is faced with a difficulty when differences in embodiment manifest between the human and the robot, for different embodiment induces different communication patterns and the communication patterns in human-human interaction cannot be directly applied to human-robot interaction. For example, we have found in our experiments that people tend to use clear, emphasized, complete,

Talking with Uncertainty

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Fig. 1. The immersive WOZ environment ICIE

and redundant expressions in human-robot interaction, as opposed to vague, subtle, incomplete, and parsimonious ones in human-human interaction [10]. It is probably because robotic agents are deemed not as competent as humans in communication and the common ground is not well-established. Under the circumstances we need to employ the WOZ (Wizard of OZ) method to observe interaction between human participants and a robot controlled by a hidden human operator (WOZ). In order for the WOZ approach to be successful, we need to overcome difficulties for manipulating a robot with many degrees of freedom. Our immersive WOZ environment ICIE [11] allows the human operator to control a robot as if s/he stayed inside it, as shown in Fig. 1. The audio-visual environment surrounding the WOZ-operated robot is captured, e.g., by an omnidirectional camera attached to the robot’s head, and is sent to the WOZ operator’s cockpit to be projected on the surrounding immersive screen and the speakers The current version of ICIE employs eight 64-inch display panels arranged in a circle with about 2.5 meters diameter. Eight surround speakers are used to reproduce the acoustic environment. Together, the immersive environment allows the WOZ operator in the center of the cockpit to grasp in detail the situation around the robot to determine exactly what to do if s/he were the robot. The WOZ operator’s behavior, in turn, is captured in real time by a collection of range sensors. Noise filters and human body model are used for robust recognition of pose, head direction and gesture. The captured motion is mapped on the robot for motion generation. The sound on each side of the WOZ operator is gathered by microphones and communicated via network so that other participants in the conversation place can hear the voice of the WOZ operator (with a modulation, when necessary). The behavioral model of the robot is generated from the collected data in four stages in the framework of learning by mimicking (Fig. 2) [12, 13]. First, the basic actions and commands are discovered on the discovery stage. A number of novel algorithms have been developed. RSST (Robust Singular Spectrum Transform) is an algorithm that calculates likelihood of change of dynamics in continuous time series without prior knowledge. DGCMD (Distance-Graph Constrained Motif Discovery) uses the result of RSST to discover motifs (recurring temporal patterns) from the given time

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Fig. 2. The framework of learning by mimicking [12, 13]

series. Second, a probabilistic model is generated to specify the likelihood of the occurrence of observed actions as a result of observed commands on the association stage. Granger causality is used to discover natural delay. Third, the behavioral model is converted into an actual controller on the controller generation stage to allow the robotic agent to act in similar situations. Finally, the gestures and actions learned from multiple interactions are combined into a single model on the accumulation stage.

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Conversational Decision Making

At the higher level, conversation can be seen as a process for realizing a joint activity. In order to participate in a conversation to help people make decisions, an artificial partner should be able to facilitate dialectic conversation so that a joint intention can be formulated incrementally during the conversational interactions. We have addressed to unveil how a skillful facilitator promotes group discussion by supporting the group's social and cognitive processes [14]. In general, facilitators are considered to allow participants to focus on substantive issues in the decision making process, by appropriately interposing based on the most important arguments of each participant. We have investigated the nonverbal and paralinguistic behavior of the participants in face-to-face discussion led by a skillful facilitator. As a result of linear discriminant analysis, we have found that four types of the facilitating behaviors can be classified with 80.0% accuracy by using six independent variables. The discriminant functions indicated that the facilitator paid attention to fairness of discussion and the participants conveyed their requests to the facilitator by using nonverbal behavior.

Talking with Uncertainty

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In the succeeding study [15], we investigated how much one could help the user formulate a preference structure on an unfamiliar subject by repeating interviews consisting of presentation of possible choices and asking for the preference. During each interview, not only verbal responses but also body movement and physiological indices (SCR and LF/HF) were measured to estimate the ordered set of features the user was emphasized. We assumed that the user’s preference structure may change as s/he changes emphasized feature. We have obtained several interesting findings, e.g., the preferential structure of the user does change during a session, our method can better track the user’s changing emphasis than a previous method, and our method resulted in user’s satisfaction more often than the previous method. A further experiment [16] suggests that critical changes of features emphasized by the user can be detected by the combination of verbal reactions, body movements, and physiological indices.

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Towards Empathic Agents for Living with Uncertainty

An artificial partner needs to be empathic in order to be successful. In sociocultural computing, building an emphatic agent for moderating communication across different cultures is deemed a key problem [17]. Pentland [4] suggests that we need to listen to honest signals, such as influence, mimicry, activity, or consistency, which come from our brain structure and physiology. Gallese, Eagle, and Migone [18] suggest that intentional attunement or embodied simulation enabled by the dynamics of our embodiment and neural system, including mirror neurons, might be a key. I suspect that sharing the abstract conceptualization with grounded symbols might be critical for the empathy to sustain. Theory of mind [19] needs to be implemented so that an artificial partner can think about the mental state of other agents. A challenging AI problem is to build an intelligent agent that can build and maintain a shared image with other participants in the conversation place.

References 1. AI evolution: From Tool to Partner, http://news.harvard.edu/gazette/story/2002/01/ ai-evolution-from-tool-to-partner/ 2. Halpern, J.Y.: Reasoning about Uncertainty. The MIT Press (2005) 3. Clark, H.H.: Using Language. Cambridge University Press, Cambridge (1996) 4. Pentland, A.: Honest Signals – How they shape our world. MIT Press, Cambridge (2008) 5. Kendon, A.: Some functions of gaze direction in social interaction. Acta Psychologica 26, 22–63 (1967) 6. Huang, H.H., Furukawa, T., Ohashi, H., Cerekovic, A., Pandzic, I., Nakano, Y., Nishida, T.: How Multiple Current Users React to a Quiz Agent Attentive to the Dynamics of Their Participation. In: Proc. 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2010), Toronto, Canada, pp. 1281–1288 (2010) 7. Ohmoto, Y., Miyake, T., Nishida, T.: A Method to Understand an Atmosphere based on Visual Information and Physiological Indices in Multi-user Interaction. In: The 8th International Workshop on Social Intelligence Design (SID 2009), Kyoto, Japan (2009)

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8. Nishida, T. (ed.): Conversational Informatics: an Engineering Approach. John Wiley & Sons Ltd., London (2007) 9. Rehm, M., Nakano, Y., André, E., Nishida, T., Bee, N., Endrass, B., Huang, H.H., Lipi, A.A., Wissner, M.: From Observation to Simulation: Generating Culture-specific Behavior for Interactive Systems. AI & Society 24(3), 267–280 (2009) 10. Ohmoto, Y., Ohashi, H., Nishida, T.: How do Constraints on Robot’s Embodiment Influence Human Robot Interaction?, 1E1-4. In: Proc. the 25th Annual Conference of the Japanese Society for Artificial Intelligence (2011) (in Japanese) 11. Ohmoto, Y., Ohashi, H., Lala, D., Mori, S., Sakamoto, K., Kinoshita, K., Nishida, T.: ICIE: Immersive Environment for Social Interaction based on Socio-Spacial Information. To be presented at TAAI Conference on Technologies and Applications of Artificial Intelligence (TAAI 2011), Taoyuan, Taiwan (2011) 12. Mohammad, Y.F.O., Nishida, T., Okada, S.: Unsupervised Simultaneous Learning of Gestures, Actions and their Associations for Human-Robot Interaction. In: Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2009), pp. 2537–2544 (2009) 13. Mohammad, Y.F.O., Nishida, T.: Learning Interaction Protocols using Augmented Baysian Networks Applied to Guided Navigation. In: Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2010), pp. 4119–4126 (2010) 14. Ohmoto, Y., Toda, Y., Ueda, K., Okada, S., Nishida, T.: The Analysis of the Facilitation Actions based on the Divergence-Convergence in Discussions and Nonverbal behavior, 2G1-OS3-5. In: Proc. the 24th Annual Conference of the Japanese Society for Artificial Intelligence (2010) (in Japanese) 15. Ohmoto, Y., Miyake, T., Nishida, T.: Estimating Preference Structure through HumanAgent Interaction, 3E2-2. In: Proc. the 25th Annual Conference of the Japanese Society for Artificial Intelligence (2011) (in Japanese) 16. Ohmoto, Y., Kataoka, M., Miyake, T., Nishida, T.: A Method to Dynamically Estimate Emphasizing Points and Degree by using Verbal and Nonverbal Information and Physiological Indices. To be Presented at the 2011 IEEE International Conference on Granular Computing (GrC 2011), Kaohsiung, Taiwan (2011) 17. Aylett, R., Paiva, A.: Computational Modelling of Culture and Affect., Emotion Review (in press, 2011) 18. Gallese, V., Eagle, M.N., Migone, P.: Intentional Attunement: Mirror Neurons and the Neural Underpinnings of Interpersonal Relations. Journal of the American Psychoanalytic Association 55, 131–176 (2007) 19. Baron-Cohen, S., Leslie, A.M., Frith, U.: Does the Autistic Child have a Theory of Mind? Cognition 21, 37–46 (1985)

Introduction to the ER Rule for Evidence Combination Jian-Bo Yang and Dong-Ling Xu Manchester Business School The University of Manchester Manchester M15 6PB, UK {jian-bo.yang,ling.xu}@mbs.ac.uk

Abstract. The Evidential Reasoning (ER) approach has been developed to support multiple criteria decision making (MCDM ) under uncertainty. It is built upon Dempster’s rule for evidence combination and uses belief functions for dealing with probabilistic uncertainty and ignorance. In this introductory paper, following a brief introduction to Dempster’s rule and the ER approach, we report the discovery of a new generic ER rule for evidence combination [16]. We first introduce the concepts and equations of a new extended belief function and then examine the detailed combination equations of the new ER rule. A numerical example is provided to illustrate the new ER rule. Keywords: Evidential reasoning, Belief function, Evidence combination, Dempster’s rule, Multiple criteria decision making.

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Basic Concepts of Evidence Theory

The evidence theory was first investigated in 1960’s [2] and formalised in 1970’s [7]. It has since been further developed and found widespread applications in many areas such as artificial intelligence, expert systems, pattern recognition, information fusion, database and knowledge discovery, multiple criteria decision making (MCDM ), audit risk assessment, etc. [1, 3, 5, 9–15]. In this section, the basic concepts of belief function and Dempster’s combination rule of the evidence theory are briefly introduced as a basis for introduction of the Evidential Reasoning (ER) approach in the next section. Suppose H = {H1 , . . . , HN } is a set of mutually exclusive and collectively exhaustive propositions, referred to as the frame of discernment. A basic probability assignment (bpa) is a belief function m : Θ → [0, 1], satisfying:  m(∅) = 0, and m(c) = 1 (1) C∈Θ

with ∅ being an empty set, C any subset of H, and Θ the power set of H, consisting of all the 2N subsets of H, or Θ = {∅, {H1 }, . . . , {HN }, {H1 , H2 }, . . . , {H1 , HN }, . . . , {H1 , . . . , HN −1 }, H} (2) Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 7–15, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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A basic probability mass m(C) measures the degree of belief exactly assigned to a proposition C and represents how strongly the proposition is supported by evidence. Probabilities assigned to all the subsets of H are summed to unity and there is no belief left to the empty set. A probability assigned to H, or m(H), is referred to as the degree of global ignorance. A probability assigned to any subset of H, except for any individual proposition Hn (n = 1, . . . , N ) and H, is referred to as the degree of local ignorance. If there is no local or global ignorance, a belief function reduces to a conventional probability function. Associated with each bpa to C are a belief measure, Bel(C), and a plausibility measure, P l(C), defined by the following equations:   Bel(C) = m(B) and P l(C) = m(B) (3) B⊆C

B∩C=∅

Bel(C) represents the exact support to C and its subsets, and P l(C) represents all possible support to C and its subsets. The interval [Bel(C), P l(C)] can be seen as the lower and upper bounds of support to C. The two functions can be connected by the following equation P l(C) = 1 − Bel(C)

(4)

where C denotes the complement of C. The difference between the belief and plausibility measures of C describes the degree of ignorance in assessment to C [7]. The core of the evidence theory is Dempster’s rule for evidence combination by which evidence from different sources is combined. The rule assumes that information sources are independent and uses the so-called orthogonal sum to combine multiple belief functions: m = m1 ⊕ m2 ⊕ . . . ⊕ mL

(5)

where ⊕ is the orthogonal sum operator. With two pieces of evidence m1 and m2 , Dempster’s rule for evidence combination is given as follows: ⎧ θ=∅ ⎨ 0,  m1 (B)m2 (C) [m1 ⊕ m2 ](θ) = (6) B∩C=θ  ⎩ 1− m1 (B)m2 (C) , θ = ∅ B∩C=∅

Note that Dempster’s rule provides a non-compensatory process for aggregation of two pieces of evidence and can lead to irrational conclusions in the aggregation of multiple pieces of evidence in conflict [4, 6, 8], in particular in cases where multiple pieces of evidence are mutually compensatory in nature. By a compensatory process of evidence combination, it is meant that any piece of evidence is not dominating but plays a relative role, which is related to its relative importance. On the other hand, the ER approach [9, 11–15] introduced in the next section provides a compensatory evidence aggregation process, which is different from Dempster’s rule in that it treats basic probability assignments as weighted belief degrees, embraces the concept of the degree of indecisiveness caused due to evidence weights, and adopts a normalisation process for combined probability masses without leaving any belief to the empty set.

Introduction to the ER Rule for Evidence Combination

2

9

The Main Steps of the ER Approach for MCDM

In the ER approach, a MCDM problem is modelled using a belief decision matrix. Suppose M alternatives (Al , l = 1, . . . , M ) are assessed on L criteria ei (i = 1, . . . , L) each on the basis of N common evaluation grades (proposition) Hn (n = 1, . . . , N ), which are required to be mutually exclusive and collectively exhaustive. If an alternative Al is assessed to a grade Hn on a criterion ei with a belief degree of βn,i (Al ), this assessment can be denoted by a belief function with global ignorance Si (Al ) = S(ei (Al )) = {(Hn , βn,i (Al )), n = 1, . . . , N, (H, βH,i (Al ))}, with βH,i (Al ) used to measure the degree of global N ignorance, i=1 βn,i (Al ) + βH,i (Al ) = 1, βn,i (Al ) ≥ 0 (n = 1, . . . , N ) and βH,i (Al ) ≥ 0. The individual assessments of all alternatives each on every criterion can be represented by a belief decision matrix, defined as follows: D = (Si (Al ))L×M

(7)

Suppose ωi is the relative weight of the ith criterion, normalised by  0 ≤ ωi ≤ 1 and ωi = 1

(8)

i

The ER approach has both the commutative and associative properties and as such can be used to combine belief functions in any order. The ER aggregation process can be implemented recursively [11–13], summarised as the following main steps. Step 1: Assignment of basic probability masses Suppose the basic probability masses for an assessment S1 (Al ) are given by: mn,1 = mH,1 = mΘ,1 = 1 − ω1

ω1 βn,1 (Al ) for n = 1, . . . , N,  N

ω1 βH,1 (Al ), and

n=1 βn,1 (Al )

 + βH,1 (Al ) = 1 − ω1

(9)

In the evidence theory, mn,1 may be interpreted as discounted belief. In MCDM, it should be interpreted as weighted belief or individual support to the assessment of Al to Hn , as it means that in assessing an alternative Al the 1st criterion only plays a limited role that is proportional to its weight. mH,1 represents the weighted global ignorance of the assessment. mΘ,1 is referred as to the degree of indecisiveness left by S1 (Al ), representing the amount of belief that is not yet assigned to any individual or any subset of grades by S1 (Al ) alone but needs to be jointly assigned in accordance with all other assessments in question. Similarly, the basic probability masses for another assessment S2 (Al ) are given by mn,2 = mH,2 = mΘ,2

ω2 βn,2 (Al ) for n = 1, . . . , N, ω2 βH,2 (Al ), and   N = 1 − ω2 n=1 βn,2 (Al ) + βH,2 (Al ) = 1 − ω2

(10)

10

J.-B. Yang and D.-L. Xu

Step 2: Combination of basic probability masses The basic probability masses for S1 (Al ) and S2 (Al ) can be combined using the following ER algorithm: {Hn } : mn,12 = k(mn,1 mn,2 + mn,1 (mH,2 + mΘ,2 ) + (mH,1 + mΘ,1 )mn,2 ) (11) for n = 1, . . . , N {H} : mH,12 = k(mH,1 mH,2 + mH,1 mΘ,2 + mH,2 mΘ,1 )

(12)

{Θ} : mΘ,12 = k(mΘ,1 mΘ,2 )

(13)

⎛ k = ⎝1 −

⎞−1

N N  

mn,1 mt,2 ⎠

(14)

n=1 t=1;t=n

In the above ER algorithm, mn,12 and mH,12 measure the relative magnitudes of the total beliefs in the individual grade Hn and the frame of discernment H, respectively, generated by combining the two belief functions S1 (Al ) and S2 (Al ). mΘ,12 is the degree of indecisiveness left by both S1 (Al ) and S2 (Al ), representing the amount of belief that needs to be re-assigned back to all subsets of grades proportionally after the combination process is completed, so that no belief is assigned to the empty set. k measures the degree of conflict between S1 (Al ) and S2 (Al ). Step 3: Generation of total belief degrees If there are more than two assessments, Step 2 can be repeated to combine an uncombined assessment with the previously-combined assessment given by mn,12 (n = 1, . . . , N ), mH,12 and mΘ,12 . After all assessments are combined recursively, the finally combined probability masses need be normalised to generate the total belief degrees βn,12 and βH,12 (for L=2) by proportionally re-assigning mΘ,12 back to all subsets of grades as follows: {Hn } : βn,12 =

mn,12 , n = 1, . . . , N 1 − mΘ,12

(15)

mH,12 1 − mΘ,12

(16)

{H} : βH,12 =

The combined assessment for Al is then given by the following belief function: S(Al ) = {(H1 , β1,12 ), (H2 , β2,12 ), . . . , (HN , βN,12 ), (H, βH,12 )}

(17)

The above belief function provides a panoramic view about the combined assessment of the alternative Al with the degrees of strength and weakness explicitly measured by the belief degrees.

Introduction to the ER Rule for Evidence Combination

11

Introduction to the ER Rule for Evidence Combination

3

In Section 2, a belief function with global ignorance was represented by Si (Al ) = S(ei (Al )) = {(Hn , βn,i (Al )), n = 1, . . . , N, (H, βH,i (Al ))}, with (Hn , βn,i (Al )) referred to as a focal element of S(ei (Al )) if βn,i (Al ) > 0. mn,1 = ω1 βn,1 (Al ) given in Equation (9) represents the individual support of the evidence S(e1 (Al )) to the hypothesis that Al is assessed to Hn . Similarly, mn,2 = ω2 βn,2 (Al ) given in Equation (10) represents the individual support of the evidence S(e2 (Al )) to the same hypothesis. As such, mn,1 mn,2 represents the joint support of both S(e1 (Al )) and S(e2 (Al )) to the same hypothesis. Generally, suppose a piece of evidence S(ei ) withthe weight ωi is represented by the following conventional belief function with θ∈Θ βθ,i = 1 S(ei ) = {(θ, βθ,i ), ∀θ ∈ Θ}

(18)

We can now show the extension of the above conventional belief function to include a special element (Θ, (1 − ωi )) for constructing a new extended belief function for S(ei ) as follows [16]: mi = {(θ, mθ,i ), ∀θ ∈ Θ, (Θ, mΘ,i )}

(19)

mθ,i = ωi βθ,i , ∀θ ∈ Θ, and mΘ,i = 1 − ωi

(20)

with Note that the following relationships between a conventional belief function and its extended belief function are always true [16]: βθ,i =

mθ,i , ∀θ ∈ Θ 1 − mΘ,i

(21)

We are now in a position to introduce the new ER rule [16] as follows. Let two pieces of independent evidence S(e1 ) and S(e2 ) with the relative weights ω1 and ω2 be represented by the conventional belief functions defined by Equation (18) with ω1 + ω2 = 1, mθ,1 = ω1 βθ,1 and mθ,2 = ω2 βθ,2 for all θ ⊆ H. Then, S(e1 ) and S(e2 ) can be combined by the following ER rule which can be used recursively for aggregating multiple pieces of evidence [16]: mθ,12 =  θ⊆H

m

θ,12 m

Θ,12 , ∀θ ⊆ H, and mΘ,12 =  m

θ,12 + m

Θ,12 m

θ,12 + m

Θ,12

(22)

θ⊆H

m

θ,12 βθ,12 =  , ∀θ ⊆ H, m

θ,12 θ⊆H



m

θ,12 = [(1 − ω2 )mθ,1 + (1 − ω1 )mθ,2 ] +

(23)

mB,1 mC,2

(24)

B,C⊆H;B∩C=θ

m

Θ,12 = mΘ,1 mΘ,2

(25)

12

J.-B. Yang and D.-L. Xu

The combined extended and conventional belief functions can then be represented as follows: m1 ⊕ m2 = {(θ, mθ,12), ∀θ ∈ Θ, (Θ, mΘ,12 )}

(26)

S(e1 ) ⊗ S(e2 ) = {(θ, βθ,12 ), ∀θ ∈ Θ}

(27)

where ⊕ is the orthogonal sum operator composed of Equations (22), (24) and (25) for generating combined extended belief functions, which can be applied recursively, and ⊗ is the ER operator consisting of Equations (23) and (24) for generating combined conventional belief functions, which can be used after extended belief functions are combined. The new ER rule results from the innovation of implementing Dempster’s rule on the new extended belief functions. It can be shown that the current ER approach as summarized in section 2 is a special case of the new ER rule. The new ER rule provides a generic process for generating total beliefs from combination of multiple pieces of independent evidence under the normal condition that each piece of evidence plays a role equal to its relative weight. The ER rule can be applied in areas where the above normal condition is satisfied, for example in multiple criteria decision making. It is important to note that the combined belief generated by using the ER rule to aggregate two pieces of evidence is composed of two parts: the bounded average of the individual support, which is the first bracketed term in Equation (24), and the orthogonal sum of the joint support, which is the last term in Equation (24). This is in contract to the partial belief generated by using Dempster’s rule on conventional belief functions, including only the orthogonal sum to count for joint support, with individual support either abandoned or assigned to the empty set, either of which is irrational.

4

Illustration of the ER Rule

We now examine a simple example to illustrate how the ER rule can be implemented and explain whether the results it generates are rational. Suppose three pieces of evidence of equal importance are given as the following three belief functions each with only its focal elements listed: S(e1 ) = {(A, 0.99), (B, 0.01)} S(e2 ) = {(B, 0.01), (C, 0.99)}

(28) (29)

S(e3 ) = {(B, 0.01), ({A, C}, 0.99)}

(30)

with ω1 = ω2 = ω3 = 0.3333. Suppose they each play a role equal to their relative weights. Note that H = {A, B, C} and Θ = {∅, A, B, C, {A, B}, {A, C}, {B, C}, {A, B, C}}

Introduction to the ER Rule for Evidence Combination

13

in this example. The extended belief functions corresponding to Equations (28)(30) are given using Equations (19) and (20) as follows m1 = {(A, 0.33), (B, 0.0033), (Θ, 0.6667)} m2 = {(B, 0.0033), (C, 0.33), (Θ, 0.6667)}

(31) (32)

m3 = {(B, 0.0033), ({A, C}, 0.33), (Θ, 0.6667)}

(33)

The calculations of the ER rule for the above example are shown in Table 1. The ER rule is applied recursively (Equations (22), (24) and (25)) to Equations (31)(33). The results of the last row, generated using Equation (23) after the second iteration, show the final combined conventional belief function, complementary to the extended belief function, shown in the last but one row, which is generated by aggregating all the three extended belief functions shown in rows 5-7. In the final results, βA,123 = βC,123 = 0.3718 are the highest total belief, which makes sense as the first evidence supports the proposition A and the second evidence supports the proposition C with the same magnitude, while the third evidence supports the proposition {A, C} with no discrimination between the two individual propositions A and C. β{A,C},123 = 0.2487 is generated rightly as the second highest total belief as the third evidence supports {A, C}, so the significant local ignorance in {A, C} should remain in the final results. The proposition B is assessed to be unlikely by all the three pieces of evidence, so it makes sense that the total belief in this proposition should also be rather small. The total belief in each of the other propositions (∅, {A, B}, {B, C} and {A, B, C}) is zero as it should be. Table 1. Illustration of the ER Rule Belief

A

B

C

βθ,1

0.99

0.01

0

{A, B} {A, C} {B, C} {A, B, C} 0

0

0

0

βθ,2

0

0.01

0.99

0

0

0

0

βθ,3

0

0.01

0

0

0.99

0

Θ

0

mθ,1 0.3300 0.0033 0.0000 0.0000 0.0000 0.0000

0.0000

0.6667

mθ,2 0.0000 0.0033 0.3300 0.0000 0.0000 0.0000

0.0000

0.6667

mθ,3 0.0000 0.0033 0.0000 0.0000 0.3300 0.0000

0.0000

0.6667

m

θ,12 0.2200 0.0045 0.2200 0.0000 0.0000 0.0000

0.0000

0.4444

mθ,12 0.2475 0.0050 0.2475 0.0000 0.0000 0.0000

0.0000

0.5000

βθ,12

0.0000 0.0000 0.0000

0.0000

m

θ,123 0.2467 0.0050 0.2467 0.0000 0.1650 0.0000

0.495

0.01

0.495

0.0000

0.3333

mθ,123 0.2475 0.0050 0.2475 0.0000 0.1655 0.0000

0.0000

0.3344

βθ,123 0.3718 0.0076 0.3718 0.0000 0.2478 0.0000 0.0000

14

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J.-B. Yang and D.-L. Xu

Conclusion

In this paper, following the discussion of Dempster’s rule and the ER approach, we reported the discovery of the new ER rule that provides a general process for combining multiple pieces of independent evidence in form of belief functions under the normal condition that every piece of evidence plays a limited role equivalent to its relative weight. The ER rule generates the total beliefs from combination of every two pieces of evidence as the addition of the bounded average of the individual support from each of the two pieces of evidence and the orthogonal sum of the joint support from the two pieces of evidence, which reveals that the orthogonal sum of the joint support from two pieces of evidence is only part of their total combined belief. A numerical example was examined in some detail to illustrate this general yet rational and rigorous ER rule for evidence combination. The new ER rule can be applied for combination of independent evidence in any cases where the normal condition is satisfied. Acknowledgments. This work was supported by the UK Engineering and Physical Science Research Council under the Grant No: EP/F024606/1 and the Natural Science Foundation of China under the Grant No: 60736026.

References 1. Beynon, M.: DS/AHP method: A mathematical analysis, including an understanding of uncertainty. European Journal of Operational Research 140(1), 148–164 (2002) 2. Dempster, A.P.: Upper and lower probabilities induced by a multi-valued mapping. Annals of Mathematical Statistics 38, 325–339 (1967) 3. Denoeux, T., Zouhal, L.M.: Handling possibilistic labels in pattern classification using evidential reasoning. Fuzzy Sets and Systems 122(3), 409–424 (2001) 4. Haenni, R.: Shedding new light on zadeh’s criticism of Dempster’s rule of combination. In: The 7th International Conference on Information Fusion (FUSION) (2005) 5. McClean, S., Scotney, B.: Using evidence theory for knowledge discovery and extraction in distributed databases. International Journal of Intelligent Systems 12, 763–776 (1997) 6. Murphy, C.K.: Combining belief functions when evidence conflicts. Decision Support Systems 29, 1–9 (2000) 7. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976) 8. Wang, Y.M., Yang, J.B., Xu, D.L.: The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees. European Journal of Operational Research 175(1), 35–66 (2006) 9. Xu, D.L., Yang, J.B., Wang, Y.M.: The ER approach for multi-attribute decision analysis under interval uncertainties. European Journal of Operational Research 174(3), 1914–1943 (2006) 10. Yager, R.R.: Decision making using minimization of regret. International Journal of Approximate Reasoning 36, 109–128 (2004)

Introduction to the ER Rule for Evidence Combination

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11. Yang, J.B., Singh, M.G.: An evidential reasoning approach for multiple attribute decision making with uncertainty. IEEE Transactions on Systems, Man, and Cybernetics 24(1), 1–18 (1994) 12. Yang, J.B.: Rule and utility based evidential reasoning approach for multiattribute decision analysis under uncertainties. European Journal of Operational Research 131, 31–61 (2001) 13. Yang, J.B., Xu, D.L.: On the evidential reasoning algorithm for multiattribute decision analysis under uncertainty. IEEE Transactions on Systems, Man, and Cybernetics Part A: Systems and Humans 32(3), 289–304 (2002) 14. Yang, J.B., Wang, Y.M., Xu, D.L.: The Evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties. European Journal of Operational Research 171(1), 309–343 (2006) 15. Yang, J.B., Xu, D.L., Xie, X.L., Maddulapalli, A.K.: Multicriteria evidential reasoning decision modelling and analysis - prioritising voices of customer. Journal of the Operational Research Society 62, 1638–1654 (2011) 16. Yang, J.B., Xu, D.L.: The evidential reasoning rule for evidence combination. Artificial Intelligence (submitted to, 2011)

Theories and Approaches to Treat Incomparability Yang Xu1 and Jun Liu2 1

2

College of Mathematics, Southwest Jiaotong University Chengdu, Sichuan 610031, P.R. China [email protected] School of Computing and Mathematics, University of Ulster, UK [email protected]

Extended Abstract As we know, when human beings try to understand and deal with practical problems, especially in their evaluation or decision making process, comparison is a way commonly used to tell something about something else - “there can be no differentiation without comparison”. Some claim that chains, i.e., totally ordered sets, can be applied in most cases. But the assumption is often an oversimplification of reality. Actually, relations in the real world are rarely linear. Incomparability is a kind of uncertainty often associated with humans’ intelligent activities in the real world, and it exists not only in the processed object itself, but also in the course of the object being processed. It is a kind of overall uncertainty of objects due to the complexity of objects themselves, associated with many factors and the inconsistency among those factors. It is caused by the complex degree of an object. The more complex for certain object, the more for attribute of certain, and the larger for incomparability. This fact implies an overall uncertainty of objects, which can be due to missing information, ambiguity or conflicting evaluations. Incomparability is an important type of uncertainties, especially inevitable in decision making and evaluation situations, but it is not easily handled through conventional methods because of its complexity. This talk is organized as follows: Section 1 as an introduction gives several examples of incomparability. Section 2 proposes the algebraic approaches to characterize incomparability, some typical algebras, such as Residuated Lattice, Pseudo-Boolean Algebra, BL-Algebra, Lattice Implication Algebra (LIA) (specially, Linguistic-Valued Lattice Implication Algebra) , are overviewed. Section 3 proposes the logical theory and methods to deal with incomparability, including the lattice-valued logic system based on LIA, the uncertainty reasoning based on lattice-valued logic with truth-values in LIAs, as well as the resolution-based automated reasoning in lattice-valued logic system with truth-values in LIAs. Section 4 proposes mathematical analysis theory and methods to treat incomparability, which is based on l ∗ -module, i.e., a lattice-ordered module with two lattice-ordered structures. Concluding remarks are presented in Section 6. In summary, some academic standpoints are pointed out as follows: Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 16–17, 2011. c Springer-Verlag Berlin Heidelberg 2011 

Theories and Approaches to Treat Incomparability

17

1. There exist in both objective world and subjective world great quantities of incomparability. 2. Algebraic method, logical method and mathematics analytical method can be applied to treat incomparability. 3. Incomparability also commonly exists in linguistic values of human language, therefore, algebraic method, logical method and mathematics analytical method can be applied to treat linguistic value structure with incomparability. 4. Generally, in the evaluation and decision-making always involved with incomparability, linguistic values and uncertainty reasoning. Therefore, it is a kind of scientific approach to apply lattice-valued logic system based on linguistic truth-valued algebra. 5. The problem about how to establish appropriate methods to handle those incomparability and complex linguistic value structure in the real world application is still open and it has been an important and worthwhile research topic. But we believe that it is feasible and reasonable to use ordering structure (especially lattice), logical reasoning and mathematics analysis as the possible solution, especially use lattice-valued algebra and lattice-valued logic to establish strict linguistic truth-valued logic and various kinds of corresponding linguistic information processing systems, based on what have been done so far about lattice-valued algebra, lattice-valued logic by different researches, also relying on a continuous work on this research direction. Acknowledgement. This work is partially supported by the National Natural Science Foundation of China (Grant No. 60875034) and the projects TIN-20090828, P08-TIC-3548 and FEDER funds.

Knowledge Science – Modeling the Knowledge Creation Process Yoshiteru Nakamori School of Knowledge Science Japan Advanced Institute of Science and Technology 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan [email protected]

Extended Abstract Knowledge science is a problem-oriented interdisciplinary field that takes as its subject the modeling of the knowledge creation process and its application, and carries out research in such disciplines as knowledge management, management of technology, support for the discovery, synthesis and creation of knowledge, and innovation theory with the aim of constructing a better knowledge-based society. This presentation considers what knowledge science should be, introducing a forthcoming book entitled “Knowledge Science - Modeling the Knowledge Creation Process” (Nakamori [1]) as well as the School of Knowledge Science at Japan Advanced Institute of Science and Technology, which is the first school established in the world to make knowledge a target of science. The first dean of the School was Professor Ikujiro Nonaka who is famous worldwide for his organizational knowledge creation model called the SECI spiral (Nonaka and Takeuchi [3]), which is in fact the key factor in establishing the School. The presentation also briefly introduces a methodology for knowledge synthesis called the theory of knowledge construction systems; its fundamental part was already published in Systems Research and Behavioral Science (Nakamori et al. [2]). Keywords: Knowledge technology, Knowledge management, Knowledge discovery, Knowledge synthesis, Knowledge justification, Knowledge construction.

References 1. Nakamori, Y. (ed.): Knowledge Science – Modeling the Knowledge Creation Process. CRC Press (2011) 2. Nakamori, Y., Wierzbicki, A.P., Zhu, Z.: A Theory of Knowledge Construction Systems. Syst. Res. Behav. Sci. 28, 15–39 (2011) 3. Nonaka, I., Takeuchi, H.: The Knowledge Creating Company: How Japanese Companies Create the Dynamics of Innovation. Oxford University Press, New York (1995)

Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, p. 18, 2011. c Springer-Verlag Berlin Heidelberg 2011 

Two Classes of Algorithms for Data Clustering Sadaaki Miyamoto Department of Risk Engineering, Faculty of Systems and Information Engineering University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan [email protected]

Abstract. The two classes of agglomerative hierarchical clustering algorithms and K-means algorithms are overviewed. Moreover recent topics of kernel functions and semi-supervised clustering in the two classes are discussed. This paper reviews traditional methods as well as new techniques. Keywords: agglomerative hierarchical clustering, K-means clustering, kernel functions, constrained clustering.

1

Introduction

Cluster analysis, also called data clustering or simply clustering, has now become a standard tool in modern data mining. Clustering techniques are divided into two classes of hierarchical and non-hierarchical methods. The major technique in the first class is the agglomerative hierarchical clustering [2, 18] which is old but has been found useful in a variety of fields of applications. The latter category has various methods. Some of them are a family of K-means algorithms [13, 14, 32], fuzzy c-means and their variations [6, 7, 12, 15, 16, 25, 40], mixture of densities [33, 43], algorithms related to SOM [27], and other heuristic techniques. To overview all of them needs a book or at least a book chapter. Instead, we discuss a class that has most frequently been studied, i.e., K-means algorithms and their variations. This paper thus considers two classes of clustering techniques: agglomerative hierarchical clustering and methods related to K-means. In the latter class we mention the mixture of distributions which is also regarded as a standard technique of clustering using the EM algorithm [11, 33, 43]. The topics discussed here is the overview of basic methods as well as more recent studies of kernel functions [44, 48] and semi-supervised clustering [3–5, 9, 50, 52]. This paper thus reviews well-known techniques as well as newly developed methods. The rest of this paper is as follows. Section 2 introduces agglomerative hierarchical clustering and standard linkage methods. Section 3 is devoted to the discussion of a class of K-means techniques. In particular fuzzy c-means and their variations are focused upon. Section 4 discusses the use of kernel functions in cluster analysis. After studying a recent topic of semi-supervised clustering in Chapter 5, The final chapter concludes the paper. Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 19–30, 2011. c Springer-Verlag Berlin Heidelberg 2011 

20

2

S. Miyamoto

Agglomerative Hierarchical Clustering

We first review the general procedure of agglomerative hierarchical clustering. Let the set of objects for clustering be X = {x1 , . . . , xN }. The object xk is generally a point in the p-dimensioanl Euclidean space Rp , unless otherwise stated. Generally a cluster denoted by Gi is a subset of X. The family of clusters is denoted by G = {G1 , G2 , . . . , GK }, where the clusters form a crisp partition of X: K 

Gi = X,

Gi ∩ Gj = ∅

(i = j).

i=1

Moreover the number of objects in G is denoted by |G|. Agglomerative hierarchical clustering uses a similarity or dissimilarity measure. We use dissimilarity here: dissimilarity between two objects x, y ∈ X is assumed to be given and denoted by D(x, y). Dissimilarity between two clusters is also used, which is denoted by D(G, G ) (G, G ∈ G) which also is called an inter-cluster dissimilarity. In the classical setting a dissimilarity measure is assumed to be symmetric: D(G, G ) = D(G , G). Let us first describe a general procedure of agglomerative hierarchical clustering [34, 36]. AHC (Agglomerative Hierarchical Clustering) Algorithm: AHC1: Assume that initial clusters are given by ˆ 2, . . . , G ˆ N0 }. where G ˆ 1, G ˆ2, . . . , G ˆ N are given initial clusters. ˆ 1, G G = {G ˆ j = {xj } ⊂ X, hence N0 = N . Generally G Set K = N0 . (K is the number of clusters and N0 is the initial number of clusters) ˆ i (i = 1, . . . , K). Gi = G Calculate D(G, G ) for all pairs G, G ∈ G. AHC2: Search the pair of maximum similarity: (Gp , Gq ) = arg min D(Gi , Gj ), Gi ,Gj ∈G

Merge: Gr = Gp ∪ Gq . Add Gr to G and delete Gp , Gq from G. K = K − 1. if K = 1 then stop and output the dendrogram. AHC3: Update similarity D(Gr , G ) for all G ∈ G. Go to AHC2. End AHC.

(1)

Two Classes of Algorithms for Data Clustering

21

Well-known linkage methods are as follows: all assume symmetric dissimilarity measures [2, 18, 34]. – single link:

– complete link:

D(G, G ) = D(G, G ) =

min

D(x, y),

(2)

max

D(x, y),

(3)

x∈G,y∈G

x∈G,y∈G

– average link: D(G, G ) =

1 |G||G |



D(x, y).

(4)

x∈G,y∈G

– centroid method: Let the centroid of a cluster G be M (G) =

1  xk . |G| xk ∈G

Using M (G) and the Euclidean norm  · , we define D(G, G ) = M (G) − M (G )2 . – Ward method: Let E(G) =



(5)

x − M (G)2 .

x∈G

We define

D(G, Gj ) = E(G ∪ G ) − E(G) − E(G).

(6)

Each of the five linkage methods have the updating formula in AHC3 whereby the computation is greatly reduced but we omit the formulas to save space [2, 34, 36]. These linkage methods are discussed later again in relation to kernel functions and semi-supervised clustering.

3

K-means Clustering

In K-means and their variations, the number of clusters, denoted by c here, is assumed to be given beforehand. We also use membership matrix U = (uki ) in which uki is the membership of xk to cluster i (or Gi ); uki may either be crisp or fuzzy. Moreover cluster centers denoted by vi (i = 1, . . . , c) are used. We first introduce an informal procedure called a c-prototypes: A c-prototype procedure. Step 0: Choose randomly initial c prototypes for clusters. Step 1: Take a set of objects and allocate each of them to the cluster of the nearest prototype.

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S. Miyamoto

Step 2: Update prototypes. Step 3: If clusters are convergent, stop. Else go to Step 1. This procedure is not a precise algorithm, since the way how prototypes are updated is not described. However, several algorithms are derived as variations of the above procedure. The simplest algorithm is that of the K-means [2, 32] which also is called crisp c-means. The basic algorithm is as follows: (I) Assume that the initial c clusters are randomly generated. Let the centroid of cluster i be vi (i = 1, . . . , c). (II) For k = 1, . . . , N , allocate xk to the cluster of the nearest center: i = arg min xk − vj 2 . 1≤j≤c

(7)

(III) Update the centroid vi for cluster i (i = 1, . . . , c). If clusters are convergent, stop. Else go to step (II). Comparing this algorithm with the c-prototype procedure, we find that the prototypes are centroids (centers of gravity) and the updating uses the recalculation of the centroids. The above algorithm reallocates all objects, but the on-line algorithm [13, 36] takes one object at a time and update the cluster centers as the centroids. SOM techniques [27] are now popular and sometimes used as methods of clustering. Let us consider the simplest algorithm of VQ [27]. The VQ clustering can be derived from the c-prototype: one object x(t), t = 1, 2, . . . , is taken and suppose vi (t) is the nearest prototype: Then vi is updated using the learning ratio α(t): vi (t + 1) = vi (t) + α(t)(x(t) − vi (t)), (8) while other prototypes remain unchanged. Each object is allocated to the cluster of the nearest prototype. Moreover fuzzy c-means and the mixture of distributions are considered to be variations of the c-prototype procedure. Fuzzy c-means are discussed later in detail, but we note that the nearest allocation is replaced by fuzzy memberships, and centroids are replaced by weighted centroids using fuzzy memberships. The mixture of distributions is also related to c-prototypes, where the nearest allocation is replaced by the probability P (Gi |xk ) by which observation xk should be allocated to cluster Gi , where the probability is calculated by the EM algorithm [33]. Hence the update of prototypes implies parameter updating in the distributions. 3.1

Fuzzy c-means and Variations

Fuzzy c-means clustering have been studied by many researchers, but there are less known and useful facts. We first introduce two basic objective functions:

Two Classes of Algorithms for Data Clustering

JB (U, V ) =

c  N 

(uki )m D(xk , vi ),

(m > 1),

23

(9)

i=1 k=1

JE (U, V ) =

c  N 

{uki D(xk , vi ) + λ−1 uki log uki },

(λ > 0),

(10)

i=1 k=1

where D(xk , vi ) = xk − vi 2 . Note that if m = 1 in JB , the function expresses that for crisp c-means (alias K-means). Function JB has been introduced by Dunn [15, 16] and Bezdek [6, 7]. On the other hand, JE using the entropy term was later discussed by a few authors [30, 31, 35]. To consider and compare these two functions is important to observe theoretical properties of fuzzy c-means. To save space, discussions of theoretical properties are omitted; readers may refer to [40]. The basic algorithm of fuzzy c-means is the following alternate optimization, where c J = JB or J = JE . Optimization with respect to U uses the constraint of i=1 uki = 1 and ukj ≥ 0 for all j, k. FCM Algorithm of Alternate Optimization. FCM1: Put initial value V¯ randomly. ¯. FCM2: Minimize J(U, V¯ ) with respect to U . Let the optimal solution be U ¯ , V ) with respect to V . Let the optimal solution be V¯ . FCM3: Minimize J(U ¯ , V¯ ) is convergent, stop. Otherwise go to FCM2. FCM4: If (U End FCM. The solutions for JB are as follows: 1

uki =

1

D(xk ,vi ) m−1 c 1 1 j=1 D(xk ,vj ) m−1

,

N

k=1 vi =  N

(uki )m xk

k=1 (uki )

m

,

whereas the solutions for JE are given by the following: uki

exp(−λD(xk , vi )) = c , j=1 exp(−λD(xk , vj ))

N

k=1 vi =  N

uki xk

k=1

uki

.

There are many variations of fuzzy c-means. For example, fuzzy c-regression models [21] that produce c regressions from input-output data {(xk , yk )}1≤k≤N using Dki = (yk − βiT xk − γi ), i = 1, . . . , c instead of D(xk , vi ). Note that the fuzzy c-regression models can also be discussed in relation to the c-prototype procedure. Another function that uses a quadratic term u2ki instead of the entropy term uki log uki in JE has been proposed in [39]. This method is theoretically interesting but requires more computation than that of JE .

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A remarkable study has been done by Ichihashi and his colleagues [24, 40]: they introduced two more variables α, S to JE in addition to U and V . The variable α = (α1 , . . . , αc ) ≥ 0 controls cluster volumes and S = (S1 , . . . , Sc ) is for clusterwise covariances. They propose the following objective function: c  N 

uik + log |Si |}. αi i=1 k=1 (11) c where i=1 αi = 1. Although we omit the optimal solutions (see, e.g., [40]), an interesting property is that the solutions are equivalent to those by the mixture of Gaussian distributions [36, 40] by choosing parameter λ = 1/2. Thus, although the fuzzy and probabilistic models are different, they lead to the same solutions. Another class of variations is the possibilistic clustering [10, 28] which uses JKL (U, V, A, S) =

JP (U, V ) =

c  N 

{uki (xk − vi )T Si−1 (xk − vi ) + λ−1 uki log

{(uki )m D(xk , vi ) + ζ −1 (1 − uki )m },

(ζ > 0).

(12)

i=1 k=1

and the FCM algorithm but without the constraint of the optimal solutions but they are easily derived. 3.2

c

i=1

uki = 1. We omit

Deciding the Number of Clusters

Frequently the number of clusters is unknown but it should be specified in Kmeans and related algorithms in this section. To decide the number is therefore important, but it is at the same time known as a difficult problem. There are two approaches: one is to use a method of model selection in statistical literature, e.g., [1]. In spite of many efforts, there is no study reporting a certain method of model selection is generally useful. The second approach is common in fuzzy literature. A cluster validity measure should be used. A problem is that although there have been many validity measures [6, 12, 22], there is no report stating a particular measure is the most useful. Hashimoto et al. [20] compared several measures using many numerical experiments, and report that, although there is no best measure for all experiments, the number of clusters can rightly be decided by comparing the results of different measures if clusters are well-separated.

