A FRAMEWORK FOR MULTI-AGENT BELIEF REVISION
By Wei Liu (M.Eng)
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY AT THE UNIVERSITY OF NEWCASTLE CALLAGHAN, NSW 2308, AUSTRALIA AUGUST 2002
c Copyright by Wei Liu (M.Eng), 2002
I hereby certify that the work embodied in this thesis is that the result of original research and has not been submitted for a higher degree to any other University or Institution.
Signature of Author
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To Hongwei, Haolin and my parents.
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Contents Abstract
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Acknowledgements
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1 Introduction
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1.1
Background and Motivation . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Issues in Multi-Agent Belief Revision . . . . . . . . . . . . . .
5
1.2.2
A Framework for Multi-Agent Belief Revision . . . . . . . . .
7
1.3
Scope and Key Assumptions . . . . . . . . . . . . . . . . . . . . . . .
9
1.4
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Intelligent Agents and Belief Revision 2.1
2.2
2.3
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Intelligent Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.1.1
The Intelligence of an Agent . . . . . . . . . . . . . . . . . . .
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2.1.2
Example: Academic Life . . . . . . . . . . . . . . . . . . . . .
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An Informal Introduction to Belief Revision . . . . . . . . . . . . . .
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2.2.1
Example: Umbrellas in the Rain . . . . . . . . . . . . . . . . .
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2.2.2
Example: Air Kangaroo and Free Beer . . . . . . . . . . . . .
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2.2.3
Three Issues of Belief Revision . . . . . . . . . . . . . . . . . .
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Belief Revision as an Epistemological Theory . . . . . . . . . . . . . .
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2.3.1
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Four elements of an epistemological theory . . . . . . . . . . .
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2.3.2
Information Economy and Criteria of Rationality . . . . . . .
3 Belief Revision Techniques under the AGM Paradigm 3.1
3.2 3.3
3.4
30 32
The AGM Postulates and Belief Set Revision . . . . . . . . . . . . . .
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3.1.1
Postulates for Expansion . . . . . . . . . . . . . . . . . . . . .
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3.1.2
Postulates for Revision . . . . . . . . . . . . . . . . . . . . . .
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3.1.3
Postulates for Contraction . . . . . . . . . . . . . . . . . . . .
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Levi Identity and Harper Identity: From Revision to Contraction and Vice Versa . . . . . . . . . . . . .
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Determining the Sentences to be Retracted . . . . . . . . . . . . . . .
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3.3.1
Epistemic Entrenchment Ordering . . . . . . . . . . . . . . . .
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3.3.2
System of Spheres . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.3
Other Important Works . . . . . . . . . . . . . . . . . . . . .
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Implementing Entrenchment Based Revision - Belief Base Revision .
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3.4.1
Finite Partial Epistemic Rankings . . . . . . . . . . . . . . . .
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3.4.2
Transmutations . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Belief Revision - The Probability and The Possibility Approach
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4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2
Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3
Belief Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.4
Possibility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.5
Probability as Special Belief Functions . . . . . . . . . . . . . . . . .
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4.6
Possibility as Special Belief Functions . . . . . . . . . . . . . . . . . .
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4.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Belief Revision in a Multi-Agent Environment
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5.1
Brief Introduction to Multi-Agent Systems . . . . . . . . . . . . . . .
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5.2
Various Frameworks for Multi-Agent Belief Revision . . . . . . . . . .
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5.2.1
Mutual Belief Revision of Van der Meyden . . . . . . . . . . .
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5.2.2
MABR of Kfir-dahav and Tennenholtz . . . . . . . . . . . . .
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5.2.3
MSBR of Dragoni et al. . . . . . . . . . . . . . . . . . . . . .
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5.2.4
DBR of Dragoni et al. . . . . . . . . . . . . . . . . . . . . . .
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5.2.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.3
Belief Revision in Multi-Agent Systems - Concepts and Terminologies
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5.4
Heterogeneities in Multi-Agent Systems . . . . . . . . . . . . . . . . .
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6 Ontology - Tackling the Heterogeneity Issues
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6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.2
Classifying Ontologies . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.3
Representing Ontologies . . . . . . . . . . . . . . . . . . . . . . . . .
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6.4
Ontologies for Multi-Agent Belief Revision . . . . . . . . . . . . . . . 100 6.4.1
Tackling the Social Character Heterogeneity . . . . . . . . . . 100
6.4.2
Tackling the Semantic Heterogeneity - Translating among various ranking systems . . . . . . . . . . . . . . . . . . . . . . . 102
6.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Trust Evaluation
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7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2
Essential Constituent Beliefs for Trusting Information Sources . . . . 108
7.3
Trustworthiness and Degree of Beliefs . . . . . . . . . . . . . . . . . . 111 7.3.1
Degree of Trustworthiness . . . . . . . . . . . . . . . . . . . . 112
7.3.2
Degree of Deception and Degree of Sincerity . . . . . . . . . . 113
7.3.3
The Evaluation of Competency and Sincerity . . . . . . . . . . 116
8 Information Pedigree 8.1
119
Evaluating Trust on Passing-On Information . . . . . . . . . . . . . . 120 8.1.1
Representing Information Pedigrees . . . . . . . . . . . . . . . 121
8.1.2
Information Pedigree Transformation using Trust Evaluation . 124
8.2
An Example in Resolving Conflicting Information . . . . . . . . . . . 126
8.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
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9 Shared Knowledge Structure and Knowledge Migration
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9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.2
Shared Knowledge Structure . . . . . . . . . . . . . . . . . . . . . . . 134
9.3
Relationship with Other Knowledge Structures . . . . . . . . . . . . . 139
9.4
Inconsistency Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.5
Knowledge Grade and Knowledge Migration . . . . . . . . . . . . . . 143
10 Multi-Agent Belief Revision – Architecture and Implementation
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10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 10.2 A Belief Revision Agent . . . . . . . . . . . . . . . . . . . . . . . . . 146 10.3 A FIPA Multi-Agent Platform Implemented in JADE 2.4 . . . . . . . 149 10.4 Design Using an Object-Oriented Approach . . . . . . . . . . . . . . 152 10.4.1 Domain Beliefs and Social Beliefs . . . . . . . . . . . . . . . . 153 10.4.2 Belief Revision Services - Use Case Study . . . . . . . . . . . . 157 10.4.3 Layered Belief Revision Engine . . . . . . . . . . . . . . . . . 162 10.5 Agent Communication Channel - JADE Agent Wrapper for Belief Revision Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 11 Discussion, Conclusions and Implications
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11.1 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 168 11.2 Limitations and Implications for Future Research . . . . . . . . . . . 173 A Agent Environment Properties
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B Logical Foundations
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C JADE Implementation of Ontologies V2.4
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D FIPA Agent Platform Components
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E Publications
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Bibliography
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Abstract Despite the fact that maintaining a consistent belief set is crucial for agents to exhibit rational behaviour, most of the agent development toolkits or frameworks use simple database insertion and deletion for the update of beliefs. One of our research aims is to investigate concepts and techniques that facilitate a service-oriented belief revision agent with social awareness. Furthermore, research in the area of belief revision mainly focuses on a single agent. Seldom are there studies into belief revision issues in a multi-agent environment. We investigated belief revision techniques for both individual agents as well as multiple agents. The research into individual agent belief revision exhibits a diverse range of revision schemes. However, they all can be described using some essential elements of an epistemological theory. Furthermore, we also show that both probabilities and possibilities are special cases of belief functions and can be represented by belief functions when certain conditions are met. With the research into multi-agent belief revision frameworks, an ontological classification of the various terminologies is presented, which provides the basis that aids the investigation into the domain. In order to address the inability of existing frameworks to deal with heterogeneities, we developed Ontologies for tackling social heterogeneities, semantic heterogeneities and syntactical heterogeneities. We identified two constituents of evaluating an information source’s trustworthiness and propose several ways of generating general or topic specific trustworthiness ix
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from the degree of sincerity and the degree of competency. This provides the crucial input required by fusion and weighted knowledge base merging, which are often assumed to be pre-computed. We also show that a pedigree of at least 3 levels of depth will enable the agent to analyse and keep track of passing-on information, whose credibility are often wrongly computed in some real application domains. Overall, the research into trust issues not only provide a clear domain description but also formulates the necessary components of agent beliefs, i.e. domain beliefs and social beliefs. After clarifying the trust issues, we proposed a shared knowledge structure that is capable of representing both shared knowledge as well as private knowledge. This leverage our research from a MSBR (Belief Revision using information from Multiple Sources) level to a MABR (Multi-Agent Belief Revision) level. The investigated concepts, such as revision schemes, trustworthiness, information pedigree, shared beliefs and ontologies, eventually come together within a flexible and general framework. The framework design herein uses JADE as the agent platform to facilitate inter-agent communication and the belief revision system - SATEN for the essential functionalities for a single belief revision agent. The so-designed framework is capable of incorporating new belief revision schemes and a viable model for a heterogenous multi-agent environment. In particular, as a layering approach is adopted, a belief revision agent designed under such framework is capable of providing services at various levels. It provides a tangible architecture of a belief revision service provider for a dynamic, heterogenous and autonomous open environment.