4

Kernel Functions in Clustering

Linear cluster boundaries are obtained from crisp and fuzzy c-means clustering. In contrast, nonlinear boundaries can be derived using kernel functions, as discussed in support vector machines [44, 47, 48]. A high-dimensional feature space H is assumed, while the original space Rp is called the data space. H is an inner product space. Assume that the inner product is ·, ·. The norm of H for g ∈ H is given by g2H = g, g.

Two Classes of Algorithms for Data Clustering

25

A mapping Φ : Rp → H is used whereby xk is mapped into Φ(xk ). Explicit representation of Φ(x) is unknown in general but the inner product Φ(x), Φ(y) is assumed to be represented by a kernel function: K(x, y) = Φ(x), Φ(y).

(13)

A well-known kernel function is the Gaussian kernel:   K(x, y) = exp − Cx − y2 , (C > 0). 4.1

Kernel Functions in Fuzzy c-means

We can use kernel functions in crisp and fuzzy c-means [37, 38]: objective functions JB and JE are used but the dissimilarity is changed as follows: D(xk , vi ) = Φ(xk ) − vi 2H ,

(14)

where vi ∈ H. When we derive a kernel-based fuzzy c-means algorithm, we cannot use the solutions for vi . Let us consider JB . We have N (uki )m Φ(xk ) vi = k=1 , N m k=1 (uki ) but function Φ(xk ) is generally unknown. Hence we cannot use FCM algorithm. Instead, we update dissimilarity measure D(xk , vi ): D(xk , vi ) = Kkk − N

N 

2

m k=1 (uki )

1 + N ( k=1 (uki )m )2

(uji )m Kjk

j=1

N  N 

(uji ui )m Kj ,

(15)

j=1 =1

where Kjk = K(xj , xk ). Note that m = 1 in (15) when JE is considered. We thus repeat (15) and 1

uki =

1

D(xk ,vi ) m−1 c 1 1 j=1 D(xk ,vj ) m−1

until convergence. To handle JE and the crisp c-means [19] is similar and omitted. 4.2

Kernel Functions in Agglomerative Hierarchical Clustering

Let us give a short remark as to how kernel functions can be used in agglomerative hierarchical clustering. It means that we consider {Φ(x1 ), . . . , Φ(xN )} instead of {x1 , . . . , xN }.

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Let us assume that the mapping from the original squared distance δij = xi − xj 2 to the squared distance in H, Δij = Φ(xi ) − Φ(xj )2 , is monotone: if δij ≤ δkl , then Δij ≤ Δkl , for all 1 ≤ i, j ≤ N and 1 ≤ k, l ≤ N . This monotone property holds for typical kernel functions such as the Gaussian kernel. Moreover, completely monotone functions [45] define a wide class of kernel functions (see, e.g., [23]). The following proposition holds: Proposition 1. The updating formulas of the single linkage, the complete linkage, the average linkage, the centroid method, and the Ward method are used for clustering {Φ(x1 ), . . . , Φ(xN )} with Δij without any change. This property has been studied in [17]. We omit the proof, as we do not show the updating formulas here. It is not strange that all the updating formulas of the linkage methods are the same even when the high-dimensional feature space is used, as the derivations of the formulas use the monotonicity and the calculation of the inner products. Note that when agglomerative hierarchical clustering is applied with a kernel function, the initial value Δij has to be calculated using the kernel: Δij = K(xi , xi ) + K(xj , xj ) − 2K(xi , xj ).

5

Semi-supervised Clustering

Many researchers are now working on semi-supervised learning [3–5, 9, 26, 29, 46, 49–52]. They are divided into two major classes: in the first class we have two sets of objects: {(xk , yk )}1≤k≤N with the class labels yk and {xl }N +1≤l≤N +L without labels. Thus, partly supervised cases should be handled. For such a case the mixture of distributions with an EM algorithm can be used [52], or the transductive support vector machines can be applied [49, 52]. A variation of fuzzy c-means for such a case has also been considered [8]. Another class is called constrained clustering [5], where two sets of constraints, called pairwise constraints, are assumed: a set M L = {(xi , xj )} ⊂ X ×X consists of must-link pairs so that xi and xj should be in a same cluster, while another set CL = {(xk , xl )} ⊂ X×X consists of cannot-link pairs so that xi and xj should be in different clusters. M L and CL are assumed to be symmetric in the sense that if (xi , xj ) ∈ M L then (xj , xi ) ∈ M L, and if (xk , xl ) ∈ CL then (xl , xk ) ∈ CL. We assume that no inconsistency (e.g., (xi , xj ) ∈ M L, (xj , xk ) ∈ M L, and (xi , xk ) ∈ CL) arises for simplicity. Methods to handle constrained clustering are as follows: 1. The method of COP K-means [50] uses the same algorithm as the crisp c-means, except that it checks the pairwise constraints are satisfied or not when each object are allocated to the nearest cluster center. If a constraint is violated, the algorithm terminates with failure. 2. Crisp c-means with penalty terms are used [29]. 3. The mixture of distributions with penalty terms has been proposed [3, 4, 46]. 4. Fuzzy c-means with constraints have also been proposed [51].

Two Classes of Algorithms for Data Clustering

27

We mention agglomerative hierarchical clustering with pairwise constraints [41, 42]. The results are summarized as follows. (i) To introduce the pairwise constraints to the single linkage, the complete linkage, and the average linkage is straightforward. We change the dissimilarity to zero for pairs in M L and +∞ for pairs in CL. (ii) To introduce the pairwise constraints to the centroid method and the Ward method needs further consideration. One way is to modify the dissimilarity using kernel functions [41]; numerical experiments showed some effects of this method. (iii) To introduce the pairwise constraints to the centroid method and the Ward method using penalties have also been studied [42] and numerical experiments showed that this method is also effective. The method in (iii) implies that (1) should be replaced by (Gp , Gq ) = arg min D(Gi , Gj ) Gi ,Gj ∈G  WCL − + x∈Gi ,x ∈Gj ,(x,x )∈CL



WML .

(16)

x∈Gi ,x ∈Gj ,(x,x )∈ML

where WCL and WML are penalties for cannot-link and must-link, respectively.

6

Conclusions

We overviewed basic algorithms of agglomerative hierarchical clustering and the class of K-means algorithms. The latter class includes the mixture of distributions and fuzzy c-means with variations. More recent studies on kernel functions and semi-supervised clustering have also been discussed. Although many studies have been done on fuzzy c-means clustering, we have still many possibilities for future research. For such a purpose, consideration of relations with probabilistic models seems promising. There are many other interesting studies of clustering, but we omit them. Moreover numerical examples are also omitted due to space limitations. They are given in the cited literature. Acknowledgment. This work has partly been supported by the Grant-inAid for Scientific Research, Japan Society for the Promotion of Science, No. 23500269.

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3. Basu, S., Bilenko, M., Mooney, R.J.: A Probabilistic Framework for SemiSupervised Clustering. In: Proc. of the Tenth ACM SIGKDD (KDD 2004), pp. 59–68 (2004) 4. Basu, S., Banerjee, A., Mooney, R.J.: Active Semi-Supervision for Pairwise Constrained Clustering. In: Proc. of the SIAM International Conference on Data Mining (SDM 2004), pp. 333–344 (2004) 5. Basu, S., Davidson, I., Wagstaff, K.L. (eds.): Constrained Clustering: Advances in Algorithms, Theory, and Applications. Chapman & Hall/CRC, Boca Raton (2009) 6. Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press (1981) 7. Bezdek, J.C., Keller, J., Krishnapuram, R., Pal, N.R.: Fuzzy Models and Algorithms for Pattern Recognition and Image Processing. Kluwer, Boston (1999) 8. Bouchachia, A., Pedrycz, W.: A Semi-supervised Clustering Algorithm for Data Exploration. In: De Baets, B., Kaynak, O., Bilgi¸c, T. (eds.) IFSA 2003. LNCS (LNAI), vol. 2715, pp. 328–337. Springer, Heidelberg (2003) 9. Chapelle, O., Sch¨ olkopf, B., Zien, A. (eds.): Semi-Supervised Learning. MIT Press, Cambridge (2006) 10. Dav´e, R.N., Krishnapuram, R.: Robust Clustering Methods: A Unified View. IEEE Trans. on Fuzzy Systems 5(2), 270–293 (1997) 11. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum Likelihood from Incomplete Data via the EM Algorithm. J. R. Stat. Soc. B39, 1–38 (1977) 12. Dumitrescu, D., Lazzerini, B., Jain, L.C.: Fuzzy Sets and Their Application to Clustering and Training. CRC Press, Boca Raton (2000) 13. Duda, R.O., Hart, P.E.: Pattern Classification and Scene Analysis. John Wiley & Sons (1973) 14. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley, New York (2001) 15. Dunn, J.C.: A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-separated Clusters. J. of Cybernetics 3, 32–57 (1974) 16. Dunn, J.C.: Well-separated Clusters and Optimal Fuzzy Partitions. J. of Cybernetics 4, 95–104 (1974) 17. Endo, Y., Haruyama, H., Okubo, T.: On Some Hierarchical Clustering Algorithms Using Kernel Functions. In: Proc. of FUZZ-IEEE 2004, CD-ROM Proc., Budapest, Hungary, July 25-29, pp. 1–6 (2004) 18. Everitt, B.S.: Cluster Analysis, 3rd edn. Arnold, London (1993) 19. Girolami, M.: Mercer Kernel Based Clustering in Feature Space. IEEE Trans. on Neural Networks 13(3), 780–784 (2002) 20. Hashimoto, W., Nakamura, T., Miyamoto, S.: Comparison and Evaluation of Different Cluster Validity Measures Including Their Kernelization. Journal of Advanced Computational Intelligence and Intelligent Informatics 13(3), 204–209 (2009) 21. Hathaway, R.J., Bezdek, J.C.: Switching Regression Models and Fuzzy Clustering. IEEE Trans. on Fuzzy Systems 1, 195–204 (1993) 22. H¨ oppner, F., Klawonn, F., Kruse, R., Runkler, T.: Fuzzy Cluster Analysis. Wiley, Chichester (1999) 23. Hwang, J., Miyamoto, S.: Kernel Functions Derived from Fuzzy Clustering and Their Application to Kernel Fuzzy c-Means. Journal of Advanced Computational Intelligence and Intelligent Informatics 15(1), 90–94 (2011) 24. Ichihashi, H., Honda, K., Tani, N.: Gaussian Mixture PDF Approximation and Fuzzy c-Means Clustering with Entropy Regularization. In: Proc. of Fourth Asian Fuzzy Systems Symposium, vol. 1, pp. 217–221 (2000)

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25. Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York (1990) 26. Klein, D., Kamvar, S.D., Manning, C.: From Instance-level Constraints to Spacelevel Constraints: Making the Most of Prior Knowledge in Data Clustering. In: Proc. of the Intern. Conf. on Machine Learning, Sydney, Australia, pp. 307–314 (2002) 27. Kohonen, T.: Self-Organizing Maps, 2nd edn. Springer, Berlin (1997) 28. Krishnapuram, R., Keller, J.M.: A Possibilistic Approach to Clustering. IEEE Trans. on Fuzzy Systems 1, 98–110 (1993) 29. Kulis, B., Basu, S., Dhillon, I., Mooney, R.: Semi-supervised Graph Clustering: A Kernel Approach. Mach. Learn. 74, 1–22 (2009) 30. Li, R.P., Mukaidono, M.: A Maximum Entropy Approach to Fuzzy Clustering. In: Proc. of the 4th IEEE Intern. Conf. on Fuzzy Systems (FUZZ-IEEE/IFES 1995), Yokohama, Japan, March 20-24, pp. 2227–2232 (1995) 31. Li, R.P., Mukaidono, M.: Gaussian Clustering Method Based on Maximum-fuzzyentropy Interpretation. Fuzzy Sets and Systems 102, 253–258 (1999) 32. MacQueen, J.B.: Some Methods of Classification and Analysis of Multivariate Observations. In: Proc. of 5th Berkeley Symposium on Math. Stat. and Prob., pp. 281–297 (1967) 33. McLachlan, G., Peel, D.: Finite Mixture Models. Wiley, New York (2000) 34. Miyamoto, S.: Fuzzy Sets in Information Retrieval and Cluster Analysis. Kluwer, Dordrecht (1990) 35. Miyamoto, S., Mukaidono, M.: Fuzzy c-means as a Regularization and Maximum Entropy Approach. In: Proc. of the 7th International Fuzzy Systems Association World Congress (IFSA 1997), Prague, Czech, June 25-30, vol. II, pp. 86–92 (1997) 36. Miyamoto, S.: Introduction to Cluster Analysis, Morikita-Shuppan, Tokyo (1999) (in Japanese) 37. Miyamoto, S., Nakayama, Y.: Algorithms of Hard c-means Clustering Using Kernel Functions in Support Vector Machines. Journal of Advanced Computational Intelligence and Intelligent Informatics 7(1), 19–24 (2003) 38. Miyamoto, S., Suizu, D.: Fuzzy c-means Clustering Using Kernel Functions in Support Vector Machines. Journal of Advanced Computational Intelligence and Intelligent Informatics 7(1), 25–30 (2003) 39. Miyamoto, S., Suizu, D., Takata, O.: Methods of Fuzzy c-means and Possibilistic Clustering Using a Quadratic Term. Scientiae Mathematicae Japonicae 60(2), 217– 233 (2004) 40. Miyamoto, S., Ichihashi, H., Honda, K.: Algorithms for Fuzzy Clustering. Springer, Heidelberg (2008) 41. Miyamoto, S., Terami, A.: Semi-Supervised Agglomerative Hierarchical Clustering Algorithms with Pairwise Constraints. In: Proc. of WCCI 2010 IEEE World Congress on Computational Intelligence, CCIB, Barcelona, Spain, July 18-23, pp. 2796–2801 (2010) 42. Miyamoto, S., Terami, A.: Constrained Agglomerative Hierarchical Clustering Algorithms with Penalties. In: Proc. of 2011 IEEE International Conference on Fuzzy Systems, Taipei, Taiwan, June 27-30, pp. 422–427 (2011) 43. Redner, R.A., Walker, H.F.: Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM Review 26(2), 195–239 (1984) 44. Sch¨ olkopf, B., Smola, A.: Learning with Kernels. MIT Press (2002) 45. Sch¨ onberg, I.J.: Metric Spaces and Completely Monotone Functions. Annals of Mathematics 39(4), 811–841 (1938)

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46. Shental, N., Bar-Hillel, A., Hertz, T., Weinshall, D.: Computing Gaussian Mixture Models with EM Using Equivalence Constraints. In: Advances in Neural Information Processing Systems, vol. 16 (2004) 47. Vapnik, V.N.: Statistical Learning Theory. Wiley, New York (1998) 48. Vapnik, V.N.: The Nature of Statistical Learning Theory, 2nd edn. Springer, New York (2000) 49. Vapnik, V.N.: Transductive Inference and Semi-supervised Learning. In: Chapelle, O., et al. (eds.) Semi-Supervised Learning, pp. 453–472. MIT Press, Cambridge (2006) 50. Wagstaff, K., Cardie, C., Rogers, S., Schroedl, S.: Constrained K-means Clustering with Background Knowledge. In: Proc. of the 9th ICML, pp. 577–584 (2001) 51. Wang, N., Li, X., Luo, X.: Semi-supervised Kernel-based Fuzzy c-Means with Pairwise Constraints. In: Proc. of WCCI 2008, pp. 1099–1103 (2008) 52. Zhu, X., Goldberg, A.B.: Introduction to Semi-Supervised Learning. Morgan and Claypool (2009)

Fusing Conceptual Graphs and Fuzzy Logic: Towards the Structure and Expressiveness of Natural Language Tru H. Cao Ho Chi Minh City University of Technology and John von Neumann Institute, VNU-HCM 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam [email protected]

Extended Abstract Natural language is a principal and important means of human communication. It is used to express information as inputs to be processed by human brains then, very often, outputs are also expressed in natural language. The capacity for humans to communicate using language allows us to give, receive, and understand information expressed within a rich and flexible representational framework. Moreover, we can reason based on natural language expressions, and make decisions based on the information they convey, though this information usually involves imprecise terms and uncertain facts. How humans process information represented in natural language is still a challenge to science, in general, and to Artificial Intelligence, in particular. However, it is clear that, for a computer with the conventional processing paradigm to handle natural language, a formalism is required. For reasoning, it is desirable that such a formalism be a logical one. A logic for handling natural language should have not only a structure of formulas close to that of natural language sentences, but also a capability to deal with the semantics of vague linguistic terms pervasive in natural language expressions. Conceptual graphs (Sowa [2,3]) and fuzzy logic (Zadeh [7,8]) are two logical formalisms that emphasize the target of natural language, each of which is focused on one of the two mentioned desired features of a logic for handling natural language. While a smooth mapping between logic and natural language has been regarded as the main motivation of conceptual graphs (Sowa [4,5,6]), a methodology for computing with words has been regarded as the main contribution of fuzzy logic (Zadeh [9,10,11]). However, although conceptual graphs and fuzzy logic have the common target of natural language, until recently they were studied and developed quite separately. Their combination would be a great advantage towards a knowledge representation language that can approach the structure and expressiveness of natural language. At this juncture, conceptual graphs provide a syntactic structure for a smooth mapping to and from natural language, while fuzzy logic provides a semantic processor for approximate reasoning with words having vague meanings. This talk presents the combined result of an interdisciplinary research Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 31–32, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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programme focused on the integration of conceptual graphs and fuzzy logic, towards a knowledge representation language that is close to natural language in both of the structure and expressiveness (Cao [1]). First, the talk summarizes the development of fuzzy conceptual graphs and their logic programming foundations, as a graph-based order-sorted fuzzy set logic programming language for automated reasoning with fuzzy object attributes and types. Second, it presents the extension of fuzzy conceptual graphs with general quantifiers and direct reasoning operations on these extended conceptual graphs, which could be mapped to and from generally quantified natural language statements. Third, it introduces recent applications of fuzzy conceptual graphs for understanding natural language queries and semantic search.

References 1. Cao, T.H.: Conceptual Graphs and Fuzzy Logic: A Fusion for Representing and Reasoning with Linguistic Information. Springer, Berlin (2010) 2. Sowa, J.F.: Conceptual graphs for a data base interface. IBM Journal of Research and Development 20(4), 336–357 (1976) 3. Sowa, J.F.: Conceptual Structures - Information Processing in Mind and Machine. Addison-Wesley Publishing Company, Massachusetts (1984) 4. Sowa, J.F.: Towards the expressive power of natural language. In: Sowa, J.F. (ed.) Principles of Semantic Networks - Explorations in the Representation of Knowledge, pp. 157–189. Morgan Kaufmann Publishers, San Mateo (1991) 5. Sowa, J.F.: Matching logical structure to linguistic structure. In: Houser, N., Roberts, D.D., Van Evra, J. (eds.) Studies in the Logic of Charles Sanders Peirce, pp. 418–444. Indiana University Press, Bloomington (1997) 6. Sowa, J.F.: Conceptual graphs. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation, pp. 213–237. Elsevier (2008) 7. Zadeh, L.A.: Fuzzy sets. Journal of Information and Control 8, 338–353 (1965) 8. Zadeh, L.A.: Fuzzy logic and approximate reasoning (In memory of Grigore Moisil). Synthese 30, 407–428 (1975) 9. Zadeh, L.A.: PRUF - a meaning representation language for natural languages. International Journal of Man-Machine Studies 10, 395–460 (1978) 10. Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems 4, 103–111 (1996) 11. Zadeh, L.A.: Precisiated natural language. AI Magazine 25, 74–91 (2004)

A MMORPG Decision-Making Model Based on Persuasive Reciprocity Helio C. Silva Neto, Leonardo F.B.S. Carvalho, F´ abio Paragua¸cu, and Roberta V.V. Lopes Federal University of Alagoas (UFAL - Universidade Federal de Alagoas), Institute of Computing (IC - Instituto de Computa¸c˜ ao), Campus A. C. Sim˜ oes, 57072-970 Macei´ o, Alagoas, Brazil {lfilipebsc,helio.hx,fabioparagua2000}@gmail.com,

Abstract. From a videogame perspective, decision-making is a crucial activity that happens at all times and at different levels of perception. Moreover, it has direct influence over a game’s performance, a fact that also concerns to RPGs as they can act as tools to enhance the improvement of the proximal development zones of the involved individuals. As the RPG has an inherent cooperative character that stimulates socialization, interaction and the improvement of communication skills, it was thought that it would be interesting to take advantage of the RPG to build a model using Petri Net, based on the Game Theory and on a application of the Theory of Persuasion to an MMORPG environment, that involves the user in some kind of plot, at the same time that favors a greater ease in decision-making. Keywords: Psychology of Persuasion, Decision Making Systems, MMORPG, RPG, Petri Net and Game Theory.

1

Introduction

It is common sense that, at all times, people have to make decisions concerning different situations and problems. At those times, they are likely to need to use past experiences, values, beliefs, knowledge or even their technical skills. In addition to it, while some people are more conservative in their actions, others will have an innovative character and will be more than willing to accept potential risks. All these assumptions are valid during a decision-making process and will represent either the success or failure of those depending on it [8]. In a game context the making of decisions is a crucial process that has direct influence on any player performance, as it is a process that is being done at all times and at different levels of importance and awareness of player. This statement clears the importance of the decision-making process in games, making it impossible to think of games without considering the constant occurrence decision-making. In fact, the decision-making can be seen even in an empirical level by simple observation of a game match [8]. To ease this process, the work presented here employs two particular methods. First, this paper uses the Game Theory that consists of models for the analyses Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 33–47, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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of conflicts and operations that are reliant on strategic behaviors in which; a player’s actions are partly dependent on the actions of other players [5][6]. Lastly and perhaps taking a more prominent approach, this paper applies the Principles of Persuasion to provide support to decision-making process, due to its emphasis of using communication skills to change attitudes, beliefs and even the behavior of other people, thus, preventing the use of force or coercion. In order words, a person who persuades others makes them accept a particular idea [3]. As to encourage players to build their own strategies and decisions for pursing their goals (i.e. to build their own knowledge), this paper presents a model that couples the two concepts described in the above paragraph for implementation in a Role Playing Game (RPG) environment. Therefore, this paper aims to build a decision-making model founded on the Reciprocity concept from the Principles of Persuasion for use on MMORPG environments and in that, taking advantage of an entertainment environment that favors the interaction of several players in order to apply concepts taken from the Principles of Persuasion and from the Game Theory to aid these players to build their own knowledge. To present this, this paper is divided into different sections. First Sec. 2 will discuss the MMORPG genre and detail some of its concepts, including those native from the more broad RPG genre and its tabletop games. Section 3 will focus on the Principles of Persuasion, particularly the principle of Reciprocity while Sec. 4 will explain the concepts and properties of decision-making systems in addition to the real conditions for making a decision. Section 4 will present the authors’ model built using a Petri Net and an analysis of its application the Principles of Persuasion, in addition to emphasize the importance of the use of the Game Theory in this model. Last, Sec. 6 will draw the paper conclusions.

2

Massively Multiplayer Online Role-Playing Game (MMORPG)

The acronym RPG that stands for Role Playing Game was first appointed by Dave Ameson in the USA at 1973. RPGs consist mainly of situations described by speech representation that rely on the imagination of players engaged in a game session. The described situations are often some kind of adventure that players take part in and that, commonly, was started at an interrupted previous session [10][11][2]. As occurs in many activities the RPG has its own language to refer to specific activities or occupations. The storyteller for example is known as master, while the ones listening and participating in the story told by the master are the players. The story itself is called an adventure. The list bellow describes the basic concepts of the RPG as is given by Debbie [7]: – Player: the individuals in charge of one or more characters of the plot (each known as PC, player character). A player has freedom of action in a game scenario, provided it meets the game’s system of rules;

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– Game Master: controls all factors concerning the game settings and plot that do not involve the actions of PCs. It is a game master responsibility to control plot characters (Non-Player Character, NPC) interacting with PCs. It is also the one responsible for controlling the game settings, being able to adjust the plot of the game according to what is needed. Additionally, the game master is also the one in charge of the plot progress and its secret objectives. A game master is not above the game’s system of rules and must obey to it, but, has the power to change things within a reasonable logic if it would benefit the game plot. – System of Rules: any action taken by PC must be addressed to the game master, who in turn verifies the game system of rules to inform the player of the results of this action under the circumstances it was taken. Therefore, there are specific rules that are applied for each different situation, as well as specific indications that must be accounted when dealing with unexpected situations. It is also common the use of a random element provided (provided by a dice) to simulate the real world uncertainty about the result of an action; – Scenario: the world where PCs are geographically and temporally located (or in some cases an astral plane or different dimension in which PC might be). Scenarios are often defined by players and the game master and chosen due to its ability to contain interesting characters and game plot. It might employ an everyday reality or a completely strange one; – Characters: the PCs. The characters are the imaginary projection of players within the game scenario. They can be built by their respective players or be provided by the game. In any case, they must respect the game system of rules and scenario. Depending of the plot some characters might be more or less attractive to players. Each PC is assign with abilities that define how it interacts with the environment and reflect the players interest in building a particular kind of character; – Plot: the game story and reason why PC have come together in some kind of adventure or action. Usually, PCs will act in accordance to the plot, meeting people and finding objects and places that are important to the progress of the plot. For this, players have no need to be sure of what is their goal or who is the one within the story that is taking the important decisions; – NPCs (Non Player character): common RPG term that indicates a character that is not controlled by any player. Thus, belonging to the game master. NPCs are usually supporting characters for the adventure; The launch of computer RPGs that allow multiplayer game mode, which allows multiple users to play via LAN, modem or the Internet, made an additional set of characteristics a common occurrence in RPGs of this instance, including: – Multiplayer interaction; – Exploration of wide worlds granted with large locations; – Existence of several sub-plots that allow players to create their own story and adventures;

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– A greater similarity with tabletop RPGs due to the possibility of creating and evolving player characters; – A large majority of these games allows users to customize their main characters, as for example, by creating adventures, items, weapons or worlds for them. An additional main characteristic of MMORPGs is the constant intervention of a team of Game Masters, whether they are NPCs or real-world human developers that work on plots and create challenges for PCs. Moreover, the plots are nonlinear and thus do not require a beginning, a middle or even an end. The real concept of this is that what exists is an open story that is presented as a virtual world to be explored. Lastly, as the genre name suggests, the idea behind a MMORPG is that of a massively multiplayer role playing game that allows for thousands of online players to coexist and interact in the game’s “virtual” world.

3

The Theory of Persuasion

According to Robert Cialdini [3] persuasion is related to the use of communication to change the attitudes, beliefs or behaviors of other people. However, this change must occur voluntarily and not through the use of force or any means of coercion. Therefore, the person using of persuasion leads others to the acceptance of a particular idea by convincing them of it. The persuasive speech seeks to represent “the whole truth” by using of linguistic resources selected as expressions of a “truth” that introduce an overlapping of one’s previous assumptions. In this sense, the ultimate goal of persuasive speech is to employ rhetorical devices to “convince or change already established attitudes or behaviors” [4]. 3.1

Reciprocity

The principle of reciprocity is one of the Principles Persuasion and is related to the meaning of a passed down obligation. It is a ubiquitous concept of human culture as states sociologists such as Alvin Gouldner [9] who affirms that, there is no human society that relinquishes to this rule. Moreover, the archaeologist Richard Leakey [13] goes as far as to attribute the concept of reciprocity to the very essence of what makes us humans: “We are human because our ancestors learned to share their food and their skills in a community network”. Competitive human societies follow the principle of reciprocity and as such expect all its members to respect and believe in this principle. Each person of these societies was taught to live by the rule connected to this principle and for this reason every one of their members knows about the social sanctions and scorn applied on those who violate it. Additionally, people who do not attend

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to the statements of the principle of reciprocity are assigned with labels such as ungrateful or deadbeat. These labels represent the discontentment of society in being faced with members who receive a “favor” and make no effort to repay it. One of the reasons for the effectiveness of the reciprocity principle is the fact that its rule is imbued with a force that, in the existence of a previous “debit” often produces a “yes” response to a request that otherwise would surely be denied. Therefore, albeit plain in appearance, the strategies connected to the principle of reciprocity are extremely efficient and almost undetectable. As a consequence, the best solution for fighting these strategies is to think before accepting any favor or concession from someone whose true intentions are not known. It is important to note that, although the mechanisms connected to the principle of reciprocity might recall those of a standard practice for exploitation by using of manipulation and influence, this principle is, in fact, a fundamental pillar of human societies and one of the reasons for the development of the very earliest human communities.

4

Decision-Making Systems

As the name suggests, decision-making systems are focused on helping people make decisions about different sorts of matter. In order to understand how these systems operate, it is necessary to know which are the elements that might lead people to make their decisions and the context in which they fit. These elements can be better perceived by looking the decision-making process from the standpoint of business management, a perspective that according to Batista [1] clears to key elements of decision-making: the information channels, responsible for defining the source for acquiring data; and communication networks that define where data should be sent. In addition, a company must have for its system what is known as a base of knowledge, which contains the sets of data, rules and relationships that its decision-making process must account to achieve a suitable result or informational value [17]. Thus, the knowledge base has the task of facilitating the reorganization of the data and information useful for achieving the objectives of an organization. A particular important feature considering that nowadays, the success of any business is connected to the speed with which it is able to assimilate information as well as the speed with which it is able to take decisions. As a final note, although it may be intuitive that the success described above is the result of a process that has its origins in the fundamental components of Information Technology, it must be clear that the context that the term “knowledge” takes here corresponds to that given by Laudon and Laudon [12]. Therefore, knowledge is seen here as the set of the conceptual tools and categories that humans use to create, collect store and share information; an observation that must be taken into account in the following sections of this paper.

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Information Systems

Information systems are designed to generate information for decision-making processes by collecting, processing and transforming data into information. Stair [17] states that information systems can be understood as a series of interconnected elements or components that collect (input), manipulate and store (process), and disseminate (output) data and information, in addition to provide a feedback mechanism for this process. To ensure the efficiency of these systems, Pereira and Fonseca [14] state that the following set o requirements must be fulfilled. – Address the real needs of users; – Keep the focus on end users (customers) instead of the professionals who created it (Developers); – Proper Customer Care; – Present compatible costs; – Constantly adapt to new information technologies; – Comply with the company’s business strategies. 4.2

Conditions for Decision-Making

The making of decisions can occur in different kinds of situations and might even involve conditions of uncertainty or risk. Additionally, previously planned decisions often present lower risk than unplanned ones. As pointed out by Stephen and Coulter [15] the ideal situation for decisionmaking is the certainty, which represents a situation that has known results for each possible alternative, thus enabling accurate decision-making in every circumstance. Moreover, when a decision includes a risk element all alternatives are designed under known probabilities and have a specific result, thus, the decision-maker knows all alternatives and is aware that the risk is unavoidable. As the above paragraph implies, the decision-making process is connected to a company’s potential of acquiring information from its Information System. In turn, the Information System of a company must provide information that is as useful as possible for the company’s needs, in order to better assist its end users in managing the business.

5

A Decision-Making Model Based on the Principle of Reciprocity from the Theory of Persuasion

Know that the concepts of the principle of Reciprocity and of the decision-making systems have been explained, this section will now present how these concepts were arranged into a model that fits the context of MMORPGs. 5.1

Circumstances for Reciprocity in MMORPGs

There are many circumstances in a MMORPG game environment for which the principle of reciprocity might be applicable, a diversity that is justified by the

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way that this genre of games combines real-world elements with those of a surreal one in a single environment in which players are able to interact with. The most common of these circumstances are: – To provide aid for someone hunting monsters in order to either increase level, locate an item or even complete a Quest1 . In such cases, the one who is receiving help will be in debt to the ones providing it and promote an exchange of knowledge; – More often than not, a MMORPG has a forum where players can interact and share their game knowledge. Therefore, by receiving tips or advices via forum one would also be in debt to the ones who provided the information or helped in some way; – To provide discount when selling some item, particularly in such cases where the said discount does not normally exists is likely to attract buyers (especially if the seller announces the discount). Unlike the previous cases, these situations might require a little “push” from sellers, or might be bounded to the acquisition of certain X amount or value of an item, thus resulting in an N discount. The same practice applies for promotions and contests; – Players that regularly buy items from a same player might get discounts by creating a bond of loyalty. Reciprocity Principle and the Flow Module Modeled in Petri Net. The principle of reciprocity acts as an exchange of favors, thus, it consists in sharing that which was received. Some of the ways to activate it in a MMORPG environment were mentioned in the previous section and were modeled according to the depiction of Fig. 1, which clearly expresses the operational flow between those concepts in a MMORPG environment. According to the flow depicted in Fig. 1 the principle of reciprocity can be activated when a player asks another for help with simple tasks such as locating an item, completing a Quest or hunting a monster. MMORPGs tend to provide tools that enable players to realize such interactions, such as to create a Party 2 or Trade 3 an item. As a consequence, the depicted flow can be started at the desire of any player that seeks reciprocity for any of its tasks, such as Help in Finding Item, Help in Hunting, Help in Quest. In this sense, should help mean assistance in finding an item that another player already has, the Trade task could be activated to perform the transference of the said item. If not, a Party could be put together to search for the desired item, or to attend any of the flow purposes. Additionally, a Party allow its members to share money among them as well as the experience points the game provides as reward for their activities. The Discount on Shopping task is a circumstance in which players apply the reciprocity principle to attain a loyalty bond from other players or a discount on 1 2 3

In RPGs the term quest denotes a mission or a purpose that must be accomplished. A term employed by RPGs for creating a hunting group or task force. A MMORPG tool that allows two players to exchange or sell items between them.

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Fig. 1. Flow of Reciprocity

a large amount of objects being negotiated. Once the items and values are set, players can use the Trade tool to transfer the intended goods. The Give Tips on Forum task is triggered when players access the virtual platform, i.e. the forum, and perform questions or searches about desired subjects. After achieving their intents the ones performing the research are able to approve the response given by other members in order to facilitate future searches on the same subject and to motivate other players to improve their reputations. When a player uses all tools contained in this principle flow the player arrives at the Goal Achieved task. As the task name suggests, it indicates that the player has reached the completion of the intended goal or mission. At this point, the player chooses whether to take a passive stand and repay the community who helped him/her and, in that, creating a cyclical movement for the principle of reciprocity, or, might choose to not repay the other players and by thus, breaking the cycle of the principle of reciprocity. Nevertheless, by choosing this last option the player is aware of the penalties that one might suffer for breaking it. In addition to the flow of 1, 2 presents the model for the principle of reciprocity using a Petri Net that allows a proper visualization of the model, including the model locations (states), transitions and set of guide arcs. The formal definition for this Petri Net as m1 as its initial marking, however, even thought it adopts a initial marking for the net, the model accepts the creation of a cyclical movement for its network, which might be extended to all other principles of persuasion. In turn, this movement can only be interrupted if the player taking the active role in the Reciprocity Principle of the Theory of Persuasion decides to not return

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that what was shared with him/her. Aside from that, the designed network also contains several final locations (final states) that are used to identify the amount of players that completed the full cycle of the reciprocity principle and the amount of players that did not. The locations and variables of this network are: – Variable m1 - Start of Reciprocity: this is the initial state of the model and corresponds to a player using the Reciprocity Principle to achieve a desired goal. Afterwards, the player activates the Requesting Support transition to change the current state of the net. The requesting player may only proceed with the desired request if a supporting player becomes available; – Variable m2 - Supporter Available: at this state, a supporting player waits for requests coming from the player who activated the Reciprocity cycle. The state also verifies whether or not a supporting player for the requested issue is available. If there is no available player to act as a passive intermediary for the reciprocity principle the state triggers a transition to Supporter Unavailable. However, if there is a supporting player available the Requesting Support transition is triggered; – Variable m3 - Waiting for Support Request : this state occurs while the active player of the principle of reciprocity is choosing what he/she whishes to ask the passive players of this persuasion principle. The Petri Net has redundant paths emerging of this state as the net itself was developed to model parallel, concurrent, asynchronous and non-deterministic systems. The reason for this is that the net was modeled after human activities. Thus, the choice of route to take depends only on the active player, who might choose either the Buy Item, Request Help or Ask in Forum transition; – Variable m5 - Waiting for Discount : an active player reaches this state after triggering the Buy Item transition. At this location, the player takes advantage of a discount due the fact of being an active costumer; be buying a large amount of items and so forth. In turn, the passive player will ensure the completion of the sale by an exchange of favors and by triggering the Discount on Item transition; – Variable m4 - Waiting for the Order : a passive player can grant a discount only in those cases in which an item is properly requested. If there is no such request, them the No Request for Purchasing Item transition is triggered. Otherwise, the item is addressed to the Discount on Item transition where the item value will be negotiated; – Variable m8 - True to Purchase: this state is attained after the value of a product is set and a reliability bond is created between the involved players. At this state either the Request Trade or Purchase Rejected transitions might be triggered; – Variable m11 - Negative Reputation for Requester by Purchase: it is one of the final states of the modeled Petri Net and is activated when a player assuming the active role of the reciprocity principle decides to not proceed with the purchase of an item. This decision creates a bond of unreliability between the involved players. At this case, as the player is breaking the

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reciprocity cycle a penalty might be applied to him/her that will remove points of his/her reputation. Even though this situation is undesirable for player as well as for the intents of this paper it may occur under these particular circumstances; Variable m9 - Waiting for Available Item: as implied by its name, this state concern the availability of an item owned by a passive player who may negotiate it. Moreover, in case the passive player does not currently have the item being negotiated the Unavailable Item transition is triggered and this condition persists until the passive player takes possession of the item in order to sell it. When this happens, the Request Trade transition is triggered; Variable m10 - Trade: is the state where the actual selling of an item occurs. Once an item becomes available for selling and its value is set the Approve Trade transition is triggered in order to check the goods being transferred, items and values alike; Variable m12 - Purchase Accomplished: it is the state where the player completes a purchase and that triggers the Score Reputation transition; Variable m13 - Score Reputation by Sale: the model final state for the analyses of the amount of sale a player accomplished by using the principle of reciprocity, thus granting the player a better reputation on future sales; Variable m14 - Score Reputation by Purchase: this state works similarly to the previous one, but focus the buyer instead of the seller; Variable m25 - Goal of Reciprocity Achieved : the principle of reciprocity does not always acts as a cycle, thus it is possible to attain a final objective. In fact, in order for this to happen a player needs only to provide a requesting player the assistance he/she needs, thus, taking the passive role of a seller or adviser for example and in that, assuming a position within the principle of reciprocity that does not place him/her in debit with someone else. This situation fits into the context of the model, while still employing the principle of reciprocity. With that in mind, it is up to the player on the active role of the principle to whether or not trigger the Activation of Reciprocity transition in order to repay the received aid; Variable m26 - Waiting for the Choice of a Requester : this states stands for a “waiting room” where players are placed while they decide what action to take. To trigger the Chooses to Not Repay or the Chooses to Repay transition; Variable m27 - Negative Reputation for Requester : it is a final state of the network that occurs when the active player decides to not return that what was shared with him/her and thus, acquires a negative score on his/her reputation; Variable m28 - Reputation for the Requester : there are two parallel routs that a player may take at the Chooses To Repay state, one of which is the Reputation for the Requester state. Here, a positive reputation is assigned to the active player, a fact that will benefit him/her on future activations of the principle of reciprocity. The other option is the Waiting Completion of Reciprocity Cycle state;

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– Variable m29 - Waiting Completion of Reciprocity Cycle: this state can be achieved when a player triggers the Chooses To Repay transition. This state poses as a “waiting room” where a player waits the activation of the Completion of Reciprocity Cycle transition;

Fig. 2. Reciprocity in Petri Net

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– Variable m30 - Waiting Start of Reciprocity: this state indicates the moment at which the Reciprocity principle starts to act. This instant effectively marks the network start and triggers the m1 and m2 states; – Variable m6 - Waiting for Help: the state that occurs when the player at the active role triggers the Request Help transition. At this state a player might trigger Request Party transition if there is another player who wants to offer help; – Variable m15 - Waiting for Available Party: the objective of this state is to assert whether or not a party assistance is available. If there is no party available, the Party Unavailable transition is triggered. Otherwise, the Request Party transition is triggered; – Variable m16 - Party: a state that occurs when players intending to assist each other create a single group of individuals. Such groups are known in RPGs as a party and in this model are created using of the homonymous Party tool. Depending of the conduct of the players in a same party the Assistance in Progress or the Party Rejected transitions might be triggered. The last of these two indicates the disbanding of the group, in general, due to a lack of commitment to provide assistance, whether it comes from the group or from the player who requested the assistance; – Variable m17 - Negative Reputation for the Requested : in case the group or the passive player asks for the disbanding of the Party, the reputation of one or several of those involved may receive a negative score as penalty; – Variable m18 - Assistance Provided : this state is the end of the helping process and is reached when the group achieves its goal. This state triggers the Score Reputation transition; – Variable m19 - Score Reputation by Assistance: grants the player with a score, which is set in respect to a previously provided assistance; – Variable m20 - Score Reputation for the Assisted : similar to the m14 state, though this final state sets a score for the player who requested the assistance (the one who received the aid). The transition that triggers this state might also trigger the m25 state, which in turn, keeps the cycle of the network; – Variable m7 - Waiting for a Reply: a state located at a different path of the network that an active player might access via the Ask in Forum transition. At this state the active player waits for the answer of a passive player. After this, the active player may chose to activate the Assert Answer transition. In case there is no given answer, the No Given Answers transition is triggered; – Variable m21 - Waiting for a Question: this state is achieved when passive players are waiting for a question that they can answer. In case no question shows up the No Given Questions state is triggered. However, after question is made and answered it is up to the active player to decide whether or not to trigger the Assert Answer transition; – Variable m22 - Tip Provided : this state stands as the consequence of asserting an answer. Thus, it follows the Assert Answer transition. This state triggers the Score Reputation transition;

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– Variable m23 - Score Reputation by Tip: a state that grants a player with a score that is based on active participation at the forums. Particularly, by providing requesting players with the information they require; – Variable m24 - Score Reputation by Question: similar to the state above. However, it grants a value to the reputation of the player who asks the question instead of the one who answers it. This state is also triggered by the Score Reputation transition, which in turn, may also lead to the m25 state in order to keep its movement. However, a Petri Net requires one more variable in addition to the above ones. This variable is given by the net weight function and is responsible for launching the network. Thus, a minimal weight value is required to effectively start the Petri Net. For this paper, the weight function corresponds to the strategy of each player, and it must also account that each player is likely to have several strategies. In order to develop those strategies, this paper also adopts some of the equations provided by the Game Theory. The scope of this theory that fits the discussion of this paper will be explained now in order to provide a better understanding of its equations. 5.2

Game Theory

The Game Theory is a mathematical theory designed to model the phenomena that can be observed when two or more “decision agents” interact. This theory also provides the language for discretion of conscious decision-making processes and for goals involving more than one individual [5][6]. When applying the principles of the Game Theory to the model proposed in this paper what it is expected to achieve is the study of the choice of the optimal decision for the activation of the Reciprocity Principle under conditions of conflict. The set of engaged players are considered the basic game elements for these circumstances. Moreover, as each player has its own set of strategies, every time a player chooses a desired strategy a new profile is created at the space that comprises all possible situations. Each of these situations also corresponds to a profile, due to every player having different interests that are focused on the outcome of different circumstances. Mathematically, it is said that each player has its own utility function that assigns a real number (the gain of the player) to every game situation [16]. With the above said, for the model proposed in this paper, a game has the following basic elements: a finite set of players given as G = {g1 , g1 , . . . , gn }, with each gi ∈ G player having a finite set of options that is given as Si = {si1 , si2 , . . . , sim } and are referred as a pure strategy for the gi (mi ≥ 2) player; a s = {s1j1 , s2j2 , . . . , snjn } vector in which siji is as a pure strategy for the gi ∈ G player and that is named a profile of pure strategies. The set of all pure strategies profiles known as the game’s space of pure strategies, which that is given by the

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Cartesian product [16] shown at 1. And finally, an utility function (shown at 2) that exists for each gi ∈ G player and that links its ui (s) gain to the appropriate s ∈ S profile of pure strategy [16]. S=

n 

Si = S1 × S2 × . . . × SN

(1)

i=1

ui :

S→R s → ui (s)

(2)

The understanding of the above principles and its mathematical functions are important for this paper because, according to the proposed model, they are what gives each player the possibility of choosing the best strategy for applying the Principle of Reciprocity and thus, to trigger it at the appropriate game moment.

6

Conclusion

There are currently few people aware of the importance of applying any of the principles within the Theory of Persuasion to the decision-making processes. In fact, there are so few researches that, during the development of this work, there were found no decision-making models using of the artifices provided by the Theory of Persuasion and thus, games are still very incipient to the use of this theory. The research for this paper also made clear that implementing the Theory of Persuasion as a decision-making tool for MMORPGs’ environments changes the way players deal with information (knowledge) as each of them try to create its own best strategy and also, as each of them tries to figure out how their own strategies relate to the interests (strategies) of other players. However, the research presented at this paper is not concluded as it can be assumed that there is still much work left to do. One directed example of this is related to the fact that, until there is no real commercial interest in developing such game environments for educational purposes or any other field of interest, the initiative showed here will be no more than a mere academic project that may never come to fruition. To change the above reality it is necessary that the ones least aware of the benefits brought by the Theory of Persuasion start to comprehend it and thus, work in the creation of environments that understand the needs of players and that encourages them to develop new strategies for gaining knowledge, including, obtaining it by learning from other players, thus, effectively transforming the game environment in an educational partner. In order to fit the perspective described above, this paper focus on the real benefits that the Theory of Persuasion (particularly its Principle of Reciprocity) may bring to the development of a decision-making environment, demonstrating its advantages by using of an existing game environment that is also used to model its architecture in accordance to the real needs of players (a problem

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that is present in conventional playable environments). A work that was done in order to demonstrate the authors’ belief that, the application of decision-making systems to MMORPGs environments that combine the principles and theories presented throughout the paper are able to accord with the settled conditions for the proposed model. Hence, giving support to games that apply this model by providing players with tools to better build their own knowledge. Therefore, the Principle of Reciprocity from the Theory of Persuasion that was discussed here will contribute in creating decision-making systems as it will transport decisions taken upon conditions of uncertainty to the ensuring scenario of conditions of certainty, trough the use of the persuasive factors contained in this principle.

References 1. Batista, E.d.O.: Information System: the conscious use of technology for management (in pt-BR). Saraiva (2004) 2. Cale, C.: The real truth about dungeons & dragons (2002), http://www.cale.com/paper.htm (last access in 10/06/2011) 3. Cialdini, R.B.: Influence: the psychology of persuasion. Harperbusiness Essentials. Collins (2007) 4. Citelli, A.: Language and persuasion (in pt-BR), 2nd edn. 5. Conway, J.: All games brigth and beautiful. The American Mathematical Monthly, 417–434 (1977) 6. Conway, J.: The gamut of game and theories. Mathematics Magazine, 5–12 (1978) 7. Debbio, M.: Arkanun (in pt-BR), 2nd edn. DAEMON EDITORA LTDA (1998) 8. Garcia, E., Garcia, O.P.: The importance of management information system for business administration (in pt-br). Social Science in Perspective Magazine, 21–32 (2003) 9. Gouldner, A.W.: The norm of reciprocity: A preliminary statement. American Sociological Review, 161–178 (1960) 10. Hughes, J.: Therapy is fantasy: Roleplaying, healing and the construction of symbolic order (1988) 11. Jackson, S.: GURPS’ RPG Basic Module (in pt-BR). Devir (1994) 12. Laudon, K.C., Laudon, J.P.: Essentials of Management Information Systems. Prentice Hall (1999) 13. Leakey, R.: People of the Lake. Anchor Press/Doubleday (1978) 14. Pereira, M.J.L.B., Fonseca, J.G.M.: Faces of Decision: the paradigm shifts and the power of decision (in pt-BR). Makron Books (1997) 15. Robbie., S.P., Coulter, M.: Management. Prentice Hall (1996) 16. Sartini, B.A., et al.: A introduction to game theory (in pt-br) (Introduction to Game Theory) 17. Stair, R.: Principles of Information Systems: A Managerial Approach, Test Item File. Course Technology (1995)

A Computing with Words Based Approach to Multicriteria Energy Planning Hong-Bin Yan1 , Tieju Ma1 , Yoshiteru Nakamori2 , and Van-Nam Huynh2 1

2

School of Business, East China University of Science and Technology Meilong Road 130, Shanghai 200237, P.R. China [email protected],[email protected] School of Knowledge Science, Japan Advanced Institute of Science and Technology 1-1 Asahidai, Nomi City, Ishikawa, 923-1292, Japan {huynh,nakamori}@jaist.ac.jp

Abstract. Exploitation of new and innovative energy alternatives is a key means towards a sustainable energy system. This paper proposes a linguistic energy planning model with computation solely on words as well as considering the policy-maker’s preference information. To do so, a probabilistic approach is first proposed to derive the underlying semantic overlapping of linguistic labels from their associated fuzzy membership functions. Second, a satisfactory-oriented choice function is proposed to incorporate the policy-maker’s preference information. Third, our model is extended to multicriteria case with linguistic importance weights. One example, borrowed from the literature, is used to show the effectiveness and advantages of our model.

1

Introduction

Efforts toward a sustainable energy system are progressively becoming an issue of universal concern and decision-makers. The objective of an energy policy toward a sustainable energy system includes efficient production, distribution and use of energy resources and provision of equitable and affordable access to energy while ensuring security of energy supply and environmental sustainability. Consequently, exploitation of sustainable or new energy alternatives (also called energy planning) is a key means of satisfying these objectives and has gained a great interest during the last decade. Energy planning endeavor involves finding a set of sources and conversion devices in order to meet the energy requirements of all the tasks in an optimal manner [4]. Making an energy planning decision involves a process of balancing diverse ecological, social, technical, and economic aspects over space and time. The complexity of energy planning and energy policy in particular makes multicriteria decision analysis (MCDA) a valuable tool in the energy planning process [6]; providing the flexibility and capacity to assess the energy supply alternatives’ implications to economy, environment and social framework. In particular, the concept of MCDM has been widely used for the design of energy and environmental policies as well as for sustainable energy planning [7,12]. However, the Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 48–59, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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assessment of innovative energy alternatives through a number of criteria is a complex and time-consuming task, since the analysis has to face a series of uncertainties such as fossil fuel price, environmental regulations, market structure, technological, and demand and supply uncertainty [11]. In addition, sustainability is an inherently vague and complex concept and the implications of sustainability development as a policy objective is difficult to be define or measured. In view of these difficulties, fuzzy set theory [13] offers a formal mathematical framework for energy planning in order to resolve the vagueness, ambiguity and subjectivity of human judgment. A realistic approach to solving such situation is the use of fuzzy linguistic approach [14], which deals with linguistic information that is represented in linguistic labels. Such a problem will be referred to as linguistic energy planning, the process of which usually creates the need for computing with words. In this context, a lot of fuzzy linguistic models have been proposed/applied for energy planning problems, such as fuzzy linguistic AHP based model [2] and fuzzy TOPSIS based model [1,10]. These fuzzy models can efficiently deal with both fuzzy uncertainty and complexity in energy planning problems. Unfortunately, it simultaneously has an unavoidable limitation of the loss of information [3], which consequently implies a lack of precision in the final result. In this sense, direct computation on linguistic variables for energy planning can be considered as a direct and adequate framework [15]. To the best of our knowledge, little attention has been paid to multicriteria approaches with direct computation on linguistic variables for the evaluation of energy alternatives. Furthermore, our another motivation comes from the fact that the experts are not necessarily the decision-makers, but only provide an advice [8]. In energy planning, the experts can be people familiar with a particular energy supply and energy problem domain; whereas the real decision-maker in energy planning is the policy-maker, whose preference information plays an important role in choice of energy supply alternatives and is missed in most research. The main focus of this paper is to propose and develop a linguistic energy planning framework with computation solely based on linguistic labels as well as considering the policy-maker’s preference information. To do so, Section 2 presents some basic knowledge of energy planning and formulates the research problems. Section 3 formulates our single criterion linguistic energy planning, which is able to capture the underlying semantic overlapping of experts’ judgments as well as the policy-maker’s target preference information. Section 4 extends single criterion to multicriteria case with linguistic weight information. Section 5 borrows one example from the literature to illustrate the effectiveness and advantages of our model. Finally, this paper is concluded in Section 6.

2 2.1

The Linguistic Assessment-Based Framework Selecting Energy Alternatives and Evaluation Criteria

Most popular energy supply resources or technologies are the alternatives based on solar energy (photovoltaic and thermal), wind energy, hydraulic energy,

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biomass, animal manure, combined heat and power, and ocean energy. Despite environmental drawbacks, nuclear and conventional energy alternatives like coal, oil and natural gas may be still included in the list of energy alternatives to be promoted. Let A = {A1 , A2 , . . . , AM } be a set of energy alternatives. The decision-making process to determine the best energy policy is multidimensional, made up of a number of aspects such as economic, technical, environmental, political, and social. It has to be noted that the criteria and the performances are dependent on the specific problem’s formulations and particularly on the country’s specific energy characteristics, its development needs and perspectives and the energy actors, and interests. Formally, the evaluation criteria are expressed as C = {C1 , C2 , . . . , CN }. 2.2

Selecting Linguistic Label Set and Their Semantics

Essentially, in any linguistic approach to solving a problem, label sets of involved linguistic variables and their associated semantics must defined first to supply users with an instrument by which they can naturally express their judgments. The concept of a linguistic variable was first introduced by [14] to model how words or labels can represent vague concepts in natural language. Definition 1. A linguistic variable is a quadruple < L, T(L), Ω, M > where L is the variable name, T(L) is the set of labels or words (i.e. the linguistic values of L), Ω is a universe of discourse, and M is the semantic rule which associates with each linguistic label its meaning. The semantic rule M is defined as a function that associates a normalized fuzzy subset of Ω with each word in T(L). We only consider the linguistic variable with a finite ordered label set L = {L0 , L1 , . . . , LG } on a continuous discourse Ω. In order to accomplish the objective of choosing the appropriate linguistic descriptors and their semantics for the label set of a linguistic variable, an important aspect needed to analyze is the granularity of uncertainty, i.e., the level of discrimination among different counts of uncertainty. Moreover, the label set must have the following characteristics: – The set presents a total order: Lg1 ≥ Lg2 if g1 ≥ g2 . – There is the negation operator: Neg(Lg1 ) = Lg2 such that g2 = G − g1 . – max(Lg1 , Lg2 ) = Lg1 if g1 ≥ g2 ; min(Lg1 , Lg2 ) = Lg1 if g1 ≤ g2 . We denote the linguistic label set for performance of energy alternatives and the one for importance weights as LI and LII , respectively. Example 1. Vahdani et al. [10] have defined the following linguistic label set with their associated fuzzy set semantics to rate different energy alternatives: LI = {LI0 = Very poor(VP), LI1 = Poor(P), LI2 = Medium poor(MP), LI3 = Fair(F), LI4 = Medium good(MG), LI5 = Good(G), LI6 = Very good(VG)} = {(0, 1, 2), (1, 2, 3), (2, 3, 4, 5), (4, 5, 6), (5, 6, 7, 8), (7, 8, 9), (8, 9, 10)}.

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Example 2. The label set and their associated fuzzy set semantics with a additive linguistic preference relation to rate the importance weights regarding different criteria can be defined as follows [10]: II II II LII = {LII 0 = Very low(VL), L1 = Low(L), L2 = Medium low(ML), L3 = Medium(M), II II LII 4 = Medium high(MH), L5 = High(H), L6 = Very high(VH)}

= {(0, .1, .2), (.1, .2, .3), (.2, .3, .4, .5), (.4, .5, .6), (.5, .6, .7, .8), (.7, .8, .9), (.8, .9, 1)}.

The value (a, b, c) or (a, b, c, d) is used to represent a triangular or trapezoidal fuzzy number. 2.3

Gathering Data and Developing Computational Models for Energy Planning

A finite set of experts, denoted as E = {E1 , E2 , . . . , EK }, is then required to assess the energy alternatives in terms of selected criteria, making use of a linguistic label set LI . In addition, the experts would be also asked to provide their opinions on the importance of different criteria, making use of a linguistic label set LII . The group of experts is assigned a weight vector ω = (ω1 , ω2 , . . . , ωK ), K k=1 ωk = 1. Without additional information, all the experts are assumed to have the same weight such that ωk = 1/K, (k = 1, 2, . . . , K). Formally, the assessment data obtained by this way can be described as follows: – The value xkmn is used to denote the rating of energy alternative Am on criterion Cn , provided by expert Ek , where m = 1, 2, . . . , M ; n = 1, 2, . . . , N ; k = 1, 2, . . . , K; and xkmn ∈ LI . – If the group of experts provides importance weights regarding each criterion directly, LII will be a linguistic label set with an additive linguistic preference. Then, ynk is used to denote the assessment of criterion Cn provided by expert Ek , where n = 1, 2, . . . , N ; k = 1, 2, . . . , K; and ynk ∈ LII .

3

Single Criterion Linguistic Energy Planning Involving Underlying Semantic Overlapping

If an expert E assesses an alternative A ∈ A using L ∈ L, it implies that the expert makes an assertion “A on C is L”. From the philosophical viewpoint of the epistemic stance [5], humans posses some kind of mechanism for deciding whether or not to make certain assertions. Furthermore, although the underlying concepts are often vague, the decisions about assertions are, at a certain level, bivalent. That is to say for an energy alternative A on a criterion C and a description L, the expert is willing to assert that “A on C is L” or not. However, the dividing line between those linguistic labels are and those are not appropriate to use may be uncertain. Therefore, if one expert assesses an energy alternative using L, other linguistic labels L ∈ L(L = L) may also be appropriate for describing A on a criterion C. Such a phenomenon is referred to as the semantic

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overlapping of linguistic data. Motivated by the epistemic stance, we assume that any neighboring linguistic labels have partial semantic overlapping in the energy evaluation framework. Also, similar with [9], the linguistic label L will be called a prototype label. 3.1

Deriving Underlying Semantic Overlapping from Fuzzy Membership Functions

Assume a linguistic label set L = {L0 , L1 , . . . , LG } and their associated membership functions are {M(L0 ), M(L1 ), . . . , M(LG )}. We first assume an expert’s judgment is a numerical value x, we obtain the linguistic description of x relative to the linguistic variable L, which is a fuzzy subset of T(L) such that fL (x) = {L0 /μM(L0 ) (x), L1 /μM(L1 ) (x), . . . , LG /μM(LG ) (x)}. For each possible x ∈ Ω, a mass assignment function mx on 2L can be   derived from the membership degrees μM(L0 ) (x), μM(L1 ) (x), . . . , μM(LG ) (x) as follows. Definition 2. Given the fuzzy subset fL (x) of a universe Ω relative to the linguistic variable L such that the range of the membership function fL (x), μfL (x) , is {π1 , π2 , . . . , πJ }, where πj > πj+1 > 0. Then the mass assignment of f , denoted as mx , is a probability distribution on 2Ω satisfying mx (∅) = 1 − π1 , mx (Fj ) = πj − πj+1 , for i = j, . . . , J − 1, mx (FJ ) = πJ , (1) where Fj = {L ∈ L|μM(L) (x)  πj } for j = 1, . . . , J, and {Fj }Jj=1 are referred to as the focal elements (sets) of the mass assignment function mx . In Definition 2, the mass function mx (F ) means one belief that F is the extensions of a value x. The notion of mass assignment mx suggests a definition of probability distribution p as follows. Definition 3. The probability distribution of x ∈ Ω on L is given by  mx (Fj ) p(L|x) = ,L ∈ L Fj :L∈Fj (1 − mx (∅))|Fj |

(2)

where {Fj } is the corresponding set of focal elements. The mass mx (∅) = 0 can be interpreted as the degree of inconsistency conveyed by x or the belief committed exactly to other hypotheses which are not included in L. The value p(L|x) reflects the probability that L ∈ L belongs to the extensions of the expert’s assessment x ∈ Ω. This notion can be extended to the case where the value given is a continuous set of Ω in which the appropriate linguistic description is defined as follows. Definition 4. Let S ⊆ Ω, then the probability distribution of S on L is  1 p(L|S) = p(L|x)dx, L ∈ L, λ(S) S

(3)

where λ is the Lebesgue measure which in the case that S is an interval corresponds to its length. The value p(L|S) reflects the probability that L ∈ L belongs to the extensions of the expert’s assessment S ⊆ Ω.

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Extending interval value to the case where a value is a fuzzy subset of Ω, the appropriate linguistic description is as follows. Definition 5. Let f ⊆f Ω (the symbol ⊆f denotes fuzzy subset), then the probability distribution of f on a linguistic label set L is  1  1 p(L|f ) = p(L|x)dxdα, L ∈ L, (4) 0 λ(fα ) fα where fα is the alpha-cut of f . The intuition underlying this definition is as follows. For each focal set F or the alpha-cut of f we average the probability of L being selected to label values in F . This is then averaged across the focal sets to give the overall probability of L. We are now able to derive the underlying semantic overlapping of an expert’s linguistic judgment. If one expert provides a linguistic label L ∈ L as his judgment, it means that the expert will choose the fuzzy subset M (L) as his judgment. Here, L will be called the prototype label. Then by using Eq. (4), the linguistic description of a prototype linguistic label is as follows. Definition 6. With a prototype label L ∈ L, then the probability distribution of L on the linguistic label set L is  1  1 p(Lg |L) = p(Lg |x)dxdα, g = 0, 1, . . . , G. (5) 0 λ(M (L)α ) M(L)α The value p(Lg |L) reflects the probability that Lg (g = 0, 1, . . . , G) belongs to the extensions of the expert’s assessment L ⊆ L. Consequently, there are G + 1 possible prototype linguistic labels with respect to L. Then we can obtain a probability distribution matrix representing the underlying semantic overlapping of the expert’s linguistic judgment. 3.2

Group Opinions Aggregation

In linguistic energy planning, a group of experts E = {E1 , E2 , . . . , EK } is chosen to assess a set of energy supplies A = {A1 , A2 , . . . , AM } on a criterion C using a linguistic variable LI = {LI0 , LI1 , . . . , LIG }. The judgment of expert Ek for alternative Am on criterion C is denoted as xkm ∈ LI . With the associated fuzzy membership functions of the linguistic labels LIg ∈ LI (g = 0, 1, . . . , G), according to Definition 6, we obtain a probability distribution pkm of xkm on LI such that        pkm = pm LI0 |xkm , pm LI1 |xkm , . . . , pm LIG |xkm , m = 1, . . . , M, k = 1, . . . , K. With the weighting vector ω = (ω1 , ω2 , . . . , ωK ) of the experts, we can obtain the collective probability distribution on LI regarding energy alternative Am on criterion C such that   K   pm LIg = ωk · pm LIg |xkm , (6) k=1

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where g = 0, 1, . . . , G and xkm ∈ LI . Therefore, we will obtain a probability distribution pm = [pm (LI0 ), pm (LI1 ), . . . , pm (LIG )], which will be referred to as the profile of energy alternative Am on criterion C. 3.3

Satisfactory-Oriented Choice Function

Most linguistic multiexpert decision making process is basically aimed at reaching a “consensus”. Consensus is traditionally meant as a strict and unanimous agreement of all the experts regarding all possible alternatives. The model presented below assumes that experts do not have to agree in order to reach a consensus. There are several explanations that allow for experts not to converge to a uniform opinion. It is well accepted that experts are not necessarily the decision-makers, but provide an advice [8]. The experts in energy planning are not necessarily the real policy-maker. Due to this observation, the linguistic judgment provided by the expert does not represent the policy-maker’s preference. In fuzzy set computation based model, an optimization procedure is usually needed to select the best energy choices. It is realized that the human behavior should be modeled as satisficing instead of optimizing. In the sequel, we shall propose a satisfactory-oriented choice function. The inferred probability distribution on a linguistic label set LI for each energy alternative Am could be viewed as a general framework of decision making under uncertainty (DUU), described as follows. Energy supplies Am (m = 1, 2, . . . , M ) represent the alternatives available to a policy-maker, one of which must be selected. There are G + 1 possible values corresponding to the so-called state space S = {S0 , S1 , . . . , SG }, which is characterized by a probability distribution pS on the state space S. pm (LIg )(g = 1, . . . , G) on LI acts as the the probability distribution on the state space S. By assuming the policy-maker has a target T in his mind and the target is independent on the set of M alternatives and the linguistic judgments provided by the experts, we define the following function: Prm  Pr(Am  T ) =

G g=0

    Pr LIg  T · pm LIg ,

(7)

  where Pr LIg  T is the probability of linguistic label LIg meeting target T . The quantity Pr(Am  T ) could be interpreted as the probability of “the performance of Am is as at least good as T ”. We refer to it as satisfactory-oriented principle. We assume there exists a probability distribution pT of target   T on the linguistic label set LI such that pT = pT (LI0 ), pT (LI1 ), . . . , pT (LIG ) , then we define the following value function:

G G Prm = u(Lg , Ll ) · pT (Ll ) · pm (LIg ), (8) g=0

l=0

where u(Lg , Ll ) is the utility level (1 or 0). Therefore, we can induce the following value function

Linguistic Multicriteria Energy Planning

 pT (LIl ) · pm (LIg ) , for a benefit criterion; g=0 l=0 

G G I I = pT (Ll ) · pm (Lg ) , for a cost criterion.

Prm = Prm

G

g=0

 g

l=g

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(9) (10)

Now let us consider two special cases. Without additional information (if the policy-maker does not assign any target), we assume the policy-maker has a tar1 get T uniformly distributed on LI such that pT (LIg ) = G+1 , g = 0, . . . , G. Then we can obtain the probability of meeting the uniformly distributed linguistic target as follows:  G g+1 · pm (LIg ), benefit criterion; G+1 Prm = g=0 (11) G G+1−g I g=0 G+1 · pm (Lg ), cost criterion. Consider the case that the policy-maker assigns a specific linguistic label LIl ∈ LI as his target. As discussed in Section 3.1, in linguistic energy planning, the linguistic judgment has a underlying semantic overlapping. In this context, we can also derive a probability distribution pT on LI such that        pT = pT LI0 |LIl , pT LI1 |LIl , · · · , pT LIG |LIl . (12) Consequently, we can obtain the probability of meeting a specific linguistic label target LIl ∈ LI for benefit and cost criteria, respectively. With the satisfactoryoriented choice function, if there exists only one criterion, we can obtain best energy alternative(s) as A∗ = arg maxm {Prm }.

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Multicriteria Linguistic Energy Planning

Extending single criterion to multicriteria, a group of experts is chosen to assess a set of energy supplies on a set of criteria C = {C1 , C2 , . . . , CN } using a linguistic set LI = {LI0 , LI1 , . . . , LIG }. The linguistic assessment of alternative Am on criterion Cn provided by expert Ek is denoted as xkmn ∈ LI . Simk k I ilarly, we  can  derive a probability  distribution pmn ofxmn on L such that pkmn = pmn LI0 |xkmn , pmn LI1 |xkmn , . . . , pmn LIG |xkmn . With the weighting vector ω = (ω1 , ω2 , . . . , ωK ), we can obtain a collective probability distribution for each energy supply Am on each criterion Cn such that   pmn = pmn (LI0 ), pmn (LI1 ), . . . , pmn (LIG ) , m = 1, 2, . . . , M, n = 1, 2, . . . , N. (13) Based on the satisfactory-oriented principle in Section 3.3, a set of linguistic targets T = {T1 , T2 , . . . , TN } regarding the criteria set C can be defined by default or by the policy-maker. With mutual independent and additive preferences of targets, we can obtain a probability vector of energy supply Am meeting target Tn for evaluation criterion Cn as follows: Prm = [Prm1 , Prm2 , . . . , PrmN ] .

(14)

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The group of experts provides their linguistic judgments for weighting in  II II formation by using a set of linguistic labels LII = LII with an 0 , L 1 , . . . , LG additive linguistic preference relation. The linguistic rating for the importance weight of criterion Cn provided by expert Ek is denoted as ynk ∈ LII . Based on the interpretation of underlying semantic overlapping of linguistic labels in k II Section 3.1, we can obtain a probability n on   IIdistribution   pw  L for the  linguis k k k II k k tic rating yn such that pwn = pwn L0 |yn , pwn L1 |yn , . . . , pwn LII , G |yn where k = 1, 2, . . . , K and n = 1, 2, . . . , N . With the weighting vector ω = (ω1 , ω2 , . . . , ωK ) of the group of experts, we can obtain a collective probability distribution on LII regarding criterion Cn such that K k pwn (LII ωk · pwn (LII (15) g)= g |yn ), g = 0, 1, . . . , G. k=1

Such a value is referred to as the weight profile of criterion Cn , denoted as pwn . Consequently, we will obtain N probability distributions pwn (n = 1, 2, . . . , N ) on LII for the criteria set C. In order to derive the weight for each criterion, we proceed as follows. First, we define the probability that the profile of criterion Cn is equivalent to that of criterion Cl using the following function G    II (16) Pr(Cn = Cl ) = pwn LII g × pwl Lg , g=0

where n, l = 1, 2, . . . , N. Second, the probability that the profile of criterion Cn is greater than that of Cl is defined as Pnl = Pr(Cn > Cl ) = Pr(Cn ≥ Cl ) − 0.5Pr(Cn = Cl ),

(17)

where n, l = 1, 2, . . . , N, and Pr(Cn ≥ Cl ) can be derived from the satisfactoryoriented choice function in Section 3.3. According to Eqs. (16)-(17), we have Pnl +Pln = 1. Such a function satisfies the properties of additive fuzzy preference relations such that – When Pnl = 0.5, it means that no difference exists between Cn and Cl . – When Pnl = 1, it indicates that Cn is absolutely better than Cl . – When Pnl > 0.5, it indicates that Cn is better than Cl . The preference matrix in AHP is generally assumed to be an additive reciprocal. Consequently, we can build a fuzzy preference matrix P by means of Eqs. (16)(17) such that C1 C2 . . . CN C1 0.5 P12 . . . P1N P = C2 P21 0.5 . . . P2N (18) .. .. .. . . . . .. . . . CN PN 1 PN2 . . . 0.5 Using the fuzzy preference relation matrix P, we can derive the importance weighting vector W = (w1 , w2 , . . . , wN ) for different criteria by using the arithmetic averaging method such that 1 N Pnl wn = , n = 1, 2, . . . , N. (19) N l=1 N m=1 Pml

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Using the derived weight vector W = [w1 , w2 , . . . , wN ] and the individual probabilities in Eq. (14), we are now able to obtain the global value for energy N supply Am as follows: V (Am ) = n=1 wn × Prmn , m = 1, 2, . . . , M.

5

An Illustrative Example

In this section, we borrow an example from the literature to illustrate the advantages and effectiveness of our model. 5.1

Problem Descriptions: Alternative-Fuel Buses Selection

Vahdani et al. [10] study the problem of alternative-fuel buses selection. In their  research, twelve energy technologies (fuel modes) are considered such that A = A1 : Conventional diesel engine, A2 : Compressed natural gas, A3 : Liquid propane gas, A4 : Fuel cell, A5 : Methanol, A6 : Electric vehicle opportunity charging,A7: Direct electric charging, A8 : Electric bus with exchangeable batteries, A9 : Hybrid electric bus with gasoline engine, A10 : Hybrid electric bus with diesel engine, A11 : Hybrid electric bus with compressed natural gas engine, A12 : Hybrid electric bus with liquid propane gas engine . They also investigate four aspects of evaluation criteria: social, economic, technological, and transportation. They establish 11 eval uation criteria such that C = C1 : Energy supply, C2 : Energy efficiency, C3 : Air pollution, C4 : Noise pollution, C5 : Industrial relationship, C6 : Implementation cost, C7 : Maintenance cost, C8 : Vehicle capability,  C9 : Road facility, C10 : Speed of traffic flow, C11 : Sense of comfort . of three experts E = {E1 , E2 , E3 } with a weighting vector ω =  1 A1 group  1 is chosen to assess each alternative-fuel mode and rate importance 3, 3, 3 weights regarding different criteria. The linguistic assessments for the performance values of 12 energy alternatives on 11 criteria by the three experts with the linguistic label set LI1 in Example (1) can be referred to [10] [pp. 1405-1406]. In addition, the linguistic assessments for criteria weights via the linguistic label C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 E1 H VH L L H L L VH H VH VH II set L in Example (2) are E H VH L L H L L VH H VH H . 2 E3 H H L L VH L L VH VH VH H 5.2

Solution Based on Our Model

Now let us apply our model to solve this problem proceeding as follows. According to Definition 6 in Section 3.1, we can derive the underlying semantic overlapping of linguistic label set LI1 in Example 1. Similarly, we can also obtain the underlying semantic overlapping of linguistic label set LII1 in Example 2. With the weight vector ω = (ω1 , ω2 , . . . , ωK ) of the group of experts, we can derive the performance profile (collective probability distribution) of alternative Am on criterion Cn , denoted as pmn . Similarly, we can derive the weight profile pwn for criterion Cn .

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With the weight profiles of all the criteria, we can obtain a probability matrix via pairwise comparison such that C1 C2 C3 C4 C P = C5 6 C7 C8 C9 C10 C11

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 0.5 0.229 1.0 1.0 0.365 1.0 1.0 0.094 0.365 0.094 0.365 0.771 0.5 1.0 1.0 0.635 1.0 1.0 0.365 0.635 0.365 0.635 0.0 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.0 0.0 0.0 0.635 0.365 1.0 1.0 0.5 1.0 1.0 0.229 0.5 0.229 0.5 0.0 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.0 0.0 0.0 . 0.0 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.0 0.0 0.0 0.906 0.635 1.0 1.0 0.771 1.0 1.0 0.5 0.771 0.5 0.771 0.635 0.365 1.0 1.0 0.5 1.0 1.0 0.229 0.5 0.229 0.5 0.906 0.635 1.0 1.0 0.771 1.0 1.0 0.5 0.771 0.5 0.771 0.635 0.365 1.0 1.0 0.5 1.0 1.0 0.229 0.5 0.229 0.5

We can then obtain a weight vector W for the criteria set via Eq. (19) as W = (0.099, 0.131, 0.033, 0.033, 0.115, 0.033, 0.033, 0.146, 0.115, 0.146, 0.115). With a set of predefined linguistic targets T = {T1 , T2 , . . . , TN } regarding the criteria set C, we can calculate the probability Prmn of energy alternative Am on criterion Cn meeting target Tn . Here as an illustration, we consider two cases: – The policy-maker does not assign any target, therefore all the criteria have the same targets T1 uniformity distributed on LI . – According to weight vector of the N criteria, we know that criteria C8 and C10 are the most two important criteria, i.e., C8 = C10 = 0.146. Therefore, we assume that the policy-maker assigns a specific linguistic label as his target T2 for criteria C8 and C10 . Other criteria have the same uniform target T1 . We consider four targets: T12 = Fair(F), T22 = Medium good(MG), T32 = Good(G), T42 = Very good(VG). According to the semantic overlapping of linguistic labels discussed in Section 3.1, we can derive a probability distribution for each specific label typed target. Consequently, we can obtain the probabilities of meeting different predefined targets for each fuel mode on each evaluation criterion. Finally, a global value for each fuel model can be obtained. It is founded that – If the policy-maker does not assign any target or assigns Good(G) as his target for criteria C8 and C10 , A3 : Liquid propane gas is the best fuel mode. – If the policy-maker assigns Fair(F) or Medium good(MG) as his targets toward C8 and C10 , A11 : Hybrid electric bus with compressed natural gas engine will be best fuel mode. – If the policy-maker assigns Very good(VG) as his targets toward C8 and C10 , A1 : Conventional diesel engine will be best fuel mode.

6

Concluding Remarks

This paper proposed a linguistic energy planning model. Essentially, a probabilistic approach was first proposed to derive the underlying semantic overlapping

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of linguistic labels from their associated fuzzy membership functions. Second, a satisfactory-oriented choice function was proposed to incorporate the policymaker’s preference information. Third, our model was extended to multicriteria case with linguistic importance information. An alternative-fuel bus selection problem was borrowed from the literature to show the effectiveness and advantages of our model. The main advantages of our model are it ability to deal with computation solely with words involving the underlying semantic overlapping as well as the real decision-maker’s preference. Acknowledgements. This study was supported by the National Natural Science Foundation of China (71101050, 70901026) and the Program for New Century Excellent Talents in University (NCET-09-0345).

References 1. Cavallaro, F.: Fuzzy TOPSIS approach for assessing thermal-energy storage in concentrated solar power (CSP) systems. Appl. Energ. 87(2), 496–503 (2010) 2. Heo, E., Kim, J., Boo, K.J.: Analysis of the assessment factors for renewable energy dissemination program evaluation using fuzzy AHP. Renew. Sustain Energ. Rev. 14(8), 2214–2220 (2010) 3. Herrera, F., Mart´ınez, L.: A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 8(6), 746–752 (2000) 4. Hiremath, R., Shikha, S., Ravindranath, N.: Decentralized energy planning; modeling and application–a review. Renew. Sustain Energ. Rev. 11(5), 729–752 (2007) 5. Lawry, J.: Appropriateness measures: An uncertainty model for vague concepts. Synthese 161(2), 255–269 (2008) 6. Løken, E.: Use of multicriteria decision analysis methods for energy planning problems. Renew. Sustain Energ. Rev. 11(7), 1584–1595 (2007) 7. Poh, K.L., Ang, B.W.: Transportation fuels and policy for Singapore: an AHP planning approach. Comput Ind. Eng. 37(3), 507–525 (1999) 8. Shanteau, J.: What does it mean when experts disagree? In: Salas, E., Klein, G.A. (eds.) Linking Expertise and Naturalistic Decision Making, pp. 229–244. Psychology Press, USA (2001) 9. Tang, Y., Lawry, J.: Linguistic modelling and information coarsening based on prototype theory and label semantics. Int. J. Appr. Reason 50(8), 1177–1198 (2009) 10. Vahdani, B., Zandieh, M., Tavakkoli-Moghaddam, R.: Two novel FMCDM methods for alternative-fuel buses selection. Appl. Math. Model 35(3), 1396–1412 (2011) 11. Venetsanos, K., Angelopoulou, P., Tsoutsos, T.: Renewable energy sources project appraisal under uncertainty: the case of wind energy exploitation within a changing energy market environment. Energ. Pol. 30(4), 293–307 (2002) 12. Wang, B., Kocaoglu, D.F., Daim, T.U., Yang, J.: A decision model for energy resource selection in China. Energ. Pol. 38(11), 7130–7141 (2010) 13. Zadeh, L.A.: Fuzzy sets. Inform. Contr. 8(3), 338–353 (1965) 14. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning–Part I. Inform. Sci. 8(3), 199–249 (1975) 15. Zadeh, L.A.: From computing with numbers to computing with words–From manipulation of measurements to manipulation of perceptions. IEEE Trans. Circ. Syst. Fund Theor. Appl. 46(1), 105–119 (1999)

Bipolar Semantic Cells: An Interval Model for Linguistic Labels Yongchuan Tang1, and Jonathan Lawry2 1 2

College of Computer Science, Zhejiang University, Hangzhou, 310027, P.R. China Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, UK [email protected], [email protected]

Abstract. An interval model for linguistic labels is proposed by introducing bipolar semantic cells for concept representation. According to this model, the degree to which each element is a positive case of a given linguistic expression is an interval value. Fundamental to our approach is that there is an uncertain border area associated with linguistic labels. This is modeled by assuming that there are two uncertain boundaries for each linguistic label, resulting in a bipolar semantic cell for concept representation. The calculus of lower and upper neighborhood functions of linguistic expressions is developed and investigated. This then provides a framework for modelliong the vague concepts in uncertain reasoning. Keywords: Prototype theory, Label semantics, Vagueness, Bipolarity, Interval fuzzy sets.

1 Introduction In human communication linguistic labels and expressions provide a flexible and effective mechanism for communicating information. The use of linguistic labels and expressions can permit us to express granular knowledge efficiently at a suitable level of detail, as well as being robust to small changes in attribute measurements and noisy data. This paper presents an interval-valued formalization of linguistic labels. The underlying philosophy of our approach is that the boundary of the extension of a linguistic label may not be a single borderline but rather a border area. This kind of philosophical viewpoint is similar to the epistemic stance of vagueness [4] where there is an uncertain but crisp division between those labels which are, and those which are not appropriate to describe a given element. The proposed bipolar model also assumes that there is a set of prototypes which is described by the linguistic label certainly. The latter assumption is the basic viewpoint of prototype theory proposed by Rosch [8,9]. Based on these assumptions, the bipolar model proposed in this paper uses a transparent cognitive structure, referred to as a bipolar semantic cell, to represent a linguistic label. Intuitively speaking, a linguistic label L on domain Ω can be expressed as about P , similar to P or close to P where P ⊆ Ω is a set of prototypes of L. We use the term bimembrane of L to refer to two uncertain boundaries, corresponding to 

Corresponding author.

Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 60–71, 2011. c Springer-Verlag Berlin Heidelberg 2011 

Bipolar Semantic Cells: An Interval Model for Linguistic Labels

61

a pair of distance thresholds (1 , 2 ) to the prototypes P , together with an associated probability density function δ on [0, +∞)2 . Then from this bipolar semantic cell we can define two functions, the lower and upper neighborhood functions, to quantity the appropriateness of this linguistic label to describe elements from Ω. According to the bipolar semantic cell model, the appropriateness of linguistic label L with prototypes P is an interval value. Hence, it is an extension of work by Lawry and Tang [6,7] where the appropriateness is a single value generated by an uncertain boundary. In this paper we will also show that the proposed bipolar semantic cells are different from the intuitionistic fuzzy sets proposed by Atanassov [1]. However, as shown in the recent work by Lawry [5] it is also possible to develop the same calculus as intuitionistic fuzzy sets theory from prototype theory and random set theory. In the sequel we explore the use of bipolar semantic cells to model linguistic labels, introduce lower and upper neighborhood functions of compound expressions, and discuss the semantic bipolarity of linguistic expressions.

2 Bipolar Semantic Cells Let Ω denote the underlying universe of discourse and LA = {L1 , . . . , Ln } be a finite set of linguistic labels for describing elements of Ω. For each label Li there is a set of prototypical elements Pi ⊆ Ω, such that Li is certainly appropriate to describe any prototypical elements Pi ⊆ Ω. In other words, the linguistic label Li can be considered as a word with meaning ‘about Pi ’, ‘similar to Pi ’ or ‘close to Pi ’ where Pi ⊆ Ω. In the following we firstly introduce a cognitive structure for the semantic representation of vague concept Li . This structure is referred to as a bipolar semantic cell and has three components: a prototype set Pi (⊆ Ω) of Li , a distance function di defined on Ω and a density function δi defined on [0, +∞)2 . Definition 1 (Bipolar Semantic Cell). Bipolar semantic cell is a structural and semantic unit of vague concepts. It is the smallest unit of a vague concept, and is the building brick of concept representation. More formally, A bipolar semantic cell ‘Li ’ is a triple Pi , di , δi , where Pi ⊆ Ω is a set of prototypes of Li , di is a distance metric on Ω such that di (x, y) ≥ di (x, x) = 0 and di (x, y) = di (y, x) for all x, y ∈ Ω, and δi is a probability density function on [0, +∞)2 such that δi ([0, +∞)2 ) = 1 and δi (1 , 2 ) ≥ 0 for any 1 , 2 ≥ 0, which represents the joint probability distribution of neighborhood sizes of Li . Intuitively speaking, a bipolar semantic cell Li is composed of a semantic nucleus and a semantic bimembrane. The semantic nucleus represents the set of prototypes of Li . The semantic bimembrane represents two uncertain boundaries of Li , where uncertainty is modelled by a 2-dimensional probability density function on two distance thresholds to Pi . Here we assume that for all x ∈ Ω di (x, Pi ) = inf y∈Pi di (x, y). According to this definition, we can see that bipolar semantic cell Pi , di , δi  can provide an effective model of a concept Li with the prototypes Pi . In this structure, Pi is a prototype set without uncertainty, while the uncertainty is associated with the unknown

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δ(ε1,ε2)

neighborhood boundaries of Li . Intuitively speaking, the semantic nucleus corresponds to the prototype set Pi and the semantic bimembrane corresponds to the resilient constraint ‘about’. An interesting component is the semantic bimembrane which consists of two crisp but uncertain boundaries of the semantic cell. This means that the distance thresholds 1 (≥ 0) and 2 (≥ 0) from two corresponding boundaries to the prototype set Pi are two random variables with a joint density function δi on [0, +∞)2 . See for example, the illustration of a bipolar semantic cell on a 2-dimensional domain in figure 1.

ε1

Pi

ε2 ε2

ε1

Fig. 1. The bipolar semantic cell Li = Pi , d, δ on a 2-dimensional domain of course

In this paper we assume that all linguistic labels Li ∈ LA share the same density function δ on [0, +∞)2 . For each linguistic label Li the lower and upper extensions of Li are defined to be those elements of Ω with distance from Pi less than or equal to a lower and an upper threshold value respectively. Formally speaking, the lower extension of Li is taken to be {x ∈ Ω : di (x, Pi ) ≤ } and the upper extension of Li is taken to be {x ∈ Ω : di (x, Pi ) ≤ }, where  ≤ . In the following, one way of defining the lower and upper extensions of Li is given. Definition 2 ((1 , 2 )-Lower and Upper Neighborhoods). For any 1 , 2 ≥ 0 the (1 ,2 ) ( , ) (1 , 2 )-lower and upper neighborhoods N Li1 2 and N Li are respectively defined as follows: ( , ) N Li1 2 = {x : di (x, Pi ) ≤ min{1 , 2 }} (1) (1 ,2 )

N Li

= {x : di (x, Pi ) ≤ max{1 , 2 }}

(2)

In this definition we take the lower distance threshold  = min{1 , 2 } and the upper distance threshold  = max{1 , 2 }. We will see that this kind of definition results in a non-truth-functional calculus system. In the recent work by Lawry [5] another way of defining the lower and upper distance thresholds was proposed, which result in a truth-functional calculus system and a new interpretation of intuitionistic fuzzy sets [1]. From the lower and upper extensions of linguistic label Li we can obtain the lower and upper neighborhood functions according to the following definition.

Bipolar Semantic Cells: An Interval Model for Linguistic Labels

63

Definition 3 (Lower and Upper Neighborhood Functions). For any x ∈ Ω and Li ∈ LA the lower and upper neighborhood functions of Li , μL (x) and μLi (x), are defined i respectively as follows:   ( , ) μL (x) = δ (1 , 2 ) : x ∈ N Li1 2 (3) i

μLi (x) = δ



(1 ,2 )

(1 , 2 ) : x ∈ N Li

 (4)

Here μL (x) quantifies the belief that the linguistic label is definitely appropriate to dei scribe x, and μLi (x) quantifies the belief that the linguistic label is possibly appropriate to describe x. It is easy to show that μL (x) ≤ μLi (x), so we can also assert that the i degree to which element x belongs to the extension of Li is at least μL (x) and at most i μLi (x). In other words the membership degree of x belonging to Li is an interval value. Proposition 4. For any Li ∈ LA the following hold:  μL (x) = i

di (x,Pi )

 μLi (x) = μL (x) + i



+∞

+∞ 1

di (x,Pi )



0

 δ(1 , 2 )d2 d1 +

+∞ di (x,Pi )



+∞

di (x,Pi )

 δ(1 , 2 )d2 d1 +

+∞

2

di (x,Pi ) 0



δ(1 , 2 )d1 d2

+∞ di (x,Pi )

δ(1 , 2 )d1 d2

If δ(1 , 2 ) = δ1 (1 )δ2 (2 ) and let Δk () = δk ([, +∞)) for k = 1, 2 then  μL (x) = i

+∞ di (x,Pi )

 δ1 (1 )Δ2 (1 )d1

+∞

di (x,Pi )

δ2 (2 )Δ1 (2 )d2

μLi (x) = μL (x) + (1 − Δ1 (di (x, Pi )) Δ2 (di (x, Pi )) + (1 − Δ2 (di (x, Pi )) Δ1 (di (x, Pi )) i

Example 1. Assume that δ(1 , 2 ) = δ1 (1 )δ2 (2 ) and δi (·) = δ(· | ci , σi ) for i = 1, 2 are the normalized normal density functions, δ( | ci , σi ) = 2

G( | ci , σi ) 1 − F (0 | ci , σi )

(5)

i) exp (−c is the normal density function with mean ci −2σi2 0 and standard variance σi and F (0 | ci , σi ) = −∞ G( | ci , σi )d. Figure 2(a) shows lower and upper neighborhood functions for L1 = {0}, d, δ and L2 = {1}, d, δ where d is the Euclidean distance function and δ(1 , 2 ) = δ(1 | c1 , σ1 )δ(2 | c2 , σ2 ) with c1 = 0.4, σ1 = 0.2, and c2 = 0.3, σ2 = 0.1. Figure 2(b) shows the lower and upper neighborhood functions for L1 and L2 again with c1 = 0.4, σ1 = 0.3, and c2 = 0.5, σ2 = 0.2.

where G( | ci , σi ) =

√ 1 2πσi

64

Y. Tang and J. Lawry 1

1 L1

L

L1

2

0.5

0.5

0 −1

L2

0 −1

2

1

0

1

0

x

x

(a)

(b)

2

Fig. 2. The lower and upper neighborhood functions of linguistic labels L1 = {0}, d, δ and L2 = {1}, d, δ where δ(1 , 2 ) = δ(1 | c1 , σ1 )δ(2 | c2 , σ2 ). (a) The lower and upper neighborhood functions for L1 and L2 with c1 = 0.4, σ1 = 0.2, and c2 = 0.3, σ2 = 0.1. (b) The lower and upper neighborhood functions for L1 and L2 with c1 = 0.4, σ1 = 0.3, and c2 = 0.5, σ2 = 0.2.

3 Lower and Upper Neighborhood Functions of Linguistic Expressions We consider linguistic expressions corresponding to compound expressions generated by recursive application of the connectives ∧, ∨ and ¬ to the labels in LA. Let LE denote the set of linguistic expressions. Then LE has the following definition: Definition 5 (Linguistic Expressions). The set of linguistic expressions, LE, is defined recursively as follows: 1. LA ⊆ LE; 2. If θ ∈ LE, ϕ ∈ LE, then θ ∧ ϕ ∈ LE, θ ∨ ϕ ∈ LE, ¬θ ∈ LE. In this definition θ ∧ϕ means ‘θ and ϕ’, θ ∨ϕ means ‘θ or ϕ’, and ¬θ means ‘not θ’. For example, ¬Li may means ‘dissimilar to Pi ’. LE is actually a T -free algebra where T = {∧, ∨, ¬}. The lower and upper neighborhoods of linguistic expressions are then defined recursively as follows: Definition 6 (Lower and Upper Neighborhoods). For any θ ∈ LE and any 1 , 2 ≥ 0 (1 ,2 ) ( , ) are defined respecthe (1 , 2 )-lower and upper neighborhoods N θ 1 2 and N θ tively in the following manner: ( ,2 )

Nθ 1

(1 ,2 )

Nθ ( ,2 )

Nθ 1

(1 ,2 )

Nθ

( ,2 )

= N Li1

(6)

if θ = Li

(7)

1 ,2 ) ∩ N ( if θ = φ ∧ ϕ ϕ

(8)

(1 ,2 )

= N Li

( ,2 )

= Nφ 1

(1 ,2 )

= Nφ

if θ = Li

(1 ,2 )

∩ Nϕ

if θ = φ ∧ ϕ

(9)

Bipolar Semantic Cells: An Interval Model for Linguistic Labels ( ,2 )

Nθ 1

(1 ,2 )

Nθ

( ,2 )

= Nφ 1

(1 ,2 )

= Nφ

1 ,2 ) ∪ N ( if θ = φ ∨ ϕ ϕ

(1 ,2 )

∪ Nϕ

65

(10)

if θ = φ ∨ ϕ

(11)

( ,2 )

 ( , ) c 1 2 = Nφ if θ = ¬φ

(12)

(1 ,2 )

 c ( , ) = Nφ 1 2 if θ = ¬φ

(13)

Nθ 1 Nθ

Since the distance threshold pair (1 , 2 ) has a density function δ on [0, +∞)2 , the lower and upper neighborhood functions of expression θ ∈ LE for element x ∈ Ω are then ( , ) given by the probability of a value of (1 , 2 ) such that x ∈ N θ 1 2 and the probability (1 ,2 )

of pair (1 , 2 ) such that x ∈ N θ

.