Acknowledgements I would like to express my deep appreciation to Mary-Anne Williams, my supervisor, for her introduction to the area, for her many suggestions, insightful guidance, broad interest and constant support during this research. I am very grateful to have her as both my supervisor and a personal friend. Thanks to her kind and cheerful nature, she has shown great understanding and support to my study as well as my family. I had the pleasure of meeting with Salem Benferhat and Daniel Le Berre. They are very insightful researchers and willing to share their profound knowledge in mathematics and software engineering with me. I learned a lot from them, no only technically, but also a systematic way of thinking. I also had the pleasure of meeting with J´erˆome Lang, who has kindly shared his knowledge in probability theory, possibility theory, belief functions and decision making theories. I also owe a great “thank you” to the Australian Government for sponsoring me with the International Postgraduate Research Scholarship during this research, my research in this area will not be possible without this. The University of Newcastle Research Scholarship, which was awarded to me in 1999, was crucial to the successful completion of this project. The staff in the Research Higher Degree of Newcastle University are extremely supportive. They are wonderful people and especially, thanks to Mr. John Sanderson the scholarship manager for his explanation of the rules during the special period of my off-campus study. I should also thank the support from RMIT university for financing me for travelling back to work with my supervisor and to prestigious international conferences etc. Special thanks to Dr. Phil Vines, Sheila Howell, A. Prof. Lin Padgham, A. Prof. Vic Ciesielski, Margaret Pagonis, Dr. Faye Liu, Dr. Hugh Williams, Dr. James Harland, Dr. Isaac Balbin, Feng Lu and many others for their mentoring and kind support. It is my great pleasure to be working with you. xi
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I should also mention all my girl friends in the School of Management, Christine Bruff, Kim Cowley, Merrilyn Stone, Dot Etheridge, Janice Carroll, Christine Toms, Susan Robinson, Tong Hong Miew, Kath Bill and April May Kerr. You are my dearest friends, the most beautiful and caring people I have ever seen. You can not imagine how your kind nature, your caring about me and my family has made my first couple of years in Australia such a wonderful life-long memory. All I can say is I am lucky to meet you girls. I am very grateful, of course, to my parents for sacrificing their time and energy to look after us, and their endless love for Hongwei, Haolin and me. My appreciation to mum, dad and my only brother is beyond words. Without them this work would never have come into existence (literally). Thanks to my husband, the best cook in my view, without you squeezing time from your busy schedule to cover up my duty in the house, this work will not be possible. Finally, to my little boy, Haolin, thanks to helping me completely relax my mind when you are around. You are our endless pleasure. Wei Liu August, 2002
Chapter 1 Introduction “Be prepared, they are coming... Computer scientists and technology professionals are preparing a brave new world of software-based, intelligent agents that will act as virtual support staff for any human beings willing to trust them. The main difference: they will work 24/7, won’t take a lunch break and never utter a gripe.” — Mark Harrington from Newsday.com1 Jan 15, 2002
1.1
Background and Motivation
The past two decades have seen a global information surge facilitated by the widely accessible Internet technology for both industry and individuals. The world is connected via this vast open network, with new nodes connecting to it and disconnecting from it every second anywhere in the world. The vision of the world has changed irreversibly forever, terms such as “global village” are not longer science fiction, everything becomes as close as at your finger-tips. 1
In online news article: Plugged In/Cyber Emissaries To Serve Online
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The Internet and the World Wide Web have fundamentally changed the way people access information. In both the computer science domain and everyday-life, more and more, collecting information from several information sources has become an exceedingly common and often critical activity to achieve our goals. Actually, the World Wide Web is an unprecedented and enormous information repository, browsing through it to locate relevant pieces of information becomes a tedious job for human beings even with the help of the state of the art search engines. Web services and web applications are important research topics in the artificial intelligence community, so too agent technologies. As agent technologies gain maturity, it becomes widely accepted that agent-oriented approaches represent the mainstream software engineering, particularly for tackling issues presented in large complex systems[63]. Agent-oriented software engineering has gained tremendous attention in recent times. Its practical and theoretical elements are proving to be helpful in understanding and modelling complex systems, and as a result they have lead to new and innovative solutions. The multi-agent designs are especially applicable to Web applications which are highly heterogenous and autonomous in the sense that no single system has global control or holds all the global data. The first generation of web agents were shopbots[67] that compare prices (e.g. Bargain Finder from Anderson Consulting[71]) and provide recommendations etc. It is also predicated that with the help of research in agent research communities, in three or four years time, the web will be populated by autonomous agents that have more intelligence, that act on human’s behalf for information gathering, and participate in on-line transactions, such as buying, selling and bidding. To realise the agent-oriented vision, agents need to be intelligent so that they
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can achieve an effective balance between proactive and reactive behaviours which allow them to survive in a dynamic non-deterministic environment such as the Web. In addition to this, agents need to have social-awareness which will enable them to communicate, cooperate and/or negotiate under certain interaction protocols. Belief revision is a ubiquitous process underlying many forms of intelligent behaviour[110]. Most importantly, it is an essential skill for an autonomous agent. It provides the agent an ability to revise its beliefs in a coherent and rational fashion when it decides to accept some new information. Most belief revision research, however, has been produced with a single agent in mind, ie, only one problem solver using the belief revision service. SATEN[97, 107]2 - a web based belief revision system which incorporates several revision strategies is a good example of a single belief revision agent. Although it is able to clone its current state, the clones and their ancestors can not communicate. In other words, they act independently as single agents without awareness of other agents’ existence. Multi-Agent Systems (MAS) are distributed computing systems composed of a number of interacting computational entities (possibly from various vendors). One important characteristic distinguishing MAS from traditional distributed systems is that both MAS and its components (agents) are intelligent[99]. As MAS become increasingly attractive for solving larger and more complex problems, the need for adequate belief revision technology in the MAS paradigm arises. As we have stated before, it is becoming increasingly common that agents acquire information from more than one information source. Further complicating the situation, quite often, 2
http://infosystems.newcastle.edu.au/webworld/saten
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during tasks such as organising a business trip, retrieving information from multiple databases, managing a large supply chain network, investigating witnesses at a crime scene can not be completed without information sharing and exchange across platforms and companies. Therefore, we are increasingly confronted by issues such as, what an agent should do in cases when it receives: • Contradictory information from the same information source; • Contradictory information from different information sources; In shared information scenarios or cooperative information systems, what should an agent do or how should agents cooperate to update/revise the shared/common knowledge in an heterogenous open system? • Buyers and sellers are only willing to expose a certain subset of their beliefs but not their entire belief bases, what should be shared/common knowledge, how should shared/common knowledge be elaborated? • The agents involved may use different belief revision techniques, some may just delete and/or insert facts to belief base, some may use sophisticated transmutation algorithms[108] to decide the new degree of beliefs. • Which agent should go about changing the shared/common beliefs, what are the implications for other agents involved? The research herein is an attempt to answer the above questions in theoretical and computational ways. The aim is to design a framework which will facilitate the transformation of a single belief revision agent like SATEN into a multi-agent belief
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revision system. Meanwhile, the so-designed framework should be general enough to accommodate existing legacy systems and flexible enough to assemble existing revision techniques into a new multi-agent belief revision system. As far as applications are concerned, the framework proposed herein will enhance the current development and deployment of robust coherent multi-agent systems, which could be widely applied to e-commerce, e-business, electronic markets, information gathering and collective ratings (e.g. articles, books etc) on the web, bandwidth allocation and the like (resource allocations), vehicle routing among independent dispatch centers, multi-enterprise agile manufacturing and scheduling, large supply-chain networks, and multi-robot domain etc.
1.2
Objectives
Belief revision is regarded as an important research area that bridges the often separate research activities[24], such as philosophy, logic, probability (statistics), possibility and theory of evidence etc. Similarly, studying belief revision in a multi-agent environment links areas such as logic, social science, game theory, group decision making, cooperative information systems etc.
1.2.1
Issues in Multi-Agent Belief Revision
One of the distinguishing features of multi-agent systems as compared to other types of distributed problem solving systems is the autonomy of their constituent entities, namely, the intelligent agents. Autonomy in general is understood to be the property that agents can achieve designed and delegated goals without human
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intervention[113]. A further interpretation is that agents may be developed by different vendors [27][92] under different standards, using different platforms, technologies and toolkits, bearing different assumptions. A good example that illustrates the diversity of agent technologies, Agenticities3 - a large European project, is presented on the Agentcities network4 . It gives an intuitive flavour of the true meaning of the phrase “large heterogenous open networks”. Among all the networks on the web it is the largest. Some other heterogenous networks in real life are supply chain networks for large enterprises, where hundreds of individual companies cooperate to meet the requirement of delivering the right amount of goods to the right location at the right time (on the right devices). It is not surprising to see that one of the major issues in managing large supply chains is the fact that each supplier has their own unique product identifier for a product which may not be exactly the same as another suppliers. There are two ways of looking at the various heterogeneities presented in multiagents systems. On one hand, it is a result of the fact that no single authority has the global control of the system. This is one of the key incentives that attracts the participation of individual entities - enjoying the benefit of joining the network without sacrificing their individual uniqueness and preferences. On the other hand, the existence of heterogeneities is a major obstacle to system inter-operation and intercommunication. Open environments are highly desirable in e-business domains and exist in real life at a human level. Therefore, in facing the challenge of heterogeneities and understanding the types and their forms of manifestations, it is important to work with an multi-agent domain. The same is true in the field of belief revision. 3 4
http://www.agentcities.org http://www.agentcities.net
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Although a few belief revision frameworks claim to be suited for MAS applications, they each in turn adopt different terminologies. For example, “mutual belief revision”[85], “arbitration”[74], “distributed belief revision”[20], “multi-agent belief revision”[68] and “belief revision using information from different sources”[19] etc. To understand the differences and similarities in our proposed research, we first need to clarify and classify these terminologies. This is our first attempt at addressing the heterogeneity issues. A close look at these systems also reveals a common shortcoming, the lack of adequate and efficient methodologies in tackling heterogeneity issues during information exchange, such as trust of heterogenous information sources, heterogenous representation on shared/common knowledge, and the heterogeneity that exists in various families of revision techniques etc. Our research goal is not to replace existing systems but rather to digest and treat them as legacy systems so that we can have a framework that is general and flexible enough to integrate them but also leaves room for accommodating new revision techniques.