Definition 7 (Lower and Upper Neighborhood Functions). For any θ ∈ LE the lower and upper neighborhood functions, μθ (x) and μθ (x), are defined respectively as follows:   ( , ) μθ (x) = δ (1 , 2 ) : x ∈ N θ 1 2 (14)   (1 ,2 ) μθ (x) = δ (1 , 2 ) : x ∈ N θ (15) In the following we will investigate the relationships between lower and upper neighborhoods of any linguistic expressions. Firstly, we will show that the lower neighborhood is indeed a subset of the upper neighborhood. Lemma 8. For any θ ∈ LE and 1 , 2 ≥ 0 the following holds: ( ,2 )

Nθ 1

(1 ,2 )

⊆ Nθ

(16)

Proof. Let LE 1 = LA and LE m = LE m−1 ∪ {φ ∧ ϕ, φ ∨ ϕ, ¬φ : φ, ϕ ∈ LE m−1 }, m then LE = ∪∞ m=1 LE . We now carry out induction on m. If θ = Li then it holds (1 ,2 ) since N Li = {x : di (x, Pi ) ≤ min{1 , 2 }} ⊆ {x : di (x, Pi ) ≤ max{1 , 2 }} = (1 ,2 )

N Li . Assume that for any θ ∈ LE m it holds, then for any θ ∈ LE m+1 either θ ∈ m LE , in which case the result holds trivially, or θ ∈ {φ ∧ ϕ, φ ∨ ϕ, ¬φ : φ, ϕ ∈ LE m } for which one of the following holds: ( ,2 )

1. For θ = φ ∧ ϕ where φ, ϕ ∈ LE m N θ 1 (1 ,2 )

Nφ

2. 3.

(1 ,2 )

∩ Nϕ

(1 ,2 )

= N φ∧ϕ

(1 ,2 )

= Nθ

( , )

( ,2 )

1 2 = N φ∧ϕ = Nφ 1

1 ,2 ) ∩ N ( ⊆ ϕ

.

( , ) (1 ,2 ) ( , ) 1 ,2 ) For θ = φ ∨ ϕ where φ, ϕ ∈ LE N θ 1 2 = N φ∨ϕ = N φ 1 2 ∪ N ( ϕ (1 ,2 ) (1 ,2 ) (1 ,2 ) (1 ,2 ) Nφ ∪ Nϕ = N φ∨ϕ = N θ .  ( , ) c  c 1 2 (1 ,2 ) ( , ) ( , ) m For θ = ¬φ where φ ∈ LE N θ = N ¬φ1 2 = N φ ⊆ Nφ 1 2 (1 ,2 ) (1 ,2 ) N ¬φ = Nθ . m

⊆ =

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Y. Tang and J. Lawry

Theorem 9. For any θ ∈ LE the following holds: μθ (x) ≤ μθ (x)

(17) ( ,2 )

Actually for any linguistic expression θ the lower neighborhood N θ 1 neighborhood

(1 ,2 ) Nφ

and upper

are the random sets taking values as subsets of Ω, and μθ and (1 ,2 )

( , )

μθ are the single point coverage functions of N θ 1 2 and N θ respectively. Hence, μθ (x) and μθ (x) can be considered as membership values of x in the lower and upper extensions of θ respectively. In other words, the membership value of x in the extension of θ is an interval value [μθ (x), μθ (x)]. Theorem 10. For any θ ∈ LE the following formulas hold: μ¬θ (x) = 1 − μθ (x)

(18)

μ¬θ (x) = 1 − μθ (x)

(19)

Proof. For any θ ∈ LE the following hold:     ( , ) c  1 2 ( , ) μ¬θ (x) = δ (1 , 2 ) : x ∈ N ¬θ1 2 = δ (1 , 2 ) : x ∈ N θ   (1 ,2 ) = 1 − δ (1 , 2 ) : x ∈ N θ = 1 − μθ (x)    c  (1 ,2 ) ( , ) (1 , 2 ) : x ∈ N ¬θ = δ (1 , 2 ) : x ∈ N θ 1 2   ( , ) = 1 − δ (1 , 2 ) : x ∈ N θ 1 2 = 1 − μθ (x)

μ¬θ (x) = δ



These two formulas describe the relationship between the lower and upper neighborhood functions. Furthermore, they are related to each other in a bipolar manner as described in [2] such that a linguistic expression θ is definitely appropriate to describe elements x if and only if ¬θ is not possibly appropriate to describe x. Lemma 11. For any θ ∈ LE and 1 , 2 ≥ 0 the following hold: ( , )

1 2 N θ∧¬θ =∅

(20)

(1 ,2 )

N θ∨¬θ = Ω ( , )

( ,2 )

1 2 Proof. For any θ ∈ LE, N θ∧¬θ = Nθ 1

(21) ( ,2 )

∩ N ¬θ1

(1 ,2 ) ( , ) since N θ 1 2 ⊆ N θ according to lemma 8. And ( , ) ( , ) ( , ) ( , ) (N ¬θ1 2 )c ∪ (N θ 1 2 )c = (N ¬θ1 2 ∩ N θ 1 2 )c =

( ,2 )

= Nθ 1

(1 ,2 ) c

∩ (N θ

(1 ,2 ) (1 ,2 ) N θ∨¬θ = N θ (1 ,2 ) c (N θ∧¬θ ) = Ω.

) =∅

(1 ,2 ) ∪ N ¬θ

=

Theorem 12. For any θ ∈ LE the following hold: μθ∧¬θ (x) = 0

(22)

μθ∨¬θ (x) = 1

(23)

Bipolar Semantic Cells: An Interval Model for Linguistic Labels

67

This theorem shows that the linguistic expression θ ∧ ¬θ is not definitely appropriate to describe any element x, and the linguistic expression θ ∨ ¬θ is possibly appropriate to describe any element x. Lemma 13. For any φ, ϕ ∈ LE, and any 1 , 2 ≥ 0, the following hold: ( , )

( ,2 )

1 2 – N ¬(¬φ) = Nφ 1

– –

(1 ,2 ) N ¬(φ∧ϕ) (1 ,2 ) N ¬(φ∨ϕ)

= =

(1 ,2 )

(1 ,2 )

and N ¬(¬φ) = N φ

(1 ,2 ) N ¬φ∨¬ϕ (1 ,2 ) N ¬φ∧¬ϕ

and and

(1 ,2 ) N ¬(φ∧ϕ) (1 ,2 ) N ¬(φ∨ϕ)

= =

.

(1 ,2 ) N ¬φ∨¬ϕ . (1 ,2 ) N ¬φ∧¬ϕ .

Proof. For any φ, ϕ ∈ LE, and any 1 , 2 ≥ 0, we have  ( , ) c  c c (1 ,2 ) 1 2 (1 ,2 ) ( , ) ( , ) – N ¬(¬φ) = N ¬φ = Nφ 1 2 = N φ 1 2 . Similarly N ¬(¬φ) =  c  ( , ) c c (1 ,2 ) 1 2 ( , ) N ¬φ1 2 = Nφ = Nφ .  ( , ) c  ( , ) c  ( , ) c  ( , ) c ( 1 2 1 2 1 ,2 ) 1 2 1 2 (1 ,2 ) – N ¬(φ∧ϕ) = N φ∧ϕ = Nφ ∩ Nϕ = Nφ ∪ Nϕ = ( ,2 )

( , )

( ,2 )

( , )

1 2 1 ,2 ) ∪ N ( = N ¬φ∨¬ϕ . ¬ϕ  c  c  c (1 ,2 ) (1 ,2 ) ( , ) ( , ) 1 ,2 ) Similarly N ¬(φ∧ϕ) = N φ∧ϕ = N φ 1 2 ∩ N ( = Nφ 1 2 ∪ ϕ  c (1 ,2 ) (1 ,2 ) (1 ,2 ) (1 ,2 ) Nϕ = N ¬φ ∪ N ¬ϕ = N ¬φ∨¬ϕ .  ( , ) c  ( , )  ( , ) c  ( , ) c  (1 ,2 ) c 1 2 1 2 1 2 1 2 (1 ,2 ) – N ¬(φ∨ϕ) = N φ∨ϕ = Nφ ∪ Nϕ = Nφ ∩ Nϕ =

N ¬φ1

1 2 1 ,2 ) ∩ N ( = N ¬φ∧¬ϕ . ¬ϕ  c  c  c (1 ,2 ) (1 ,2 ) ( , ) ( , ) 1 ,2 ) Similarly N ¬(φ∨ϕ) = N φ∨ϕ = N φ 1 2 ∪ N ( = Nφ 1 2 ∩ ϕ  c (1 ,2 ) (1 ,2 ) (1 ,2 ) 1 ,2 ) N ( = N ¬φ ∩ N ¬ϕ = N ¬φ∧¬ϕ . ϕ

N ¬φ1

Theorem 14. For any φ, ϕ ∈ LE the following hold: μ¬(¬φ) (x) = μφ (x)

(24)

μ¬(¬φ) (x) = μφ (x)

(25)

μ¬(φ∧ϕ)) (x) = μ¬φ∨¬ϕ (x)

(26)

μ¬(φ∧ϕ)) (x) = μ¬φ∨¬ϕ (x)

(27)

μ¬(φ∨ϕ)) (x) = μ¬φ∧¬ϕ (x)

(28)

μ¬(φ∨ϕ)) (x) = μ¬φ∧¬ϕ (x)

(29)

Theorem 15. For any θ ∈ LE the following hold: μφ∨ϕ (x) = μφ (x) + μϕ (x) − μφ∧ϕ (x)

(30)

μφ∨ϕ (x) = μφ (x) + μϕ (x) − μφ∧ϕ (x)

(31)

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Y. Tang and J. Lawry

Proof. For any θ ∈ LE we have ( , )

( ,2 )

1 2 μφ∨ϕ (x) = δ({(1 , 2 ) : x ∈ N φ∨ϕ }) = δ({(1 , 2 ) : x ∈ N φ 1

( ,2 )

= δ({(1 , 2 ) : x ∈ N φ 1

( ,2 )

δ({(1 , 2 ) : x ∈ N φ 1

1 ,2 ) ∪ N ( }) ϕ

1 ,2 ) }) + δ({(1 , 2 ) : x ∈ N ( }) − ϕ 1 ,2 ) ∩ N ( }) ϕ

= μφ (x) + μϕ (x) − μφ∧ϕ (x) (1 ,2 )

(1 ,2 )

(1 ,2 ) Nφ })

+ δ({(1 , 2 ) : x ∈

(1 ,2 ) Nϕ })

(1 ,2 ) Nφ

(1 ,2 ) Nϕ })

μφ∨ϕ (x) = δ({(1 , 2 ) : x ∈ N φ∨ϕ }) = δ({(1 , 2 ) : x ∈ N φ = δ({(1 , 2 ) : x ∈ δ({(1 , 2 ) : x ∈

∩

(1 ,2 )

∪ Nϕ

})

−

= μφ (x) + μϕ (x) − μφ∧ϕ (x) Notice that in general we cannot expect that μφ∨ϕ (x) = max{μφ (x), μϕ (x)} and μφ∨ϕ (x) = max{μφ (x), μϕ (x)}. The calculus of lower and upper neighborhood functions is essentially non-truth-functional. Corollary 16. For any θ ∈ LE the following hold: μθ∨¬θ (x) = μθ (x) + μ¬θ (x)

(32)

μθ∧¬θ (x) = μθ (x) − μθ (x)

(33)

Formula (32) means that the degree of linguistic expression θ ∨ ¬θ being definitely appropriate to describe element x is exactly the summation of degrees of linguistic expressions θ and ¬θ being definitely appropriate to describe element x. Formula (33) means that the difference between the upper and lower neighborhood functions is exactly the degree of linguistic expression θ ∧ ¬θ being possibly appropriate to describe the underlying element. We could not expect μθ∧¬θ (x) to be zero in general, since the border area of θ may be non-null. In the following we consider other concise representations of sets {(1 , 2 ) : x ∈ (1 ,2 ) (1 ,2 ) Nθ } and {(1 , 2 ) : x ∈ N θ } determined by linguistic expression θ and element x. Definition 17. For any θ ∈ LE I(x, θ) ⊆ [0, +∞)2 and I(x, θ) ⊆ [0, +∞)2 are defined recursively as follows: I(x, θ) = {(1 , 2 ) : di (x, Pi ) ≤ min{1 , 2 }} if θ = Li

(34)

I(x, θ) = {(1 , 2 ) : di (x, Pi ) ≤ max{1 , 2 }} if θ = Li

(35)

I(x, θ) = I(x, φ) ∩ I(x, ϕ) if θ = φ ∧ ϕ

(36)

Bipolar Semantic Cells: An Interval Model for Linguistic Labels

I(x, θ) = I(x, φ) ∩ I(x, ϕ) if θ = φ ∧ ϕ

(37)

I(x, θ) = I(x, φ) ∪ I(x, ϕ) if θ = φ ∨ ϕ

(38)

I(x, θ) = I(x, φ) ∪ I(x, ϕ) if θ = φ ∨ ϕ

(39)

I(x, θ) = (I(x, φ))c θ = ¬φ

(40)

I(x, θ) = (I(x, φ))c θ = ¬φ

(41)

( ,2 )

It is very easy to show that I(θ, x) = {(1 , 2 ) : x ∈ N θ 1 x∈

(1 ,2 ) Nθ },

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} and I(θ, x) = {(1 , 2 ) :

which then immediately result in the following theorem.

Theorem 18. For any θ ∈ LE the following hold: μθ (x) = δ(I(x, θ))

(42)

μθ (x) = δ(I(x, θ))

(43)

Let LE ∧,∨ be the set of linguistic expressions generated by recursively applying connectives ∧ and ∨ to linguistic labels in LA. Then the lower and upper neighborhood functions of any θ ∈ LE ∧,∨ have relatively simple forms. Definition 19. For any θ ∈ LE ∧,∨ the real number lb(θ) is defined recursively as follows: lb(θ) = di (x, Pi ) if θ = Li

(44)

lb(θ) = max(lb(φ), lb(ϕ)) if θ = φ ∧ ϕ

(45)

lb(θ) = min(lb(φ), lb(ϕ)) if θ = φ ∨ ϕ

(46)

Theorem 20. For any θ ∈ LE ∧,∨ the following hold: I(x, θ) = {(1 , 2 ) : lb(θ) ≤ min{1 , 2 }}

(47)

I(x, θ) = {(1 , 2 ) : lb(θ) ≤ max{1 , 2 }}

(48)

∧,∨ ∧,∨ ∧,∨ Proof. Let LE0∧,∨ = LA, LEm+1 = LEm ∪ {φ ∧ ϕ, φ ∨ ϕ : φ, ϕ ∈ LEm } +∞ ∧,∨ ∧,∨ for m > 0, then LE = ∪m=0 LEm . We now carry out the induction on m. If ∧,∨ θ = Li then the results hold immediately. Assume that θ ∈ LEm the results hold. ∧,∨ ∧,∨ Then for any θ ∈ LEm+1 either θ ∈ LEm , for which case the results hold trivially, ∧,∨ or θ ∈ {φ ∧ ϕ, φ ∨ ϕ : φ, ϕ ∈ LEm } for which one of the following holds:

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∧,∨ 1. If θ = φ∧ϕ where φ, ϕ ∈ LEm then I(x, θ) = I(x, φ∧ϕ) = I(x, φ)∩I(x, ϕ) = {(1 , 2 ) : lb(φ) ≤ min{1 , 2 }} ∩ {(1 , 2 ) : lb(ϕ) ≤ min{1 , 2 }} = {(1 , 2 ) : max{lb(φ), lb(ϕ)} ≤ min{1 , 2 }} = {(1 , 2 ) : lb(θ) ≤ min{1 , 2 }}. Similarly, I(x, θ) = I(x, φ ∧ ϕ) = I(x, φ) ∩ I(x, ϕ) = {(1 , 2 ) : lb(φ) ≤ max{1 , 2 }} ∩ {(1 , 2 ) : lb(ϕ) ≤ max{1 , 2 }} = {(1, 2 ) : max{lb(φ), lb(ϕ)} ≤ max{1 , 2 }} = {(1 , 2 ) : lb(θ) ≤ max{1 , 2 }}. ∧,∨ 2. If θ = φ∨ϕ where φ, ϕ ∈ LEm then I(x, θ) = I(x, φ∨ϕ) = I(x, φ)∪I(x, ϕ) = {(1 , 2 ) : lb(φ) ≤ min{1 , 2 }} ∪ {(1 , 2 ) : lb(ϕ) ≤ min{1 , 2 }} = {(1 , 2 ) : min{lb(φ), lb(ϕ)} ≤ min{1 , 2 }} = {(1 , 2 ) : lb(θ) ≤ min{1 , 2 }}. Similarly, I(x, θ) = I(x, φ ∨ ϕ) = I(x, φ) ∪ I(x, ϕ) = {(1 , 2 ) : lb(φ) ≤ max{1 , 2 }} ∪ {(1 , 2 ) : lb(ϕ) ≤ max{1 , 2 }} = {(1 , 2 ) : min{lb(φ), lb(ϕ)} ≤ max{1 , 2 }} = {(1 , 2 ) : lb(θ) ≤ max{1 , 2 }}.

Theorem 21. For any θ ∈ LE ∧,∨ the following hold: μθ (x) = min(μφ (x), μϕ (x)) if θ = φ ∧ ϕ

(49)

μθ (x) = min(μφ (x), μϕ (x)) if θ = φ ∧ ϕ

(50)

μθ (x) = max(μφ (x), μϕ (x)) if θ = φ ∨ ϕ

(51)

μθ (x) = max(μφ (x), μϕ (x)) if θ = φ ∨ ϕ

(52)

Proof. The proof is relative easy. 1. If θ = φ ∧ ϕ where φ, ϕ ∈ LE ∧,∨ then μθ (x) = δ(I(x, θ)) = δ({(1 , 2 ) : lb(θ) ≤ min{1 , 2 }}) = δ({(1 , 2 ) : max{lb(φ), lb(ϕ)} ≤ min{1 , 2 }}) = min{δ({(1 , 2 ) : lb(φ) ≤ min{1 , 2 }}), δ({(1 , 2 ) : lb(ϕ) ≤ min{1 , 2 }})} = min(μφ (x), μϕ (x)).

Similarly, μθ (x) = δ(I(x, θ)) = δ({(1 , 2 ) : lb(θ) ≤ max{1 , 2 }}) = δ({(1 , 2 ) : max{lb(φ), lb(ϕ)} ≤ max{1 , 2 }}) = min{δ({(1 , 2 ) : lb(φ) ≤ max{1 , 2 }}), δ({(1 , 2 ) : lb(ϕ) ≤ max{1 , 2 }})} = min(μφ (x), μϕ (x)). 2. If θ = φ ∨ ϕ where φ, ϕ ∈ LE ∧,∨ then μθ (x) = δ(I(x, θ)) = δ({(1 , 2 ) : lb(θ) ≤ min{1 , 2 }}) = δ({(1 , 2 ) : min{lb(φ), lb(ϕ)} ≤ min{1 , 2 }}) = max{δ({(1 , 2 ) : lb(φ) ≤ min{1 , 2 }}), δ({(1 , 2 ) : lb(ϕ) ≤ min{1 , 2 }})} = max(μφ (x), μϕ (x)).

Similarly, μθ (x) = δ(I (x, θ)) = δ({(1 , 2 ) : lb(θ) ≤ max{1 , 2 }}) = δ({(1 , 2 ) : min{lb(φ), lb(ϕ)} ≤ max{1 , 2 }}) = max{δ({(1 , 2 ) : lb(φ) ≤ max{1 , 2 }}), δ({(1 , 2 ) : lb(ϕ) ≤ max{1 , 2 }})} = max(μφ (x), μϕ (x)).

Although the lower and upper neighborhood functions is non-truth-functional in general, they are truth-functional for the linguistic expressions in LE ∧,∨ . Hence bipolar semantic cells are partially consistent with intuitionistic fuzzy sets [1].

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4 Conclusions An interval model of linguistic labels is proposed based on bipolar semantic cells. The important aspect of this model is that the linguistic label and its negation share an uncertain border area. The bipolar semantic cell formalizes this idea by assuming that there are two uncertain boundaries for the linguistic label. The calculus of lower and upper neighborhood functions developed from this interval model is non-truth-functional in essence. Future possible work may include the extension of theory of label semantics [3] using this bipolar model, and bipolar rule-based reasoning and decision-making. In addition, it could be interesting to incorporate the bipolar into applications of semantic cells to machine learning and control [10]. Acknowledgment. Yongchuan Tang is funded by the National Natural Science Foundation of China (NSFC) under Grant No. 61075046 and Zhejiang Natural Science Foundation under Grant No. Y1090003.

References 1. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, 87–96 (1986) 2. Dubois, D., Prade, H.: An introduction to bipolar representations of information and preference. International Journal of Intelligent Systems 23, 866–877 (2008) 3. Lawry, J.: A framework for linguistic modelling. Artificial Intelligence 155, 1–39 (2004) 4. Lawry, J.: Appropriateness measures: an uncertainty model for vague concepts. Synthese 161(2), 255–269 (2008) 5. Lawry, J.: A Random Set and Prototype Theory Interpretation of Intuitionistic Fuzzy Sets. In: H¨ullermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010, Part I, CCIS, vol. 80, pp. 618–628. Springer, Heidelberg (2010) 6. Lawry, J., Tang, Y.: Uncertainty modelling for vague concepts: A prototype theory approach. Artificial Intelligence 173, 1539–1558 (2009) 7. Lawry, J., Tang, Y.: Granular knowledge representation and inference using labels and expressions. IEEE Trans. Fuzzy Syst. 18(3), 500–514 (2010) 8. Rosch, E.: Natural categories. Cognitive Psychology 4(3), 328–350 (1973) 9. Rosch, E.: Cognitive representations of semantic categories. Journal of Experimental Psychology: General 104(3), 192–233 (1975) 10. Tang, Y., Lawry, J.: A prototype-based rule inference system incorporating linear functions. Fuzzy Sets and Systems 161, 2831–2853 (2010)

A Fuzzy Rule-Based Classification System Using Interval Type-2 Fuzzy Sets Min Tang1 , Xia Chen1 , Weidong Hu1 , and Wenxian Yu2 1

ATR Key Lab, National University of Defense Technology, Changsha 410073, China [email protected] 2 School of Electronic, Information and Electrical Engineering, Shanghai Jiaotong University, Shanghai 200030, China

Abstract. The design of type-2 fuzzy rule-based classification systems from labeled data is considered in this study. With the aid of interval type-2 fuzzy sets, which can effectively capture uncertainties in the data, a compact and interpretable interval type-2 fuzzy rule base with fewer rules is constructed. Corresponding type-2 fuzzy reasoning method for classification is also presented. The validity of this classification system is shown through experimental results on several data sets. Keywords: fuzzy rule-based classification system, interval type-2 fuzzy set, type-2 fuzzy rule base.

1

Introduction

Fuzzy rule-based systems[1] are a popular tool to solve pattern classification problems, which use linguistic variables and fuzzy logic to build interpretable models. The fuzzy rule-based classification systems have been widely employed to real applications such as intrusion detection, medical diagnosis, and shewhart control charts. The generation of fuzzy classification rule base is a critical problem in rule-based systems design. There are many approaches for extracting fuzzy classification rules from data, such as heuristic approach[2, 3] clustering methods [4, 5], nero-fuzzy approaches[6, 7], and genetic algorithms based schemes[8–10]. One of the main advantages of the fuzzy rule-based systems is the high interpretability of the rules. However, the inflexibility of the concept of linguistic variable imposes restrictions to the fuzzy rule structure[11], which degrades the system classification accuracy when dealing with some complex systems. For example, in the case that the classes overlap, the exact knowledge about the membership degree of some elements to the fuzzy sets that characterize the attributes defining the class is not possible. It has been shown that type-2 fuzzy sets (T2 FSs)[12] can manage the uncertainty of patterns more effectively than conventional fuzzy sets (T1 FSs) due the extra degree of freedom. This fact suggests that desirable results may be obtained by representing fuzzy attribute characterizations (of objects) with T2 FSs in rule-based classification context. Besides, with T2 FSs, the apparent paradox of modeling imprecise concepts using precise membership functions faced by T1 FSs is naturally addressed. Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 72–80, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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The design of type-2 fuzzy rule-based classification system has been limitedly discussed[13, 14]. In these works, the type 2 fuzzy sets used are constrained to a special class of Gaussian-form, which can not describe the uncertainty accurately. Besides, there is also a limitation in the number of classes for the problem to be resolved. In this paper, we design a fuzzy classification system from multiclass sample data using general interval type-2 fuzzy sets. A compact type-2 fuzzy rule base with fewer rules is generated with the aid of the representing effectiveness of type-2 fuzzy sets. In particular, interval type-2 fuzzy sets are deployed due to the relative small computational complexity for operations on them. A type2 fuzzy reasoning method is proposed accordingly. Simulation results show the validity of the proposed system generation method.

2

Type-2 Fuzzy Sets and Fuzzy Classification Rules

As a generalization of T1 FSs, the T2 FSs can effectively describe uncertainties in complex uncertain situations. A secondary membership function modeling the  uncertainty of the membership of exact T1 FSs is used in T2 FSs. A T2 FS A in the universal of discourse X is characterized by a type-2 fuzzy membership function µA (x) x ∈ X = A

 x∈X



 µA (x)/x =

x∈X

u∈Jx

 fx (u)/u /x Jx ⊆ [0, 1]

(1)

Where µA (x) is the type-2 fuzzy membership grade of element x,which is represented by the secondary membership function fx (u)[12]. Jx is the primary membership representing the domain of the secondary membership function. By extending the extra degree of freedom of primary membership, the T2 FSs are useful in modeling uncertainty. But the operations with regards to the general T2 FSs, like intersection, union, and type-reduction, require undesirably large amount of computations. However, the computational complexity can be reduced drastically by dealing with interval type-2 fuzzy sets (IT2 FSs). IT2 FSs are special T2 FSs whose secondary membership functions are interval sets    = A 1/u /x (2) x∈X

u∈Jx

Where Jx is a subinterval of [0,1]. In this case, the Jx x ∈ X completely deter Two T1 FSs, µ  (x) = maxu∈Jx (u) and µ (x) = minu∈Jx (u) mines the IT2 FS A.  A A  are called the upper memx ∈ X,which together are equivalent to the IT2 FS A, bership function and lower membership function respectively. Type-2 fuzzy classification rules can be constructed accordingly by extending their type-1 counterparts, with type-1 fuzzy sets replaced by type-2 fuzzy sets. We use type-2 fuzzy rules of the following form for a M classes problem, j , Rj : If x is A

j then (r1j , · · · , rM )

(3)

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Where Rj is the label of the j th rule, x = (x1 , · · · , xn ) is an n-dimensional j is an antecedent fuzzy set in the n-dimensional pattern space pattern vector, A j and rq is the certainty degree for the rule j to predict the class Cq q = 1, · · · , M for a pattern belonging to the fuzzy region represented by the antecedent of the rule. The type-1 counterpart of Rj is used in[15].

3

Design of Type-2 Fuzzy Rule-Based Classification System

The fuzzy rule base consisting of fuzzy classification rules establishes a relationship between the space of the features and the space of the classes. We generate type-2 fuzzy rules by using interval type-2 fuzzy C means algorithm for the derivation of IT2 FSs from labeled patterns in this study. A fuzzy reasoning method is proposed accordingly. 3.1

Generation of Interval Type-2 Fuzzy Rules from Data

Given a set of labeled patterns, we use the interval type-2 fuzzy C means (IT2 FCM) algorithm[16] to generate the IT2 FSs for the fuzzy rules to be derived. By allowing a variation of the fuzzifier in FCM, the IT2 FCM can manage the uncertainty in data more effectively. For the given set of labeled patterns, the IT2 FCM algorithm can be implemented in a supervised manner to derive a type-2 fuzzy partition of the pattern space. Since the number of clusters for each class and corresponding initial prototypes can be estimated from the labeled patterns. Each resulting cluster of the IT2 FCM algorithm describes a type-2 fuzzy subspace contained in the region of pattern space formed by the patterns for one class. A fuzzy classification rule is generated from each resulting cluster accordingly. For example, we can partition the pattern space into type-2 fuzzy subspaces with the number of subspaces the same as the number of classes if the patterns from different classes overlap but are separable to certain extent. Then each cluster represents a class. Fig.1 gives a bidimensional example of pattern distribution for three classes, in which each class can be effectively described by a IT2 FS. When the patterns from one class are better described by more than one cluster, we can include this requirement in the IT2 FCM by setting the corresponding initial cluster prototypes for patterns. Let the Mnumber of clusters for class Ck be lk . Then the total cluster number is C = k=1 lk . The IT2 FCM is implemented after the C cluster prototypes are initialized. Suppose that the resulting C cluster prototypes of the IT2 FCM be vj j = 1, . . . C. Then C type-2 fuzzy sets in the n-dimensional pattern space can be formulated. The membership grade bounds of any pattern x for cluster j can be calculated as follows.  1 1 1 C C C 2/(m1 −1) if 2/(m1 −1) > 2/(m2 −1) k=1 (dj /dk ) k=1 (dj /dk ) k=1 (dj /dk ) µj (x) =  (4) 1 otherwise C 2/(m2 −1) k=1 (dj /dk )  1 1 1 C C C 2/(m1 −1) if 2/(m1 −1) ≤ 2/(m2 −1) k=1 (dj /dk ) k=1 (dj /dk ) µj (x) =  k=1 (dj /d1k ) (5) otherwise C 2/(m −1) 2 (d /d ) k=1

j

k

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10

8

6

x

2

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0

−2

−4 −5

0

5 x

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Fig. 1. A bidimensional sample dataset

Where dj is the distance between x and the cluster prototype vj j = 1, . . . C, m1 and m2 are two fuzzifiers. j j = 1, . . . C, then µ  (x) = Let the j th IT2 FS obtained in this manner be A Aj [µj (x), µj (x)]. We construct a type-2 fuzzy rule base consisting of C rules. j , If x is A Where rqj =



c(xk )=Cq Uj (xk ) N k=1 Uj (xk )

j then (r1j , · · · , rM ) j = 1, . . . , C.

(6)

, with c(xk ) denotes the class label of training pat-

µ (xk )+µj (xk )

tern xk and Uj (xk ) = j . N is the number of training patterns. The 2 antecedent type-2 fuzzy sets model uncertainty in data more precisely and effectively. A compact rule base consisting of fewer rules is consequently constructed, which gives a robust description to the patterns by taking into account the certainty degrees for all classes in each rule. The three IT2 FSs generated by IT2 FCM from patterns in Fig.1 are shown in Fig.2-4 by setting the number of prototypes for each class be one and m1 = 2, m2 = 5. The upper and lower membership functions are plotted to describe the IT2 FSs in these figures. 3.2

A Fuzzy Reasoning Method

Given an input pattern, conclusions can be derived using the fuzzy reasoning method based on the set of fuzzy if-then rules. We give a reasoning method to match the generated type-2 fuzzy rules to realize class discrimination.

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1 0.8 0.6 0.4 0.2 0 10 10

5 5 0 x

0 −5

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−5

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Fig. 2. The interval type-2 fuzzy set corresponding with class “o”

1 0.8 0.6 0.4 0.2 0 10 10

5 5 0 x

0 −5

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Fig. 3. The interval type-2 fuzzy set corresponding with class “+”

Considering a new pattern x, the reasoning steps are the following: 1) Matching degree calculation. The matching degree is the strength of activation of the if-part for all rules in the rule base with the pattern x. Since the antecedent fuzzy sets are IT2 FSs, an interval matching degree can be obtained for each rule, i.e., µA j (x) = [µj (x), µj (x)].

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1 0.8 0.6 0.4 0.2 0 10 10

5 5 0 x

0 −5

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−5

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Fig. 4. The interval type-2 fuzzy set corresponding with class “×”

2) Association degree computation. The association degre bqj of pattern x with class Cq under the j th rule is obtained by combining µA j (x) and rqj with an aggregation operator h, bqj = h(µA j (x), rqj ). The “product” operator is used for h in this study. Therefore the association degree bqj is also an interval, which can be expressed as bqj = [bqjl , bqjr ] = [rqj · µj (x), rqj · µj (x)]. 3) Pattern classification soundness degree for all classes. Computing the soundness degree Yq of pattern x with class Cq , Yq = f (bqj , j = 1, . . . , C, bqjr > 0), f being a aggregation operator verifying min ≤ f ≤ max. The Quasiarithmetic mean s operator[15] is adopted in this study, i.e., f (a1 , . . . , as ) = H −1 [ 1s i=1 H(ai )], where H(a) = a20 . Yq still assumes a interval form due to the monotonous and continuous properties of H. Let Yq expressed as Yq = [Yql , Yqr ]. 4) Classification. Interval soundness degrees need to be treated after previous step 3). A simple discrimination criteria is used: assign x to the class Cq such Y +Y kr that ql 2 qr = maxk=1,...,M Ykl +Y . 2

4

Numerical Examples

The utility of the interval type-2 fuzzy classification system is demonstrated on two data sets. One is the dataset shown in Fig.1, and the other is the iris dataset from the UCI repository of machine learning databases. The dataset in Fig.1 contains 180 bidimensional patterns from three classes, with 60 patterns in each class, denoted by three point types. The iris dataset is composed of 150 4-dimensional patterns uniformly distributed among three classes.

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For each problem, we compare the presented interval type-2 fuzzy classification system with the type-1 and triangular type-2 fuzzy systems addressed in[17]. Ten runs of the methods are performed on each dataset. In each run, half of the patterns (90 patterns in the bidimensional problem, 75 patterns in the iris problem) are randomly chosen as training patterns, and the rest are used for testing the systems. In our presented method, the number of clusters for each class is chosen as one in both these experiments. Three type-2 rules are generated in each experiment. The parameters m1 and m2 in the supervised IT2 FCM algorithm are selected as 2 and 5 respectively. The mean of the training patterns in each class is used as the initial prototype for each cluster. The best, the worst as well as the average classification results for the ten independent runs of the three methods are shown in Table 1. For an intuitive perception, the classification result of one run for the sample dataset is shown in Fig. 5, where the patterns belonging to three classes are denoted by three types of points,the black points and red points represent training patterns and testing patterns respectively and the points covered by blue squares are misclassified patterns. It can be observed that the proposed interval type-2 classification system outperforms the type-1 and triangular type-2 systems, improvements in sometimes are significant. The classification accuracy could be enhanced through a further refinement to the system design process, such as the certainty degree derivation and class discrimination criteria determination. However, the aim of this study is mainly to stress the usability of type-2 fuzzy sets in constructing fuzzy classification rules from labeled patterns. Moreover, the effectiveness of IT2 FSs in representing uncertainty in data is also validated to be incorporable in the fuzzy reasoning process. 10

8

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Fig. 5. The classification result of the proposed interval type-2 system in one run

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Table 1. Classification accuracy of the type-1, triangular and interval type-2 methods Problem Result Type-1 (%) Triangular type-2 (%) Interval type-2 (%) Sample dataset Best 95.56 93.33 95.56 in Fig.1 Worst 86.67 86.67 87.78 Average 90.67 89.89 92.00 Iris Best 94.67 97.33 97.33 Worst 70.67 72.00 72.00 Average 84.00 85.07 88.80

5

Conclusions

A type-2 fuzzy rule-based classification system has been developed in this paper. The IT2 FSs are generated from labeled patterns using IT2 FCM algorithm, which give effective descriptions to the uncertainty in data in a compact form. A interpretable type-2 fuzzy rule base is constructed using the resulting IT2 FSs accordingly, which has fewer rules and models the relationship between feature space and class space effectively. A fuzzy reasoning method is also presented to match the generated type-2 fuzzy rules. Simulation results show that classification accuracy is improved by the interval type-2 fuzzy system as compared with its type-1 and triangular type-2 counterparts. The generated interval type-2 fuzzy system takes advantage of the effectiveness of IT2 FSs in representing the uncertainty in data and the classical fuzzy classification reasoning form, which is considered to be a very powerful tool for solving complex problems. Continuing work can be done to improve the presented interval type-2 method in applications, such as the determination of the number of prototypes for each class, identification of the fuzzifiers and refinement of fuzzy reasoning method.

References 1. Ishibuchi, H., Nakashima, T., Nii, M.: Classification and Modeling with Linguistic Information Granules: Advanced Approaches to Linguistic Data Mining. Springer, Heidelberg (2004) 2. Ishibuchi, H., Nozaki, K., Tanaka, H.: Distributed Representation of Fuzzy Rules and Its Application to Pattern Classification. Fuzzy Sets and Systems 52, 21–32 (1992) 3. Mansoori, E.G., Zolghadri, M.J., Katebi, S.D.: A Weighting Function for Improving Fuzzy Classification Systems Performance. Fuzzy Sets and Systems 158, 583–591 (2007) 4. Abe, S., Thawonmas, R.: A Fuzzy Classifier with Ellipsoidal Regions. IEEE Transactions on Fuzzy Systems 5, 358–368 (1997) 5. Roubos, J.A., Setnes, M., Abonyi, J.: Learning Fuzzy Classification Rules from Labeled Data. Information Sciences 150, 77–93 (2003) 6. Chakraborty, D., Pal, N.R.: A Neuro-Fuzzy Scheme for Simultaneous Feature Selection and Fuzzy Rule-Based Classification. IEEE Transactions on Neural Networks 15, 110–123 (2004)

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7. Nauck, D., Kruse, R.: A Neuro-Fuzzy Method to Learn Fuzzy Classification Rules from Data. Fuzzy Sets and Systems 89, 277–288 (1997) 8. Ishibuchi, H., Yamamoto, T., Nakashima, T.: Hybridization of Fuzzy GBML Approaches for Pattern Classification Problems. IEEE Transactions on Systems, Man, and Cybernetics, Part B 35, 359–365 (2005) 9. Mansoori, E.G., Zolghadri, M.J., Katebi, S.D.: SGERD: A Steady-State Genetic Algorithm for Extracting Fuzzy Classification Rules from Data. IEEE Transactions on Fuzzy Systems 16, 1061–1071 (2008) 10. Ho, S.Y., Chen, H.M., Ho, S.J., Chen, T.K.: Design of Accurate Classifiers with a Compact Fuzzy-Rule Base using An Evolutionary Scatter Partition of Feature Space. IEEE Transactions on Systems, Man, and Cybernetics, Part B 34, 1031– 1044 (2004) 11. Bastian, A.: How to Handle the Flexibility of Linguistic Variables with Applications. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2, 463–484 (1994) 12. Mendel, J.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice Hall (2001) 13. Liang, Q., Mendel, J.: MPEG VBR Video Traffic Modeling and Classification Using Fuzzy Technique. IEEE Transactions on Fuzzy Systems 9, 183–193 (2001) 14. Wu, H., Mendel, J.: Classification of Battlefield Ground Vehicles Using Acoustic Features and Fuzzy Logic Rule-Based Classifiers. IEEE Transactions on Fuzzy Systems 15, 56–72 (2007) 15. Cordn, O., del Jesus, M.J., Herrera, F.: A Proposal on Reasoning Methods in Fuzzy Rule-Based Classification Systems. International Journal of Approximate Reasoning 20, 21–45 (1999) 16. Hwang, C., Rhee, F.: Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy Approach to C-means. IEEE Transactions on Fuzzy Systems 15, 107–120 (2007) 17. Starczewski, J.T.: Efficient Triangular Type-2 Fuzzy Logic Systems. International Journal of Approximate Reasoning 50, 799–811 (2009)

Rough Approximations in General Approximation Spaces Keyun Qin1 , Zheng Pei2 , and Yang Xu1 1

2

College of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China School of Mathematics & Computer Engineering, Xihua University, Chengdu, Sichuan, 610039, China {keyunqin,pqyz}@263.net, [email protected]

Abstract. This paper is devoted to the discussion of rough approximations in general approximation space. The notions of transitive and Euclidean uncertainty mapping were introduced. The properties of some rough approximations were derived based on transitive and Euclidean uncertainty mappings. Additionally, it is pointed out that some existing approximation mappings are not suitable candidate for rough approximations. Keywords: Rough set, general approximation space, transitive uncertainty mapping, Euclidean uncertainty mapping, rough approximations.

1

Introduction

The theory of rough sets was firstly proposed by Pawlak [13, 14]. It is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of upper and lower approximations in rough set theory, knowledge hidden in information systems may be unravelled and expressed in the form of decision rules. In Pawlak’s rough set model, an equivalence relation is a key and primitive notion. The equivalence classes are the building blocks for the construction of the lower and upper approximations. This equivalence relation, however, seems to be a very stringent condition that may limit the application domain of the rough set model. To solve this problem, generalizations of rough sets were considered by some scholars. One generalization approach is to consider a similarity or tolerance relation [3, 8, 12, 17, 19, 20, 25–27] rather than an equivalence relation. Another generalization approach is to extend the partition of the universe to a cover [1, 2, 10, 15, 28–32]. The equivalence relation is replaced by a fuzzy relation to deal with data sets with both vagueness and fuzziness and the rough sets are generalized to fuzzy rough sets [4, 5, 9, 11, 16, 18, 21–24]. Gomolinska [6] provided a new approach for the study of rough approximations where the starting point is a generalized approximation space. The rough approximation operator was regarded as set-valued mapping, called approximation mapping. Two pairs of basic approximation mappings were defined typically Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 81–89, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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and generalized approximation mappings were constructed by the compositions of these basic approximation mappings. Some axioms for approximation mappings were proposed. Based on these axioms, the best low-approximation mapping was studied. This paper is devoted to the discussion of rough approximations in general approximation space. The notions of transitive and Euclidean uncertainty mapping were introduced. The properties of some rough approximations which are compositions of basic approximation mappings were derived based on transitive and Euclidean uncertainty mappings. Additionally, it is pointed out that, in accordance with Gomolinska’s axioms, the rough approximations f8 and f9 presented in [6] are not suitable candidate for rough approximations.

2

Preliminaries

This section presents a review of some fundamental notions of Pawlak’s rough sets. We refer to [13, 14] for details. Let U be a finite set, the universe of discourse, and R an equivalence relation on U , called an indiscernibility relation. The pair (U, R) is called a Pawlak approximation space. R will generate a partition U/R = {[x]R ; x ∈ U } on U , where [x]R is the equivalence class with respect to R containing x. For each X ⊆ U , the upper approximation R(X) and lower approximation R(X) of X are defined as [13, 14]: R(X) = {x; [x]R ∩ X = ∅},

(1)

R(X) = {x; [x]R ⊆ X}.

(2)

Alternatively, in terms of equivalence classes of R, the pair of lower and upper approximation can be defined by R(X) = ∪{[x]R ; [x]R ∩ X = ∅},

(3)

R(X) = ∪{[x]R ; [x]R ⊆ X}.

(4)

Let ∅ be the empty set and ∼ X the complement of X in U , the following conclusions have been established for Pawlak’s rough sets: (1) (2) (3) (4) (5) (6) (7) (8) (9)

R(U ) = U = R(U ). R(∅) = ∅ = R(∅). R(X) ⊆ X ⊆ R(X). R(X ∩ Y ) = R(X) ∩ R(X), R(X ∪ Y ) = R(X) ∪ R(Y ). R(R(X)) = R(X), R(R(X)) = R(X). R(X) =∼ R(∼ X), R(X) =∼ R(∼ X). X ⊆ Y ⇒ R(X) ⊆ R(Y ), R(X) ⊆ R(Y ). R(∼ R(X)) =∼ R(X), R(∼ R(X)) =∼ R(X). R(R(X)) ⊆ X ⊆ R(R(X)).

It has been shown that (3), (4) and (8) are the characteristic properties of the lower and upper approximations [25, 27].

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A General Notion of Rough Approximation Mapping

A general approximation space [6] is a triple A = (U, I, k), where U is a nonempty set called the universe, I : U → P (U ) is an uncertainty mapping, and k : P (U ) × P (U ) → [0, 1] is a rough inclusion function. In general approximation space A = (U, I, k), w ∈ I(u) is understood as w is in some sense similar to u and it is reasonable to assume that u ∈ I(u) for every u ∈ U . Then {I(u); u ∈ U } forms a covering of the universe U . The role of the uncertainty mapping may be played by a binary relation on U . In what follows, we suppose that u ∈ I(u) for every u, v ∈ U . We consider mappings f : P (U ) → P (U ). We can define a partial ordering relation, ≤, on the set of all such mappings as follows: f ≤ g if and only if ∀x ⊆ U (f (x) ⊆ g(x)), for every f, g : P (U ) → P (U ). By id we denote the identity mapping on P (U ). g ◦ f : P (U ) → P (U ) defined by g ◦ f (x) = g(f (x)) for every x ⊆ U , is the composition of f and g. We call g dual to f , written g = f d , if g(x) =∼ f (∼ x). The mapping f is monotone if and only if for every x, y ⊆ U , x ⊆ y implies f (x) ⊆ f (y). 3.1

Axioms for Rough Approximation Mappings

Theoretically speaking, every rough approximation operator is a mapping from P (U ) to P (U ), we call it approximation mapping. Gomolinska [6] proposed some fundamental properties that any reasonable rough approximation mapping f : P (U ) → P (U ) should possibly possess. They are the following axioms: (a1) Every low-mapping f is decreasing, i.e., f ≤ id. (a2) Every upp-mapping f is increasing, i.e., id ≤ f . (a3) If f is a low-mapping, then (∗)∀x ⊆ U ∀u ∈ f (x)(I(u) ⊆ x). (a4) If f is a upp-mapping, then (∗∗)∀x ⊆ U ∀u ∈ f (x)(I(u) ∩ x = ∅). (a5) For each x ⊆ U , f (x) is definable in A, i.e., there exists y ⊆ U such that f (x) = ∪{I(u); u ∈ y}. (a6) For each x ⊆ U definable in A, f (x) = x. The motivation behind these axioms was analyzed in [6]. Also, it is noticed that finding appropriate candidates for low- and upp-mappings satisfying these axioms is not an easy matter in general case. 3.2

The Structure of Rough Approximation Mappings

Let A = (U, I, k) be a general approximation space. The approximation mappings f0 , f1 : P (U ) → P (U ) were defined as [6]: for every x ⊆ U ,  f0 (x) = {I(u); u ∈ x}, (5) f1 (x) = {u; I(u) ∩ x = ∅}.