1.2.2
A Framework for Multi-Agent Belief Revision
Ontologies are becoming widely accepted as a powerful tool for bridging the gap between legacy systems and enabling communication and inter-operability in a heterogeneous environment[50]. Therefore, an ideal belief revision system for multi-agent environments should provide the necessary ontologies that enable the inter-operation between revision agents that are deployed under various assumptions and that use various revision techniques. Some of the ontologies used by a multi-agent belief revision system are implicit and embedded in the designed framework, some are explicit
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and allow interaction with end-users. Grounded by our research in classifying the types of belief revision in multi-agent systems and taking into account the necessity of evaluating information sources’ trustworthiness, we propose a layered architecture for designing a single belief revision agent. Layer-1: Domain Belief Revision Layer-2: Social Belief Revision Layer-3: Shared/Common Belief Generation based on Interaction Protocols Each layer is connected to the external environment via a communication channel. Typically, in the proposed system, an agent needs to maintain two separate knowledge bases, i.e., a domain knowledge base and a social knowledge base. There also needs to be a mechanism which will dynamically generate a set of shared/common domain knowledge with other agents according the data stored in its social knowledge base. As we will see in the thesis, the proposed system provides the essential building blocks for modelling belief revision agents at different levels using a 3-layered architecture: • Combining the bottom Layer-1 and the communication channel, we will have the simplest agent, that is able to carry out single belief revision tasks or provide single belief revision services. This would be an agent with SATEN-like functionalities and communication capabilities.
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• Combining the two bottom layers, i.e. Layer-1 and Layer-2, and the communication channel, we will have a more sophisticated agent, capable of revising beliefs received from multiple sources. For example, using Dempster-Shafer’s Rule of Combination[98] or some fusion techniques[4]. Meanwhile, the agent can also dynamically update and revise its social knowledge base, e.g. its record of another agent’s trustworthiness value. • Combing all three layers and a communication channel, we have an agent who is not only capable of revising its own knowledge base, but also able to participate in revising shared/common knowledge. Moreover, in order to address the heterogeneity issues, firstly, we propose a message structure for information exchange that allows agents to evaluate the information sources’ trustworthiness. Secondly, we propose a shared knowledge structure in order to capture the different interpretation of shared/common knowledge in the existing systems. Lastly, we use a FIPA[33] compliant agent platform - JADE[61] - to facilitate agent communication, where domain ontologies can be added by the end-user if needed. The design is described using UML (Unified Modelling Language)[90] and various existing design patterns[39] are applied.
1.3
Scope and Key Assumptions
Firstly, we adopt the essential assumption of belief revision, i.e., we assume during the revision process, the world is static. In other words, we do not investigate cases when the new information to be added to the belief base is a result of change in the world. Belief update[66] is required to model agents belief change in a dynamic world.
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Secondly, we will assume an open decentralised loosely coupled environment. By decentralised we mean that both control and data are logically and often geographically distributed. There is neither global control nor global data. Loosely coupled means that individual agents spend most of their time in computation rather than communication. Open means that the platform is free for agents to join and leave, there is no obligation enforced. Thirdly, we assume that each agent has special expertise in specific areas. The expertise may overlap, but is not identical. Each agent has local memory and local access to certain data. Often agents need to cooperate to achieve designed goals. Cooperation here is in the usual sense that none have sufficient information to solve the entire problem: information must be shared to allow the group as a whole to produce a solution. Finally, we assume agents are interconnected so that every agent can communicate with every other agent by sending messages. The communication between agents is facilitated by the agent platforms5 and is always successful. In other words, we assume the message sent from the sender arrives at the receiver without distortion. In addition, the term information, knowledge and belief are used interchangeably throughout the thesis. Except we sometimes distinguish knowledge from belief by emphasising that finite flat knowledge takes the format of a sentence or a formula only, while belief is a piece of graded knowledge and consists of flat knowledge and a degree of belief. 5
See Appendix D for description of FIPA agent platform components.
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1.4
Outline
In Chapter 2, the research starts with an intuitive example which attempts to motivate the importance of the role of belief revision for an agent to be intelligent. Then in the same chapter we review the various techniques and issues researchers have used to address belief revision. In Chapter 3, by presenting a widely accepted belief revision standard - the AGM paradigm and works related to it, we then will have described the main research methodologies in the field. In Chapter 4, we conduct a comparative study, which gives a clear picture of how the different approaches relate to each other. These include the traditional Bayesian probabilities, the theory of evidence and the possibility theories. As a result, we conclude our description of the major research efforts in belief revision for single agent in a single agent environment. This sets the foundation for realising an individual revision service agent, which may adopt various revision strategies. In Chapter 5, we begin our investigation of belief revision in a multi-agent domain by looking at some existing systems. We suggest an ontological belief revision hierarchy which clarifies the terminologies adopted in the current research. By investigating the characteristics of multi-agent systems, we identify three major types of heterogeneities that need to be addressed when one models multi-agent belief revision. In Chapter 6, ontologies are introduced as a powerful approach to tackle the heterogeneity issues. We discuss suitable representations of ontologies and identify the types of ontologies needed for the three kinds of heterogeneities identified in Chapter 5.
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In Chapter 7, we address the social heterogeneities by looking at the trustworthiness of information sources. An agent’s ability to evaluate the trustworthiness of other agents becomes crucial for it to survive, and indeed thrive, in an open heterogenous information sharing environment. In this chapter, we identify two essential dimensions of beliefs for evaluating sources’ trustworthiness, namely, competency and sincerity. Moreover, a couple of evaluation procedures are proposed and investigated. In Chapter 8, we investigate a number of complicated cases of passing-on information, where the same information may reach a receiver agent via different routes. In order to keep information about the source agent we propose the use of an information pedigree as the means of maintaining the history of communicated information. The evaluation of overall trustworthiness prepares social knowledge to drive data fusion, weighted knowledge base merging, and multiple source conflict resolution. In Chapter 9, we address semantic heterogeneities, namely, various definitions of social/shared knowledge. To facilitate the sharing of knowledge between agents and dynamically maintain their private knowledge, a shared knowledge structure is proposed and investigated. In the same chapter, we also investigate possible revision mechanisms for such a structure - the knowledge migration. Based on the inconsistency principle, and the knowledge grade defined by the shared knowledge structure, the process of multi-agent belief revision becomes the process of knowledge migration. In Chapter 10, the design of a flexible framework for multi-agent belief revision is presented using software engineering approach. The research concludes in Chapter 11 with discussions of the results and implications for future research.
Chapter 2 Intelligent Agents and Belief Revision “Every time we contemplate the leftovers in the refrigerator, trying to figure out what else needs to be fetched from the grocery store before fixing a dinner, we are exercising an aspect of intelligence not seen in even the smartest ape.” — William H. Calvin in How Brains Think[10], 1996
2.1
Intelligent Agents
This section will briefly answer the following three key questions: • What are agents and intelligent agents? • What is belief revision? • Why belief revision is an ubiquitous process underlying intelligent behaviours? Formally, an agent is a computer system that is situated in some environment, and is capable of autonomous action in order to achieve its design objectives[112]. 13
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Although this is not the ultimate universally accepted definition of agent, it is flexible and generic enough to include almost all the automatic control systems and software daemons into the agent family, it is also restrictive enough to exclude those software tools (such as a word processor) which do not have autonomy but rely on human interventions to accomplish their tasks. The key presupposition of an agent system is the ability to interact with (perceive and act upon) the environment. The uncertain, non-deterministic, partially accessible nature of the environment[96] makes autonomous actions not only desirable but also necessary. This is due to the fact that such complexity and unpredictability of the environment results in the inability of human designers and developers to hard-code all possible situations that an agent is likely to meet. Autonomy is the feature that differentiates an agent system from other computer systems. Informally, autonomous action involves carrying out action without human intervention. This implies that an agent system is expected to have more robust behaviour such as recovery from failure and more reliable behaviour such as coping with an uncertain and changing environment.