(6)

Observe that f0d and f1d satisfy: f0d (x) = {u; ∀w(u ∈ I(w) ⇒ w ∈ x)},

(7)

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f1d (x) = {u; I(u) ⊆ x}.

(8) f0d

f1d

If {I(u); u ∈ U } is a partition of U , then f0 = f1 , = and they are the classical rough approximation operators. Based on f0 , f1 and their dual mappings, several approximation mappings were defined in [6] by means of operations of composition and duality as follows: for every x ⊆ U ,  . f2 = f0 ◦ f1d : i.e., f2 (x) =  {I(u); I(u) ⊆ x}, . f3 = f0 ◦ f1 : i.e., f3 (x) = {I(u); I(u) ∩ x = ∅}, . f4 = f0d ◦ f1 = f2d : i.e., f4 (x) = {u; ∀w(u ∈ I(w) ⇒ I(w) ∩ x = ∅)}, . f5 = f0d ◦ f1d = f3d : i.e., f5 (x) = {u; ∀w(u ∈ I(w) ⇒ I(w) ⊆ x)}, . f6 = f1d ◦ f1d : i.e., f6 (x) = {u; ∀w(w ∈ I(u) ⇒ I(w) ⊆x)}, . f7 = f0 ◦ f6 = f0 ◦ f1d ◦ f1d = f2 ◦ f1d : i.e., f7 (x) = {I(u); ∀w(w ∈ I(u) ⇒ I(w) ⊆ x)}, . f8 = f1d ◦ f1 : i.e., f8 (x) = {u; ∀w(w ∈ I(u) ⇒ I(w) ∩x = ∅)}, . f9 = f0 ◦ f8 = f0 ◦ f1d ◦ f1 = f2 ◦ f1 : i.e., f9 (x) = {I(u); ∀w(w ∈ I(u) ⇒ I(w) ∩ x = ∅)}. Theorem 1. [6] Consider any f : P (U ) → P (U ). (1) f (x) is definable for any x ⊆ U iff there is a mapping g : P (U ) → P (U ) such that f = f0 ◦ g. (2) The condition (∗) is satisfied iff f ≤ f1d . (3) The condition (∗∗) is satisfied iff f ≤ f1 . Theorem 2. [6] For any sets x, y ⊆ U , we have that: (1) fi (∅) = ∅ and fi (U ) = U for i = 0, 1, · · · , 9. fid (∅) = ∅ and fid (U ) = U for i = 0, 1. (2) fi and fjd are monotone for i = 0, 1, · · · , 9 and j = 0, 1. (3) fi (x ∪ y) = fi (x) ∪ fi (y) for i = 0, 1, 3. (4) fi (x ∩ y) = fi (x) ∩ fi (y) and fjd (x ∩ y) = fjd (x) ∩ fjd (y) for i = 5, 6 and j = 0, 1. Theorem 3. [6] Let A = (U, I, k) be a general approximation space. (1) f5 ≤ f1d ≤ f2 ≤ id ≤ f4 ≤ f1 ≤ f3 . (2) f5 ≤ f0d ≤ id ≤ f0 ≤ f3 . (3) f6 ≤ f7 ≤ f1d . (4) f8 ≤ f9 ≤ f1 . (5) fi ◦ fi = fi for i = 2, 4. Example 1. Let U = {x, y}. We suppose that I : U → P (U ) is defined by: I(x) = {x}, I(y) = {x, y}. (1) By the definition, we have: f8 ({x}) = f1d (f1 ({x})) = f1d ({x, y}) = {x, y}, f8 ({y}) = f1d (f1 ({y})) = f1d ({y}) = ∅. f9 ({x}) = f0 (f8 ({x})) = f0 ({x, y}) = I(x) ∪ I(y) = {x, y}, f9 ({y}) = f0 (f8 ({y})) = f0 (∅) = ∅.

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It follows that neither f8 ≤ id nor id ≤ f8 holds in general. Similarly, we have f9  id and id  f9 . By axiom (a1) and (a2), neither f8 nor f9 is a suitable candidate for rough approximations. (2)Let f = f0 ◦ f0d . Then f ({x}) = f0 (f0d ({x})) = f0 (∅) = ∅, f ({y}) = f0 (f0d ({y})) = f0 ({y}) = I(y) = {x, y}. It follows that f  id and id  f . Consequently, f is not a suitable candidate for rough approximations. Based on f0 , f1d , we define f10 = f1d ◦ f0 , i.e., f10 (x) = {u ∈ U ; ∀v(v ∈ I(u) → ∃w ∈ x(v ∈ I(w)))} for every x ⊆ U . Theorem 4. For any sets x, y ⊆ U , we have that: (1) f10 (∅) = ∅, f10 (U ) = U . (2) id ≤ f10 . (3) x ⊆ y implies f10 (x) ⊆ f10 (y). (4) f10 (x ∩ y) ⊆ f10 (x) ∩ f10 (y), f10 (x ∪ y) ⊇ f10 (x) ∪ f10 (y). The proof of this theorem is straightforward. In view of the previous results, we summarize the rough approximations in the following table. By upp (resp., low) we denote upper (resp. lower) approximations, while ⊥ denotes that the corresponding composition is not a suitable candidate for rough approximations. Table 1. Rough approximations based on uncertainty mapping f f0 f1 f0d f1d d f0 f0 ◦ f0 (upp) f0 ◦ f1 (f3 , upp) ⊥ f0 ◦ f1 (f2 , low) d f1 f1 ◦ f0 (upp) f1 ◦ f1 (f6d , upp) f1 ◦ f0d (f10 , low) ⊥ f0d ⊥ f0d ◦ f1 (f4 , upp) f0d ◦ f0d (low) f0d ◦ f1d (f5 , low) f1d f1d ◦ f0 (f10 , upp) ⊥ f1d ◦ f0d (low) f1d ◦ f1d (f6 , low)

4

The Transitive and Euclidean Uncertainty Mapping

In view of the previous results and in accordance with the axioms, any lowor upp-mapping should have the form f0 ◦ g, where g : P (U ) → P (U ) satisfies f0 ◦ g ◦ f0 = f0 and, moreover, f0 ◦ g ≤ f1d in the lower case, while id ≤ f0 ◦ g ≤ f1 in the upper case [6]. Clearly, ≤ −maximal among the low-mappings

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and ≤ −minimal among the upp-mappings would be the best approximation operators. The greatest element among the low-mappings just described is the mapping h : P (U ) → P (U ) where for any x ⊆ U , h(x) = ∪{(f0 ◦ g)(x); g : P (U ) → P (U ) ∧ f0 ◦ g ◦ f0 = f0 ∧ f0 ◦ g ≤ f1d }.

(9)

It is noticed that an analogous construction, using ∩, does not provide us with the least element of the family of upp-mappings [6]. 4.1

The Transitive Uncertainty Mapping

Let A = (U, I, k) be a general approximation space. I is said to be a transitive uncertainty mapping [7], if u ∈ I(v) implies I(u) ⊆ I(v) for every u, v ∈ U . Theorem 5. [7] Consider any f : P (U ) → P (U ). f satisfies (a5) and (a6) if and only if there is a mapping g : P (U ) → P (U ) such that f = f0 ◦ g and f 0 ◦ g ◦ f0 = f0 . Theorem 6. [7] Let A = (U, I, k) be a general approximation space and I a transitive uncertainty mapping. (1) f0 ◦ f1d = f1d , (2) f1d ◦ f0 = f0 . (3) fi ◦ fi = fi for i = 0, 1. (4)f2 = f6 = f7 = f1d . (5) f4 = f1 . Theorem 7. [7] Let A = (U, I, k) be a general approximation space. There exists a lower approximation f which satisfy (a1), (a3), (a5) and (a6) if and only if I is a transitive uncertainty mapping. In this case, f1d is ≤ −maximal among the lower approximations which satisfy (a1), (a3), (a5) and (a6). This theorem shows that f1d is the best lower approximation if I is transitive. In this case, the rough approximations can be summarized in the following table: Table 2. Rough approximations based on transitive uncertainty mapping f f0 f1 f0d f1d d f0 f0 (upp) f0 ◦ f1 (f3 , upp) ⊥ f1 (f2 , low) d f1 f1 ◦ f0 (upp) f1 (f6d , upp) f0d (f10 , low) ⊥ f0d ⊥ f1 (f4 , upp) f0d (low) f0d ◦ f1d (f5 , low) f1d f0 (f10 , upp) ⊥ f1d ◦ f0d (low) f1d (f6 , low)

4.2

The Euclidean Uncertainty Mapping

In this subsection, we concentrate on properties specific for Euclidean uncertainty mapping.

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Definition 1. Let A = (U, I, k) be a general approximation space. (1) I is said to be a symmetric uncertainty mapping, if u ∈ I(v) implies v ∈ I(u) for every u, v ∈ U . (2) I is said to be an Euclidean uncertainty mapping, if u ∈ I(v) implies I(v) ⊆ I(u) for every u, v ∈ U . By this definition, if I is Euclidean, then I is symmetric. Theorem 8. Let A = (U, I, k) be a general approximation space. I is symmetric if and only if f0 = f1 . Proof. Assume that I is symmetric. Consider any x ⊆ U and u ∈ U . Then u ∈ f0 (x) ⇔ ∃w ∈ x(u ∈ I(w)) ⇔ ∃w ∈ x(w ∈ I(u)) ⇔ I(u) ∩ x = ∅ ⇔ u ∈ f1 (x). It follows that f0 = f1 . Conversely, assume that f0 = f1 . Let u, v ∈ U and u ∈ I(v). Then u ∈ I(v) = f0 ({v}) = f1 ({v}) = {w ∈ U ; I(w) ∩ {v} = ∅}. It follows that I(u) ∩ {v} = ∅ and hence v ∈ I(u) as required. Theorem 9. Let A = (U, I, k) be a general approximation space. I is Euclidean if and only if f1d ◦ f1 = f1 . Proof. Assume that I is Euclidean. Consider any x ⊆ U and u ∈ U . If u ∈ f1 (x), then I(u) ∩ x = ∅. For every v ∈ I(u), we have I(v) ∩ x ⊇ I(u) ∩ x = ∅. It follows that v ∈ f1 (x) and hence I(u) ⊆ f1 (x). Consequently, u ∈ {w ∈ U ; I(w) ⊆ f1 (x)} = f1d (f1 (x)) = (f1d ◦ f1 )(x), and hence f1 (x) ⊆ (f1d ◦ f1 )(x). By f1d ≤ id we know that f1 (x) ⊇ (f1d ◦ f1 )(x), and f1 (x) = (f1d ◦ f1 )(x) as required. Conversely, assume that f1d ◦ f1 = f1 . Let u, v ∈ U and u ∈ I(v). For every w ∈ I(v), by f1 ({w}) = {t ∈ U ; I(t) ∩ {w} = ∅} = {t ∈ U ; w ∈ I(t)}, we know that v ∈ f1 ({w}) = f1d (f1 ({w})) and hence I(v) ⊆ f1 ({w}). By u ∈ I(v), u ∈ f1 ({w}) followed and consequently w ∈ I(u). So we have I(v) ⊆ I(u) as required. Corollary 1. Let I be an Euclidean uncertainty mapping. Then (1) f1 ◦ f1d = f1d . (2) f0d ◦ f0 = f0 . (3) f0 ◦ f0d = f0d . Corollary 2. Let I be an Euclidean uncertainty mapping. Then (1) f2 = f0d . (2) f4 = f8 = f9 = f10 = f0 . (3) f3 = f0 ◦ f0 . (4) f5 = f6 = f7 = f0d ◦ f0d . We notice that an Euclidean uncertainty mapping need not necessarily to be transitive. So, in this case, the lower approximation which satisfy (a1), (a3), (a5) and (a6) does not exist in general according to Theorem 8.

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Acknowledgements. This work has been supported by the National Natural Science Foundation of China (Grant No. 60875034) and the Fundamental Research Funds for the Central Universities of China (Grant No. SWJTU09ZT37).

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Multi-agents and Non-classical Logic Systems Chenfang Zhao and Zheng Pei School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China [email protected]

Abstract. To model voting machine by internet, valuation of classical propositional calculus is extended, and multi-agents valuation of propositional calculus is proposed. Then formal concept analysis is used to express uncertainty of statements, i.e., degrees of truth value, the conclusion points out that non-classical logic systems is necessary to process uncertain information.

1

Introduction

Classical logic is used as a main tool in inference and decision making field in real world application. In addition, many non-classical logic systems such as Lukasiewicz logic, Goguen logic, G¨odel logic, fuzzy logic, etc [1]-[5] and random set theory, fuzzy set theory, rough set theory, etc [6]-[11] are presented and used to describe and deal with uncertainty. In this paper, a kind of uncertainty which is generated by multi-agents assignation, will be discussed. The application background of this uncertainty is voting machine by internet. As far as our knowledge concerned, voting machine by internet is widely used. In many cases, information of voting machine by internet is certain. However, when all information given by all voting agents (or voting users) is considered, uncertainty will generate, e.g., abstractly, for a statement p, when the question about “truth value (true or false) of p?” is asked, voting agent A maybe assign “true” to p; voting agent B maybe assign “false” to p, · · · . Now, considering all information assigned by voting agents, then p is “true” or “false”? From valuation of classical propositional calculus, the problem can be expressed as 1. For an voting agent, e.g., A, valuation of A is true or false, i.e. TA (p) = true or TA (p) = f alse ; 2. For all voting agents, valuation of voting agents is T(A,B,··· ) (p) = (true, f alse, · · · ). Uncertainty is inherent in extending one-dimension valuation true or f alse to multi-valuation (true, f alse, · · · ). To solve the meaning of fuzzy truth values or membership functions of fuzzy sets, a voting mechanism for fuzzy logic is proposed, and used to the fuzzy predicate calculus in literature [12]-[14]. Formal concept analysis(FCA) is a discipline that studies the hierarchical structures induced by a binary relation between Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 90–97, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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a pair of sets in literature [15]-[25]. In this paper, based on a voting mechanism for fuzzy logic, valuation of classical propositional calculus is extended, and multi-agents valuation of propositional calculus is proposed. Then FCA is used to express uncertainty of statements, i.e., degrees of truth value, the conclusion points out that non-classical logic systems is necessary to process uncertain information.

2

Preliminaries

In literature [12], a voting mechanism for fuzzy logic is proposed to extend notion of binary valuation for fuzzy concepts and fuzzy predicate. Formally, let L be a countable language of the propositional calculus consisting of a countable set of propositional variables P V L together with the connectives ∧, ∨ and ¬. Let SL denote the sentences of L. Definition 1. [12] A fuzzy valuation of L is a function F : SL×[0, 1] −→ {t, f } such that ∀θ ∈ SL, ∀0 ≤ y < y  ≤ 1, F (θ, y) = f =⇒ F (θ, y  ) = f and satisfies the following: ∀θ, φ ∈ SL, y ∈ [0, 1], 1)F (θ ∧ φ, y) = t ⇐⇒ F (θ, y) = t and F (φ, y) = t; 2)F (θ ∨ φ, y) = t ⇐⇒ F (θ, y) = t or F (φ, y) = t; 3) F (¬θ, y) = t ⇐⇒ F (θ, 1 − y) = f . In the Definition, y ∈ [0, 1] is viewed as the scepticism level of the voting agent. Based on the voting model, many interesting and important subjects of fuzzy logic have been discussed. Information systems and formal concept analysis(FCA) are widely used in data mining and knowledge discovering, etc [13]-[25]. Definition 2. A formal context is an ordered triple T = (G, M, I), where G, M are nonempty sets and I : G × M −→ {0, 1} is a binary relation. The elements of G are called objects and the elements of M attributes. I(g, m) = 1 means that object g has attribute m. Binary relation I of a formal context can be naturally represented by an twovalued table. The following two set-valued functions are used to define the formal concept of T = (G, M, I) ↑ : 2G → 2M , X ↑ = {m ∈ M ; ∀g ∈ X, I(g, m) = 1}, ↓ : 2M → 2G , Y ↓ = {g ∈ G; ∀m ∈ Y, I(g, m) = 1}.

(1) (2)

Definition 3. A formal concept of a context T = (G, M, I) is a pair (A, B) ∈ 2G × 2M such that A↑ = B and B ↓ = A. The set A is called its extent, the set B its intent.

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Multi-agents Valuation of Simple Statements

The propositional calculus considers ways in which simple statements may be combined to form more complex statements by using connectives ∧, ∨, → and ¬. Let L = {p1 , p2 , · · · , pn } be the set of simple statements, AG = {ag1 , ag2 , · · · , agm } the set of Multi-agents. Definition 4. A multi-agents valuation of propositional calculus is F : AG × L −→ {true, f alse}, (agi , pj ) −→ true(orf alse).

(3)

In which, F (agi , pj ) = true(orf alse) means that agent agi assigns “true (t)” (or “false (f )”) to simple statement pj ∈ L. It is noticed that a multi-agents valuation of propositional calculus can be expressed as in Table 1. Table 1. A multi-agents valuation table ag1 ag2 .. . agi .. . agm

p1 t (or f ) t (or f ) .. . t (or f ) .. . t (or f )

p2 t (or f) t (or f) .. . t (or f) .. . t (or f)

··· ··· ··· .. . ··· .. . ···

pj t (or f ) t (or f ) .. . t (or f ) .. . t (or f )

··· ··· ··· .. . ··· .. . ···

pn t (or f ) t (or f ) .. . t (or f ) .. . t (or f )

Comparing a multi-agents valuation with classical valuation of propositional calculus, for L = {p1 , p2 , · · · , pn }, 1. There are N = 2mn valuations instead of N = 2n ; 2. Table 1 can be understood as classical formal context, in which, L = {p1 , p2 , · · · , pn } is understood as set of attributes, AG = {ag1 , ag2 , · · · , agm } as set of objects. From this point of view, for every simple statement pj ∈ L, truth value of pj , denoted by Tpj , is an vector, i.e., Tpj = (t (or f ), t (or f ), · · · , t (or f )) ↑ ↑ ··· ↑ ag1 ag2 ··· agm

(4)

instead of Tpj = t (or f ) in classical propositional calculus 3. In multi-agents valuation, degree of truth value is necessary for every simple statement pj . Based on (4), it can be found that it is meaningless that considering valuation of pj is t or f under multi-agents environment. In our opinion, if classical logic system is understood as one-agent valuation, due to valuation “t or f ” in one-agent

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valuation is converted by vector (t (or f ), · · · , t (or f )) in multi-agents valuation, uncertainty (incomparability between two vectors) is created. It maybe a good idea to use non-classical logic systems to solve the problem. In this paper, the uncertainty is expressed by degree of truth value. Definition 5. For a fixed multi-agents valuation F and every pj ∈ L, degree of truth value of pj is DTpj =

|p↓j | , |AG|

(5)

in which, p↓j = {agi |F (agi , pj ) = t}. As a special case, if DTpj = 0, i.e., p↓j = {agi |F (agi , pj ) = t} = ∅ (every agent assigns f to pj ), then pj is called absolutely false under multi-agents environment. If DTpj = 1, i.e., p↓j = {agi |F (agi , pj ) = t} = AG (every agent assigns t to pj ), then pj is called absolutely true under multi-agents environment. In other cases, DTpj expresses some uncertainty of valuation of pj . Example 1. Let L = {p1 , p2 , · · · , p7 } be the set of simple statements, and AG = {ag1 , · · · , ag5 } the set of multi-agents. A multi-agents valuation is Table 2. Table 2. A multi-agents valuation table ag1 ag2 ag3 ag4 ag5

p1 t t t t t

p2 f t f f t

p3 t f f t t

p4 t f t t t

p5 f t f f f

p6 f t t f f

p7 f f f f f

According to Table.2, the following degrees of truth value can be computed |p↓ |

1 DTp1 = |AG| = 55 = 1, DTp2 = 25 , DTp3 = 35 , DTp4 = 45 , DTp5 = 15 , DTp6 = 25 and DTp7 = 0. In the valuation, truth value of p1 and p7 are absolutely true and absolutely false, respectively.

4

Degree of Truth Value of Complex Statements

From the viewpoint of logic, let L = {p1 , p2 , · · · , pn } be the set of simple statements, then all statements can be recursively generated by using connectives ∧, ∨, → and ¬as following: 1) Simple statement is a statement; 2) If pj is statement, then ¬pj is a statement; 3)If pj and pk are statement, then pj ∗ pk is a statement, in which, ∗ is ∧, ∨ or →; 4)All statements are generated finitely by using above three steps. In classical logic system, truth values of complex

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statements can be obtained by truth values of simple statements which is used to generate complex statements, i.e.,   t, if pj = f t, if pj = t and pk = t T¬pj = , Tpj ∧pk = (6) f, if pj = t f, otherwise  f, if pj = f and pk = f Tpj ∨pk = , Tpj →pk = T(¬pj )∨pk . (7) t, otherwise Similarly, degrees of truth value of complex statements under multi-agents environment are consider in this Section. Definition 6. Let L = {p1 , p2 , · · · , pn } be the set of simple statements, F be defined as in (3). For pj , pk ∈ L, ag1 ag2 agm F (¬pj ) = (T¬p , T¬p , · · · , T¬p ), j j j

F (pj ∧ pk ) = = F (pj ∨ pk ) = = F (pj → pk ) = =

(8)

1 m 1 m F (pj ) ∧ F (pk ) = (Tpag , · · · , Tpag ) ∧ (Tpag , · · · , Tpag ) j j k k ag1 ag2 agm (Tpj ∧pk , Tpj ∧pk , · · · , Tpj ∧pk ), 1 m 1 m F (pj ) ∨ F (pk ) = (Tpag , · · · , Tpag ) ∨ (Tpag , · · · , Tpag ) j j k k ag1 ag2 agm (Tpj ∨pk , Tpj ∨pk , · · · , Tpj ∨pk ), 1 m 1 m F (pj ) → F (pk ) = (Tpag , · · · , Tpag ) → (Tpag , · · · , Tpag ) j j k k ag1 ag2 agm (Tpj →pk , Tpj →pk , · · · , Tpj →pk ),

agi i in which, Tpag is truth value of pj assigned by i-th agent agi , T¬p , j j agi and Tpj →pk are obtained by (6) and (7), respectively.

i Tpag , j ∧pk

(9) (10) (11)

i Tpag j ∨pk

Example 2. As a continuation of Example 1. According to Table 1 and (8)ag1 ag5 (11), we have F (¬p1 ) = (T¬p , · · · , T¬p ) = (f, f, f, f, f ), F (¬p7 ) = (t, t, t, t, t), 1 1 1 5 F (¬p2 ) = (t, f, t, t, f ), F (¬p6 ) = (t, f, f, t, t), F (p2 ∧p6 ) = (Tpag , · · · , Tpag )= 1 ∧p6 1 ∧p6 (f, t, f, f, f ), F (p1 ∧ p7 ) = (f, f, f, f, f ), F (p3 ∧ p4 ) = (t, f, f, t, t), F (p3 ∧ p6 ) = 1 5 (f, f, f, f, f ), F (p2 ∨ p6 ) = (Tpag , · · · , Tpag ) = (f, t, t, f, t), F (p1 ∨ p7 ) 1 ∨p6 1 ∨p6 = (t, t, t, t, t), F (p3 ∨ p4 ) = (t, f, t, t, t), F (p3 ∨ p6 ) = (t, t, t, t, t), F (p2 → p6 ) = 1 5 (Tpag , · · · , Tpag ) = (t, t, t, t, f ), F (p1 → p7 ) = (f, f, f, f, f ), F (p3 → p4 ) = 1 →p6 1 →p6 (t, t, t, t, t), F (p3 → p6 ) = (f, t, t, f, f ). Property 1. Let F be a multi-agents valuation. ∀pj ∈ L, DT¬pj =

|p↓j | |(¬pj )↓ | =1− = 1 − DTpj , |AG| |AG|

(12)

where, DT¬pj is degree of truth value of ¬pj and (¬pj )↓ = {agi ∈ AG|F (agi , ¬pj ) = t}. agi i Proof. According to (6), F (agi , ¬pj ) = t ⇐⇒ T¬p = t ⇐⇒ Tpag = f ⇐⇒ j j ↓ F (agi , pj ) = f , this means that (¬pj ) = {agi|F (agi , ¬pj ) = t} ⇐⇒ {agi |F (agi , pj ) = f } ⇐⇒ AG − (pj )↓ = AG − {agi |F (agi , pj ) = t}, hence,

DT¬pj

|AG| − |p↓j | |p↓j | |(¬pj )↓ | = = = 1− = 1 − DTpj . |AG| |AG| |AG|

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Property 2. Let F be a multi-agents valuation. ∀pj , pk ∈ L, DTpj ∧pk = DTpj ∨pk

|(pj ∧ pk )↓ | |{pj , pk }↓ | = , |AG| |AG|

|p↓j ∪ p↓k | |(pj ∨ pk )↓ | = = , |AG| |AG|

(13) (14)

where, DTpj ∧pk and DTpj ∨pk are degree of truth value of pj ∧ pk and pj ∨ pk , respectively. (pj ∧ pk )↓ = {agi ∈ AG|F (agi , pj ∧ pk ) = t} and (pj ∨ pk )↓ = {agi ∈ AG|F (agi , pj ∨ pk ) = t}. i i Proof. According to (6) and (7), F (agi , pj ∧pk ) = t ⇐⇒ Tpag = t ⇐⇒ Tpag =t j ∧pk j i and Tpag = t ⇐⇒ F (ag , p ) = t and F (ag , p ) = t, F (ag , p ∨ p ) = t ⇐⇒ i j i k i j k k i i i Tpag = t ⇐⇒ Tpag = t or Tpag = t ⇐⇒ F (agi , pj ) = t or F (agi , pk ) = t, j ∨pk j k these mean that (pj ∧ pk )↓ = {agi ∈ AG|F (agi , pj ) = t and F (agi , pk ) = t} = {pj , pk }↓ , (pj ∨ pk )↓ = {agi ∈ AG|F (agi , pj ) = t or F (agi , pk ) = t} = {agi ∈ AG|F (agi , pj ) = t}∪ {agi ∈ AG|F (agi , pk ) = t} = p↓j ∪ p↓k .

Property 3. Let F be a multi-agents valuation. ∀pj , pk ∈ L, DTpj →pk =

|(AG − p↓j ) ∪ p↓k | |(¬pj )↓ ∪ p↓k | |(pj → pk )↓ | = = , |AG| |AG| |AG|

(15)

where, DTpj →pk is degree of truth value of pj → pk . (pj → pk )↓ = {agi ∈ AG|F (agi , pj → pk ) = t}. agi i Proof. According to (7), F (agi , pj → pk ) = t ⇐⇒ Tpag = t ⇐⇒ T¬p = t or j →pk j agi Tpk = t ⇐⇒ F (agi , ¬pj ) = t or F (agi , pk ) = t, these mean that (pj → pk )↓ = {agi ∈ AG|F (agi , ¬pj ) = t or F (agi , pk ) = t} = (¬pj )↓ ∪ p↓k = (AG − p↓j ) ∪ p↓k .

Example 3. As a continuation of Example 1. In Table 1, p↓1 = AG, p↓2 = {ag2 , ag5 }, p↓3 = {ag1 , ag4 , ag5 }, p↓5 = {ag2 }, p↓4 = {ag1 , ag3 , ag4 , ag5 }, p↓6 = {ag2 , ag3 }, p↓7 = ∅. DTp1 ∧p7 = 0, DTp2 ∧p6 = 15 , DTp3 ∧p4 = 35 , DTp3 ∧p6 = 0, DTp1 ∨p7 = 1, DTp2 ∨p6 = 35 , DTp3 ∨p4 = 45 , DTp3 ∨p6 = 1, DTp1 →p7 = 0, DTp2 →p6 = 45 , DTp3 →p4 = 1, DTp4 →p3 = 45 , DTp3 →p6 = 25 . In the example, it can be noticed that DTpj ∧pk = min{DTpj , DTpk } and DTpj ∨pk = max{DTpj , DTpk } do not always hold, e.g., for DTp2 ∧p6 and DTp2 ∨p6 , DTp2 ∧p6 = min{DTp2 , DTp6 } and DTp2 ∨p6 = max{DTp2 , DTp6 }. On the other hand, it is difficult to connect → in multi-agents valuation with → in many existing valued logic systems. However, in some special case, a multi-agents valuation is Lukasiewicz logic system. Corollary 1. If a multi-agents valuation F satisfies that ∀pj , pk ∈ L, either p↓j ⊆ p↓k or p↓j ⊇ p↓k , then DTpj ∧pk = min{DTpj , DTpk }, DTpj ∨pk = max{DTpj , DTpk } and DTpj →pk = min{1, 1 − DTpj + DTpk }.

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Proof. According to Proposition 2, if ∀pj , pk ∈ L, either p↓j ⊆ p↓k or p↓j ⊇ p↓k , it is easy to prove DTpj ∧pk = min{DTpj , DTpk } and DTpj ∨pk = max{DTpj , DTpk }. For DTpj →pk , according to Proposition 3 and set operation, (AG − p↓j ) ∪ p↓k = AG − (p↓j ∩ (AG − p↓k )), if p↓j ⊆ p↓k , then p↓j ∩ (AG − p↓k ) = ∅. Hence, DTpj →pk =

↓ |(AG−p↓ j )∪pk | |AG|

↓ |AG−(p↓ j ∩(AG−pk ))| = |AG| = 1, if p↓j ⊇ p↓k , then p↓j ∩(AG−p↓k ) = |AG| |AG| ↓ ↓ ↓ |(AG−p↓ |AG−(p↓ |AG|−|p↓ j )∪pk | j ∩(AG−pk ))| j |+|pk | DTpj →pk = = = |AG| |AG| |AG|

=

p↓j − p↓k . Hence, = 1 − DTpj + DTpk , finally, it can be obtained that DTpj →pk = min{1, 1 − DTpj + DTpk }.

Example 4. In Example 3, p↓3 = {ag1 , ag4 , ag5 } ⊆ p↓4 = {ag1 , ag3 , ag4 , ag5 }, and DTp3 ∧p4 = 35 = DTp3 = min{DTp3 , DTp4 }, DTp3 ∨p4 = 45 = DTp4 = max{DTp3 , DTp4 }, DTp3 →p4 = 1 = min{1, 1− 53 + 45 } = min{1, 1−DTp3 +DTp4 }, DTp4 →p3 = 45 = min{1, 1 − 45 + 35 } = min{1, 1 − DTp3 + DTp4 }.

5

Conclusion

In this paper, multi-agents valuation of propositional calculus is discussed. Then FCA is used to express uncertainty of statements, i.e., degrees of truth value. In a special case, multi-agents valuation of complex statements can be processed by Lukasiewicz logic system. Acknowledgments. The authors would like to thank the research fund of sichuan key laboratory of intelligent network information processing (SGXZD1002-10), key laboratory of the radio signals intelligent processing (Xihua university) (XZD0818-09) and the fund of Key Disciplinary of Computer Software and Theory, Sichuan. Grant No.SZD0802-09-1.

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9. Fortin, J., Dubois, D., Fargier, H.: Gradual numbers and their application to fuzzy interval analysis. IEEE Transactions on Fuzzy Systems 16(2), 388–402 (2008) 10. Cooman, G.: A behavioural model for vague probability assessments. Fuzzy Sets and Systems 154, 305–358 (2005) 11. Zadeh, L.A.: Toward a theory of fuzzy information granulation and its centrality in houman reasoning and fuzzy logic. Fuzzy Sets and Systems 90, 103–111 (1997) 12. Lawry, J.: A voting mechanism for fuzzy logic. International Journal of Approximate Reasoning 19, 315–333 (1998) 13. Lawry, J.: A methodology for computing with words. International Journal of Approximate Reasoning 28, 51–89 (2001) 14. Lawry, J.: A framework for linguistic modelling. Artificial Intelligence 155, 1–39 (2004) 15. Wille, R.: Concepe lattices and conceptual knowledge systems. Comput. Math. Apll. 23(6-9), 493–515 (1992) 16. Lawry, J., Tang, Y.: Granular knowledge representation and inference using labels and label expressions. IEEE Transactions on Fuzzy Systems 18(3), 500–514 (2010) 17. Pei, Z., Ruan, D., Liu, J., Xu, Y.: Linguistic Values based Intelligent Information Processing: Theory, Methods, and Application. In: Atlantis Computational Intelligence Systems, vol. 1. Atlantis press & World Scientific (2009) 18. Pei, Z., Xu, Y., Ruan, D., Qin, K.: Extracting complex linguistic data summaries from personnel database via simple linguistic aggregations. Information Sciences 179, 2325–2332 (2009) 19. Pei, Z., Resconi, G., Van Der Wal, A.J., Qin, K., Xu, Y.: Interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference. Information Sciences 176, 1869–1897 (2006) 20. Jin, J., Qin, K., Pei, Z.: Reduction-Based Approaches Towards Constructing Galois (Concept) Lattices. In: Wang, G.-Y., Peters, J.F., Skowron, A., Yao, Y. (eds.) RSKT 2006. LNCS (LNAI), vol. 4062, pp. 107–113. Springer, Heidelberg (2006) 21. Stumme, G., Taouil, R., Bastide, Y., Pasquier, N., Lakhal, L.: Computing iceberg concept lattices with TITANIC. Data & Knowledge Engineering 42, 189–222 (2002) 22. Berry, A., Sigayret, A.: Representing a concept lattice by a graph. Discrete Applied Mathematics 144, 27–42 (2004) 23. Berry, A., SanJuan, E., Sigayret, A.: Generalized domination in closure systems. Discrete Applied Mathematics 154, 1064–1084 (2006) 24. Diday, E., Emilion, R.: Maximal and stochastic Galois lattices. Discrete Applied Mathematics 127, 271–284 (2003) 25. Kim, M., Compton, P.: Evolutionary document management and retrieval for specialized domains on the web. Int. J. Human-Computer Studies 60, 201–241 (2004)

An Information Processing Model for Emotional Agents Based on the OCC Model and the Mood Congruent Effect Chao Ma, Guanghong Gong, and Yaofei Ma Beijing University of Aeronautics and Astronautics, Advanced Simulation Technology Lab, Dept. of ASEE, Xueyuan Road 37, 100191, Beijing, China [email protected]

Abstract. Emotional Agents can be regarded as traditional ones with emotional factors. There are differences between emotional Agents and traditional Agents in information perception and processing. This paper mainly deals with the design of cognitive module (information processing module) for emotional Agents. The design contains mathematical approaches to human information processing, and also takes account of the achievements in modern Psychology. The cognitive module is easy to be applied in engineering, which makes the design suitable for most circumstances. Keywords: Emotional Agent, OCC Model, Mood Congruent Effect, Cognition, Information Processing.

1

Introduction

Agent is an important concept in Artificial Intelligence(AI). An Agent is anything that can be viewed as perceiving its environment through sensors and acting upon that environment through actuators [1]. The intelligence of Agents reflects the state of the art of AI directly. During the early days, researchers paid much attention to building up rational machines which can make proper decisions according to the environment and the problems they have to face. However, they failed to consider the irrational factors such as emotion and affect. One of the founders of AI, Marvin Minsky has stated: “The question is not whether intelligent machines can have any emotions, but whether machines can be intelligent without any emotions [2].” This statement figured out the relations between emotions and intelligence: Agents cannot achieve true intelligence without emotions. Constructing emotional Agents is to equip the Agents with some emotional factors and take them as influences during the information process. Agents thus will appear to be more human-like. AI researchers have plumbed into emotional Agent construction in different application fields [3] [4]. Emotional machines have been realized in game industry [5], intelligent traffic [6] and etc. Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 98–108, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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99

The Structure of Emotional Agents

Emotional Agents are traditional ones with emotional factors. The structure of emotional Agents is depicted in Fig.1. The cognitive module containing information processing model which is the main topic in this paper is a substantial part of emotional Agents.

Fig. 1. The structure of emotional Agents

Sensor: in charge of perceiving information (the occurrence of events, the state change of other Agents, the parameter change of the objects) from the outside world. Cognitive module: processes the information taking into account of emotional states of the Agents. The information processed will then be used as factors to produce emotional states and to make decisions. The principle of this module is mood congruent effect [7], and the mathematical model is the information processing model, which we will cover later. Emotional module: produces emotional states of the Agents, taking information processed by the cognitive module as a factor. It’s based on OCC model [8]. Performance element module: makes decision of what to do next for the Agents, taking information from cognitive module and the emotional states produced by emotional module as input. Knowledge refresh module: refreshes the knowledge base of performance element module to bring about new knowledge to the system. The cognitive module is the main focus of this paper. This module imitates human information perception in the real world. Human beings do not make use of all the information perceived by the perceptional organs. In other words, human beings “choose” the information according to certain laws. And emotion plays an important role in this process. To make the Agents more like a human, we simulate the effects of emotions on information processing.

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When the Agent is working, the emotional states produced by the emotional module using OCC model will be transferred to the cognitive module, and this module will process the information in the guide of the information processing model and the mood congruent effect.

3

OCC Model

There are two main distinguished viewpoints in the modeling of emotions: cognitive theories and dimensional theories of emotion [9]. OCC model belongs to the former. OCC Model is brought forward by A. Ortony, G.Clore and A.Collins in 1988 [8]. It hypotheses that emotions are aroused by the occurrence of events, the appraise of Agents and the liking of objects. Hence, we can inspire artificial emotions using events, states of Agents and attributes of objects in the emotional module. The OCC Model defined 22 basic emotions, tabled below [10] in Table 1: Table 1. OCC’s pairs of emotions Groups of Emotion

Positive Emotion Negative Emotion

Fortunes of Others

Happy for

Pity

Gloating

Resentment

Hope

Fear

Satisfaction

Fears Confirmed

Relief

Disappointment

Well-being

Joy

Distress

Attribution

Pride

Shame

Admiration

Reproach

Prospect Based

Well-being/Attribution Gratification

Attraction

Remorse

Gratitude

Anger

Love

Hate

The model divided the 22 emotions into 11 pairs. In each pair, there is a positive emotion and a negative one. There are also other methods to divide emotions, which are beyond the scope of this paper. In the cognitive module discussed later, we will mark the emotions as positive or negative according to this table.

4

Mood Congruent Effect

The Mood Congruent Effect is the basic rule the Agents have to obey when processing information and also the principle for the design of cognitive module.

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We use the term mood congruent effect to refer to the phenomenon that people tend to choose the information that is congruent with their emotional states, which indicates some startup effect of emotions (i.e. people with positive emotions tend to believe the positive information from outside world and vise versa). Sometimes, we also call it the emotion congruent effect [7]. The Mood Congruent Effect can be recognized as mood congruent memory and mood congruent processing. For the reason that the cognitive module we built does not have a memory part, we only implement mood congruent processing which means the test-takers with a certain emotion will be apt to choose the information congruent with it. This effect depicts the relations between human emotions and human information processing. In order to imitate these relations, the emotional Agents should embody this effect. If the Agent is in a positive emotional state, it will probably choose more information that will better its emotion rather than choose the information that will bring it to a negative state. We can divide the information into positive and negative groups. Positive information refers to the information that will probably bring the Agent to a positive emotional state and negative information the information does the opposite. There is a third information group, which will be best described as neutral information because it refers to the information that will hardly ever arouse any emotions.

5

Information Processing Model

The information processing model is the key part of the cognitive module. This model is based on the OCC Model. 5.1

Descriptions of Emotion

We have to classify and quantitate different emotions to make it easier to be used in the information processing model. There are varieties of methods to quantitate emotions, and we will adopt the n-dimensional method mentioned by Schlosberg and Lang [11]. We instantiate n as 1. The magnitude refers to intensity and the scale to direction. We can consider the emotions of the same row in table1 as a pair, for example gratitude and anger. Variable S ∈ (−1, 1) is introduced to describe them. In the case that S > 0, we explain the emotion as gratitude, and the intensity of the emotion is |S|; otherwise, we explain the emotion as anger, the intensity is also |S|. If S → 1, the emotion of gratitude tends to be greatest in intensity, while when S → −1, the emotion of anger tends to be greatest. When S = 0, the emotional state is calm. In the information processing model, we only care about whether the emotional state is positive or negative without drilling into certain concrete emotions such as gratitude and anger. Based on this simplification, it can be understood as when S > 0, the emotional state is positive with an intensity of |S| and when S < 0, the emotional state is negative also with an intensity of |S|.

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Common Information Processing

The common information mentioned here is the information that will not have strong effects on the existence or the main goal of the Agents, which can also be expressed as information that could be ignored in the decision-making process. We can draw some conclusions according to the Mood Congruent Effect: 1. Humans can only deal with part of the information from the outside world; and 2. the choosing of information will be influenced by emotional factors. Considering these characteristics, we can describe the information processing stochastically. If the Agent is in a positive emotion, it will choose positive information with higher possibilities, and vise versa. We define this possibility as the possibility of believing (PB). A General Mathematical Model. In a general condition, when S ∈ (−1, 1), we set P (S)as the PB of positive information and P  (S)as the PB of negative information. We make two assumptions based on the fact that the ability of human information processing is limited and emotional factors will influence the PB of the information: 1. Positive emotions and negative emotions have the same mechanism which influences the PB, but the effect is opposite. The PB of positive information influenced by emotion S equals the PB of negative information influenced by emotion −S. In mathematical form, we have P (S) = P  (−S). 2. The aggregate possibility of the PB will be a constant, and we set it as D. D is also the PB for the neutral information. According to the assumptions, we can infer: xP (S) + (1 − x)P  (S) = xP (S) + (1 − x)P (−S) = D

(1)

Where S ∈ (−1, 1), x stands for the proportion of positive information in non-neutral information. Check (1), we can get P (0) = D. It is easy to be observed that when D is constant and x varies, P (S) and P (−S) will have different values, too. 5.3

Key Information Processing

Key information refers to the information should not be ignored in decisionmaking process. In the real world, this information should not be neglected and has much greater importance and priority. The method we use to deal with key information is different from the one with common information. We adopt stochastic method to imitate the processing of common information for the reason that it can be ignored without bringing

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about calamitous results. However, if we adopted the same method for key information, the Agent would be insensitive to the substantial information from the environment, which will lead to the decline of the adaptation ability of the Agents. Thus, we do not consider stochastic method for key information. We have to make sure the Agent will get the key information immediately. This approach reflects the priority in human information-process. D. A. Norman and et al have also introduced different methods for different information with varieties of priorities [12], which is an imitation of the actions we human will take when processing information.

6

Cognitive Module

Cognitive module is an implement of the information processing model in Agent structure. 6.1

The Structure of Cognitive Module

There are three kinds of information from the outside world (occurrence of events, states of Agents, attributes of objects). Cognitive module accepts these three arouses as input. We only take the occurrence of events as an example to explain the work of cognitive module and the processing of the other two kinds of information are actually similar. The structure of cognitive module is displayed in Fig.2. Input: The input comes from the sensors. The sensors perceive the information outside and then transfer it to the cognitive module for further processing. Output: The output becomes the inputs for the emotional module and the performance element module. The inner structure: Cognitive module can be divided into two sub-modules, namely the appraise sub-module and the main-processor sub-module. Appraise: Provide elementary process of information and evaluate what emotional state will be aroused by the current event. Main processor: Take the emotional states and information from appraise sub-module as inputs and use the information processing model to decide which information to take and which to drop. 6.2

Main Processor Sub-module

This sub-module is the implement of the information processing model. First of all, we have to declare some definitions: Event attribute: the category that the event belongs to, such as positive, negative and neutral.

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Fig. 2. The structure of the cognitive module

Event attribute value (θ): 1 for positive event, −1 for negative and 0 for neutral. Event aggregate: the aggregate of events perceived by the sensors in one time step. Event aggregate attribute value (Θ):  Θ= kθ (2) Where k stands for the ratio between a certain kind of believed events and the events in the event aggregate. Assume that the ratio between positive events and the event aggregate is lp , the ratio between negative events and the event aggregate is ln , the ratio between neutral events and the event aggregate is lneu , then we get Θ = lp P (S) · 1 + ln P (−S) · (−1) + lneu D · 0 = lp P (S) − ln P (−S)

(3)

Θ > 0 reflects the fact that the Agent takes the environment as positive while Θ < 0 as negative. The attribute value defined as 0, the neutral events have no effect on the event aggregate attribute value. Hence, divide equation (3) by (lp + ln ) and consider lp x = (lp +l we will get n) Θ =

Θ = xP (S) − (1 − x)P (−S) (lp + ln )

(4)

When the event aggregate is given, the difference between Θ and Θ is just a constant ratio. Thus Θ is also capable of describing the attribute of the event aggregate. What’s more, equation (4) is much easier to be used together with equation (1). According to the research by Eric J. Johnson and et al [13], people will overevaluate risks by 74% when in a negative emotion (three groups of experiments

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were carried out, and over-evaluation was observed in each group. The percentages are 133%, 56% and 50%, average 74%). This risk over-evaluation is also caused by the mood congruent effect so it is sufficient to embody the strength of the effect. Other psychological experiments also brought us similar results [14] [15]. In the information processing model, we take the effect of mood congruent effect as aggrandizing the event aggregate attribute (compared with the situation in a calm state). And the percentage of aggrandizement will be around 74%. The event aggregate attribute in a calm state is xP (0) − (1 − x)P (−0) = (2x − 1)P (0) = (2x − 1)D, so we can get the equation as follows: When x ∈ [0.5, 1], S ∈ (0, 1) (positive information & positive emotion) or x ∈ [0, 0.5], S ∈ (−1, 0) (negative information & negative emotion), there exists: xP (S) − (1 − x)P (−S) = 1.74(2x − 1)D The simultaneous equations of (5) & (1):  xP (S) + (1 − x)P (−S) = D xP (S) − (1 − x)P (−S) = 1.74(2x − 1)D The result is:



P (S) = (1.74x−0.37)D x P (−S) = (1.37−1.74x)D 1−x

(5)

(6)

(7)

Considering a value of possibility will be constraint to [0, 1], we illustrate P (S) and P (−S) in Fig.3. When D and S vary, we get the curves of P (S) in Fig.4. Different Ds mean different considerations of the information perceived by the Agents. The greater the D, the more information will be taken into account by the Agents so that they can make more accurate judgments on the situation they have to face; and the smaller the D, the less sensitive the Agents. In practice, the users of this model can set the D themselves. To test this emotional model, we built up a testing framework called emotional model testing framework. In this framework, we can view documents and details of the models, take tests of them, record the data and make some analysis as well. What’s more, we apply this model in some scenarios such as flight route planning demo. In this demo, Agents with different emotions (Hope and Fear) have to choose their own route between two places separated by a high mountain. See Fig.5. From Fig.5, we can conclude that Flight Agents with positive emotions tend to choose short routes with high risks (higher mountains etc.) like route A; while Flight Agents with negative emotions tend to choose longer routes with lower risks like route B. This means that Agents with positive emotions will pay more attention to the positive information such as shorter routes while Agents with negative emotions will pay more attention to negative information such as danger when making decisions.