2.1.1
The Intelligence of an Agent
What is an intelligent agent then? It is easy to say that an intelligent agent is an agent endowed with intelligence. The precondition is that we know the precise definition of intelligence. But unfortunately, we do not. Often, we describe ourselves as intelligent human beings. Rarely do we stop and question ourselves: what exactly is intelligence? Are we intelligent by nature? Such questions have not been of much interest to the research community until people started to investigate the nature of intelligence by building intelligent machines[103],
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e.g. intelligent agents. The following quotation states a neuro-physiologist’s view of human intelligence[10]: Piaget used to say that intelligence is what you use when you do not know what to do. If you are good at finding the one right answer to life’s multiple-choice questions, you are smart. But there is more to being intelligent - a creative aspect, whereby you invent something new “on the fly”. Indeed, various answers occur to your brain, some better than others. Every time we contemplate the leftovers in the refrigerator, trying to figure out what else needs to be fetched from the grocery store before fixing dinner, we are exercising an aspect of intelligence not seen in even the smartest ape. The best chefs surprise us with interesting combinations of ingredients, things we would ordinarily never think “went together”. Poets are particularly good at arranging words in ways that overwhelm us with intense meaning. Yet we are all constructing brand-new utterances hundreds of times every day, recombining words and gestures to get across a novel message. Whenever you set out to speak a sentence that you’ve never spoken before, you have the same creativity problem as the chefs and poets. Furthermore, you do all your trial-and-error inside your brain, in the last second before speaking aloud. What we can obtain from the excerpt above is an intuitive explanation of the creative aspect of the reasoning process. More precisely, intelligence implies the human behaviour that attempts to determine consequences by thinking rather than acting, i.e., by reasoning about the world rather than taking action in it. Most intelligent agents try to model or replicate some part of a human being’s reasoning process, so we
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might expect that, an intelligent agent should exhibit creative behaviour (although not necessarily as sophisticated as human creativity). Within the agent research community, Wooldridge and Jennings’ definition[113] of intelligent agent is widely accepted: an intelligent agent is one that is capable of flexible autonomous action in order to meet its design objectives, where flexibility means three things: • reactivity: the ability to perceive their environment, and respond in a timely fashion to changes that occur in it in order to satisfy their design objectives. The subsumption architecture developed by Rodney Brooks is the best-known reactive agent architecture. One of the important characteristics of such a system is that the agent behaviours are implemented simply by mapping perceptual input directly to actions without symbolic reasoning at all. • pro-activeness: the ability to exhibit goal-directed behaviour by taking the initiative in order to satisfy their design objectives; e.g. a procedure in PASCAL, a function in C, or a method in JAVA is the basic building block for a system that exhibits goal directed behaviour. • social ability: the ability to interact with other agents (possibly humans) in order to satisfy their design objectives. The social ability means things beyond exchanging binary code, but more related to the human world, in terms of sharing resource and goals, and thus agent can negotiate and cooperate with others. Purely reactive behaviour(event-driven) is ideal for agents that can operate in a time-constrained (constantly changing) environment. Since agents must constantly
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check the environment and reconsider the validity of their goals. However, such an agent may waste time and computational resources unnecessarily in a comparably static environment, so proactive (goal-driven) behaviour is more appropriate for a static, unchanging environment whereas the assumption (pre-condition) of executing the procedure remains true while the procedure is executing. Through a number of experiments, Kinny and Georgeff [69] show that the key parameter that affects an agent’s decision about being more proactive or more reactive is the the rate of world change, γ. Further to the above definition and to my understanding, being either proactive or reactive only is not intelligent. Whereas being both proactive and reactive is. That is, the ability to achieve an effective balance between being proactive and reactive is the key feature that differentiates intelligent agents from other computing entities. The thinking/reasoning process that attempts to achieve the balance is the key that underpins an agents intelligent behaviour. Therefore, the Wooldridge&Jenning’s definition on intelligent agent can be put in a stronger form: Definition 2.1.1. Intelligent software agents are the software systems that are not only both proactive and reactive, but also capable of deciding the effective balance between pro-activeness and reactiveness in a given situation, and at the same time capable of social interaction with others. In the following section, an example is used to describe the balance between proactiveness and reactiveness. The analysis and investigation of that example sheds light on the importance of belief revision for intelligent agents.
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2.1.2
Example: Academic Life
An academic, John, has two intentions1 : being a good researcher and a good lecturer. For the time being, John has two tasks to carry out concurrently, preparing a paper for an international conference and managing a postgraduate subject. The two intentions are not mutually exclusive, but as they consume the same resource, a good balance is needed to achieve both. Imagine that John sits in front of the computer writing his paper while the phone rings and the email program alerts: “you have a mail”. What should John do? Continue writing the paper or answer the phone calls and emails? If John constantly answers the phone calls and replies to the emails, there is little chance that he will finish the paper on time. But if John simply ignores the phone calls and emails from his students, eventually, he will not be recognised as a responsible and approachable lecturer. Writing a paper is one of the proactive procedures he adopted to achieve his good researcher intention, while answering phone calls and replying to emails are the proactive procedures for achieving his other intention of being a good lecturer. Instead of blindly carrying out the procedures for one intension and dropping the other, naturally, John will prioritise his tasks. Suppose John’s philosophy of time management is as follows: • If the paper’s due date is still relatively far away, but the students’ assignment due date is approaching, and he expects many urgent phone calls and emails, then the teaching duty takes priority. • If the paper’s due date is approaching but not the assignment due date, then the paper writing takes priority. 1
Goals if you like, goals can be seen as committed intentions.
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• If there is a conflict between the precondition, ie, both the paper and assignment due dates are approaching, the two intentions become mutually exclusive at that period of time, then a decision has to be made to drop one intention temporarily. We can see from the above example, as a human being, an academic can achieve an effective balance between proactive behaviours (writing a paper, answering phone calls and emails) and reactive behaviours(reconsidering the priority according to due dates). The flexibility resides in the change (the evolution) of the environment (e.g. how close the due date is) that triggers the rearrangement of the priority of the current intentions. This is essentially the same intelligence underlying the ability of working out different plans for dinner based on the different left-overs in the fridge everyday. To implement this in an intelligent software system, we need knowledge representation2 . One necessity is the agent knowledge base, which contains the agent’s knowledge of the world, be it the recipes for dinner or the “recipes” for achieving research and teaching success. The other necessity is the agent belief base, which reflects the current state of the world, e.g. the beliefs about what is left in the fridge, the beliefs about how many days left before the paper due etc. The result of observing the left-overs in the fridge and checking the calendar for today’s date are the new pieces of information respectively that trigger the changes of the belief base. Belief revision thus comes into play. As can be seen, for any intelligent agents that are proposed or designed to be both proactive and reactive, belief revision is a crucial capability. This argument can be taken even further, that is, belief revision is a ubiquitous process underlying many forms of intelligent behavior[110]. 2
Knowledge representation can be viewed as “the surrogate - the substitute for the thing itself, used to enable a computational entity to reason about the world”. from http://www.medg.lcs
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Belief revision is an essential skill that an autonomous agent should possess in order to revise its beliefs in a coherent and rational fashion when it receives new information. The concept will be explained further in the following sections.
2.2
An Informal Introduction to Belief Revision
Belief revision concerns the process of modifying an agent’s existing belief base in a rational way by the introduction of new information that the agent has decided to accept. Before discussing belief revision within the paradigm of epistemological theories, two examples are presented informally to illustrate: • four elements of an epistemological theory (i.e. epistemic state, epistemic attitude, epistemic input and epistemic change); • two different ways of changing belief systems (i.e. direct/immediate belief revision for foundation theory and logic-constrained belief revision for coherence theory); • three essential issues that concern the belief revision theorists, see Section 2.2.3 for more detail.
2.2.1
Example: Umbrellas in the Rain
John is just about to go out. Given that he believes it is raining outside now, he decided to take an umbrella with him. But someone who just came in from outside tells him that it is not raining now.
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Let r stand for the fact that “it is raining” and ¬r for the opposite that “it is not raining”. Similarly, u for “to take an umbrella” and ¬u for “not to take an umbrella”. Therefore, John’s current state of belief can be represented as in Table 2.1. It is raining. r r→u If it is raining, I will take an umbrella. If it is not raining, I will not take an umbrella. ¬r → ¬u I will take a umbrella. u Table 2.1: Raining and Umbrella It is not rational for John to believe that it is raining and it is not raining at the same point of time. Therefore, in the light of the new information (i.e. epistemic input), ¬r, John is left with at least three options for his new belief state (assuming John is only capable of three types of epistemic attitude towards beliefs, accepted, rejected and unknown): 1) to continue believing r (e.g. John believed that person was just joking) 2) to stop believing in r but to believe in ¬r (e.g. John trusted that person completely) 3) to be ignorant about r (e.g. John was neither sure of that person nor himself) In case 1), John can simply keep his old belief state and do nothing. While in both case 2) and 3), so-called epistemic changes are involved. Simply deleting r from the database then adding ¬r into the database is one way of doing the changes, which is the approach taken in foundation theory[57]. The trivial deletion and addition operation is often accompanied by a sophisticated and in general nonmonontonic inference operation to deduce the new beliefs[42] (e.g. NMR approach[36] or TMS[70,
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59] approach). This is called direct/immediate belief revision as compared to the logic-constrained belief revision. In logic-constrained settings, changes to the database are no longer trivial but governed by set of rationality postulates. Such an approach is often taken by coherence theory[57]. In the above example, because r ↔ u, using one of the AGM postulates in Section 3.1, u will be removed when r is removed and ¬u will be added automatically when ¬r is accepted. In other words, ¬u is added as a result of the change function, but not as a result of any form of logical deduction as normally found in direct belief revision. In this thesis, the focus is on logic-constrained belief revision, which is most eminently exemplified by the Alchourr´on-G¨ardenfors-Makinson (AGM) paradigm. In Table 2.1, beliefs above the horizontal line (r, r → u and ¬r → ¬u in this example) are normally explicitly represented and so called explicit beliefs. The belief below the horizontal line (u) is derivable from the explicit beliefs, and is called an implicit belief. For computational reasons, implicit beliefs are not normally stored explicitly in a database because they are always infinite. They include all tautologies which are infinite (e.g. p ∨ ¬p, p ∨ ¬p ∨ ¬p, p ∨ ¬p ∨ ¬p ∨ ¬p ...). Further discussion of explicit beliefs v.s. implicit beliefs is given in Section 2.2.3 and also in [42] and[41]. One may also find that the above process of changing the beliefs only makes sense when the world is static. That is, during the change of beliefs, it is assumed the environment does not change from a state of “raining” to “not raining”. Belief revision investigates the issues of changing beliefs when the environment is static, whereas belief update concentrates on the issues raised when the environment is changing while beliefs change as well. Belief update is beyond the scope of this thesis, for
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interested readers, please refer to [66]. It is important to note however that update can be defined in terms of revision[95].