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Fig. 3. Distributions of P (S) and P (−S)

Fig. 4. Curves of P (S) and P (−S)

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Fig. 5. Emotional Model Testing Framework & Flight Route Planning Demo

7

Conclusion

In this paper, we weaken the effects of neutral information. We only put positive and negative information in the event aggregate attribute but this kind of approach is enough for engineering and the calculation is comparably easy. If we take neutral information into account, we have to consider the ratio between neutral event and the event aggregate to decide the attribute of the aggregate. All in all, the method we offer in the paper is easy to be realized and able to embody the effect of emotion in the Agents. What’s more, the psychological foundation is firm. This method can be used in most Agent systems to increase the similarity between human information processing and the Agent information processing. However, the human information processing is a complicated, nonlinear process, so more accurate consideration will need much more efforts of AI researchers.

References 1. Russell, S.J., Norvig, P.: Artificial Intelligence: A Modern Approach, 2nd edn. Prentice Hall, New Jersey (2003) 2. Minsky, M.: The Society of Mind. Simon & Schuster, New York (1988) 3. Turkia, M.: A Computational Model of Affects. CoRR, Vol.abs/081 1.0123 (2008) 4. Barteneva, D., Lau, N., Reis, L.P.: A Computational Study on Emotions and Temperament in Multi-Agent Systems. CoRR, Vol.abs/0809.4784 (2008) 5. Qiang, J., Lan, P., Looney, C.: A Probabilistic Framework for Modeling and Realtime Monitoring Human Fatigue. IEEE Trans.Systems, Man and Cybernetics, Part A: Systems and Humans 36, 862–875 (2006) 6. Slater, S., Moreton, R., Buckley, K., Bridges, A.: A Review of Agent Emotion Architectures. Eludamos Journal for Computer Game Culture 2, 203–214 (2008) 7. Zhuang, J.: The Psychology of Decision-Making. Shanghai Educational Press, Shanghai (2006)

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8. Ortony, A., Clore, G.L., Collins, A.: The Cognitive Structure of Emotions. Cambridge University Press, New York (1988) 9. MacDorman, K.F., Ishiguro, H.: Generating Natural Motion in an Android by Mapping Human Motion. In: Proceeding IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, pp. 3301–3308. IEEE Press, New York (2005) 10. Trabelsi, A., Frasson, C.: The Emotional Machine: A Machine Learning Approach to Online Prediction of User’s Emotion and Intensity. In: 2010 10th IEEE International Conference on Advanced Learning Technologies, pp. 613–617. IEEE Press, New York (2010) 11. Wang, Z.: Artificial Emotion. China Machine Press, Beijing (2007) 12. Norman, D.A., Ortony, A., Russell, D.M.: Affect and Machine Design: Lessons for the Development of Autonomous Machines. IBM Systems Journal 42(1), 38–44 (2003) 13. Johnson, E.J., Tversky, A.: Affect, Generalization and the Perception of Risk. Journal of Personality and Social Psychology 45(1), 20–31 (1983) 14. Eysenck, M.W.: Anxiety: The Cognitive Perspective. Erlbaum, Hove (1992) 15. Lerner, J.S., Keltner, D.: Beyond Valence: Toward a Model of Emotion-specific Influences on Judgement and Choice. Cognition and Emotion 14, 473–494 (2000)

On Distributive Equations of Implications and Contrapositive Symmetry Equations of Implications Based on a Continuous t-Norm Feng Qin1,2 and Meihua Lu2 1

College of Mathematics and Information Science, Nanchang Hangkong University, 330063, Nanchang, P.R. China 2 College of Mathematics and Information Science, Jiangxi Normal University, 330022, Nanchang, P.R. China [email protected]

Abstract. In this paper, we summarize the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T1 (y, z)) = T2 (I(x, y), I(x, z)) and characterize all solutions of the functional equations consisting of I(x, T1 (y, z)) = T2 (I(x, y), I(x, z)) and I(x, y) = I(N(y), N(x)), when T1 is a continuous but not Archimedean triangular norm, T2 is a continuous and Archimedean triangular norm, I is an unknown function, N is a strong negation. We also underline that our method can apply to the three other functional equations closely related to the above-mentioned functional equations. Keywords: Fuzzy connectives, Fuzzy implications, Continuous Archimedean t-norms, Continuous t-norms, Distributive equations of implications, Contrapositive symmetry equations of implications.

1

Introduction

The ability to build complex commercial and scientific fuzzy logic applications has been hampered by what is popularly known as the combinatorial rule explosion problem, which is associated with the conventional fuzzy rule configuration and its accompanying rule matrix. Since all the rules of an inference engine are exercised during every inference cycle, the number of rules directly affects the computational duration of the overall application. To reduce complexity of fuzzy “IF-THEN” rules, Combs and Andrews [8-10] required of the following classical tautology (p ∧ q) → r = (p → r) ∨ (q → r). They refer to the left-hand side of this equivalence as an intersection rule configuration (IRC) and to its right-hand side as a union rule configuration (URC). Subsequently, there were many discussions [9-11,18], most of them pointed out 

This work is supported by National Natural Science Foundation of China (Nos. 60904041, 61165014) and Jiangxi Natural Science Foundation (No.2009GQS0055).

Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 109–120, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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the need for a theoretical investigation required for employing such equations, as concluded by Dick and Kandel [11], “Future work on this issue will require an examination of the properties of various combinations of fuzzy unions, intersections and implications” or by Mendel and Liang [18], “We think that what this all means is that we have to look past the mathematics of IRC⇔URC and inquire whether what we are doing when we replace IRC by URC makes sense.” And then, Trillas and Alsina[26], in the standard fuzzy theory, turned the about requirement into the functional equation I(T (x, y), z) = S(I(x, z), I(y, z)) and obtained all solutions of T when I are special cases of R-implications, Simplications and QL-implications, respectively. Along the lines, Balasubramaniam and Rao[6] investigated the other three functional equations interrelated with this equation. In order to study it in more general case, Ruiz-Aguilera[22,23] and Qin[21], in their own papers, generalized the above equation into uninorm. On the other hand, from fuzzy logical angle, Turksen[27] posed and discussed the equation I(x, T (y, z)) = T (I(x, y), I(x, z)),

x, y, z ∈ [0, 1],

(1)

and then, got the necessary conditions for a fuzzy implication I to satisfy Eq.(1) when T = TP . Later, Baczy´ nski[1] generalized some Turksen’s results into strict t-norm T and obtained the sufficient and necessary conditions of functional equations consisting of Eq.(1) and the following equation I(x, I(y, z)) = I(T (x, y), z),

x, y, z ∈ [0, 1].

(2)

Moreover, he[2] also studied the functional equations composed of Eq.(1) and the following equation I(x, y) = I(N (y), N (x)),

x, y, z ∈ [0, 1].

(3)

After this, Yang and Qin in [28] got the full characterizations of the functional equations composed of Eq.(1) and Eq.(3) when T is a strict t-norm. Recently, many people[2,4,5,20], including Baczy´ nski and Qin, investigate again the distributivity of fuzzy implications over nilpotent or strict triangular t-norms or t-conorms. Specially, we in [19], in the most general case, explored and got the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T1 (y, z)) = T2 (I(x, y), I(x, z)),

x, y, z ∈ [0, 1].

(4)

And then, we in [18], characterized all solutions of the functional equations consisting of Eq.(3) and Eq.(4). Along the above line, in this paper, we summarize the sufficient and necessary conditions of solutions for Eq.(4) and the functional equations consisting of Eq.(3) and Eq.(4), when T1 is a continuous but not Archimedean triangular norm, T2 is a continuous and Archimedean triangular norm, I is an unknown function, N is a strong negation. We also underline that our method can apply to the three other functional equations closely related to the above-mentioned functional equations.

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The paper is organized as follows. In section 2, we present some results concerning basic fuzzy logic connectives employed in the sequel. In Section 3, we recall all solutions of Eq.(4) when T1 is a continuous but not Archimedean triangular norm, T2 is a continuous Archimedean triangular norm and I is an unknown function. In section 4, we investigate the functional equations consisting of Eq.(3) and Eq.(4) when T1 is a continuous but not Archimedean triangular norm, T2 is a strict triangular norm. In section 5, we do the same investigation except that T2 is a nilpotent triangular norm. Finally, a simple conclusion is in Section 6. Unless otherwise stated, we always assume that T1 is only continuous but not Archimedean, which means that T1 must have at least one non-trivial idempotent element, because the case that T1 is continuous and Archimedean has been studied by Baczy´ nski[1,2,4,5], Qin[5,20,28], and so on.

2

Preliminaries

In this section, we recall basic notations and facts used in the sequel. Definition 2.1[12-14]. A binary function T : [0, 1]2 → [0, 1] is called a triangular norm (t-norm for short), if it fulfills, for every x, y, z ∈ [0, 1], the following conditions (1) T (x, y) = T (y, x), (commutativity) (2) T (T (x, y), z) = T (x, T (y, z)), (associativity) (3) T (x, y) ≤ T (x, z), whenever y ≤ z, (monotonicity) (4) T (x, 1) = x. (boundary condition) Definition 2.2[12,14]. A t-norm T is said to be (1) Archimedean, if for every x, y ∈ (0, 1), there exists some n ∈ N such that xnT < y, where xnT = T (x, x, · · · , x);    n times

(2) strict, if T is continuous and strictly monotone, i.e., T (x, y) < T (x, z) whenever x ∈ (0, 1] and y < z; (3) nilpotent, if T is continuous and if for each x ∈ (0, 1) there exists some n ∈ N that xnT = 0. Remark 2.1. If T is strict or nilpotent, then it must be Archimedean. The converse is also true when it is continuous. (see Theorem 2.18 in [14]) Theorem 2.1[14,16]. For a function T : [0, 1]2 → [0, 1], the following statements are equivalent: (1) T is a continuous Archimedean t-norm. (2) T has a continuous additive generator, i.e., there exists a continuous, strictly decreasing function t : [0, 1] → [0, ∞] with t(1) = 0, which is uniquely determined up to a positive multiplicative constant, such that it holds T (x, y) = t(−1) (t(x) + t(y)) for all x, y ∈ [0, 1], where t(−1) is the pseudo-inverse of t, given  −1 t (x), x ∈ [0, t(0)], by t(−1) (x) = 0, x ∈ (t(0), ∞].

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Remark 2.2. (1) Without the pseudo-inverse, the representation of a t-norm in Theorem 2.1 can be rewritten as T (x, y) = t−1 (min(t(x) + t(y), t(0))),

x, y ∈ [0, 1].

(5)

(2) A t-norm T is strict if and only if each continuous additive generator t of T satisfies t(0) = ∞. (3) A t-norm T is nilpotent if and only if each continuous additive generator t of T satisfies t(0) < ∞. Theorem 2.2[7,12]. T is a continuous t-norm, if and only if (1) T = TM , or (2) T is continuous Archimedean, or (3) there exists a family {[am , bm ], Tm }m∈A such that T is the ordinal sum of this family denoted by T = (< am , bm , Tm >)m∈A . In other words,  y−am m am + (bm − am )Tm ( bx−a ) (x, y) ∈ [am , bm ]2 ; m −am bm −am T (x, y) = (6) min(x, y) otherwise, where {[am , bm ]}m∈A is a countable family of non-over lapping, closed, proper subintervals of [0, 1]with each Tm being a continuous Archimedean t-norm, and A is a finite or countable infinite index set. For every m ∈ A, [am , bm ] is called the generating subinterval of T , and Tm the corresponding generating t-norm of T on [am , bm ]. In some literatures we can find several diverse definitions of fuzzy implications (see [7], [14], [16], [25]). But, in this article, we will use the following one, which is equivalent to the definition introduced by Fodor and Roubens (see [12]). Definition 2.3[3,12]. A function I : [0, 1]2 → [0, 1] is called a fuzzy implication, if I fulfills the following conditions I1: I is decreasing with respect to the first variable; I2: I is increasing with respect to the second one; I3: I(0, 0) = I(0, 1) = I(1, 1) = 1, I(1, 0) = 0. (7) In virtue of the above definition, it is obvious that each fuzzy implication satisfy I(0, x) = I(x, 1) = 1 for all x ∈ [0, 1]. But we can say nothing about the value of I(x, 0) and I(1, x) for all x ∈ (0, 1). Definition 2.4[14,17,19,24]. A continuous function N : [0, 1] → [0, 1] is called a strong negation, if it is strictly decreasing, involutive and satisfies N (0) = 1 and N (1) = 0. Specially, when N (x) = 1 − x, we call it the standard negation, denoted by N0 .

3

Solutions to Eq.(4) When T1 Is a Continuous t-Norm and T2 Is a Continuous Archimedean t-Norm

In this section, we recall the characterizations of function I satisfying Eq.(4) when T1 is a continuous t-norm and T2 is a continuous Archimedean t-norm.

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For any given continuous t-norm T1 and binary function I, and fixed x ∈ [0, 1], we define U(T1 ,I,x) = {y ∈ [0, 1] | I(x, y) = 0, y is an idempotent element of T1 }, μ(T1 ,I,x) = sup U(T1 ,I,x) , and V(T1 ,I,x) = {y ∈ [0, 1] | I(x, y) = 1, y is an idempotent element of T1 }, ν(T1 ,I,x) = inf V(T1 ,I,x) . For a more precise presentation, we must underline the relation between (T1 and I) and (μ(T1 ,I,x) and ν(T1 ,I,x) ). Note that U(T1 ,I,x) and V(T1 ,I,x) actually determined by T1 , I and x. U(T1 ,I,x) and V(T1 ,I,x) may be different when either T1 or I are different. We stipulate here that sup ∅ = 0 and inf ∅ = 1 and obtain from Lemma 3.3 in [18] that μ(T1 ,I,x) ≤ ν(T1 ,I,x) for any T1 , I and x ∈ [0, 1]. Now, by the order between μ(T1 ,I,x) and ν(T1 ,I,x) , we need to consider two cases: μ(T1 ,I,x) = ν(T1 ,I,x) and μ(T1 ,I,x) < ν((T1 ,I,x) . Theorem 3.1. Let T1 be a continuous t-norm, T2 a continuous Archimedean t-norm, I: [0, 1]2 → [0, 1] a binary function and assume that μ(T1 ,I,x) = ν(T1 ,I,x) for some fixed x ∈ [0, 1]. Then the following statements are equivalent: (1) The triple of functions (T1 , T2 , I(x, ·)) satisfies Eq.(4) for any y, z ∈ [0, 1]; (2) The vertical section I(x, ·) has the following forms: (i) If μ(T1 ,I,x) ∈ U(T1 ,I,x) , then  0, y ≤ μ(T1 ,I,x) , y ∈ [0, 1]. (8) I(x, y) = 1, y > μ(T1 ,I,x) , (ii) If ν(T1 ,I,x) ∈ V(T1 ,I,x) , then  0, y < ν(T1 ,I,x) , I(x, y) = 1, y ≥ ν(T1 ,I,x) ,

y ∈ [0, 1].

Next, let us consider the case μ(T1 ,I,x) < ν(T1 ,I,x) .

(9)



0, y < μ(T1 ,I,x) , 1, y > ν(T1 ,I,x) , for any x ∈ [0, 1], when μ(T1 ,I,x) < ν(T1 ,I,x) . But we do say nothing about the value of I(x, y) for any y ∈ [μ(T1 ,I,x) , ν(T1 ,I,x) ]. We will solve this problem next, considering the different assumptions on t-norm T2 . At first, we recall characterizations of fuzzy implications I satisfying Eq.(4) when T1 is a continuous t-norm and T2 is a strict t-norm. Remark 3.1. We know from Remark 3.11 in [18] that I(x, y) =

Theorem 3.2. Let T1 be a continuous t-norm, T2 a strict t-norm, I: [0, 1]2 → [0, 1] a continuous binary function except the vertical section I(0, y) = 1 for y ∈ [0, a], which satisfies Eq.I3. Then the following statements are equivalent: (1) The triple of functions (T1 , T2 , I) satisfies Eq.(4) for all x, y, z ∈ [0, 1].

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(2) T1 admits the representation (6), there exist two constants a < b ∈ [0, 1] such that μ(T1 ,I,x) = a, ν(T1 ,I,x) = b for all x ∈ [0, 1], and there exist continuous, strictly decreasing functions ta , t2 : [0, 1] → [0, ∞] with ta (1) = t2 (1) = 0, ta (0) = t2 (0) = ∞, which are uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] and T2 admit the representation (5) with ta and t2 respectively, and there exists a continuous function c: (0, 1] → (0, ∞), c(0) = 0, uniquely determined up to a positive multiplicative constant depending on constants for ta and t2 , such that I has the form ⎧ 1, x = 0, y ∈ [0, a], ⎪ ⎪ ⎨ 0, x = 0, y ∈ [0, a], I(x, y) = x, y ∈ [0, 1]. (10) y−a −1 t (c(x)t ( )), x ∈ [0, 1], y ∈ [a, b], ⎪ a b−a ⎪ ⎩ 2 1, x ∈ [0, 1], y ∈ [b, 1]. Next, we recall characterizations of fuzzy implications I satisfying Eq.(4) when T1 is a continuous t-norm and T2 is a nilpotent t-norm. Theorem 3.3. Let T1 be a continuous t-norm, T2 a nilpotent t-norm, there exist two constants a < b ∈ [0, 1] such that μ(T1 ,I,x) = a, ν(T1 ,I,x) = b for all x ∈ [0, 1], and the corresponding generating t-norm Ta of T1 on the generating subinterval [a, b] is strict. I: [0, 1]2 → [0, 1] a continuous binary function except the vertical section I(0, y) = 1 for y ∈ [0, a], which satisfies Eq. I3. Then the following statements are equivalent: (1) The triple of functions (T1 , T2 , I) satisfies Eq.(4) for all x, y, z ∈ [0, 1]. (2) T1 admits the representation (6), and there exist continuous, strictly decreasing function ta , t2 : [0, 1] → [0, ∞] with ta (1) = t2 (1) = 0, ta (0) = ∞, t2 (0) < ∞, which is uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] and T2 admit the representation (5) with ta and t2 respectively, and there exists a continuous function c: (0, 1] → (0, ∞), c(0) = 0, uniquely determined up to a positive multiplicative constant depending on constants for ta and t2 , such that I has the form ⎧ 1, x = 0, y ∈ [0, a], ⎪ ⎪ ⎨ 0, x = 0, y ∈ [0, a], I(x, y) = (11) y−a −1 t (min(c(x)t ( ), t (0))), x ∈ [0, 1], y ∈ [a, b], ⎪ a b−a 2 ⎪ ⎩ 2 1, x ∈ [0, 1], y ∈ [b, 1]. Theorem 3.4. Let T1 be a continuous t-norm, T2 a nilpotent t-norm, there exist two constants a < b ∈ [0, 1] such that μ(T1 ,I,x) = a, ν(T1 ,I,x) = b for all x ∈ [0, 1], and the corresponding generating t-norm Ta of T1 on the generating subinterval [a, b] is nilpotent. I: [0, 1]2 → [0, 1] a continuous binary function except the vertical section I(0, y) = 1 for y ∈ [0, b], which satisfies Eq. I3. Then the following statements are equivalent: (1) The triple of functions (T1 , T2 , I) satisfies Eq.(4) for all x, y, z ∈ [0, 1].

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(2) T1 admits the representation (6), and there exist continuous, strictly decreasing function ta , t2 : [0, 1] → [0, ∞] with ta (1) = t2 (1) = 0, ta (0) < ∞, t2 (0) < ∞, which is uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] and T2 admit the representation (5) with ta and t2 respectively, and there exists a continuous function c: (0, 1] → [ tta2 (0) , ∞), c(0) = 0, uniquely deter(0) mined up to a positive multiplicative constant depending on constants for ta and t2 , such that I has the form ⎧ 1, x = 0, y ∈ [0, b], ⎪ ⎪ ⎨ 0, x = 0, y ∈ [0, a], I(x, y) = (12) y−a −1 t (min(c(x)t ( ), t (0))), x ∈ (0, 1], y ∈ [a, b], ⎪ a b−a 2 ⎪ ⎩ 2 1, x ∈ [0, 1], y ∈ [b, 1].

4

Solutions to the Functional Equations Consisting of Eq.(3) and Eq.(4) When T1 Is a Continuous t-Norm and T2 Is a Strict t-Norm

In this section, we characterize all solutions to the functional equations consisting of Eq.(3) and Eq.(4) when T1 is a continuous t-norm and T2 is a strict t-norm. Remark 4.1. From the Remark 4.1 in [18], we can draw a conclusion that if I is not continuous at point (0, 0), then I is also not continuous on the partly vertical section I(0, y) for all y ∈ [0, a]. On the other hand, note that we mainly investigate the functional equations consisting of Eq.(3) and Eq.(4), and Eq.(3) is the contrapositive symmetry equation of implication. Hence, we get that I is yet not continuous on the partly horizontal section I(x, 1) for all x ∈ [N (a), 1], where N is a strong negation. Next, let’s find all solutions to the functional equations consisting of Eq.(3) and Eq.(4). To this end, we need to consider the following several cases. Theorem 4.1. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm, N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except at the points (0, 0) and (1, 1), which satisfies Eq.(7). Then the quaternion of functions (T1 , T2 , I, N ) does not satisfy the functional equations consisting of Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1]. Theorem 4.2. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm. There exist two constants a < b ∈ [0, 1], a ∈ (0, 1) such that μ(T1 ,I,x) = a, ν(T1 ,I,x) = b for all x ∈ [0, 1], and two continuous, strictly decreasing functions ta , t2 : [0, 1] → [0, ∞] with ta (1) = t2 (1) = 0, ta (0) = t2 (0) = ∞, which are uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] and T2 admit the representation (5) with ta and t2 respectively. N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except

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the partly vertical section I(0, y) = 1 for y ∈ [0, a] and the partly horizontal section I(x, 1) for x ∈ [N (a), 1], which satisfies Eq.(7). Then the following statements are equivalent: (1) the quaternion of functions (T1 , T2 , I, N ) satisfies the functional equations consisting of Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1]. (2) b = 1 and there exists a constant r ∈ (0, ∞) such that I has the following form ⎧ (x, y) ∈ {0} × [0, 1] ∪ [0, 1] × {1}, ⎨ 1, N (x)−a y−a I(x, y) = t−1 (r · t ( ) · t ( )), (x, y) ∈ (0, N (a)) × (a, 1), a a 1−a 1−a ⎩ 2 0, otherwise. (13) Theorem 4.3. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm, N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except the partly vertical section I(0, y) = 1 for y ∈ [0, 1] and the partly horizontal section I(x, 1) for x ∈ [0, 1], which satisfies Eq.(7). Then the quaternion of functions (T1 , T2 , I, N ) satisfies the functional equations consisting of Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1] if and only if I has the following form  1, (x, y) ∈ {0} × [0, 1] ∪ [0, 1] × {1}, I(x, y) = (14) 0, otherwise. Remark 4.2. By contrast with the results obtained by Baczy´ nski in [2], we can find our results are more complex. In [28], we have showed that the functional equations consisting of Eq.(3) and Eq.(4) have many solutions when both T1 and T2 are strict t-norms, N is a strong negation, I is a continuous binary function except at the points (0, 0) and (1, 1). While, in this paper, Theorem 4.1 shows that the above-mentioned functional equations have no solution when T1 is a continuous t-norm, T2 is a strict t-norm, N is a strong negation and I is a continuous binary function except at the points (0, 0) and (1, 1). But there is no any contradiction because T1 has at least one idempotent element in this paper. Hence these results do not include the others.

5

Solutions to the Functional Equations Consisting of Eq.(3) and Eq.(4) When T1 Is a Continuous t-Norm and T2 Is a Nilpotent t-Norm

Similar to analysis in last section, it is enough to consider I(x, y) is not continuous on the vertical section I(0, y) for all y ∈ [0, a] and on the horizontal section I(x, 1) for all x ∈ [N (a), 1]. Again, we only investigate all solutions of the functional equations consisting of Eq.(3) and Eq.(4). To do this, we firstly consider the case the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] is strict.

On Distributive Equations of Implications

117

Theorem 5.1. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm. And there exist one constant b ∈ [0, 1] such that μ(T1 ,I,x) = 0, ν(T1 ,I,x) = b for all x ∈ [0, 1], and two continuous, strictly decreasing function t0 , t2 : [0, 1] → [0, ∞] with t0 (1) = t2 (1) = 0, t0 (0) = ∞, t2 (0) < ∞, which are uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm T0 of T1 on the generating subinterval [0, b] and T2 admit the representation (5) with t0 and t2 respectively. N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except the points (0, 0) and (1, 1), which satisfies Eq.(7). Then the quaternion of functions (T1 , T2 , I, N ) does not satisfy Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1]. Theorem 5.2. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm. There exist two constants a ∈ (0, 1), b ∈ (0, 1] such that μ(T1 ,I,x) = a, ν(T1 ,I,x) = b for all x ∈ [0, 1], and two continuous, strictly decreasing function ta , t2 : [0, 1] → [0, ∞] with ta (1) = t2 (1) = 0, ta (0) = ∞, t2 (0) < ∞, which are uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] and T2 admit the representation (5) with ta and t2 , respectively. N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except the partly vertical section I(0, y) = 1 for y ∈ [0, a] and the partly horizontal section I(x, 1) for x ∈ [N (a), 1], which satisfies Eq.(7). Then the following statements are equivalent: (1) The quaternion of functions (T1 , T2 , I, N ) satisfies the functional equations consisting of Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1]. (2) b = 1 and there exists a constant r ∈ (0, ∞) such that I has the following form ⎧ (x, y) ∈ {0} × [0, 1] ∪ [0, 1] × {1}, ⎨ 1, N (x)−a y−a I(x, y) = t−1 (r · t ( ) · t ( )), (x, y) ∈ (0, N (a)) × (a, 1), a a 1−a 1−a ⎩ 2 0, otherwise. (15) Theorem 5.3. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm, N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except the partly vertical section I(0, y) = 1 for y ∈ [0, 1] and the partly horizontal section I(x, 1) for x ∈ [0, 1], which satisfies Eq.(7). Then the quaternion of functions (T1 , T2 , I, N ) satisfies the functional equations consisting of Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1] if and only if I has the following form  1, (x, y) ∈ {0} × [0, 1] ∪ [0, 1] × {1}, I(x, y) = (16) 0, otherwise. Next, let us consider the case the generating t-norm Ta of T1 on the generating subinterval [a, b] is nilpotent.

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Theorem 5.4. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm. And there exist one constant b ∈ [0, 1] such that μ(T1 ,I,x) = 0, ν(T1 ,I,x) = b for all x ∈ [0, 1], and two continuous, strictly decreasing function t0 , t2 : [0, 1] → [0, ∞] with t0 (1) = t2 (1) = 0, t0 (0) < ∞, t2 (0) < ∞, which are uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm T0 of T1 on the generating subinterval [0, b] and T2 admit the representation (5) with t0 and t2 respectively. N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except the points (0, 0) and (1, 1), which satisfies Eq.(7). Then the quaternion of functions (T1 , T2 , I, N ) does not satisfy Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1]. Theorem 5.5. Let T1 : [0, 1]2 → [0, 1] be a continuous t-norm, T2 : [0, 1]2 → [0, 1] a strict t-norm. There exist two constants a ∈ (0, 1), b ∈ (0, 1] such that μ(T1 ,I,x) = a, ν(T1 ,I,x) = b for all x ∈ [0, 1], and two continuous, strictly decreasing function ta , t2 : [0, 1] → [0, ∞] with ta (1) = t2 (1) = 0, ta (0) < ∞, t2 (0) < ∞, which are uniquely determined up to positive multiplicative constants, such that the correspondingly generating t-norm Ta of T1 on the generating subinterval [a, b] and T2 admit the representation (5) with ta and t2 , respectively. N : [0, 1] → [0, 1] is a strong negation, I : [0, 1]2 → [0, 1] a continuous binary function except the partly vertical section I(0, y) = 1 for y ∈ [0, a] and the partly horizontal section I(x, 1) for x ∈ [N (a), 1], which satisfies Eq.(7). Then The quaternion of functions (T1 , T2 , I, N ) does not satisfy the functional equations consisting of Eq.(3) and Eq.(4) for all x, y, z ∈ [0, 1]. Remark 5.1. In fact, applying the method used by Baczy´ nski in [2], we easily show that the functional equations consisting of Eq.(3) and Eq.(4) have many solutions when T1 is a strict t-norm, T2 is nilpotent t-norm, N is a strong negation, I is a continuous binary function except at the points (0, 0) and (1, 1). While, in this paper, Theorem 5.4 shows that the above-mentioned functional equations have no solution when both T1 is a continuous t-norm , T2 is a nilpotent t-norm, N is a strong negation and I is a continuous binary function except at the points (0, 0) and (1, 1). But there is no any contradiction because T1 has at least one idempotent element in this paper. Hence these results do not include the others.

6

Conclusion

We, in this work, summarize the sufficient and necessary conditions of solutions for Eq.(4) and the functional equations consisting of Eq.(3) and Eq.(4), when T1 is a continuous but not Archimedean triangular norm, T2 is a continuous and Archimedean triangular norm, I is an unknown function, N is a strong negation. We also underline that our method can apply to the three other functional equations and show that our results and the results obtained by Baczy´ nski in [3] and Qin in [20] , do not include the others. In future works we will try to concentrate on the other cases about these functional equations that are not considered in this paper, for example, when both T1 and T2 are continuous but not Archimedean t-norms.

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References 1. Baczy´ nski, M.: On a class of distributive fuzzy implications. Internat. J. Uncertainty, Fuzziness, Knowledge-Based Systems 9, 229–238 (2001) 2. Baczy´ nski, M.: Contrapositive symmetry of distributive fuzzy implications. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 10(suppl.), 135–147 (2002) 3. Baczy´ nski, M.: On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms. Fuzzy Sets and Systems 161, 2256–2275 (2010) 4. Baczy´ nski, M., Balasubramaniam, J.: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms. IEEE Trans. Fuzzy Syst. 17(3), 590–603 (2009) 5. Baczy´ nski, M., Drewniak, J.: Conjugacy classes of fuzzy implication, in: Computational Inteligence: Theory and Applications. In: Reusch, B. (ed.) Fuzzy Days 1999. LNCS, vol. 1625, pp. 287–298. Springer, Heidelberg (1999) 6. Balasubramaniam, J., Rao, C.J.M.: On the distributivity of implication operators over T-norms and S-norms. IEEE Trans. Fuzzy Syst. 12(1), 194–198 (2004) 7. Bustince, H., Burillo, P., Soria, F.: Automorphisms, negations and implication operators. Fuzzy Sets and Systems 134, 209–229 (2003) 8. Combs, W.E., Andrews, J.E.: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 6, 1–11 (1998) 9. Combs, W.E.: Author’s reply. IEEE Trans. Fuzzy Syst. 7, 371 (1999) 10. Combs, W.E.: Author’s reply. IEEE Trans. Fuzzy Syst. 7, 478–479 (1999) 11. Dick, S., Kandel, A.: Comments on Combinational rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 7, 477–477 (1999) 12. Fodor, J.C., Roubens, M.: Fuzzy preference modeling and multi-criteria decision support. Kluwer, Dordrecht (1994) 13. Gottwald, S.: A Treatise on Many-Valued Logics. Research Studies Press, Baldock (2001) 14. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000) 15. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equations and Jensen’s Inequality. PWN-Polish Scientitic Publishers and University of Silesia, Warszawa-Krakow-Katowice (1985) 16. Ling, C.H.: Representation of associative functions. Publ. Math. Debrecen 12, 189– 212 (1965) 17. Mas, M., Monserrat, M., Torrens, J.: Modus ponens and modus tollens in discrete implications. International Journal of Approximate Reasoning 49, 422–435 (2008) 18. Mendel, J.M., Liang, Q.: Comments on Combinational rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 7, 369–371 (1999) 19. Qin, F., Baczy´ nski, M.: Distributive equations of implications based on continuous triangular norms. IEEE Trans. Fuzzy Syst. (acception) 20. Qin, F., Yang, L.: Distributive equations of implications based on nilpotent triangular norms. International Journal of Approximate Reasoning 51, 984–992 (2010) 21. Qin, F., Zhao, B.: The Distributive Equations for Idempotent Uninorms and Nullnorms. Fuzzy Sets and Systems 155, 446–458 (2005) 22. Ruiz-Aguilera, D., Torrens, J.: Distributivity of Strong Implications over Conjunctive and Disjunctive Uninorms. Kybernetika 42, 319–336 (2005) 23. Ruiz-Aguilera, D., Torrens, J.: Distributivity of Residual Implications over Conjunctive and Disjunctive Uninorms. Fuzzy Sets and Systems 158, 23–37 (2007)

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24. Trillas, E.: Sobre funciones de negacion en la teoria de conjuntos difusos. Stochastica III, 47–60 (1979) (in Spanish) 25. Trillas, E., Mas, M., Monserrat, M., Torrens, J.: On the representation of fuzzy rules. International Journal of Approximate Reasoning 48, 583–597 (2008) 26. Trillas, E., Alsina, C.: On the law [p ∧ q → r] ≡ [(p → r) ∨ (q → r)] in fuzzy logic. IEEE Trans. Fuzzy Syst. 10, 84–88 (2002) 27. Tursksen, I.B., Kreinovich, V., Yager, R.R.: A new class of fuzzy implications: Axioms of fuzzy implication revisited. Fuzzy Sets and Systems 100, 267–272 (1998) 28. Yang, L., Qin, F.: Distributive equations based on fuzzy implications. In: IEEE International Conference on Fuzzy Systems, Korea, pp. 560–563 (2009)

A Novel Cultural Algorithm Based on Differential Evolution for Hybrid Flow Shop Scheduling Problems with Fuzzy Processing Time Qun Niu, Tingting Zeng, and Zhuo Zhou Shanghai Key Laboratory of Power Station, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, China [email protected]

Abstract. Considering the imprecise or fuzzy nature of the data in realworld problems, this paper proposes a novel cultural algorithm based on differential evolution (CADE) to solve the hybrid flow shop scheduling problems with fuzzy processing time(FHFSSP). The mutation and crossover operations of differential evolution (DE) are introduced into cultural algorithm (CA) to enhance the performance of traditional CA. Experimental results demonstrate that the proposed CADE method is more effective than CA, particle swarm optimization (PSO) and quantum evolution algorithm (QA) when solving FHFSSP. Keywords: Cultural algorithm, Differential evolution, Hybrid flow shop scheduling, Makespan, Fuzzy processing time.

1

Introduction

The hybrid flow shop scheduling problem (HFSSP) [1] has been well-known as one of the hardest combinatorial optimization problems. In the most of studies concerned with the HFSSP, processing times were treated as crisp value. However, in many practical applications, information is often ambiguous or imprecise. It may be more appropriate to consider fuzzy processing time for HFSSP to reflect the real-world situations. In the past few decades, a great deal of research work has been performed on fuzzy scheduling problems. The earliest paper in fuzzy scheduling appeared in 1979 [2]. Scheduling problem with fuzzy due-dates was firstly studied by Ishii et al. [3]. In [4], Ishibuchi et al. investigated flow shop scheduling with fuzzy processing times. Kuroda [5] analyzed fuzzy job shop scheduling problem. The open shop scheduling problem with fuzzy allowable time and resource constraint was discussed by Konno [6]. Lei [7] proposed an efficient Pareto archive particle swarm optimization for multi-objective fuzzy job shop scheduling. Recently, Pengjen [8] presented ant colony optimization to minimize the fuzzy makespan and total weighted fuzzy completion time in flow shop scheduling problems. Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 121–132, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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Zhengyi [9] introduced a hybrid particle swarm optimization for solving flow shop scheduling problem with fuzzy due date. In recent years, as a relatively new member in the family of evolutionary algorithms, cultural algorithms (CA) was first proposed by Reynolds [10] as a vehicle for modeling social evolution and learning in agent-based societies. Since it’s simple to implement when little tuning on its parameters, CA has been successfully used to solve many diverse optimization problems. For instance, evolutionary neural fuzzy network, timetabling problems and constrained optimization. However, for scheduling problems, there are only few applications. Daniel [11] and Tieke [12] employed CA to solve the job shop scheduling problem. This paper presents a novel CA based on differential evolution (CADE) to tackle the fuzzy hybrid flow shop scheduling problems (FHFSSP). Due to lack of a direction track toward the best solution and development burden for different applications in CA, the mutation operator and nonuniform crossover with DE are introduced into CA so as to overcome the premature convergence and increase the performance of traditional CA to make it competitive with other approaches.

2 2.1

Hybrid Flow Shop Scheduling Problems with Fuzzy Processing Time Problem Description

The hybrid flow shop (HFS) system is defined by the set M = {1, ..., i, ..., m} of m processing stages. There is a set Mi = {1, ..., l, ..., mi} of mi identical parallel machines at each stage i. The set J = {1, ..., j, ..., n} of n independent jobs has to be processed in the system defined by the sets M and Mi . It can be seen that each job j is composed of m operations and operation Oij illustrates the ith operation of job Jj . The processing time of operation Oij is delegated as a triangular fuzzy number (TFN) pij = (a1ij , a2ij , a3ij ). The ith operation of a job is processed at the ith stage and can begin only after the completion of former operation from the same sequence. Notations are given as follows: i: The number of the stages. j: Index of job, j = 1, 2, ..., n. sij : The fuzzy starting time of job j in stage i. pij : The fuzzy processing time of job j in stage i. cij : The fuzzy completion time of job j in stage i.  Then, the HFSSP can be formulated as followed with the above notation. M in.

n  j=1

Subject to cij = sij + pij , i = 1, ..., n; j = 1, ..., m.  cij ≤ si(j+1) , i = 1, ..., n; j = 1, ..., m. 

j C

A Novel CADE for FHFSSP m  i=1 n  l=1 n  i=1

123

xij = 1, j = 1, 2, ..., n. xil = 1, i = 1, 2, ..., n. xil sij ≤

n  i=1

xi(l+1) sij , i, l = 1, ..., n; j = 1, ..., m.

In this paper, maximum fuzzy completion time (makespan) is considered as a criterion which is described: max = C

max

j=1,2,...,n

j C

(1)

j and C max denotes where the fuzzy completion time of job Jj is shown as TFN C the maximum fuzzy completion time. 2.2

Operations on Fuzzy Processing Time

The fuzzy sets theory was proposed by A. Lotfi Zadeh in 1965[13]. In fuzzy context, some operations of fuzzy number are required to be redefined to build a  and fuzzy maximum schedule. These operations include the fuzzy addition + max  of two fuzzy numbers as well as the ranking methods of fuzzy numbers. Fuzzy addition is used to calculate the fuzzy completion time of operation, fuzzy maximum is used to determine the fuzzy beginning time of operation and the ranking method is for the maximum fuzzy completion time [14]. For two triangular fuzzy numbers: s = (a1 , b1 , c1 ) and  t = (a2 , b2 , c2 ), where a1 (a2 ) and c1 (c2 ) are lower and upper bounds, while b1 (b2 ) is the modal value of the triangle. We adopt the following fuzzy addition and fuzzy maximum in order to conserve the triangular form of the obtained result: s +  t = (a1 + a2 , b1 + b2 , c1 + c2 )

(2)

max(  s+ t) = (max(a1 + a2 ), max(b1 + b2 ), max(c1 + c2 )) (3) The following criteria are adopted to rank s = (a1 , b1 , c1 ) and  t = (a2 , b2 , c2 ): Criterion 1: If ct1 ( s) = a1 +2b41 +c1 > ( (  t,  of s and t is approximated   then s  t = s; else s  t= t. The criterion s  t ≈ (a1 a2 , b1 b2 , c1 c2 ) is first used by Sakawa and Mori [15] and named Sakawa criterion for simplicity. Compared with the Sakawa criterion, the new criterion has the following features:

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(1) For s and  t, their approximate max is either s or  t; (2) Only three pairs of special points (si , ti ) are compared in the Sakawa criterion and three criteria to rank them are used in the new criterion. The approximate max of the new criterion approaches the real max better than that of the Sakawa criterion. Assume that a number of job sequences are constructed. The question is how to evaluate their fuzzy makespans.

3

The Proposed Cultural Algorithm Based on Differential Evolution

In this section, basic concepts concerning the cultural algorithm and a novel cultural algorithm differential evolution (CADE) method are described. Next, using CADE to solve fuzzy hybrid flow shop scheduling problem (FHFSSP) are further represented. 3.1

Cultural Algorithm Differential Evolution (CADE)

CA [10] involves acquiring the belief space from the evolving population space and then exploiting that information to guide the search. CA is dual inheritance systems which utilize a belief space and a population space. The belief space is where the knowledge, acquired by individuals through generations, is preserved. The population space consists of a set of possible solutions to the problem, and can be modeled using any population-based approach. CA provides self-adaptive capabilities which can generate the helpful information for the FHFSSP. However, state-of-the-art CAs exhibit drawbacks related to the lack of a direction track toward the best solution and the development burden (time and cost) for different applications. For the FHFSSP, the diverse population has a more desirable impact on the search ability in CA, therefore, embedding the mutaion and crossover operators with DE into CA can improve the performance of CA. The difference between the proposed CADE and the previous version of cultural differential evolution (CDE) [16] is that CDE uses DE in the population space while CADE introduces the operators of DE into influence function. To enhance the ability of searching for a near-global optimal solution, a novel CADE method is proposed, which combines the cooperative DE and CA to reproduce its search logic capacity and to increase diversity of the population. Fig. 1 shows the flowchart of the proposed CADE method. 3.2

Implementation of CADE for FHFSSP

The CADE method process is described step-by-step as follows. Step 1: Create initial populations The initial population which is represented by a real number encoding can be generated randomly using the following equation: xij,G = rand ∗ mi + 1

(5)

A Novel CADE for FHFSSP Start

125

End

Create initial vectors

Yes No

Create initial belief space

Termination? Yes

Update every position (YDOXDWHWKHSHUIRUPDQFHIXQFWLRQ 8SGDWHWULDOYHFWRUDQGWDUJHWYHFWRU $GMXVWHDFKEHOLHIVSDFH *HQHUDWHHDFKQHZYHFWRU

p=p+1 No

Is this the last Vectorp? 1d p d P

Fig. 1. Flowchart of the proposed CADE method

where rand is a random number between [0, 1) and mi means the number of the identical parallel machines at each stage i. Step 2: Create initial belief space The belief space is the information storage vault in which the vectors can preserve their experiences for other vectors to learn from them indirectly. Create P belief space, Bp (p = 1, 2, ..., P ). Each initial Bp is defined as an empty set. Step 3: Update every target vector Step 3.1: Evaluate the performance objective of each V ectori In this paper, FHFSSP with maximum fuzzy completion time (makespan) is considered as the objective and the criterion. Step 3.2: Update the trial vector Up,i and target vector Vp,i In this step, the first thing should update the trial vector. Compare the fitness value of each initial vector with that of trial vector. If the fitness value of the trial vector is better than that of initial vector, then the trial vector is replaced with the value of the initial vector, such as Eq.6. Then, updating the target vector. Compare the fitness value of all vectors with that of the target vector. If the fitness value of the target vector exceeds those of trial vectors, then the next generation of target vector is replaced with the trial vector, as shown in Eq.7.  Up,i(G) , if f (Up,i(G) ) < f (xij,G ) Up,i(G+1) = (6) xij,G , otherwise  Vp,i(G+1) =

Up,i(G+1) , if f (Up,i(G+1) ) < f (Vp,i(G) ) Vp,i(G) , otherwise

(7)

where G is the current generation. Step 4: Acceptance function The acceptance function yields the number of vectors that are applied to each belief space as Eq.8. The number of accepted vectors decreases when the number of generations increases.

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Naccepted = ap% × I +

ap% ×I t

(8)

where ap% is a parameter that is set by the user, and must specify the top performing 20% [17], I and t denote the number of vectors and the tth generation. Step 5: Adjust each belief space Bp using an acceptance function This step sorts these vectors in each V ectorp in the order of increasing fitness. Then the best value of each V ectorp is put into the belief space Bp using an acceptance function. The region of belief space BIp is described as BIp = [lp , up ] = {x|lp ≤ x ≤ up , x ∈ R}, where lp and up are represent the lower bound and the upper bound on Bp , respectively. Then, compare the solution of each vector in Bp with lp . If the solution of the vector is smaller than lp , then lp is replaced with the current solution, such as Eq.9. Moreover, compare the solution of each vector in the Bp with up . If the solution of the vector is greater than up, then up is replaced with the current solution, such as Eq.10.  lp =  up =

xp,i , if xp,i ≤ lp lp , otherwise

(9)

xp,i , if xp,i ≥ up up , otherwise

(10)

Step 6: Generate new V ectorp using lp , up , Up,i and Vp,i Using an influence Eq.11 to adjust every solution of each V ectorp is the first step which can change the direction of each vector in solution space so that it is not easily being trapped at a local optimum. Then, according to Eq.12, 13 and 14 update the trial vector and target vector to generate the each new V ectorp .  xp,i(G) =

xp,i(G) + |rand × (up − lp )|, if xp,i < lp xp,i(G) − |rand × (up − lp )|, if xp,i > lp

up,i(G+1) = xp,r1 (G) + F ∗ (xp,r2 (G) − xp,r3 (G) )  Vp,i(G) =

up,i(G+1) , if rand ≤ CR xp,i(G) , otherwise

xp,i(G+1) = xp,i(G) + vp,i(G+1)

(11)

(12)

(13)

(14)

where F denotes scaling factor, CR represents crossover rate and r1 , r2 , r3 th vectors are three parameter vectors which are chosen in a random fashion from the current population.