2.2.2
Example: Air Kangaroo and Free Beer
The Raining and Umbrella example only illustrates a very simple case of revision. Revision can be more complicated and involve many choices among the alternative resultant states, which is shown by this example. Table 2.2 illustrates the knowledge base (i.e. belief state or epistemic state). Air Kangaroo is a start-up airline. Air Kangaroo is an Australian airline. All Australian start-up airlines serve free beer. Air Kangaroo servers free beer.
a b a∧b→c c
Table 2.2: Air Kangaroo and Free Beer Suppose John took Air Kangaroo and discovered that Air Kangaroo does not provide beer for free (¬c). The consistency of the above knowledge base is thus challenged. To keep this knowledge base consistent, John needs to revise it. For example, simply retracting a will retain consistency when ¬c is added. Actually, removing any belief that is above the horizontal line in Table 2.2 will recover a consistent knowledge base. Questions are, shall we remove all beliefs or just some/one of them? If not all beliefs, then which one should be given up and which should be kept?
2.2.3
Three Issues of Belief Revision
Although simple, the above two examples do illustrate important issues associated with belief revision. Most commonly, a development of a new belief revision mechanism needs to address the following three key issues.
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1) How are the beliefs in the database represented? In our examples, beliefs are represented in the simplest and most intuitive way, by sentences. There are other options available, such as possible worlds (also called interpretations); one may also use mathematical frameworks such as probability theory for a quantitative representation, or possibility theory or belief functions for a comparatively qualitative representation. Different schemes of representation will be introduced in the sections that follow. Clearly, a belief revision mechanism is sensitive to the formalism chosen to represent the beliefs. Whatever belief representation models a revision system may choose, it satisfies the following Principle of Categorical Matching (PCM)[42], The representation of a belief state after a belief change has taken place should be of the same format as the representation of the belief state before the change. In other words, belief change happens within the formalism of representing the beliefs but shall not change it. 2) What is the relation between the elements explicitly represented in the database and the beliefs that may be derived from these elements? This issue occurs when choosing sentences as the representation formalism. Which sentences should we assume to be explicitly stored in an agent’s knowledge base? What do we do with the logical consequence of a set of sentences? Basically, there are two school of thought, one school prefers to take both the set of sentences and its logical consequences (so-called belief set) into consideration, the other prefers to operate on the explicitly stored sentences only (so-called
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belief base) and resorts to some external logic to further the deduction. There are pros and cons for both option. Belief set representation requires an agent to be logical omniscient, i.e., it not only knows its explicit beliefs, but also all their logical consequences. This is an ideal case and good for investigating theoretical issues. The most important contribution to the belief revision field - the AGM framework (more in section 3.1 - is based on this representation. Let L denote a countable language which is closed under a complete set of Boolean Connectives. We will denote sentences in L by lower case Greek letters. We assume L is governed by a logic that is identified with its consequence relation `, that is, a sentence α is logically valid iff it is a consequence of the empty set. The relation ` is assumed to satisfy the following conditions[41]: (a) If α is a truth-functional tautology, then ` α. (b) Modus ponens. If α ` β and ` α, then ` β. (c) Not `⊥. That is, ` is consistent. (d) Satisfies the deduction theorem. That is, ` α → β iff α ` β). (e) Is compact. That is, if α is a consequence of some set X, then α is a consequence of some finite subset of X. It follows that ` contains classical propositional logic. The belief base approach has been criticised as syntax sensitive[12], which is not desirable for rational changes. For example, consider a belief base B1 = {α, β}. In order to retract α ∨ β, both α and β have to be retracted. But after
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adding α ∨ β back again, we don’t get back α and β. But if we represent the belief base as B2 = {α, β, α ∨ ¬β, ¬α ∨ β}, retract α ∨ β and add it back we will obtain α and β by logical deductions. Note that B1 and B2 generate the same belief set. However, the representation of belief sets requires double-exponential (for propositional logic) or infinite (for first-order logic) space. Computationally, belief bases are a more appealing representation as compared to belief sets. Section 3.4 focuses on rational base revision. 3) How are the choices concerning what beliefs to retract made? Retraction always involves more than one sentence as shown in the Air Kangaroo example. The guiding criterion adopted by most researchers is the principle of minimal change. By specifying an underlying preference relation defined on either sentences or possible worlds, one can express that a sentence (world) is more preferable than another in the face of change. The preference relation can either be a purely subjective preference such as Epistemic Entrenchment Ordering advocated by G¨ardenfors et al. [40], or it might possess some mathematical meaning such as probability, possibility and belief function.
2.3
Belief Revision as an Epistemological Theory
The theoretical framework that provides a conceptual apparatus for investigating problems about changes of knowledge and belief, is called an epistemological theory[41]. The following four epistemic factors are the essential characteristics that form the core of epistemological theories:
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2.3.1
Four elements of an epistemological theory
Belief State The first and most fundamental characteristic is a class of models of epistemic states or states of belief. An epistemic state is a representation of a person’s knowledge and belief at a certain point of time, for example, John’s state of belief at the moment he discovered that Air Kangaroo does not serve free beer.3 Epistemic states are the central entities in epistemological theories. Different models of epistemic states often determine the representation of the other three core entities. Here three popular approaches are briefly outlined to illustrate the different ways of representing epistemic states, which paves the way for introducing different types of belief revision techniques. According to G¨ardenfors[41], the best-known approaches are the Bayesian models which are widely used in decision theory and game theory. It is assumed in Bayesian theory that all information that is relevant for decision making is conveyed by a probability measure, which is defined over some object language or over some space of events. In such approaches, a state of belief is represented by a probability measure. A dominant rationality criterion governing this type of model is that of coherence. A second kind of approach represents an epistemic state as a set of propositions, 3
The epistemic states here are not seen as psychological states. This means that a state in a computer program may also be seen as a model of an epistemic state.
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expressed by sentences from some given object language4 . Such approaches are certainly simpler than Bayesian approaches but also less informative (see the epistemic attitudes that are representable in the two kinds of models). An epistemic state contains those propositions that the agent accepts at a certain point of time. The essential rationality criterion for such models is consistency. Logical omniscience is also often assumed in the sense that the set of propositions representing an epistemic state is supposed to be closed under logical consequences. A third way of representing epistemic states is by a set of possible worlds. The interpretation of such a set is that the agent associated with the state knows that the “actual world” is a member of a set of possible worlds, and that any world in the set could be the actual one, i.e. they are all consistent with his beliefs. In the approaches based on sets of propositions, the valuation is given by a membership relation; and in probabilistic models the valuation is given by a probability measure. Epistemic Attitude A second characteristic of an epistemological theory is a classification of the epistemic attitudes that describe the status of various elements of belief that are contained in an epistemic state. For example, a person may accept or not accept a particular fact as true or he may judge it to be certain, probable, or possible. These are different attitudes toward the same fact. In an epistemic state based on a set of sentences, one can distinguish three different kinds of belief: 4
One may differentiate propositions from sentences if needed: propositions are the contents of the sentences, ie, propositions concern the semantic of the sentences, while sentences are the syntactic representation.
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1. A proposition α may be accepted, which means that α belongs to the set representing the epistemic state; 2. α may be rejected, which is to say that the negation of α is accepted in the state; 3. α may be kept in suspense or undetermined, which means that neither α nor its negation is an element of the relevant set of sentences. In a model of an epistemic state based on a probability measure P, a richer typology of beliefs is possible: one may say that a sentence α is more probable than β, meaning that P (α) > P (β); that α is likely, meaning that P (α) > 0.5, and so on. In this type of model we can also say that α is accepted if P (α) = 1 and similarly that α is rejected if P (α) = 0 and undetermined if 0 < P (α) < 1. Epistemic Inputs A third characteristic is an account of the epistemic inputs that may lead to changes of epistemic states. These inputs can be thought of as the deliverance of experience or as linguistic (or other symbolic) information provided by other individuals (or machines). For example, when John discovered that Air Kangaroo doesn’t serve free beer, this observation served as an epistemic input that lead to a revision of his state of belief. Epistemic Changes A fourth characteristic involves a classification of epistemic changes or changes of belief. Different kinds of epistemic input result in different types of change in the
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epistemic states. The most important epistemic changes are expansions, revisions, and contractions, which we will discuss further in Section 3.1.
2.3.2
Information Economy and Criteria of Rationality
In addition to the above four characteristics, on the meta-level, an epistemological theory should also contain criteria of rationality, which are used for evaluating the above four factors. As we know, epistemic states are used for representing an actual or a possible cognitive state of some individual at a given point of time. One can think of the idealised epistemic states as being equilibrium states. For example, if a set of beliefs is not consistent or if a probability assignment to a field of beliefs is not coherent, then the individual should, if it is to fulfill the rationality criteria, adjust the state of belief until it reaches an equilibrium that satisfies the criteria of consistency or coherence. To achieve a consistent belief set or a coherent probability assignment is the driving force of belief revision and also the fundamental rationality criteria. On the other hand, information is in general not gratuitous, and unnecessary losses of information are therefore to be avoided. When we change our beliefs, we want to retain as much as possible of our old beliefs. This heuristic criterion is called the criterion of informational economy, which guides the selection of change operators. This criterion is used to evaluate the changes of belief in the AGM paradigm and is called the principle of minimal change, that is, the change is required to be minimal such that it accommodates the epistemic input which initiates the change. It can been seen in Section 3.1 that the minimal change criterion is main driver behind the AGM postulates.
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Different interpretations of the minimal change principle appear in the literatures, such as absolute minimal change due to Williams[108], relative minimal change (conditionalization) due to Sphon[100]. Williams[111] furthers the understanding of minimal change while bearing the preference relation in mind. According to her, the principle of minimal change says that, as much information should be conserved as is possible in accordance with an underlying preference relation. In this sense, the number of sentences to be given up is not necessarily the rational measure of the magnitude of the change. For example, rational agents might choose to throw out several weakly believed sentences to achieve consistency rather than throw out one that is very strongly believed. The question of explicitly defining a rational minimal change is still open and is often related to the application domain in practice.
Chapter 3 Belief Revision Techniques under the AGM Paradigm “Beliefs The bird caught in the trap is a swan The bird caught in the trap comes from Sweden Sweden is part of Europe All European swans are white Consequences The bird caught in the trap is white New Information The bird caught in the trap is black Which sentence(s) would you give up?” — Peter G¨ardenfors and Hans Rott (1995) in [42]
3.1
The AGM Postulates and Belief Set Revision
A major source of inspiration that has made belief revision theory one of the most active research areas in philosophical logic is the development and investigation of two sets of postulates in a 1985 paper[1] by Alchourr´on, G¨adenfors, and Makinson(AGM). Named after their founders, the postulates (one set for revision, the other for contraction)1 are often called the AGM postulates. 1
For the sake of completion, in Section 3.1.1 we listed the AGM postulates for expansion as well.
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In the AGM paradigm, an agent’s epistemic states (or belief states) are represented by a deductively closed set (K) of sentences, i.e. a belief set or a theory. The logic governing the deduction is ` as defined in Section 2.2.3. The changes to the epistemic state are thus modelled as processes involving the addition and removal of sentences. Please note, the AGM belief change and the other change operations discussed in this thesis do not consider the possibility of explicitly modifying individual facts. For example, weakening the following sentence: All Australian startup airlines serve free beer. into All Australian startup airlines except Air Kangaroo serve free beer. will not be considered. For recent work on weakening beliefs in belief bases, see [6]. In AGM’s 1988 paper [40], the three kinds of theory change that were introduced informally in section 2.2 are described as follows: 1)Expansion: A new sentence together with its logical consequences is added to a theory K regardless of the consequences of the larger set formed, i.e. the resulted set is not necessarily consistent. The theory that results from expanding K by a sentence α will be denoted Kα+ . 2)Revision: A new sentence, possibly inconsistent with the theory K is added, but in order that the resulting theory be consistent some of the sentences in K are removed. The result of revising K by a sentence α will be denoted Kα∗ 3)Contraction: Some sentence in K is retracted without adding any new facts. In order that the resulting system satisfies (I) some other sentences from K must be given up. The result of contracting K with respect to the sentence α will be
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denoted Kα− . The above description illustrates the AGM’s fundamental view of the theory change problem. That is, computationally, theory change can be handled by a function which maps a belief set K and a sentence α to a new belief set Kα+ (or Kα∗ or Kα− ). Respectively, we have for such purposes, expansion functions, revision functions and contraction functions. Therefore, the ultimate goal of belief change is to develop algorithms for computing appropriate revision and contraction functions for an arbitrary theories. Sets of postulates presented were developed and first introduced in [1], which serves as the standard that the candidate change functions should comply with. In the following subsections, the postulates are listed and briefly discussed.
3.1.1
Postulates for Expansion
(K + 1) Kα+ is a belief set. (Closure) Expansion functions should take one belief set to another when new information α is introduced and in so doing abide by the principle of categorical matching. (K + 2) α ∈ Kα+ (Success) The epistemic input should be accepted. Note the resulting belief set could be inconsistent. (K + 3) K ⊆ Kα+ . (Informational Economy) As information is valuable, all the old beliefs should be retained in the expansion of K by α. (K + 4) If α ∈ K, Kα+ = K is a belief set.
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If α is already accepted in K, a requirement to accept α should have no effect on the belief state. (K + 5) If K ⊆ H, then Kα+ ⊆ Hα+ (Monotonicity2 ) Kα+ should not contain any beliefs that are not also included in Hα+ . (K + 6) For all belief sets K and all sentences α, Kα+ is the smallest belief set that satisfies (K + 1)-(K + 5). It turns out that the expansion function (+) satisfies (K + 1)-(K + 5) iff Kα+ = Cn(K) ∪ α. An important result from the above is that the expansion postulates (K + 1)-(K + 5) uniquely determine the expansion of K by α as the set of all logical consequences of K together with α. In other words, there is an unique expansion function Cn(K ∪{α}) that satisfies the AGM expansion postulates. But as will be shown in the following sections, using the postulates presented below alone will not guarantee an unique function for revision (contraction). This brings us to the fore the third issue we discussed in Section 2.2.3. Extra logical information will be required to help single out one revision (contraction) function. Two standard ways of doing that will be introduced in Section 3.3.
3.1.2
Postulates for Revision
(K ∗ 1) Kα∗ is a belief set. (Closure) The revision operators take one belief set to another when new possibly conflicting information is incorporated. 2
0
0
An inference operation or logic Cn is monotonic iff H ⊆ H implies Cn(H) ⊆ Cn(H ). Otherwise it is nonmonotonic.
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(K ∗ 2) α ∈ Kα∗ (Success) It guarantees that the epistemic input α is accepted in Kα∗ . (K ∗ 3) Kα∗ ⊆ Kα+ (Expansion 1) (K ∗ 4) If ¬α 6∈ K, then Kα+ ⊆ Kα∗ (Expansion 2) (K ∗ 3) and (K ∗ 4) together say that in the consistent case, revision is equivalent to expansion. (K ∗ 5) Kα∗ = K⊥ only if ` ¬α (Consistency preservation) The purpose of revision is to produce a new consistent belief set. Thus Kα∗ should be consistent, unless α is logically inconsistent itself. (K ∗ 6) If ` α ↔ β, then Kα∗ = Kβ∗ (Extensionality) Logically equivalent sentences should lead to identical revisions. Shown in the Raining and Umbrella Example in Section 2.2.1. In other words revision is syntax insensitive. ∗ (K ∗ 7) Kα∧β ⊆ (Kα∗ )+ β (Conjunction 1) ∗ It is shown in [40] that (K ∗ 7) is equivalent to Kα∗ ∩ Kβ∗ ⊆ Kα∨β ∗ (K ∗ 8) If ¬β 6∈ Kα∗ , then (Kα∗ )+ β ⊆ Kα∧β (Conjunction 2, Rational Monotony)
(K ∗ 2) is often called “Priority of the New Information” by some researchers (e.g. Dragoni[19]). It is sometimes criticised as an unfavourable property in an open environment where information may come from multiple sources with various reliability. We argue that taking postulate (K ∗ 2) as giving priority to the new information is a misconception. In an open environment, where information may need to be judged
37
according to the source reliability, belief revision should be treated as a two step process. Firstly, the agent needs to make a decision about whether to accept the new information or not, or accept with a certain degree of plausibility. Then, in the second step, if the agent decides to accept the new information, postulate (K ∗ 2) becomes applicable. This can be illustrated by the example in Section 2.2.1. Before John changes his current beliefs, he normally will give it a second thought to see whether the information source is trustworthy or not (maybe the source is only joking about the matter). In other words, the raw information from the source will be filtered prior to the revision of the epistemic state. If the source is trustworthy, John will then apply existing belief revision techniques to revise his belief state. However, Benferhat[4] has a valid argument when considering revision as a special process of fusion. He supports the idea that a fusion process is symmetric as opposed to a revision process where the new information is prioritised. As a result, in fusion, the new information may not necessarily be accepted in the result belief set.