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4

127

Experimental Results

To illustrate the effectiveness of our approach, 30 sets of instances are randomly generated using the method in [18] which can fuzzify some of the crisp benchmarks to evaluate the performance of the proposed CADE. For each crisp duration x, a three-point triangular fuzzy number is built. The first point is drawn randomly from the interval [δ1 x, x], where δ1 < 1. The center point is set equal to x, and the third point is drawn randomly from the interval [x, δ2 x], where δ2 > 1, as show in Fig. 2. In this paper, we set δ1 = 0.85 and δ2 = 1.3. Taking the first instance for example, the notation of j10s5fp1 means a 10-job, 5-stage and 1-fuzzy problem. The letters j, s and fp denote job, stage and fuzzy problem, respectively. The combinations of the 3 factors gave a total of 30 sets of problems. Then three other algorithms including cultural algorithm (CA), quantum evolution algorithm (QA) and particle swarm algorithm (PSO) are tested to compare with CADE in the experimental results. All methods were implemented using Matlab software and run on a PC with a Pentium (R) Dual 1.6 GHz processor with 2GB of RAM.

P A (t )

1.0

[G1 x, x]

x

[ x, G 2 x ]

time

Fig. 2. Fuzzification of a crisp data(x)

4.1

Parameter Setting

The population size is 20 and maximum number of iteration G is 100, so that the total fitness evaluations are the same for all the compared methods. Thirty independent runs are performed on each instance. In this paper, the parameter configurations for CA, PSO and the proposed CADE are over a large amount of trials. For CA, the probability to apply the situational knowledge p = 0.2 [17]. For PSO, inertia weight w = 0.8 and the acceleration coefficient c1 = c2 = 2 [19]. For the proposed CADE, the operators embedded from DE are crossover rate CR = 0.8, the scaling factor F = 0.6 [20] and the parameter from CA is the probability to apply the situational knowledge p = 0.2 [17].

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Table 1. Comparison of the four algorithms with 10 jobs on 5 stages Instance Algorithm Best solution j10s5fp1 CADE 97.75 115.00 149.50 CA 97.75 115.00 149.50 QA 97.75 115.00 149.50 PSO 97.75 115.00 149.50 j10s5fp2 CADE 102.00 120.00 156.00 CA 102.00 120.00 156.00 QA 102.00 120.00 156.00 PSO 102.00 120.00 156.00 j10s5fp3 CADE 107.95 127.00 165.10 CA 107.95 127.00 165.10 QA 107.95 127.00 165.10 PSO 107.95 127.00 165.10 j10s5fp4 CADE 86.70 102.00 132.60 CA 86.70 102.00 132.60 QA 86.70 102.00 132.60 PSO 87.55 103.00 133.90 j10s5fp5 CADE 96.05 113.00 146.90 CA 96.05 113.00 146.90 QA 96.05 113.00 146.90 PSO 99.45 117.00 152.10 j10s5fp6 CADE 62.90 74.00 96.20 CA 63.75 75.00 97.50 QA 66.30 78.00 101.40 PSO 67.15 79.00 102.70 j10s5fp7 CADE 67.15 79.00 102.70 CA 67.15 79.00 102.70 QA 69.70 82.00 106.60 PSO 73.10 86.00 111.80 j10s5fp8 CADE 67.15 79.00 102.70 CA 68.00 80.00 104.00 QA 68.85 81.00 105.30 PSO 73.95 87.00 113.10 j10s5fp9 CADE 68.00 80.00 104.00 CA 68.00 80.00 104.00 QA 72.25 85.00 110.50 PSO 73.10 86.00 111.80 j10s5fp10 CADE 68.85 81.00 105.30 CA 70.55 83.00 107.90 QA 73.10 86.00 111.80 PSO 76.50 90.00 117.00

Average value 97.86 115.13 149.67 98.86 116.23 151.10 98.86 116.23 151.10 103.76 122.07 158.69 102.00 120.00 156.00 103.02 121.20 157.56 102.06 120.07 156.09 106.34 125.10 162.63 108.21 127.30 165.49 108.91 128.13 166.57 108.23 127.33 165.53 110.70 130.10 169.13 88.06 103.57 134.64 90.19 106.07 137.89 88.09 103.63 134.72 95.26 112.07 145.47 98.57 115.83 150.58 100.27 117.83 153.18 98.60 116.00 150.80 103.25 121.57 158.04 66.36 78.00 101.36 67.21 78.80 102.44 70.61 83.03 107.94 74.04 87.10 113.06 70.83 83.33 108.33 71.00 83.53 108.59 75.40 88.20 114.57 78.60 92.30 119.99 70.24 82.33 106.64 70.32 82.50 106.73 73.87 86.90 112.97 77.61 91.30 118.69 71.51 83.27 108.20 71.91 84.60 109.98 76.98 90.40 117.13 81.20 95.33 123.89 73.64 86.63 112.32 74.29 87.33 112.84 77.83 91.57 118.99 81.86 96.27 124.80

Time 0.69 0.61 0.70 0.68 0.67 0.61 0.71 0.68 0.67 0.60 0.73 0.66 0.66 0.60 0.73 0.67 0.66 0.61 0.73 0.69 0.67 0.60 0.69 0.68 0.71 0.62 0.73 0.66 0.68 0.60 0.73 0.66 0.70 0.61 0.73 0.66 0.72 0.61 0.74 0.66

A Novel CADE for FHFSSP

Table 2. Comparison of the four algorithms with 10 jobs on 10 stages Instance Algorithm Best solution j10s10fp1 CADE 118.15 139.00 180.70 CA 119.00 140.00 182.00 QA 120.70 142.00 184.60 PSO 125.80 148.00 192.40 j10s10fp2 CADE 135.15 159.00 206.70 CA 136.00 160.00 208.00 QA 138.55 163.00 211.90 PSO 141.95 167.00 217.10 j10s10fp3 CADE 126.65 149.00 193.70 CA 129.20 152.00 197.60 QA 127.50 150.00 195.00 PSO 130.05 154.00 200.20 j10s10fp4 CADE 128.35 151.00 196.30 CA 133.45 157.00 196.30 QA 133.45 157.00 196.30 PSO 131.75 155.00 201.50 j10s10fp5 CADE 125.80 148.00 192.40 CA 126.65 149.00 193.70 QA 126.65 149.00 193.70 PSO 130.90 154.00 200.20 j10s10fp6 CADE 109.65 129.00 167.70 CA 111.35 131.00 170.30 QA 110.50 130.00 169.00 PSO 117.30 144.00 179.40 j10s10fp7 CADE 109.65 129.00 167.70 CA 112.20 132.00 171.60 QA 117.30 138.00 179.40 PSO 119.00 140.00 182.00 j10s10fp8 CADE 107.95 127.00 165.10 CA 118.15 139.00 171.60 QA 113.90 134.00 174.20 PSO 117.30 138.00 179.40 j10s10fp9 CADE 114.75 135.00 175.50 CA 116.45 137.00 178.10 QA 119.00 140.00 182.00 PSO 122.40 144.00 187.20 j10s10fp10 CADE 119.85 141.00 184.60 CA 129.20 152.00 187.20 QA 122.40 144.00 187.20 PSO 129.20 152.00 197.60

Average value 120.76 142.07 183.73 123.02 144.60 186.03 127.27 149.77 192.53 132.29 155.20 201.59 139.06 163.37 212.38 139.94 164.10 212.55 142.69 167.70 217.23 148.10 174.30 226.63 128.35 151.00 196.30 129.00 151.30 196.56 130.05 153.00 198.90 138.61 162.63 210.95 131.13 152.93 198.64 131.41 153.57 199.33 132.37 155.40 201.80 138.10 161.73 209.78 128.72 151.00 195.69 129.00 151.07 195.74 132.18 154.87 201.11 136.62 160.77 208.43 114.55 134.07 173.81 116.37 136.00 175.59 120.05 140.43 181.91 123.42 144.93 188.15 115.63 136.03 176.50 117.73 137.43 178.06 122.83 144.53 187.24 126.76 149.57 193.40 115.23 135.43 175.46 116.25 136.43 176.84 122.66 143.03 185.21 124.13 145.80 189.67 120.55 141.83 184.38 120.85 142.17 184.82 125.97 148.27 191.40 128.27 151.10 196.13 125.80 146.87 189.58 126.31 148.17 191.18 130.67 153.47 199.25 136.00 160.13 207.18

Time 1.33 1.17 1.34 1.25 1.32 1.16 1.34 1.24 1.33 1.17 1.34 1.24 1.33 1.18 1.34 1.26 1.33 1.17 1.34 1.24 1.39 1.24 1.40 1.25 1.43 1.25 1.40 1.25 1.30 1.24 1.40 1.24 1.31 1.24 1.40 1.25 1.38 1.24 1.40 1.24

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Table 3. Comparison of the four algorithms with 15 jobs on 10 stages Instance Algorithm Best solution j15s10fp1 CADE 200.60 236.00 306.80 CA 201.45 237.00 308.10 QA 201.45 237.00 308.10 PSO 201.45 237.00 308.10 j15s10fp2 CADE 173.40 204.00 265.20 CA 174.25 205.00 266.50 QA 175.10 206.00 267.80 PSO 180.20 212.00 275.60 j15s10fp3 CADE 168.30 198.00 257.40 CA 170.00 200.00 260.00 QA 170.00 200.00 260.00 PSO 177.65 209.00 271.70 j15s10fp4 CADE 191.25 225.00 292.50 CA 194.65 229.00 297.70 QA 194.65 229.00 297.70 PSO 196.35 231.00 300.30 j15s10fp5 CADE 155.55 183.00 237.90 CA 156.40 184.00 239.20 QA 157.25 185.00 240.50 PSO 164.05 193.00 250.90 j15s10fp6 CADE 188.70 222.00 288.60 CA 189.55 223.00 289.90 QA 189.55 223.00 289.90 PSO 190.40 224.00 291.20 j15s10fp7 CADE 158.95 187.00 243.10 CA 159.80 188.00 244.40 QA 159.80 188.00 244.40 PSO 160.65 189.00 245.70 j15s10fp8 CADE 188.70 222.00 288.60 CA 189.55 223.00 289.90 QA 189.55 223.00 289.90 PSO 190.06 223.30 290.29 j15s10fp9 CADE 187.85 221.00 287.30 CA 188.70 222.00 288.60 QA 188.70 222.00 288.60 PSO 189.55 223.00 289.90 j15s10fp10 CADE 170.85 201.00 261.30 CA 171.70 202.00 262.60 QA 175.95 203.00 263.90 PSO 175.10 206.00 267.80

Average value 200.71 236.10 306.93 202.24 237.93 309.01 202.24 237.93 309.01 210.26 247.37 321.58 177.88 209.20 271.96 178.44 209.93 272.91 181.45 213.47 277.51 187.51 220.60 286.78 171.98 202.07 262.47 173.60 204.07 265.29 178.19 209.23 271.79 186.58 219.47 285.31 195.36 229.83 298.22 196.72 231.43 300.86 195.73 230.27 298.78 201.17 236.67 307.67 159.72 187.80 243.92 160.65 188.83 245.18 163.80 192.53 249.69 171.08 201.27 261.56 189.98 223.50 290.20 190.37 223.77 290.46 192.12 225.83 293.63 196.78 231.17 299.95 159.46 187.53 243.79 160.08 188.10 244.53 163.20 191.97 248.73 166.54 196.13 254.06 188.76 222.00 288.60 189.55 223.00 289.90 190.06 223.30 290.29 193.21 226.87 294.93 187.99 221.17 287.43 188.67 221.73 287.47 190.29 224.00 290.59 193.35 227.43 295.49 172.47 202.83 263.68 172.78 202.97 263.86 176.46 207.13 269.23 178.33 209.77 272.35

Time 1.47 1.38 1.51 1.42 1.45 1.40 1.51 1.41 1.45 1.37 1.53 1.42 1.46 1.40 1.59 1.43 1.45 1.40 1.59 1.42 1.43 1.33 1.56 1.42 1.43 1.35 1.49 1.42 1.43 1.34 1.48 1.43 1.43 1.33 1.48 1.42 1.42 1.34 1.48 1.42

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4.2

131

Comparative Study

In the following Tables, “Best solution” denotes the best makespan found in 30 runs, “Average value” indicates the average value of the best solutions found in all runs and “Time” expresses CPU average time. Tables 1, 2 and 3 summarize the comparison results obtained by CADE, CA, QA, and PSO. Each Table consists of 10 instances. From Table 1-3, it can be concluded that CADE performs better than CA, QA and PSO. For all instances, the average values obtained by CADE are always smaller than the corresponding results for CA, QA and PSO. Moreover, CADE can find the best solution of all instances, CA, QA and PSO do not approximate the best solutions of most instances and the computational times for CADE are always slightly longer than those for CA, QA and PSO. In particular, from Table 1, for ten 10 × 5 simple problems, four algorithms find the same “Best solution” for five 10 × 5 problems and CADE outperforms the other three algorithms for the other five 10 × 5 instances. From Table 2 and 3, for ten 10 × 10 instances and ten 15 × 10 instances, the “Best solution” generated by CADE are better than those of CA, QA and PSO.

5

Conclusion

This paper proposes a novel cultural algorithm based on differential evolution (CADE) method to solve FHFSSP with makespan objective. Since processing times were modeled as triangular fuzzy numbers, the makespan is a triangular fuzzy number as well. Combining CA and DE reasonable, CADE method has the ability to obtain better solution for the FHFSSP. The performance of CADE is evaluated in comparison with CA, QA and PSO for 30 instances. Computational results demonstrate the effectiveness of the proposed CADE. With respect to the application, CADE can be applied to some other scheduling problems such as parallel machine scheduling in the future work. Acknowledgments. This work is supported by the National Natural Science Foundation of China (grant no.60804052),Chen Guang project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, Shanghai University “11th Five-Year Plan” 211 Construction Project.

References 1. Linn, R., Zhang, W.: Hybrid flow shop scheduling: A survey. Computers & Industrial Engineering 37, 57–61 (1999) 2. Prade, H.: Using fuzzy set theory in a scheduling problem: a case study. Fuzzy Sets and Systems 2, 153–165 (1979) 3. Ishii, H., Tada, M., Masuda, T.: Two scheduling problems with fuzzy due-dates. Fuzzy Sets and Systems 46, 339–347 (1992) 4. Ishibuchi, H., Yamamoto, N., Murata, T., Tanaka, H.: Genetic algorithms and neighborhood search algorithms for fuzzy flow shop scheduling problems. Fuzzy Sets and Systems 67, 81–100 (1994)

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5. Kuroda, M., Wang, Z.: Fuzzy job shop scheduling. International Journal of Production Economics 44, 45–51 (1996) 6. Konno, T., Ishii, H.: An open shop scheduling problem with fuzzy allowable time and fuzzy resource constraint. Fuzzy Sets and Systems 109, 141–147 (2000) 7. Lei, D.M.: Pareto archive particle swarm optimization for multi-objective fuzzy job shop scheduling problems. Int. J. Adv. Manuf. Technol. 37, 157–165 (2007) 8. Peng-Jen, L., Hsien-Chung, W.: Using ant colony optimization to minimize the fuzzy makespan and total weighted fuzzy completion time in flow shop scheduling problems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, 559–584 (2009) 9. Zhengyi, J., Shanqing, L., Jianmin, Z.: A new hybrid particle swarm optimization for solving flow shop scheduling problem with fuzzy due date. Advanced Materials Research, 189-193, 2746–2753 (2011) 10. Reynolds, R.G.: An introduction to cultural algorithms. In: Sebald, A.V., Fogel, L.J. (eds.) Proceedings of the Third Annual Conference on Evolutionary Programming, pp. 131–139. World Scientific, River Edge, New Jersey (1994) 11. Daniel, C.R., Ricardo, L.B., Carlos, A.C.: Cultural algorithms, an alternative heuristic to solve the job shop scheduling problem. Engineering Optimization 39, 69–85 (2007) 12. Tieke, L., Weiling, W., Wenxue, Z.: Solving flexible job shop scheduling problem based on cultural genetic algorithm. Computer Integrated Manufacturing Systems 16, 861–866 (2010) 13. Deming, L.: Solving fuzzy job shop scheduling problems using random key genetic algorithm. Int. J. Adv. Manuf. Technol. 49, 253–262 (2010) 14. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965) 15. Sakawa, M., Wang, Z.: An efficient genetic algorithm for job shop scheduling problems with fuzzy processing time and fuzzy due date. Comput. Ind. Eng. 36, 325–341 (1999) 16. Storn, R., Price, K.V.: Differential evolution: A simple and efficient adaptive scheme for global optimization over continuous spaces. J. Global Optimization 11, 341–359 (1997) 17. Li, B.B., Wang, L.: A hybrid quantum-inspired genetic algorithm for multiobjective flow shop scheduling. IEEE Trans. on Systems, Man, and CyberneticsPart B: Cybernetics 37, 576–591 (2007) 18. Omar, A.G.: A bi-citeria optimization: minimizing the integral value and spread of the fuzzy makespan of job shop scheduling problems. Applied Soft Computing 2, 197–210 (2003) 19. Shi, Y., Eberhart, R.C.: Parameter selection in particle swarm optimization. In: Porto, V.W., Waagen, D. (eds.) EP 1998. LNCS, vol. 1447, pp. 591–600. Springer, Heidelberg (1998a) 20. Liu, J., Lampinen, J.: On setting the control parameter of the differential evolution algorithm. In: Proceedings of the 8th International Mendel Conference on Soft Computing, pp. 11–18 (2002a)

An Over-Relaxed (A, η, m)-Proximal Point Algorithm for System of Nonlinear Fuzzy-Set Valued Operator Equation Frameworks and Fixed Point Problems Heng-you Lan1, , Xiao Wang2 , Tingjian Xiong1 , and Yumin Xiang1 1

2

School of Science, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, P.R. China School of Computer and Science, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, P.R. China [email protected]

Abstract. In order to find the common solutions for nonlinear fuzzy-set valued operator equations and fixed point problems of Lipschitz continuous operators in Hilbert spaces, the purpose of this paper is to construct a new class of over-relaxed (A, η, m)-proximal point algorithm framework with errors by using some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. Further, the variational graph convergence analysis for this algorithm framework is investigated. Finally, some examples of applying the main result is also given. The results presented in this paper improve and generalize some well known results in recent literatures. Keywords: (A, η, m)-maximal monotonicity, nonlinear fuzzy-set valued operator equation and fixed point problem, Over-relaxed (A, η, m)proximal Point Algorithm with errors, variational graphical convergence.

1

Introduction

It is well known that variational inequalities and variational inclusions have been widely used as a mathematical programming tool in modeling many optimization and decision making problems. However, facing uncertainty is a constant challenge for optimization and decision making, see, for example, [1-6] and the references therein. In 1989, Chang and Zhu [1] introduced the concepts of the variational inequalities for fuzzy-set valued operators. Several kinds of variational inequalities, variational inclusions and complementarity problems for fuzzy-set valued operators were considered and studied by many authors, see, for example, [2-5] and the references therein. On the other hand, in order to solve the variational inclusions and related optimization problems, the generalized resolvent operator techniques, which are 

Corresponding author.

Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 133–142, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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extended and modified the projection method and its variant forms including the Wiener-Hopf equations, have been in use for a while and are being applied to several other fields, for stance, equilibria problems in economics, global optimization and control theory, operations research, management and decision sciences, and mathematical programming. See the following example, and [4-17] and the references therein. Example 1. ([18]) Let H be a real Hilbert space, and M : dom(M ) ⊂ H → H be an operator such that M is monotone and R(I + M ) = H, Then based on the Yosida approximation Mρ = 1ρ (I − (I + ρM )−1 ), for each given u0 ∈ dom(M ), there exists exactly one continuous function u : [0, 1) → H such that the first  u (t) + M u(t) = 0, order evolution equation holds for all t ∈ (0, ∞), where the u(0) = u0 , derivative u (t) exists in the sense of weak convergence, that is, u(t+h)−u(t)  h u (t) as h → 0. In [12], Lan first introduced a new concept of (A, η)-monotone (so called (A, η, m)maximal monotone [14]) operators, which generalizes the (H, η)-mono-tonicity, A-monotonicity and other existing monotone operators as special cases, and studied some properties of (A, η)-monotone operators and defined resolvent operators associated with (A, η)-monotone operators. Further, some (systems of) variational inequalities, nonlinear (random or parametric) operator inclusions, nonlinear (set-valued) inclusions, complementarity problems and equilibrium problems have been studied by some authors in recent years because of their close relations to Nash equilibrium problems. See, for example, [6,13,14,19] and the references therein. ˆ An operator T from a real Hilbert spaces H to the collection F(H) = {E : H → [0, 1] a function} of fuzzy sets over H is called a fuzzy-set valued operator, which means that for each x ∈ H a fuzzy set T (x), denoted by Tx in the sequel, is a function from H to [0, 1]. For each y ∈ H, Tx (y) denotes the membership-grade ˆ of y in Tx . A fuzzy-set valued operator T : H → F(H) is said to be closed if for each x ∈ H, the function y → Tx (y) is upper semicontinuous, that is, for any given net {yk } ⊂ H, satisfying ys → y0 ∈ H, we have lim sup Tx (ys ) ≤ Tx (y0 ). s

ˆ Let E ∈ F(H), α ∈ [0, 1]. Then the set (E)t = {x ∈ H : E(x) ≥ t} is called a ˆ t-cut set of E. Let T : H → F(H) be a fuzzy-set valued operator satisfying the following condition (I): There exists an operator a : H → [0, 1], such that for all x ∈ H, we have (Tx )a(x) ∈ CB(H). Remark 1. ([4]) Let X be a normed vector space. If T is a closed fuzzy-set valued operator satisfying the condition (I), then for all x ∈ X, the set (Tx )a(x) belongs to the collection CB(X) of all nonempty closed and bounded subsets of X. Let H be a real Hilbert space, A : H → H, η : H × H → H be any nonlinear operators, M : H × H → 2H be an (A, η, m)-maximal monotone operator,

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B : H × H → H, f, g : H → H be single-valued operators, λ,  > 0 are two constants, a : H → [0, 1] be an operator and T : H → Fˆ (H) be a fuzzy-set valued operator satisfying the condition (I). In this paper, we shall consider the following nonlinear fuzzy-set valued operator equation: Find x, u ∈ H such that Tx (u) ≥ a(x) (i.e., u ∈ (Tx )a(x) ) and η,M g(x) − Jρλ,A [(1 − λ)A(g(x)) + λ(A(f (x)) − ρB(x, u))] = 0,

(1)

η,M = (A + ρλM )−1 is the where , ρ, λ are three positive constants, and Jρλ,A resolvent operator associated with the set-valued operator M . Based on the definition of the resolvent operators associated with (A, η, m)maximal monotone operators, Eqn. (1) can be written as

0 ∈ A(g(x)) − A(f (x)) + ρ[B(x, u) + M (g(x))].

(2)

We remark that for appropriate and suitable choices of H, A, η, B, M, f, g, T and λ, , one can know that the problem (1) or (2) includes a number (systems) of quasi-variational inclusions, generalized (random or parametric) quasivariational inclusions, quasi-variational inequalities, implicit quasi-variational inequalities as special cases. See, for example, [6,9,10,16] and the references therein. Motivated and inspired by these recent algorithmic developments in [4,19], especially, the approach based on the variational graph convergence for approximating the solutions of the nonlinear variational inclusions [6,11,17], in this paper, by using some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity, we shall study the variational graph convergence analysis for a new class of over-relaxed (A, η, m)-proximal point algorithm framework with errors in the context of finding the common solutions for equation (1) and fixed point problems of Lipschitz continuous operators in Hilbert spaces

2

Preliminaries

Let H be a real Hilbert space endowed with a norm  ·  and an inner product ·, · , CB(H) denote the family of all nonempty closed bounded subsets of H and 2H denote the family of all the nonempty subsets of H. In the sequel, we give some concept and lemmas needed later. Definition 1. Let A : H → H be a single-valued operator. Then an operator B : H × H → H is said to be (i) (π, υ)-relaxed cocoercive with respect to A in the first argument, if there exist positive constants π and υ such that for x, y, w ∈ H, B(x, w) − B(y, w), A(x) − A(y) ≥ −πx − y2 + υB(x, w) − B(y, w)2 . (ii) Lipschitz continuous with constant σ in the first argument if there exists a constant σ > 0 such that B(x, z) − B(y, z) ≤ σx − y, ∀x, y, z ∈ H.

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In a similar way, we can define (relaxed) cocoercivity and Lipschitz continuity of the operator B(·, ·) in the second argument. Remark 2. The notion of the cocoercivity is applied in several directions, especially to solving variational inequality problems using the auxiliary principle and projection methods [15], while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoercivity. Several classes of relaxed cocoercive variational inequalities and variational inclusions have been studied in [7, 11-17]. Definition 2. A single-valued operator η : H × H → H is said to be τ -Lipschitz continuous if there exists a constant τ > 0 such that η(x, y) ≤ τ x − y for all x, y ∈ H. Definition 3. Let η : H×H → H and A : H → H be two single-valued operators. Then set-valued operator M : H → 2H is said to be (i) m-relaxed η-monotone if there exists a constant m > 0 such that u − v, η(x, y) ≥ −mx − y2 ,

∀x, y ∈ H, u ∈ M (x), v ∈ M (y);

(ii) (A, η, m)-maximal monotone if M is m-relaxed η-monotone and R(A + ρM ) = H for every ρ > 0. Similarly, we can define strictly η-monotonicity and strongly η-monotonicity of nonlinear operators. Remark 3. (1) If m = 0 or A = I or η(x, y) = x − y for all x, y ∈ H, (A, η, m)maximal monotonicity (so-called (A, η)-monotonicity [12], (A, η)-maximal relaxed monotonicity [7]) reduces to the (H, η)-monotonicity, H-monotonicity, Amonotonicity, maximal η-monotonicity, classical maximal monotonicity (see, for example, [7,9, 12-16, 20]). Further, we note that the idea of this extension is so close to the idea of extending convexity to invexity introduced by Hanson in [21], and the problem studied in this paper can be used in invex optimization and also for solving the variational-like inequalities as a direction for further applied research, see, related works in [22,23] and the references therein. (2) Moreover, operator M is said to be generalized maximal monotone (in short GMM-monotone) if: (i) M is monotone; (ii) A + ρM is maximal monotone or pseudomonotone for ρ > 0. Example 2. ([7]) Suppose that A : H → H is r-strongly η-monotone, and f : H → R is locally Lipschitz such that ∂f , the subdifferential, is m-relaxed ηmonotone with r − m > 0. Clearly, we have u − v, η(x, y) ≥ (r − m)x − y2 , where u ∈ A(x) + ∂f (x) and v ∈ A(y) + ∂f (y) for all x, y ∈ H. Thus, A + ∂f is η-pseudomonotone, which is indeed, η-maximal monotone. This is equivalent to stating that A + ∂f is (A, η, m)-maximal monotone. Definition 4. Let A : H → H be a strictly η-monotone operator and M : H → 2H be an (A, η, m)-maximal monotone operator. Then the corresponding general ρ,A solvent operator Jη,M : H → H is defined by ρ,A (x) = (A + ρM )−1 (x), Jη,M

∀x ∈ H.

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Remark 4. The (A, η, m)-resolvent operators include the corresponding resolvent operators associated with (H, η)-monotone operators, maximal η-monotone operators, H-monotone operators, A-monotone operators, η-subdifferential operators, the classical maximal monotone operators [7,9, 12-14, 20]. Lemma 1. ([12]) Let η : H × H → H be τ -Lipschitz continuous, A : H → H be r-strongly η-monotone and M : H → 2H be (A, η, m)-maximal monotone. Then ρ,A τ the resolvent operator Jη,M : H → H is r−ρm -Lipschitz continuous. Definition 5. Let T : H → Fˆ (H) be a closed fuzzy-set valued operator satisfying ˆ the condition (I). Then, T is said to be ξ-H-Lipschitz continuous if ˆ x )a(x) , (Ty )a(y) ) ≤ ξx − y, H((T

∀x, y ∈ H,

ˆ is the Hausdorff metric on CB(H). where ξ > 0 is a constant and H Definition 6. Let M n , M : H → 2H be (A, η, m)-maximal monotone operators for n = 0, 1, 2, . . . . Let A : H → H be r-strongly η-monotone and β-Lipschitz A−G continuous. The sequence M n is graph-convergent to M , denoted M n −→ M , if for every (x, y) ∈ graph(M ) there exists a sequence (xn , yn ) ∈ graph(M n ) such that xn → x, yn → y as n → ∞. Based on Definition 6 and Theorem 2.1 in [17], we have the following lemma. Lemma 2. Let M n , M : H → 2H be (A, η, m)-maximal monotone operators for A−G n = 0, 1, 2, · · · . Then the sequence M n −→ M if and only if ρ,A ρ,A Jη,M n (x) → Jη,M (x), ∀x ∈ H, ρ,A ρ,A where Jη,M = (A + ρM n )−1 , Jη,M = (A + ρM )−1 , ρ > 0 is a constant, and A : H → H is r-strongly η-monotone and β-Lipschitz continuous.

3

Proximal Point Algorithm and Graph-Convergence

In this section, by using some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity, we shall develop a new perturbed iterative algorithm framework based on the variational graph convergence for approximating the common solutions for nonlinear fuzzy-set valued operator equations and fixed point problems for Lipschitz continuous operators in Hilbert spaces. Firstly, we note that x ∈ H is called a fixed point of a fuzzy-set valued operator (as generalization of fixed point of a set-valued (multivalued) operator) if T : H → Fˆ (H) is a fuzzy-set valued operator satisfying the condition (I) and x ∈ (Tx )a(x) ∈ CB(H).

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Further, if (x∗ , u∗ ) is a solution of Eqn. (1) and g(x∗ ) ∈ F (Q), where F (Q) is the set of fixed points of Q, that is, F (Q) = {x ∈ H : Q(x) = x}, then we note that the Eqn. (1) can be rewritten as  λρ,A ∗ g(x∗ ) = Q(g(x∗ )) = Q(Jη,M (z )), z ∗ = (1 − λ)A(g(x∗ )) + λ(A(f (x∗ )) − ρB(x∗ , u∗ )). This formulation allows us to construct the following perturbed iterative algorithm framework with errors for finding a common element of two different sets, that is, the set of the solutions for nonlinear fuzzy-set valued operator equation (1) and the set of fixed point for a Lipschitz continuous operator. Algorithm 1. Step 1. For an arbitrary initial point x0 ∈ H, take z0 ∈ H such that λρ,A g(x0 ) = Jη,M (z0 ).

Step 2. For all b = c, d, e, choose sequences {bn } ⊂ H is error sequence to take into account a possible inexact computation of the operator points, which satisfy the following conditions: lim bn = 0,

n→∞

∞ 

bn − bn−1  < ∞.

n=1

Step 3. Let the sequence {(xn , zn )} ⊂ H×H, a : H → [0, 1] and Tx (un ) ≥ a(xn ) satisfy  λρ,A xn+1 = (1 − α)xn + α[xn − g(xn ) + Q(Jη,M n (zn ))] + αdn + en , (3) zn = (1 − λ)A(g(xn )) + λ(A(f (xn )) − ρB(xn , un )) + cn . λρ,A n −1 where α, λ, ρ,  > 0 are constants and Jη,M for all n ∈ N. n = (A + λρM ) Step 4. Choose Tx (un+1 ) ≥ a(xn+1 ) such that (see [24])

un − un+1  ≤ (1 +

1 ˆ xn )a(x ) , (Txn+1 )a(x ) ). )H((T n n+1 n+1

(4)

Step 5. If xn , un , zn , cn , dn and en satisfy (3) and (4) to sufficient accuracy, stop; otherwise, set n := n + 1 and return to Step 2. Now we prove the existence of a solution of the problem (1) and the convergence of Algorithm 1. Theorem 1. Let H be a real Hilbert space, η, A, M, B and T , g, f be the same as in the Eqn. (1). Also suppose that the following conditions hold: (H1 ) η is τ -Lipschitz continuous, Q : H → H is κ-Lipschitz continuous, and A is r-strongly η-monotone and σ-Lipschitz continuous; (H2 ) g is δ-strongly monotone and υ-Lipschitz continuous, f is ς-Lipschitz ˆ continuous, T is a closed ξ-H-Lipschitz continuous fuzzy-set valued operator satisfying the condition (I) with a function a : H → [0, 1]; (H3 ) B is (π, ι)-relaxed cocoercive with respect to f in the first argument and Lipschitz continuous with constants β and  in the first and second variable,

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respectively, where f : H → H is defined by f (y) = A ◦ f (y) = A(f (y)) for all y ∈ H; (H4 ) for n = 0, 1, 2, · · · , M n : H → 2H is (A, η, m)-maximal monotone operA−G ators with M n −→ M ; (H5 ) there exist positive constants λ,  and ρ such that √ ⎧ k = 1 − 2δ + υ 2 < 1, h = ξ + m(1−k) < β, ⎪ λκτ ⎪ ⎪ συ r l ⎨ l = r(1−k)−(1−λ)κτ < σς, ρ < min{ , }, λκτ  m h (5) 2 2 2 ς 2 − l 2 )(β 2 − h2 ), ιβ > π + lh + (σ ⎪ √ 2 ⎪ ⎪ 2 ⎩ (ιβ −π−lh)2 −(σ2 2 ς 2 −l2 )(β 2 −h2 ) ρ − ιββ 2−π−lh . −h2  < β 2 −h2 Then the iterative sequence (xn , un ) defined by Algorithm 1 converges strongly to a solution (x∗ , u∗ ) of the problem (1). Proof. By the assumptions of the theorem, (3) and (4), we have  xn − xn−1 − [g(xn ) − g(xn−1 )] ≤ 1 − 2δ + υ 2 xn − xn−1 , B(xn , un ) − B(xn , un−1 ) ≤ un − un−1  ˆ xn )a(x ) , (Txn−1 )a(x ) ) ≤ ξ(1 + n−1 )xn − xn−1 , ≤ (1 + n−1 )H((T n

n−1

A(f (xn )) − A(f (xn−1 )) − ρ[B(xn , un−1 ) − B(xn−1 , un−1 )]2 ≤ A(f (xn )) − A(f (xn−1 ))2 + ρ2 B(xn , un−1 ) − B(xn−1 , un−1 )2 −2ρ B(xn , un−1 ) − B(xn−1 , un−1 ), A(f (xn )) − A(f (xn−1 ))

≤ (σ 2 2 ς 2 − 2ριβ 2 + 2ρπ + ρ2 β 2 )xn − xn−1 2 , and zn − zn−1  ≤ (1 − λ)A(g(xn )) − A(g(xn−1 )) + ρλB(xn , un ) − B(xn , un−1 ) +λA(f (xn )) − A(f (xn−1 )) − ρ[B(xn , un−1 ) − B(xn−1 , un−1 )] +cn − cn−1  ≤ ϑn xn − xn−1  + cn − cn−1 ,

 where ϑn = (1−λ)συ +ρλξ(1+n−1 )+λ σ 2 2 ς 2 − 2ριβ 2 + 2ρπ + ρ2 β 2 . Thus, by Lemma 1, now we know xn+1 − xn  ≤ (1 − α)xn − xn−1  + αxn − xn−1 − [g(xn ) − g(xn−1 )] λρ,A λρ,A λρ,A λρ,A +ακJη,M n (zn ) − Jη,M (zn ) + ακJη,M (zn ) − Jη,M (zn−1 ) λρ,A λρ,A +ακJη,M n−1 (zn−1 ) − Jη,M (zn−1 ) + αdn − dn−1  + en − en−1   ακτ ≤ (1 − α + α 1 − 2δ + υ 2 )xn − xn−1  + zn − zn−1  r − ρm +ακ(εn + εn−1 ) + αdn − dn−1  + en − en−1 

≤ θn xn − xn−1  + ωn ,

(6)

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√ where θn = 1 − α + α 1 − 2δ + υ 2 +

ακτ ϑn r−ρm ,

ωn =

ακτ r−ρm cn − cn−1  + ακ(εn + ρλ,A ρλ,A Jη,M p (zp ) − Jη,M (zp ) for

εn−1 ) + αdn − dn−1  + en − en−1  and εp = p = n − 1, n.  + ρλξ + λ σ 2 2 ς 2 − 2ριβ 2 + 2ρπ + ρ2 β 2 , θ = 1 − α + √Let ϑ = (1 − λ)συ ακτ ϑ α 1 − 2δ + υ 2 + r−ρm . Then, we know that θn ↓ θ as n → ∞. The condition (5) implies that 0 < θ < 1 and so there exist n0 > 0, θ0 ∈ (θ, 1) such that θn ≤ θ0 for all n ≥ n0 . Hence, it follows from (6) that xn+1 − xn  ≤ θ0 xn − xn−1  + ωn ≤ θ0n−n0 xn0 +1 − xn0  +

n−n 0

θ0i−1 ωn−(i−1) ,

i=1

which implies that for any m ≥ n > n0 , xm − xn  ≤

m−1 

xj+1 − xj 

j=n

≤

m−1  j=n

θ0j−n0 xn0 +1 − xn0  +

m−1  j−n 0

θ0i−1 ωj−(i−1) .

(7)

j=n i=1

From the hypothesis of Algorithm 1, Lemma 2 and (7), it follows that {xn } is a Cauchy sequence, that is, there exists x∗ ∈ H such that xn → x∗ as n → ∞. Next, we prove that un → u∗ ∈ (Tx∗ )a(x∗ ) . In fact, condition (H2 ) implies that {un } is also Cauchy sequence in H. Let un → u∗ . In the sequel, we will show that u∗ ∈ (Tx∗ )a(x∗ ) . Noting un ∈ (Txn )a(xn ) , from the results in [24], we have d(u∗ , (Tx∗ )a(x∗ ) ) = inf{un − y : y ∈ (Tx∗ )a(x∗ ) } ≤ u∗ − un  + d(un , (Tx∗ )a(x∗ ) )

ˆ xn )a(x ) , (Tx∗ )a(x∗ ) ) ≤ u∗ − un  + d(un , (Txn )a(xn ) ) + H((T n ≤ u∗ − un  + 0 + ξxn − x∗  → 0.

Hence d(u∗ , (Tx∗ )a(x∗ ) ) = 0 and therefore u∗ ∈ (Tx∗ )a(x∗ ) . By continuity and the hypothesis of Algorithm 1, we know that (x∗ , u∗ ) satisfies the Eqn. (1). This completes the proof. 2 Remark 5. Condition (H5 ) in Theorem 1 holds for some suitable value of constants, for example, λ = ς = 0.8, ρ = 4.5,  = 1, δ = 0.3, υ = 0.6, σ = τ = 0.5, β = 0.08, ξ = 0.01, κ = 0.4,  = 0.05, π = m = 0.02, ι = 10, r = 0.2. Remark 6. If M is a (H, η)-monotone operator, H-monotone operator, Amonotone operator, maximal η-monotone operator and classical maximal monotone operator and dn = 0 or en = 0 or cn = 0 for all n ≥ 0 in Algorithm 1, then we can obtain the corresponding results of Theorem 3.1. Our results improve and generalize the corresponding results of [6,9,16,19] and many other recent works.

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Example 3. Assume that H is a real Hilbert space, A : H → H is r-strongly η-monotone, and ϕ : H → R is a locally Lipschitz functional such that ∂ϕ, the subdifferential, is m-relaxed η-monotone with r − m > 0. This is equivalent to stating that A + ∂ϕ is (A, η, m)-maximal monotone. Thus, if all the conditions for Theorem 1 are satisfied, one can apply Theorem 1 to the approximationsolvability of the operator inclusion problem of finding x ∈ H and u ∈ (Tx )a(x) such that A(f (x)) ∈ (1 + ρ)A(g(x)) + ρB(x, u) + ρ∂ϕ(g(x)).

4

Conclusions

In this paper, we first introduce a class of nonlinear fuzzy-set valued operator equations. Then, by using some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity, we construct a new class of over-relaxed (A, η, m)-proximal point algorithm framework with errors and investigate the variational graph convergence analysis for this algorithm framework in the context of finding the common solutions for the nonlinear equations and fixed point problems of Lipschitz continuous operators in Hilbert spaces. Furthermore, we also give some examples of applying the main result. The results presented in this paper improve and generalize some well known results in recent literatures. Acknowledgments. This work was supported by the Sichuan Youth Science and Technology Foundation (08ZQ026-008), the Open Foundation of Artificial Intelligence of Key Laboratory of Sichuan Province (2009RZ001) and the Scientific Research Fund of Sichuan Provincial Education Department (10ZA136). The authors are grateful to the editors and referees for valuable comments and suggestions.

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8. Cai, L.C., Lan, H.Y., Zou, Y.Z.: Perturbed algorithms for solving nonlinear relaxed cocoercive operator equations with general A-monotone operators in Banach spaces. Commun. Nonlinear Sci. Numer. Simulat. 16(10), 3923–3932 (2011) 9. Fang, Y.P., Huang, N.J.: H-Monotone operator and resolvent operator technique for variatonal inclusions. Appl. Math. Comput. 145, 795–803 (2003) 10. He, X.F., Lou, J., He, Z.: Iterative methods for solving variational inclusions in Banach spaces. J. Comput. Appl. Math. 203(1), 80–86 (2007) 11. Lan, H.Y., Cai, L.C.: Variational convergence of a new proximal algorithm for nonlinear general A-monotone operator equation systems in Banach spaces. Nonlinear Anal. TMA. 71(12), 6194–6201 (2009) 12. Lan, H.Y.: A class of nonlinear (A, η)-monotone operator inclusion problems with relaxed cocoercive mappings. Adv. Nonlinear Var. Inequal. 9(2), 1–11 (2006) 13. Lan, H.Y.: Approximation solvability of nonlinear random (A, η)-resolvent operator equations with random relaxed cocoercive operators. Comput. Math. Appl. 57(4), 624–632 (2009) 14. Lan, H.Y.: Sensitivity analysis for generalized nonlinear parametric (A, η, m)maximal monotone operator inclusion systems with relaxed cocoercive type operators. Nonlinear Anal. TMA. 74(2), 386–395 (2011) 15. Verma, R.U.: A-monotononicity and applications to nonlinear variational inclusion problems. J. Appl. Math. Stochastic Anal. 17(2), 193–195 (2004) 16. Verma, R.U.: A-monotone nonlinear relaxed cocoercive variational inclusions. Central European J. Math. 5(2), 386–396 (2007) 17. Verma, R.U.: A generalization to variational convergence for operators. Adv. Nonlinear Var. Inequal. 11(2), 97–101 (2008) 18. Komura, Y.: Nonlinear semigroups in Hilbert space. J. Math. Society Japan 19, 493–507 (1967) 19. Petrot, N.: A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems. Appl. Math. Letters 23(4), 440–445 (2010) 20. Zeidler, E.: Nonlinear functional analysis and its applications, vol. I. Springer, New York (1986) 21. Hanson, M.A.: On sufficiency of Kuhn-Tucker Conditions. J. Math. Anal. Appl. 80(2), 545–550 (1981) 22. Soleimani-damaneh, M.: Generalized invexity in separable Hilbert spaces. Topology 48(2-4), 66–79 (2009) 23. Soleimani-damaneh, M.: Infinite (semi-infinite) problems to characterize the optimality of nonlinear optimization problems. European J. Oper. Res. 188(1), 49–56 (2008) 24. Nadler, S.B.: Muliti-valued contraction mappings. Pacific J. Math. 30, 475–488 (1969)

Reliability-Based Route Optimization of a Transportation Network with Random Arc Capacities and Time Threshold Tao Zhang, Bo Guo, and Yuejin Tan College of Information Systems and Management, National University of Defense Technology, 410073 Changsha, China

Abstract. The classical route optimization problem of a network focuses on the shortest or fastest route mainly under the assumption that all roads will not fail. In fact, the capacities of roads in a transportation network are not determinate but random because of the traffic accidents, maintenance or other activities. So a most reliable route from source to sink under the time threshold may be more important than the shortest or fastest route sometimes. This paper describes a stochastic Petri net-based simulation approach for reliability-based route optimization of a transportation network. The capacities of arcs may be in a stochastic state following any discrete or continuous distribution. The transmission time of each arc is also not a fixed number but stochastic according to its current capacity and demand. To solve this problem, a capacitated stochastic colored Petri net is used for modeling the system behavior. By the simulation, the optimal route with highest reliability can be obtained. Finally, an example of a transportation network with random arc capacities is given. Keywords: Route optimization, Reliability, Multi-state, transportation network, Petri net, time threshold.