3.1.3
Postulates for Contraction
(K − 1) Kα− is a belief set.(Closure) Contraction functions take a belief set to another with the concerned information to be retracted. (K − 2) Kα− ⊆ K (Inclusion) The sentence to be retracted should not be in Kα− . (K − 3) If α 6∈ K, then Kα− = K (Vacuity) If α is not in K, nothing should be retracted from K, and all the existing beliefs
38
should be left unchanged, according to minimal change criterion. (K − 4) If not ` α, then α 6∈⊆ Kα− (Success) Contraction is successful when the sentence to be retracted is no longer a logical consequence of the new belief state Kα− unless α is a tautology, in which case it can never be removed. (K − 5) If α ∈ K, then K ⊆ (Kα− )+ α (Recovery) This requires that all beliefs in K are recovered after first contracting and then expanding with respect to the same belief. (K − 6) If ` α ↔ β, then Kα− = Kβ− (Extensionality) Logically equivalent sentences should lead to identical contractions, same motivation as in (K ∗ 6). − (K − 7) Kα− ∩ Kβ− ⊆ Kα∧β (Conjunction 1) − The beliefs that are in both Kα− and Kβ− are also in Kα∧β . − − (K − 8) If α 6∈ Kα∧β , then (Kα∧β ) ⊆ Kα− (Conjunction 2, Rational Monotony)
The minimal change necessary to give up α ∧ β is closely related to the minimal change necessary to reject α itself. Recovery (K − 5) is the most controversial postulate of all. Although it is a valid principle in non-probabilistic contexts for belief set revisions. It is questionable in probabilistic contexts, especially in the case of contracting a sentence that has causal consequences. It is also questionable in the context of base revision as shown in Section 2.2.3. For detail discussion on the recovery postulate in base revision, see [55].
39
3.2
Levi Identity and Harper Identity: From Revision to Contraction and Vice Versa
It is widely accepted that [1, 73, 38] contraction operators and revision operators are inter-definable. A revision of a belief set can be seen as a composition of a contraction and an expansion, this is known as the Levi Identity, formally written as:
Kα∗ = (Kα− )+ α On the other hand, a contraction operator can be defined using a revision operator, this is known as the Harper Identity, below,
∗ Kα− = K ∩ K¬α
3.3
Determining the Sentences to be Retracted
On seeing that the AGM expansion postulates uniquely define an expansion operator Kα+ = Cn(K ∪ {α}), one may wonder whether a unique revision and contraction operator will be circumscribed by the corresponding revision and contraction postulates or not. Unfortunately, it turns out that both contraction and revision are nonunique operations. This is due to the fact that they both involve the retraction of information, which in general presents a nonunique choice when determining which sentence is to be given up. Pure logical and set theoretical information is not sufficient and extralogical information is needed for singling out a unique revision or contraction function[40, 106]. A variety of mechanisms for selecting what to retract have been tried. Some of these mechanisms select among all the maximal consistent subsets with respect to K and
40
α, others among the possible worlds or models - the semantic extension of the concerned set of sentences[45]. Some select among the candidates for deletion, others among candidates for being retained. Some select by means of a choice function, others select by preference orderings (unit intervals [0,1] or ordinals). Among these, two prevailing ordering mechanisms are AGM’s epistemic entrenchment orderings[40] and Grove’s systems of spheres[45]. They are briefly described in this section.
3.3.1
Epistemic Entrenchment Ordering
An epistemic entrenchment ordering is intended to capture the importance of a sentence in the face of change. Given a belief set K of L, an epistemic entrenchment related to K is any binary relation ≤ on L satisfying (EE1)-(EE5) below: (EE1) For any α, β, and γ, if α ≤ β and β ≤ γ, then α ≤ γ. (Transitivity) (EE2) For any α and β, if α ` β, then α ≤ β. (Dominance) (EE3) For all α, β ∈ K, if α ` β then α ≤ β. (Conjunctiveness) (EE4) When K 6= K⊥ , α 6∈ K iff α ≤ β for all β. (Minimality) (EE5) If β ≤ α for all β, then ` α. (Maximality) The notation α ≤ β is a shorthand for ‘β is at least as epistemologically entrenched as α’. (EE1) requires the ordering relation ≤ to be transitive. (EE2) says that if α entails β and either α or β must be retracted from the belief set K, then it is a smaller change to give up α and retain β rather than to give up β, because then α must also be retracted if we want the revised theory to be closed under logical consequences. (EE3) prescribes that retracting α ∧ β from the belief set K can be achieved only by
41
giving up either α or β, but not necessarily both. (EE4) states that sentences that are not in belief set K are the least entrenched and thus minimal in the ordering ≤. Similarly, (EE5) simply says that only tautologies can be maximal in the ordering ≤. G¨ardenfors and Makinson[40] have shown that for every contraction operator − there exists an epistemic entrenchment ≤ related to K such that the condition E − below, is true for every α ∈ L, and conversely.
−
(E )
3.3.2
Kα−
=
(
{β ∈ K : α < α ∨ β} if 6` α K
otherwise
System of Spheres
In contrast to an epistemic entrenchment ordering, where the ordering is defined on sentences, Grove’s system of spheres[45] focuses on the set Ω of possible worlds that can be described in a language L. Any belief set K can be represented by the subset [K] of Ω that consists of all possible worlds that contain the sentences in K. A system of spheres centered on [K] [45] is a collection S of subsets of Ω that satisfies the following conditions: (S1) S is totally ordered by ⊆; that is, if S and S 0 are in S, then S ⊆ S 0 or S 0 ⊆ S. (S2) K is the ⊆-minimum of S; that is, if [K] ∈S, and, if S ∈S, then K ⊆ S. (S3) Ω is in S and so the largest element of S. (S4) If α is a sentence and there is any sphere is S intersecting [α], then there is a smallest sphere in [S] intersecting [α].
42
[ S] S Sα S' [K] C(α )
[α]
Figure 3.1: System of Spheres Centered on [K] Condition (S4) ensures that for any sentence α, if [α] intersects any sphere at all in S, there is some sphere Sα that intersects [α] and is smaller than any other sphere with this property. If [α]=∅ and does not intersect any sphere, then Sα = Ω. The set C(α) = [α] ∩ Sα is the set of “closest” element in Ω to [K] in which α is an element. The set C(α) is marked as the hatched area in Figure 3.1, which is adapted from Figure 4.2 on page 85 in [41]. Grove[45] proves that Definition 3.3.1. Let S be any system of spheres in M centered on [K] for some belief set K. If, for any α, Kα∗ is defined to be KC(α) , where KC(α) is the belief set represented by set C(α) of possible worlds, KC(α) = ∩C(α). The resulting revision function satisfies (K ∗ 1) − (K ∗ 8). Conversely, let ∗ be any revision function satisfying (K ∗ 1) − (K ∗ 8). Then for any (fixed) belief set K there is a system S of spheres that is centered on K and that satisfies Kα∗ = KC(α) . The key idea Grove presents is that the revision Kα∗ can be represented by KC(α) , which is the belief set represented by the set C(α) of worlds. That is, the following condition (S ∗ ) can be used to construct a revision function from a system of spheres.
43
∗
(S )
3.3.3
Kα∗
=
(
∩C(α) if α is consistent ⊥
otherwise
Other Important Works
Due to its simplicity, the AGM framework has inspired a lot of significant work in the area of belief change. Among others, there is Peppas’[94] contribution of identifying the class of well-behaved revision operators based on a well-ordered system of spheres. Furthermore, Peppas and Williams[95] identify another representation – a nice preorder on models and importantly they prove that under condition (ES) below, an epistemic entrenchment, a system of spheres and a nice pre-order represent the same revision function: (ES) For every consistent φ, ψ ∈ L such that 6` φ and 6` ψ, φ ≤ ψ if and only if C(¬φ) ⊆ C(¬ψ) where C(φ) represents the smallest sphere in S intersecting [φ]. In terms of implementing a revision system, epistemic entrenchment orderings have better computational properties than pre-orders on models for large knowledge bases since the number of possible worlds are exponential in the size of sentences. Consequently it is considered as a practical way of representing preference of epistemic states in a computing system. However, such choice is particularly domain and application dependent. An example of implementing a system of spheres using database technology is described by Williams in [111]. In addition to the ordering techniques stemming from an epistemic entrenchment ordering and a system of spheres, which we call qualitative rankings, there are quantitative orderings for specifying orderings between epistemic states based on probability
44
theory, belief functions or possibility theory, which will be introduced in Chapter 4. The AGM paradigm provides an elegant general framework for describing revision and contraction operators. However, two major problems arise when using it as a foundation theory for a computer-based implementation of belief revision[105]. • The epistemic entrenchment orderings operate on belief sets, which requires an ideal agent who knows all the logical consequences of his current beliefs. The epistemic state thus represented by belief sets may be infinite in size, while a computer implementation needs a finite representation. • As AGM operators map epistemic entrenchment orderings of a belief set to a new belief set, there is no specification of an entrenchment ordering of the resulting set. Although this contributes to the flexibility of the AGM framework, it presents difficulties when iterated revision is desired in an implementation. Therefore, a means of propagating an epistemic entrenchment ordering to another is needed. Extensions to the AGM framework that address these problems can be found in works on iterated revision[72], works on belief base change [105] and some combination of the two [108].
3.4
Implementing Entrenchment Based Revision Belief Base Revision
Just as a theory does not uniquely determine a theory change operator, so too a theory base does not uniquely determine a theory base change operator. In the case of theory change additional structure in the form of an epistemic entrenchment ordering is used
45
to construct unique operators, and for theory base change Williams proposed finite partial entrenchment rankings [105, 111].