1

Introduction

The classical route optimization problems are the variants of the shortest path problem, which focus on how to obtain the shortest route, least cost route, quickest route or route with some combination of multiple criteria when goods or commodities are transmitted from one node (source) to another node (sink) through the network[1–4]. In a transportation network, it is an important issue to reduce the total transportation time through the network. Hence, the quickest route optimization problem, a time version of the shortest path problem, is proposed. This problem is for finding a quickest route with minimum transmission time to send a given amount of data from the source to the sink, where each arc has two attributes; the capacity and the duration time[5]. In most of the studies, the capacity and the duration time of each arc are both assumed to be deterministic. However, in many real transportation networks, the capacity of each road (arc) is stochastic due to failure, partial failure, traffic accident, maintenance, etc. The state of each road may not be fully working either fully Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 143–156, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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failed, but be in an intermediate state. Such a network is named a multi-state network, also called as a stochastic-flow network. In this network, the shortest or quickest route may not be the most reliable route, but sometimes it would be preferred to obtain the most reliable route under a given time threshold. To obtain the most reliable route, we need to calculate the reliability of a given route firstly. The classical network reliability problem is the two-terminal reliability (2T R) problem which assumes both network and components have only binary states, fully working or fully failed states. However, the traditional binary reliability theory and models fail to characterize the behavior of a multi-state network[6]. Hence, some multi-state reliability analysis approaches and models have been proposed to cope with this problem. The multi-state two-terminal reliability at demand level d (M 2T Rd) is the original one which is defined as the probability that d units of demand can be supplied from source to sink through multi-state arcs[7, 8]. Then it is extended to some more complex problems, such as multi-commodity reliability[9–13], M 2T Rd under time threshold (M 2T R(d,T ))[14, 15] and under budget constraint (M 2T R(d,c))[16, 17]. Many previous models for computing M 2T Rd mostly based on minimal cut (MC) or minimal path (MP). In such approaches, the key problem is to obtain all possible d-MCs or d-MPs under d units of demand. Then the reliability can be computed directly if d-MCs or d-MPs are given. These extended problems, i.e. M 2T R(d,T ) and M 2T R(d,c), can also be solved by the MC or MP based approaches where the difference is the algorithm of obtaining MCs or MPs. Considering a stochastic-flow network, Lin Y.K.[18] extended the fastest route problem to evaluating the system reliability that d units of data could be sent under the time constraint T . In his study, the lead time of each arc is related to its current capacity and demand d, and the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time. A simple algorithm is proposed to generate all (d, T )-MPs and the system reliability can then be computed in terms of (d, T )-MPs. Moreover, the optimal pair of minimal paths with highest system reliability could be obtained. The MC/MP-based approach is often a tedious process. It is also hard to be applied when there exist some arcs whose states or lead times follow different continuous probability distributions. In fact, each road of a transportation network may be in a stochastic state following a probability distribution and the total transportation time of a given route is not determinate but related with the type of vehicle, the current states of roads and the transportation demand. Hence, more practical algorithms have been proposed to solve this problem. Simulation is a very effective method which trade off accuracy for execution time. Ramirez-Marquez J.E. et al.[6] presented a Monte-Carlo simulation approach for approximating the M 2T Rd. Janssens G.K. et al.[19] considered the uncertainty in travel times caused by congestion on roads and applied a methodology in which a heuristic was used to find a solution for routing and the time Petri net was used to evaluate the sensitivity of the route solution. This paper describes a capacitated stochastic colored Petri net (CSCPN) based simulation approach for reliability-based route optimization of a transportation

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network with arcs whose capacities are stochastic following any discrete or continuous distribution. The rest of the paper is organized as follows. Section 2 describes the problem. Section 3 presents the CSCPN model and how to describe the dynamic behavior of a multi-state transportation network. Section 4 gives how to obtain the most reliable route by the CSCPN-based simulation. Section 5 presents a computational experiment. The final section makes a conclusion.

2

Problem Description

There is a transportation network G = (N, A) with a required transportation demand d from source node s to sink node t in the time threshold T by the specified vehicles. The required minimal capacity of each road is c so that the vehicles can pass by. N = {n1 , n2 , ..., nm } represents the set of nodes, m is the number of nodes and ni represents the ith node. A = {a1 , a2 , ..., an } represents the set of arcs, n is the number of arcs and ai represents the ith arc. The current state (capacity) of arc ai is represented by ci . It is stochastic with a given probability distribution. Let li and ui be the smallest and largest capacities of arc ai respectively, so li ≤ ci ≤ ui . Let δ(ai , ci , d) be the transmission time to transmit d units of demand through arc ai under its capacity ci . If ci < c, δ(ai , ci , d) = ∞. It is also stochastic and described by a given probability distribution. The aim is to obtain the most reliable route when the demand d, the time threshold T and the required minimal capacity c are given. The assumptions in this study are as below. (1) Each node is perfectly reliable. (2) The capacities of different arcs are statistically independent.

3

Capacitated Stochastic Colored Petri Net

Due to the uncertainties in a multi-state network, the Petri-net method is suitable for describing the dynamic behavior of the system. First created in 1962 and reported in the thesis of Petri[20], Petri-net are an adaptable and versatile, yet simple, graphical modeling tool used for dynamic system representation. In this study, CSCPN, a kind of advanced stochastic colored Petri net (SCPN), is advanced for analyzing the dynamic behavior of multi-state transportation network. 3.1

Petri Net

A basic Petri net (PN) is a kind of directed graph which has four elements: places, transitions, arcs and tokens. The arcs connect the places to the transitions and the transitions to the places. Each place is marked with m tokens where m is a non-negative integer, called marking. Such a token can be used to stand for state, data, items or resources. After the enabling condition of a transition is satisfied, some tokens may move out or in the corresponding places. The number of tokens moving out or in is related to the weight of the corresponding arc[21].

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Formally, a PN is a five-tuple P N = P, T, P re, P ost, M0  ,

(1)

where[22]: P = {P1 , P2 , ..., PL } is a finite, nonempty set of places, L > 0. T = {T1 , T2 , ..., TJ } is a finite, nonempty set of transitions, J > 0. P re : P × T → N + is the input incidence function such that P re(Pi , Tj ) is the weight of the arc directed from place Pi to transition Tj . N + is the set of non-negative integers. P ost : P × T → N + is the output incidence function such that P ost(Pi , Tj ) is the weight of the arc directed from transition Tj to place Pi . M0 : P → N + is the initial marking function that associates zero or more tokens to each place. 3.2

SCPN

To describe the duration of behavior, timed Petri net (TPN) was defined by associating time with the firing of transitions in PN. A special case of TPN is stochastic Petri net (SPN) where the firing times are considered random variables. In many systems, they present similar processes, and differ from each other only by their inputs and outputs[23]. To reduce the quantity of places, transitions and arcs in a PN, a more compact representation of a Petri net , called as colored Petri net (CPN), was developed[24]. In CPN, each token moving between places and transitions is assigned with a color. The concept of color is analogous to the concept of type, common among the programming languages. It is easy to know that SCPN have combined the characteristics of both SPN and CPN. SCPN is defined as a directed graph represented by a eight-tuple SCP N = P, T, Co, H, P re, P ost, F T, M0  ,

(2)

where: P = {P1 , P2 , ..., PL } is a finite, nonempty set of places, L > 0. T = {T1 , T2 , ..., TJ } is a finite, nonempty set of transitions, J > 0. Co : P ∪ T → C is a color function that associates with each element in P ∪ T a non empty set of colors. C is the set of all possible token colors. In this study, a token color is represented by not an integer but a structure with multi-values. Co(Pi ) is the set of possible token colors in place Pi ∈ P . Co(Ti ) is the set of possible token colors in transition Ti ∈ T . H : P × T → CMS is defined for an inhibitor arc that connects a place to a transition. An inhibitor arc between Pi ∈ P and Tj ∈ T (i.e. H(Pi , Tj ) = {c}, c ∈ C) implies that as soon as there are a token whose color is c in Pi , the arc inhibits the firing of Tj . CMS is a multiset over the non-empty set C. P re is the pre-incidence matrix. P re(Pi , Tj ) : Co(Tj ) → Co(Pi )MS is the input incidence mapping function from the set of occurrence colors of Tj ∈ T to the set of multisets over Co(Pi ), i.e. P re(P1 , T1 ) : c1 → {c1 , c3 } implies that if

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there are token colors c1 , c3 in the place P1 , the firing condition of transition T1 on the place P1 is satisfied. P ost is the post-incidence matrix. P ost(Pi , Tj ) : Co(Tj ) → Co(Pi )MS is the output incidence mapping function from the combination of Tj and Pi to the multiset over the set of colors of Co(Pi ). F T : T × C → R+ is the firing time function such that F T (Tj , ci ) is the duration of the firing of transition Tj when the enabled token color is ci ∈ Co(Tj ). R+ is the set of non-negative real numbers. F T is according to a kind of given distribution and the token colors. M0 : P → CMS is the initial token color set function that associates zero or more token colors to each place. M0 (Pi ) = {c1 , c2 , ..., cn }, ci ∈ C, i = 1, 2, ..., n, implies that the initial token color set of place Pi is {c1 , c2 , ..., cn }. For example, there are two components whose failure and maintenance time follow different distribution types. The requirements of maintenance resources are also different. One repairman of type A should be available for the maintenance of the first component and one repairman of each type A and B should be available for second one. Fig.1(a) shows a SPN model for this system. The places P1 , P3 respectively stand for the working condition of the first and second component. The places P2 , P4 respectively stand for the failed condition of the first and second component. The places P5 , P6 respectively stand for the state of repairman of type A and B. The transitions T1 , T2 respectively stand for the failure and maintenance process of the first component. The transitions T3 , T4 respectively stand for the failure and maintenance process of the second component. Although their time distributions and requirements are different, their failure and maintenance process are similar. Hence, the SPN shown in Fig.1(a) can be modified to the SCPN shown in Fig.1(b). The token color marked with and respectively stand for the first and second component. The token color marked with and respectively stand for the repairman of type A and B. So,

Fig. 1. SPN and SCPN of the failure and maintenance process of two different components

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in the SCPN shown in Fig.1(b), P = {P1 , P2 , P3 }, T = {T1 , T2 }, Co(P1 ) = {1, 2}, Co(P2 ) = {1, 2}, Co(P3 ) = {3, 4}, Co(T1 ) = {5, 6}, Co(T2 ) = {7, 8} , P re(P1 , T1 ) = {{5} → {1}, {6} → {2}}, P re(P2 , T2 ) = {{7} → {1}, {8} → {2}}, P re(P3 , T2 ) = {{7} → {3}, {8} → {3, 4}}, P re(P2 , T1 ) = {{5} → {1}, {6} → {2}}, P re(P1 , T2 ) = {{7} → {1}, {8} → {2}}, P ost(P2 , T1 ) = {{5} → {1}, {6} → {2}}, P ost(P1 , T2 ) = {{7} → {1}, {8} → {2}}, P ost(P3 , T2 ) = {{7} → {3}, {8} → {3, 4}}, M0 (P1 ) = {1, 2}, M0 (P3 ) = {3, 4}. F T (T1 , 5), F T (T1 , 6), F T (T2 , 7), F T (T2 , 8) may follow any kind of probability distribution according to the failure and maintenance process. 3.3

CSCPN for Transportation Network

In this study, to describe the dynamic behavior of a multi-state transportation network, SCPN is modified to be more universal basing on the original definition. Original transition is extended to capacitated transition which has a stochastic capacity. CSCPN is defined by a nine-tuple CSCP N = P, T, Co, H, P re, P ost, F T, CP, M0  ,

(3)

where CP is added into the definition of SCPN and F T is extended as below. CP : T → R+ is the capacity function such that CP (Tj , ci ) is the capacity of transition Tj when the enabled token color is ci ∈ Co(Tj ). CP is related to a kind of given distribution. F T : T × C → R+ is the firing time function such that F T (Tj , cpj , ci ) is the duration of the firing of transition Tj when its current capacity is cpj and the enabled token color is ci ∈ Co(Tj ). For a multi-state transportation network, the nodes and arcs of network are respectively represented by the places and transitions in CSCPN model. The one way and two way arcs in a transportation network will be represented by one transition and two transitions respectively as shown in Fig. 2 and Fig. 3. The firing durations of the transition T1 and T2 are respectively the transmission time from node 1 to 2 and from node 2 to 1. In fact, their transmission time would be following different kinds of probability distribution.

Fig. 2. SCPN model for a one way arc

Fig. 3. SCPN model for a two way arc

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The transitions whose input/output places include Pi are called as the output/input transitions of place Pi . Because of the characters of a transportation network, each transition in the CSCPN of this kind network has only one input place and one output place. Transitions with more than one input or output places shown in Fig. 4 are not permitted in the CSCPN of a transportation network.

Fig. 4. Impossible transitions in the SCPN of a transportation network

For example, there is a transportation network with eight crossings and fifteen roads. When the source and sink node are node 1 and 8 respectively, this network can be represented by a graph as shown in Fig. 5. Basing on the description rule of CSCPN, it is easy to be transformed from a network graph into CSCPN as shown in Fig. 6.

Fig. 5. An example of transportation network

In order to analyze the route and provide enough information for evaluation of two-terminal reliability, A complex token color in CSCPN of a multi-state network is defined as a structure as below. Struct T okenColor { string RouteInf o; // the information of the route that this token has passed int SourceN o; // the number of the source node place int SinkN o; // the number of the sink node place double Demand; // the demand of transportation, it is set to be d double Capacity; // the minimal requirement of the road capacity double T imestamp; //the time stamp }

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Fig. 6. The CSCPN of the transportation network shown in Fig.5

The color of a token can be also described by a six-tuple co=(RouteInf o, SourceN o, SinkN o, Demand, Capacity, T imestamp). The aim is to record the route and time information and to guide this token to the sink node place. If a token is labelled by co, co.[propertyname] is used to get or set the value of corresponding property, i.e. co.SinkN o represents the sink node place number of token co. If a token co = (”[P 1] − [T 2] − [P 3] − [T 11] − [P 6] − [T 20] − [P 8]”, 1, 8, 50, 10, 60.8) appears in place P8 , it stands for that 50 units of demand of need to be sent from source node (P1 ) to sink node (P8 ), the required minimal capacity is 10 and it arrives at the sink node P8 at the time of 60.8 through the nodes of P3 and P6 . In the SCPN of this study, the enabling condition of the transition Tj is that the input place of Tj has more than one token whose color co satisfies three conditions as below. (1) co.RouteInf o does not contain the name of the output place of Tj . It guarantees that a token will not be circulating in the network. (2) co.Capacity is less than the current capacity of Tj which will be given by the sampling function. (3) co.SinkN o is not the number of the input place. As shown in Fig.7, the transition Tj is enabled by the token co in its input  place Pi and a new token color co will be created as below. 

co .RouteInf o = co.RouteInf o + ” − [Tj ] − [Pk ]”

(4)

co .T imestamp = co.T imestamp + F T (Tj , cpj , co.Demand) ,

(5)



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where Pk is the output place of Tj and F T (Tj , cpj , co.Demand) is the firing time function which stands for the transportation time under the current capacity cpj and demand d given, F T = δ(aj , cpj , d). Then it will move into the transition  Tj . Analogously, the token color co will be created and move into the transition Tj as shown in Fig. 7(b). The token co will not be removed until all output transitions of place Pi have been checked out. Anyhow, the token colors in the  sink node place will not be moved. In the example of Fig. 7, co .T imestamp <    co .T imestamp. So when it is the time of co .T imestamp, the token color co in the transition Tj will move into the output place Pk as shown in Fig. 7(c).  Analogously, when it is the time of , the token color co will move into the  output place Pn as shown in Fig. 7(d). Because the token color co is moved  from the place Pi and co .RouteInf o contains the name of the place Pi , the enabling condition of the transition Tm will not be satisfied by this token. It is shown that the token will be broadcast out until it arrives at the sink node. It helps to find all possible routes from the source node to the sink node under the time threshold.

Fig. 7. An example of tokens moving in CSCPN of a transportation network

4

Simulation for Reliability-Based Route Optimization

The reliability in this problem is defined as the probability that a demand of d units can be transported from source to sink node in given time threshold T and the required minimal capacity c considering a multi-state transportation network (M 2T R(d,T,c)). Basing on the simulation of SCPN, all possible routes

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which may satisfy the requirement of the transportation can be found. The route with highest reliability will be taken as the best route. Let N be the number of simulations. In each simulation, the simulation will be over when the current simulation time is more than T . Let Ci = {ci1 , ci2 , ..., ciMi } , Mi respectively be the set and number of token colors in the sink node place after the ith simulation, i = 1, 2, ..., N . They are obtained when each simulation is fini ished. Let Ci .RouteInf o be {C1i .RouteInf o, C2i .RouteInf o, ..., CM .RouteInf o}. i So if the number of simulations is big enough, the set of all possible routes is given by N  AllRoutes = {r1 , r2 , ..., rM } = Ci .RouteInf o , (6) i=1

where M is the total number. For each route rj in AllRoutes, the route reliability can be calculated by R(rj ) =

N 1  ( Count(Ci , rj )) , N

(7)

i=1

where

 0, if rj is not in Ci .RouteInf o, Count(Ci , rj ) = 1, if rj is in Ci .RouteInf o.

For a transportation network, the reliability can be calculated by M 2T R(d,T,c) = max(R(r1 ), R(r2 ), ..., R(rj ), ..., R(rM )), rj ∈ AllRoutes

(8)

The route with highest reliability which will be taken can be given by Route = rj ,

(9)

where R(rj ) = M 2T R(d,T,c), rj ∈ AllRoutes.

5

Experimental Results

The experiment are basing on the example shown in Fig. 5. The source and sink respectively are the 1st and 8th nodes. The capacity and the transmission time distribution parameters of each arc are all shown in Table 1. Table 2 presents the experiment result when different combinations of the time threshold T and the required capacity c are given and the transportation demand d is 50. Table 3 presents the experiment result when the transportation demand d is 100. Each experiment result includes the best route and its reliability M 2T R(d,T,c). Hence, after the time threshold and required capacity are given, the reliabilitybased best route can be obtained in these tables which support the planners to do a fast decision making.

Reliability-Based Route Optimization of a Transportation Network Table 1. Network data of the example Arc(T ransition) Capacity distribution (ci ) Transmission time δ(ai , ci , d), ci > 0

1 (T1 )

Capacity Probability 0 0.063 5 0.162 10 0.236 20 0.549

2 (T2 )

Mean distribution Min=0, Max=50

3 (T3 )

4 (T4 , T5 )

5 (T6 , T7 ) 6 (T8 ) 7 (T9 , T10 ) 8 (T11 )

Triangular distribution Min=0,Mean=20, Max=25 Capacity Probability 0 0.053 5 0.257 8 0.434 12 0.256 Triangular distribution Min=0,Mean=15, Max=25 Mean distribution Min=0, Max=40 Triangular distribution Min=0,Mean=16, Max=40 Triangular distribution Min=0,Mean=40, Max=50

9 (T12 , T13 )

Mean distribution Min=0, Max=50

10 (T14 )

Mean distribution Min=0, Max=35

11 (T15 , T16 )

12 (T17 , T18 )

Triangular distribution Min=0,Mean=15, Max=40 Capacity Probability 0 0.012 4 0.203 8 0.331 12 0.454

13 (T19 )

Mean distribution Min=0, Max=30

14 (T20 )

Mean distribution Min=0, Max=50

15 (T21 )

Triangular distribution Min=0,Mean=15, Max=20

Lognormal distribution, StandardDeviation=6, if d | <page> <page> → <system> <page> | <system> <system> → <stave> <system> | <stave> <system> → <part name> <stave> <system> | <part name> <stave> <part name> → Flute | Piano | etc. <stave> → beginning-barline | → | → | → <measure> | <measure> <measure> → | → | → | → <stem> | <stem> <stem> → <note stem> | <note stem> | <note stem> → left-beam | right-beam → left-beam | right-beam → flag | flag <note stem> → note-head <note stem> | note-head Music notation can be described by different grammars. Construction of such grammars may reflect various aspects of spaces of music notation, e.g. geometrical or logical structuring, c.f. [4]. The above description is constructed from the point of view of geometrical properties. The nonterminals < page > and < system > define items of music notation strictly related to paginated music notation. The nonterminal defines items related to paginated music notation, but it is also important as a logical element of music information. The grammar presented here is a simplified version of a real grammar fully describing paginated music notation. Due to limitation of the paper, in this grammars many details are skipped. For instance, pitch and duration of notes, articulation and ornamentation symbols, dynamics, placement of graphical elements on the page etc. are not outlined. On the other hand, expansion of the grammar is not difficult in its essence and could be easily done based on this

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Fig. 3. A fragment of the derivation tree of Polonaise from Sonata for Flute and Piano by L. v. Beethoven. Derivation of the first beamed group in the second stave of the second system is expanded. Ellipsis indicate other parts of the score. The tree includes attributes of terminals: number of beams for left-beam and right-beam, pitch for head and duration for stem.

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study. Also, discussing all details of the subject can neither better explain the concept of the subject, nor bring new values. In Figure 3 a fragment of derivation tree of paginated music notation is shown. Terminal symbols have attributes associated. Namely, number of beams is associated with terminals left-beam and right-beam, the terminal note-head has pitch defined and the terminal stem has duration (of notes tied to this stem) defined. For the sake of simplicity, these attributes are not included in the grammar outlined above. Expansion of grammar’s productions with these and similar attributes is a matter of utilizing methods of well elaborated attribute grammars. Since attributes are not important for construction of derivation tree, they are not discussed in details here. 2.2

The Lexicon

Lexicon is the key concept of syntactical structuring. Lexicon is a set of language constructions, which describe objects and local as well as global structures of objects. Roughly speaking, lexicon elements are phrases generated in a grammar; note that a score (a full piece of music) is the phrase as well. Phrases with parts of derivation tree built on them create elements of lexicon. Such parts of the derivation tree, that correspond to the given phrase, should be the minimal ones covering the phrase and forming a subtree. Thus, a lexicon element has its upper vertex, which is its root. Such subtree can be extended by adding a lower part of the path from derivation-tree-root to the subtree-root. Four different lexicon elements are shown in Figure 4. All these elements are built on the same phrase of music notation: eighth note placed on the second space of the stave. The above grammar is intended to cover music notation, i.e. to generate correct notation with incorrect notations admitted. Due to complexity of music notation construction of a grammar generating strictly correct notations would be very difficult, if possible at all in practice. Therefore, we assume that the grammar is used for valid paginated music notations, c.f. methods in [1].

3

Semantics, Understanding

Lexicon elements describe objects in a real world. In the case of music information the real world includes sounds produced when a piece of music is performed. For simplicity, we assume that sounds are described by three parameters: pitch, beginning time and duration of the note. Such triples may be supplemented by e.g. volume (loudness), articulation (legato, staccato etc.). Semantics outlined in section 1.3 casts the lexicon in the space of sounds/notes. Such a cast is a relation defined in the Cartesian product of the lexicon and the space of sounds/notes. Note that lexicon elements shown in parts a), b) and c) of Figure 4 may refer to many objects in the world. The case a) defines all eighth notes a in a score (one-lined a in treble clef, with different beginning times). Notes at given position in chords and measures are defined in cases b) and c). In case d) the lexicon element begins with the root of the derivation tree and defines the unique element.

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Fig. 4. Lexicon elements and their interpretation: a) an eighth note, b) the lower eighth note in a vertical event, c) the lower eighth note in the first vertical event of a measure, d) the lower eighth note in the first vertical event of the third measure of a stave. The part of the derivation tree of the Bach’s score associated with the phrase (the eighth beamed note) is indicated by thick edges. Extra nodes are added for the sake of clarity.

Summarizing - semantics is a relation V in the lexicon L and the world W , called valuation relation: V ⊂L×W The world W includes objects and sets of objects described by language texts. In fact, objects of the world W usually create meaningful sets and - moreover - create structures with internal relations between objects. Such structures are

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Fig. 5. An example of selection: a rectangle is drawn with the mouse

also subjects described by language constructions. Therefore, the world W is a structured space of objects or structured space of information: a piece of music includes notes that are strictly related to each other.

4

Querying

According to the Cambridge Dictionaries Online query is a question, often expressing doubt about something or looking for an answer from an authority, c.f. [3]. In this paper we assume that answer from an authority is also understood as accomplishment of an operation for a given request. We will be querying in terms of selecting, searching, copying, replacing, pasting, transposing etc. 4.1

Selecting

Technically, drawing a rectangle with mouse or marking a sequence of symbols with keyboard or mouse are the most popular methods of selection. Possibly, a multi selection can be done, usually with mouse moves or clicks while the key Ctrl is pressed. This common operation in computer applications, is usually performed on raw, low level data. Examples of low level data selections are texts in text editors and raster images (rectangles of pixels), c.f. [5]. An example of a raster image selection is shown in Figure 5: the second and third measures of the lower stave are highlighted. A questions can be raised - what is the meaning of such selections? Is it a part of an image or a part of a printing? It depends on context of performance of the operation. In image editor selection is done on a raster bitmap. However, in context of our discussion, this selection cannot be considered as a part of a screen, i.e. as a raster bitmap. It is a part of displayed paginated music notation rather than a part of an image. In our discussion such a selection is interpreted as an element of the lexicon. Of course, a selection is usually related to many elements of the lexicon, as it is shown in Figure 4. Mouse-dragged rectangle is a simple form of selection. For complex spaces of data methods of selection mentioned above are far inadequate to needs. Music spaces of information may serve as examples of needs of more complex selection methods. An example of such selections is explains in Figure 7. There are three voice lines on the first stave of the Bach’s score. In lower part of this Figure

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Fig. 6. J. S. Bach, Suite (Ouverture) No. 3 in D major, BWV 1068, transposition of voice lines: in the middle part of the image the upper voice line is transposed by one octave up (12 halftones), the lower one - by one octave down; in the bottom part of the image the middle voice line in the third measure is moved down by forth (seven halftones).

three voice lines are displayed in separate staves. It is clear that The rectangle dragging tool is not appropriate for a voice line selection. This is why a more suitable tool should be used for such a selection. 4.2

Searching

Searching is an operation of locating instance(s) matching a given pattern, i.e. locating instance(s) identical or similar to the pattern. The operation Search in a space of music information concerns a pattern, which is a structure of music information. Searched pattern is usually a result of selection. Now we discuss the operation Search with a couple of examples. The first example is based on Figure 4. We assume that the eighth a’ note is the pattern selected. If the selection is interpreted as in the case a) of this

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Fig. 7. J. S. Bach, Suite (Ouverture) No. 3 in D major, BWV 1068, identification of voice lines

Figure, i.e. an eighth a’, then any eighth a’ note matches this pattern. In Figure 2 the lower note in the fourth vertical event in the fifth measure of the first stave matches the pattern. The pattern interpreted as in cases b), c) and d) of Figure 4 matches no instance. The second example concerns Figure 3. Let us assume that the first beamed group of four sixteenths of the first measure is selected and that only minimal part of derivation tree creates the lexicon element. This selection matches next two beamed groups of sixteenths in the same measure and last two beamed groups of sixteenths in the fourth measure of the same stave. If the lexicon element reaches the node <measure>, then the pattern the first beamed group of four sixteenths in a measure does not match any instance. However, if we admit notes to be moved up or down by the same interval, then the first beamed group of four sixteenths in the third measure match the pattern; notes are moved down by three scale steps. 4.3

Transposing

Transposition is a fundamental operation in music. Semantic interpretation of transposition is simple: it moves sounds/notes up or down by a number of halftones preserving pitch intervals. Syntactic interpretation of transposition is more difficult due to properties of the music scale. Placement of notes on staves is governed by rules, which are irregular, i.e. placement of notes is not strictly proportional to pitch. Moreover, rules are ambiguous sometimes. Therefore, this

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operation is firmly based on semantical interpretation. Namely, a valuation relation V moves a given element of the lexicon to the world of sounds/notes, then the semantic image of the lexicon element is moved by the specified number of halftone and - finally - an inverse relation V −1 moves the semantic item back to the lexicon. Examples of transposition are given in Figure 6. 4.4

Converting

Conversions between different formats of music information are inherently based on semantic representation of music information. Formats of music representation are usually differently designed and direct conversion is highly difficult. Therefore a conversion from one format to another is usually done with an intermediate semantic analysis of converted information. As mentioned above, firstly a valuation relation V1 moves converted information to a world of objects (usually, to the world of sounds/notes) and then an inverse V2−1 casts semantic image on the target format, c.f. [6].

5

Conclusions

This study concerns methods of domain-dependent music information processing. The general idea of syntax/semantics/understanding is applicable to other information domains. Nevertheless, concretization of the general idea seems to be strongly domain-dependent. Despite similarities of methods applicable to different domains (e.g. music, medical images, natural language, geographical information) a universal solution relevant to many domains might be impossible to find. This study opens some research topics. The most important are: a) construction of grammars covering music information (the grammar given in section 2.1 is geometrically oriented while logical orientation would be more suitable in some applications), b) developing details of such grammars, c) semantics, i.e. developing methods of construction of valuation relation is a key issue in automation of querying, d) developing methods of Select and Search/Replace operations for paginated music notation, e) studying inherent imperfectness of music information, i.e. incompleteness, uncertainty and (sometimes) incorrectness, which are hidden under the level of syntax and semantics.

References 1. Bargiela, A., Homenda, W.: Information structuring in natural language communication: Syntactical approach. Journal of Intelligent & Fuzzy Systems 17(6), 575–581 (2006) 2. Castan, G., et al.: Extensible Markup Language (XML) for Music Applications: An Introduction. In: Hewlett, W.B., Selfridge-Field, E. (eds.) The Virtual Score: Representation, Retrieval, Restoration. The MIT Press (2001) 3. Collins Cobuild English Language Dictionary, http://dictionary.cambridge.org/

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4. Homenda, W.: Breaking Accessibility Barriers: Computational Intelligence in Music Processing for Blind People. SCI, vol. 107, pp. 207–232. Springer, Heidelberg (2008) 5. Homenda, W.: Automatic data understanding: A necessity of intelligent communication. In: Rutkowski, L., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2010. LNCS (LNAI), vol. 6114, pp. 476–483. Springer, Heidelberg (2010) 6. Homenda, W., Sitarek, T.: Notes on automatic music conversions. LNCS (LNAI), vol. 6840, pp. 533–542. Springer, Heidelberg (2011) 7. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley Publishing Company, 1979 (2001) 8. Krolick, B.: How to Read Braille Music, 2nd edn. Opus Technologies (1998) 9. MIDI 1.0, Detailed Specification, Document version 4.1.1 (February 1990) 10. Tadeusiewicz, R., Ogiela, M.R.: Automatic Image Understanding A New Paradigm for Intelligent Medical Image Analysis. Bioalgorithms and Med-Ssystems 2(3), 5–11 (2006) 11. Tadeusiewicz, R., Ogiela, M.R.: Why Automatic Understanding? In: Beliczynski, B., Dzielinski, A., Iwanowski, M., Ribeiro, B. (eds.) ICANNGA 2007. LNCS, vol. 4432, pp. 477–491. Springer, Heidelberg (2007)

Automatic Music Transcription: An Experiment with Simple Tools Wladyslaw Homenda Faculty of Mathematics and Information Science, Warsaw University of Technology Plac Politechniki 1, 00-660 Warsaw, Poland and Faculty of Mathematics and Computer Science, University of Bialystok ul. Sosnowa 64, 15-887 Bialystok, Poland

Abstract. This study is focused on an automatic music transcription, i.e. converting acoustic music, stored in the form of WAVE files, to MIDI like data. An experiment on this topic is the key point of the paper. The experiment is aimed on determining what kinds of music could be successfully recognized with a simple tool rather than on searching for new recognition methods. Fourier transform was used as the recognition tool. It was utilized for analysis of music of simple harmony played with basic musical instruments. Keywords: live music, music transcription, Fourier transform, Hanning window, spectral analysis.

1

Introduction

Automatic recognition and understanding of aspects of a real world is an important area of computing. This area includes such topics as all sorts of conversions of information to computer memory and processing of acquired information. This paper is focused on automatic recognition of played music and its transcription to a MIDI similar form. The experiment, solely based on Fourier transform as the tool, tests ability of recognition of chords of single musical instrument in time. The paper includes description of the experiment and selected results. Namely, sound modelling and recognition of chords is described in Section 2. In Section 3 problems raised by chords’ linking are outlined. Results of the experiment are shown in several figures in Appendix. These results are discussed in Section 4. It is worth to say that discussion is skipped on tools utilized in the experiment: Fourier Preprocessing (signal windowing), Fourier Transform, Fast Fourier Algorithms, filtering etc. Tools mentioned here are widely known and it is easy to find good description of them. Alike, many other problems, for instance frequency-time resolution, spectral leakage, phantom components etc., are also dropped in this paper. The reader can find these topics in a wide spectrum of sources, among them those used in experiment preparing: [1–3, 10]. Technical information can be found in [4, 6, 7]. It is worth to underline that automatic music transcription requires analysis of uncertainty: analyzed data are usually incomplete, uncertain and ambiguous. Y. Tang, V.-N. Huynh, and J. Lawry (Eds.): IUKM 2011, LNAI 7027, pp. 256–267, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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Such imperfectness of information is hidden under the level of syntax and semantics. Due to limitation of the paper this problem is not touched. Thus, this study concerns such topics as data mining, knowledge discovery as well as information fusion and knowledge integration in uncertain environments. Digital signal processing methods are utilized as tools rather then subjects of the study.

2

Chords

2.1

Sound Modelling

Sound analysis is based on an assumed sound model. Non-percussion musical instruments generate sounds that are periodic vibrations having the fundamental frequency and overtones, also called harmonics. Overtones are frequencies equal to integer multiple of the fundamental frequency. The fundamental frequency f0 defines pitch of a tone. Presence of overtones depend on many factors, a musical instrument is the most important one. Usually the fundamental frequency has the maximal amplitude or at least its amplitude is not much less than amplitude of a few lower harmonics. Sometimes this is not true and amplitude of one or a few harmonics are much higher than amplitude of the fundamental frequency. Since the latter does not happen for typical musical instrument, this problem is not discussed here. We simply assume that amplitude of the fundamental frequency f0 is high. We assume the model of sound as a series of amplitudes corresponding to consecutive harmonics: Sound(f ) = {f ; Af , A2f , A3f , A4f , . . .} where: f is the fundamental frequency and Akf is amplitude of the kth harmonics, k = 1, 2, 3, . . . The above model of sound depends on time. Firstly, amplitude of sound changes in time. Secondly, proportions between amplitudes of harmonics change in time. We do not attempt to discuss the latter property because it does not affect our experiment. Contour of amplitude in time is called envelope. Envelope depends on musical instrument as well as a way of playing. In our experiment this property is used to distinguish consecutive sounds of the same pitch, e.g. two consecutive hits on the same key of piano. Real sound is approximated with so called ADSR model: – A - attack time is the time taken for initial run-up of level from nil to peak, beginning when the key is first pressed, – D - decay time is the time taken for the subsequent run down from the attack level to the designated sustain level, – S - sustain level is the level during the main sequence of the sound’s duration, until the key is released, – R - release time is the time taken for the level to decay from the sustain level to zero after the key is released.

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Fig. 1. ADSR model: sound envelope of piano (left part) and violin (right part) sounds

Fig. 2. Spectral characteristic of piano sound E3, 8192 samples in window

In Figure 1 ADSR envelope of piano (left) and violin (right) is shown. The spectral characteristic of the example piano sound E3 is shown in Figure 2. In this case window has width 8192 samples. Height of the peak of fundamental frequency is four times less than the one of second harmonics. 2.2

Filtering and Windowing

Sampled sound is distorted and noised. To minimize influence of noise and distortions sound is filtered with use of well known low-pass and high-pass filters. A special statistical filter allows for reducing noise. Recognizing the set of peaks in Fourier transform is the most basic element of music recognition. The set of peaks is taken in short time intervals. This short time part of signal is then subjected preprocessing, so called windowing. Windowing is used for lowering border effects and improving measurement capabilities. In the experiment Hanning window is utilized. 2.3

Recognizing Chords

Quality of peaks’ identification decides about quality of the whole recognition process. Data concerning peaks are: frequency of the maximal value, minimal and maximal frequencies and height of the peak. The idea of the algorithm is simple: a peak is identified when values of Fourier transform increases rapidly

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Fig. 3. Spectral characteristic of piano sound C3, Sustain part

Fig. 4. Spectral characteristic of piano chord C3, C4, G4, Sustain part

to get a maximal value and then decreases rapidly. However, this obvious and simple method faces problems with distortions. It happens that sides of peaks are disrupted by a sort of small peaks. Finally, the algorithm is as follows: 1. going right find a frequency with the value of Fourier transform not less than a given minimal value or being a local minimum, 2. go right as far as values of Fourier transform increase, set the frequency, for which Fourier transform reaches the local maximal value, 3. go right as far as values of Fourier transform decrease, set the frequency, for which Fourier transform reaches a minimal value or a local minimum value, 4. if differences between the maximal value and minimal values are greater than the given threshold, include to the spectrum the frequency corresponding to the maximal value,

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5. otherwise, when at least one difference is less than the given threshold, ignore the maximal value (it is a distortion) and continue with point 2 or 3, respectively, 6. repeat points 2 - 5 until new harmonics are found.

3

Sequences of Chords

Recognition of chords’ sequences is the second fundamental step in acoustic music recognition. The algorithm is based on recognition of single chords. The WAVE file is split up into short time intervals, so called windows. In the experiment, length of windows is set to 4096 samples. Then, every window is subjected to chord recognition. Consecutive windows with the same structure of chords are linked to create chords of acoustic music. Effectiveness and correctness of this simple and obvious method of recognition of sequences of chords depends on three factors: correctness of chords’ recognition (discussed in previous section), finding chords’ borders and differentiating consecutive notes of the same pitch. 3.1

Splitting

In practice, Fourier transform is applied to short-time signals assuming uniform frequency characteristic. Fourier analysis on nonuniform signals, with a change of the sound, produces frequency spectrum difficult for interpretation. The spectrum of such a signal includes frequencies absent in both sounds. Moreover, ending sound may raise its frequencies in the spectrum, what may be interpreted as the attack time of a new note. Wave graph and spectra of two sounds A2 and E3 are displayed in Figure 5. Top part of the Figure shows wave graph and placement of three windows: (1) covering end of A2 , (2) covering beginning of E3 and (3) covering connection of both sounds. Middle parts display spectra of windows (1) and (2). The spectrum of the windows (3) including change of these sounds is displayed in bottom part of this Figure. It is worth to underline, the bottom part of the Figure shows the spectrum of two consecutive tones and not a chord of two tones. 3.2

Notes’ Duration

The algorithm based on analysis of sound changes will not distinguish consecutive sounds with the same characteristic, for instance repeated notes of the same pitch. Detection of such notes is being done on the basis of the ADSR model. Analysis of dynamics of sounds allows for detection of consecutive parts of a sound: attack time, decay time, sustain time and release time. The ADSR model strongly depends on an instrument. Therefore, such an analysis should involve instrument’s parameters. Effectiveness of this method is rather low for a piece of music with frequent full range of dynamics’ changes.

Automatic Music Transcription: An Experiment with Simple Tools

Fig. 5. Wave graph and spectra of consecutive piano sounds A2 and E3

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Results Data and Methodology

Results presented in Appendix come from the experimental computer programs prepared in frames of M.Sc. Theses [5, 8, 9]. First, MIDI files were created by second hand software as the result of conversion from music notation. Then, WAVE files, representing acoustic music, were generated by a computer equipment based on these original MIDI files. WAVE files were produced by a typical sound card of a computer. Finally, WAVE files were subjected to recognition and uncovered notes were stored as MIDI files. In Figures 6, 7, 8 and 9 original MIDI files are displayed in the top part. Middle and bottom part of every Figure shows result of recognition of WAVE files. MIDI-to-WAVE conversion was done for different groups of (electronic) instruments: piano, violin, viola, cello, double bass, guitar. Thanks to this synthetic production of data, created acoustic music was correct and highly regular. In this way, a usual anarchy of live performance: an influence of musician’s subjective interpretation of music, irregularities in beginning and ending of played notes, errors etc. were avoided. Hence, recognition of acoustic music as well as conversion of recognized notes to MIDI files were easier. On the other hand, synthetical data subjected to the study makes it less reliable for live music. In Appendix results of analysis of the following pieces of music are presented: – Breve Regnum - the student’s song, Jagiellonian University, XV century, – Gaude Mater Polonia - medieval song, XIII century, anonymous adaptation in XVIII century, – Goodnite Sweetheart, Goodnite - music and lyric by Calvin Carter and James Hudson. Vocal voices of these songs were replaced by instruments, as indicated in Figures 6, 7, 8 and 9. Note: the mentioned Figures show only such parts of data, which are representative for problems and discussion. Errors are indicated with a markers arcs and cicrcled. 4.2

Analysis and Discussion

Analysis of results shows that monophonic music is recognized almost perfectly, c.f. Figure 6. Errors in 6th measure (ending time of C# note) may be caused just by imperfection of the experimental software. Other parts of this example are equally well recognized. Recognition of polyphonic music is not as good as recognition of monophonic music. Recognition is quite good when pitches of notes played simultaneously are rather close. For notes that are far away harmonics overlay each other and the higher notes is usually lost. This problem is outlined in Figures 7 and 8. Upper D# in the 5th (first displayed) measure is missing in both Piano and Strings playbacks (absence is indicated by red marker). Pitch of the lost note is just one octave higher than another simultaneous note. For instance, the following notes have harmonics covering each other: C2 , C3 , G3 , C4 , E4 , G4 , C5 , D5 , E5 , . . ..

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Next problem is raised by different instruments playing melodic lines. Distortions at borders of played notes affect Fourier transform generating images of very short phantom notes. This effect can be observed in Figure 8. On the other hand, such short phantom notes can be easily detected and removed in some post-processing. Finally, in this discussion we mention the time-frequency resolution trade-off problem. In practice, window size of 4096 samples is required for correct recognition. It corresponds to time resolution 0.1 second in case of standard sampling rate 44.1 kHz. Such the resolution is not enough for fast tempo of played music. Another factor affecting recognition quality is grown by complicated rhythmic melodic lines, especially when melodic lines much differs.

5

Conclusions

In this paper automatic transcription of acoustic music to a form of music description is studied. Recognition of sounds of musical instruments is based on Fourier analysis. The system proposed is a complete solution allowing for conversion of source acoustic music stored as WAVE file to MIDI file. Another module could convert MIDI to notation. Such a solution, implemented as a computer program with full range user interface, would be used by musicians to generate automatically music notation for performed music. Another application of alike computer program would be in music education to control pupils playing. Results of the experiment show that recognition could be acceptable for recognizing simple music with with single or a few tiny melodic lines and with simple harmony. This may be applied e.g. for controlling beginners playing basic musical instruments (piano, strings, flute, guitar etc. However, testing results of music recognition is not satisfactory for polyphonic music with reach harmony. Yet, recognition of more complex music is rather poor. It seems that Fourier analysis is the tool not strong enough for such application. Looking for improvements, with more complex music in mind, it is worth to note that a lot of research in related fields have been utilizing wavelet analysis. Despite much higher complexity of wavelets, it seems to be a proper tool for supporting simple Fourier analysis. Another factor to be utilized is a database of stored samples of sounds of musical instruments in acoustical format as well as in parameterized form. Such a database would support Fourier and wavelet analysis of acoustical music.

References 1. Berg, R.E., Stork, D.G.: The Physics of Sound. Prentice-Hall, New Jersey (1995) 2. Lyons, R.G.: Understanding Digital Signal Processing. Addison Wesley Longman Inc. (1997) 3. Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice-Hall Inc., New Jersey (1989) 4. Fastest Fourier Transform in the West, http://www.fftw.org

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5. Homenda, W., Szlenk, M.: A Practical Approach to the Chord Analysis in the Acoustical Recognition Process. In: Halgamuge, S.K., Wang, L. (eds.). SCI, pp. 221–232. Springer, Heidelberg (2005) 6. Intel Signal Processing Library - Reference Manual 7. MIDI 1.0, Detailed Specification, Document version 4.1.1 (February 1990) 8. Siminski, J.: Automatic identification of sequences of notes and chords (in polish), M.Sc. Thesis, Warsaw University of Technology, Warsaw (2004) 9. Szlenk, M.: Automatic identification of the acoustical space of the musical sound properties (in polish), M.Sc. Thesis, Warsaw Univ. of Technology, Warsaw (2000) 10. Tanguiane, A.S.: Artificial Perception and Music Recognition. Springer, Berlin (1993)

Appendix: Automatic Transcription of Selected Pieces

Original MIDI file, Soprano part

Transcription of Piano playback

Transcription of Double Bass playback

Fig. 6. Breve Regnum, Soprano part, transcription of measures 1-8

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Original MIDI file, Soprano and Alto parts

Transcription of Piano playback

Transcription of Violin and Viola playback

Fig. 7. Gaude Mater Polonia, Soprano and Alto parts, transcription of measures 5-8

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Original MIDI file, Soprano, Alto and Tenor parts

Transcription of Piano playback

Transcription of Violin, Viola and Cello playback

Fig. 8. Gaude Mater Polonia, Soprano, Alto and Tenor parts, transcription of measures 5-8

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Original MIDI file, Soprano, Alto and Tenor parts

Transcription of Piano playback

Transcription of Guitar playback

Fig. 9. Goodnite Sweetheart, Goodnite, Soprano, Alto and Tenor parts, transcription of measures 1-4

Author Index

Cao, Tru H. 31 Carvalho, Leonardo F.B.S. Chen, Jingxian 157 Chen, Xia 72 Chu, Meifen 168

33

Dashnyam, Bayarmaa 197 Destercke, S´ebastien 219 Gong, Guanghong Guo, Bo 143

98

Homenda, Wladyslaw 243, 256 Hsu, Ming-Jia 197 Hsu, Ping-Yu 197 Hu, Weidong 72 Huynh, Van-Nam 48 Ikeda, Mitsuru

231

Paragua¸cu, F´ abio 33 Pei, Zheng 81, 90 Qin, Feng 109 Qin, Keyun 81 Quost, Benjamin

219

Rybnik, Mariusz

243

Sun, Xiao-Yan

179

Takumi, Satoshi 187 Tan, Kangrong 168 Tan, Yuejin 143 Tang, Min 72 Tang, Yongchuan 60 To, Hoai-Viet 231 Tokinaga, Shozo 168

Lan, Heng-you 133 Lawry, Jonathan 60 Le, Hoai-Bac 231 Liu, Jun 16 Lopes, Roberta V.V. 33 Lu, Jianxin 157 Lu, Meihua 109

Wang, Xiao 133 Wang, Zhi-Chao 179

Ma, Chao 98 Ma, Tieju 48 Ma, Yaofei 98 Miyamoto, Sadaaki

Yan, Hong-Bin 48 Yang, Jian-Bo 7 Yu, Wenxian 72

Xiang, Yumin 133 Xiong, Tingjian 133 Xu, Dong-Ling 7 Xu, Yang 16, 81

19, 187

Nakamori, Yoshiteru 18, 48, 209 Neto, Helio C. Silva 33 Nishida, Toyoaki 1 Niu, Qun 121

Zeng, Tingting 121 Zhang, Hua-Xiang 179 Zhang, Tao 143 Zhao, Chenfang 90 Zhou, Zhuo 121