3.4.1
Finite Partial Epistemic Rankings
A finite partial entrenchment ranking maps a finite set of sentences to rational numbers in the interval [0,1]. The higher the value assigned to a sentence the more entrenched it is. Not like an epistemic entrenchment ordering which ranks a possible infinite belief set, formally defined below, it grades the content of a finite belief base according to its epistemic importance. Definition: A finite partial entrenchment ranking is a function B from a finite subset of sentences into the interval [0,1] such that the following conditions are satisfied for all φ ∈ L: (PER1) {φ ∈ L : B(φ) < B(ψ)} 6` φ. (PER2) If ` ¬φ, then B(φ) = 0. (PER3) B(φ) = 1 if and only if ` φ. (PER1) is actually a reformulation of (EE2), and says that one should not assign a higher value to an arbitrary sentence ψ than that of φ if ψ ` φ. (PER2) and (PER3) simply say that inconsistent sentences are assigned zero while tautologies are assigned 1. As noted before, a finite partial entrenchment ranking is an ordering of explicit beliefs that explicitly represent a knowledge base, denoted by Kexp . A belief set is the logical consequences of the explicit belief, the set of implicit beliefs, which is denoted by Kimp . The major contribution of Williams’ work is that she defines a way
46
of ranking implicit beliefs based on (PER1) such that a finite partial entrenchment ranking can be used to represent a finitely representable epistemic entrenchment ordering3 . Intuitively, beliefs that are not in the implicit belief set take the lowest degree of acceptance in face of change, 0 in this case. On the other hand, a sentence φ in the implicit belief set should be accepted at a degree no lower than its most accepted precedent (i.e. the belief that entails φ). The ranking on implicit beliefs can be formally captured in the definition of the function degree(B, φ) below. Definition 3.4.1. Let φ be a nontautological sentence. Let B be a finite partial entrenchment ranking. The degree of acceptance of φ is defined to be: degree(B, φ) =
max(B(ψ)), ψ ∈ Kexp ∩ {ψ : ψ ` φ} if φ ∈ Kimp 0 otherwise
Williams[105] proves that a minimal epistemic entrenchment ordering ≤B can be generated from a finite partial entrenchment ranking B if following the definition holds:
φ ≤B ψ iff ` ψ, or degree(B, φ) ≤ degree(B, ψ) It is minimal in the sense that sentences take their minimal value in the ordering and sentences that are not in the implicit belief base Kimp will take a minimal entrenchment 0.
3.4.2
Transmutations
As we can see from the revision techniques we introduced before, an epistemic entrenchment ordering is an integral part of an agent’s epistemic state and it is the crucial information for identifying a unique revision operator. Therefore, the term 3
For the condition on an epistemic entrenchment ordering to be finitely representable, see Williams’ [105].
47
knowledge system is introduced to take into account the information presented by an epistemic entrenchment ordering. A knowledge system differs from a belief set in the sense that a knowledge system contains not only the sentences as beliefs but also their epistemic entrenchment as extralogical information about relative importance of beliefs in the face of change. To facilitate iterated revision, belief revision is viewed as a transformation of a knowledge system to another knowledge system rather than that of a belief set to another belief set. Spohn[100] presents a knowledge system as an ordinal conditional function that maps the set of all consistent and complete theories4 into the class of ordinals such that there is a possible world assigned the smallest ordinal 0, which is interpreted as an accepted world. An ordinal conditional function represents a plausibility grading of possible worlds or a grading of disbelief, the worlds that are assigned the smallest ordinal are the most plausible. Belief change in such a system is realised through transmutations on ordinal conditional functions. Through transmutations, the ordering on the possible worlds is propagated into the new epistemic state when new information is incorporated. Among the family of transmutation operators, Spohn[100] claims that conditionalisation is a desirable transmutation because the worlds which are consistent with the new information and those which are not are shifted in relation to one another. Close to the idea of ordinal conditional functions[100], where an ordinal is assigned to a possible world, Williams proposes the idea of transmutations on partial entrenchment rankings[108]. Different transmutation strategies are developed and proposed based on different interpretations of the “minimal change” criterion. Two 4
For definitions on consistent and complete theory, see Appendix A
48
of the major types of transmutation are, adjustment, maxi-adjustment, and disjunctive maxi-adjustment[6]. Here we briefly introduce the process of adjustment to illustrate a basic idea of how to transmute one partial entrenchment ranking into another. Adjustments define change functions for belief bases rather than belief sets, which makes it computationally attractive for real world applications. Intuitively, an (φ, i)adjustment of B involves minimal changes to B such that φ is accepted with degree i. In particular, each sentence ψ ∈ Kexp is reassigned a number B∗ (φ, i)(ψ) closest to its original B(ψ). This is guided by the principle that if an accepted sentence φ is reduced to a degree i, each sentence that would be retracted in φ’s contraction will be reduced to i as well. Formally, the definition of adjustment is as follows, Definition 3.4.2. Let φ ∈ L and 0 ≤ i < 1. The adjustment of a finite partial entrenchment ranking B is defined to be a function ∗ such that ∗
B (φ, i) =
(
(B− (φ, i)) if i ≤ degree(B, φ) − + (B (¬φ, 0)) (φ, i) otherwise
where for all ψ ∈ Kexp , B− (φ, i)(ψ) =
i, if degree(B, φ) = degree(B, φ ∨ ψ) and B(ψ) > i B(ψ) otherwise
and for all ψ ∈ Kexp ∪ {φ}, B+ (φ, i)(ψ) =
B(ψ)
if B(ψ) > i i if φ ↔ ψ or B(ψ) ≤ i < degree(B, φ → ψ) degree(B, φ → ψ) otherwise
For example, if B(ψ → φ) = 0.7, B(φ → ψ) = 0.6 and B(ψ) = 0.5, using the definition above, the degree of implicit information φ is 0.5. If we want to accept φ at a lower degree i = 0.2, we will have the new partial entrenchment ranking, where B(ψ → φ) = 0.7, B(φ → ψ) = 0.6, B(φ) = 0.2 and B(ψ) = 0.2. ψ migrate down the ranking.
49
If we want to accept φ at a greater degree say i = 0.5, we will have B(ψ → φ) = 0.7, B(φ → ψ) = 0.6, B(ψ) = 0.5 and B(φ) = 0.5. If we want to accept φ at a greater degree say i = 0.7, we will have B(ψ → φ) = 0.7, B(φ → ψ) = 0.6, B(ψ) = 0.7 and B(φ) = 0.7. The guiding principle here is actually (EE2), no matter how the new entrenchment is generated, the logical consequence should be at least as entrenched as its precedent. As we can see from the above example, to obtain a successful (φ, i)adjustment, one must explicitly specify the ranking for both ψ → φ and φ → ψ, that is specify the degree of independence (φ ∨ ψ) between sentences, which is not as natural and easy as specifying dependence. To address this problem, Williams proposed maxi-adjustment[111], where information independence is assumed by default unless dependence is derivable from the explicitly specified information. This is based on Spohnian reasons[100], where α is reason for β iff raising the epistemic rank of α would raise the epistemic rank of β. A maxi-adjustment is done by specifying reasons and during the contraction of α, β is retained if β can not be derived as a reason for α. Readers are referred to [110, 111] for the definition of maxi-adjustment and related algorithms. It is worthwhile to point out that both adjustment and maxi-adjustment satisfy the AGM postulates. If only a base is used then extra processing is needed to satisfy recovery. Promisingly, adjustment and maxi-adjustment, and other transmutation strategies are implemented in an object-oriented web based belief revision system, SATEN. For details, see [107]. Actually, the work presented in this thesis builds on SATEN as a single belief revision agent which provides the platform to study the issues that
50
might be presented in an open heterogenous multi-agent environment, where agents can interact and exchanging information with other agents. The result of our study is a theoretical and practical framework for extending SATEN into a multi-agent belief revision test-bed.
Chapter 4 Belief Revision - The Probability and The Possibility Approach “The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man’s mind.” — James Clerk Maxwell (1850)
4.1
Introduction
G¨ardenfors [41] points out rather convincingly that the change of an epistemic state can be defined in the framework of logic as well as in traditional probability theory. As recognised by Dubios and Prade [24], this creates an exciting interaction between different fields, in particular, logic, probability theory, belief functions, possibility theory and more recently plausibility theory[35]. When it comes to devising rational change operators, these different theories fit into the epistemological framework
51
52
described in Section 2.3 rather nicely. The pure logic framework may limit the representational power to accepted, rejected and indetermined, while the probability-like theories are more expressive, for example, one can represent complete ignorance, one can also express that α is more probable/possible/plausible than β, which is achieved at the price of working out a quantitative rather than a qualitative representation of an agent’s belief state. A brief introduction to probability theory, possibility theory and belief functions is given, and followed by our discussion on their inter-relationships. We show that both probabilities and possibilities are special cases of belief functions. This will pave the way to out ontological1 view of the field.
4.2
Probability Theory
Subjective or personalistic probabilities are the best known models of epistemic states and of how epistemic inputs affect an epistemic state[41], which is widely used in decision making and game theory. The epistemic states in a probability theory can be represented by either sentences or possible worlds. A probability function defined over these sentences or possible worlds provides a measure of an individual’s degree of belief in the sentence or propositions, which enriches the epistemic attitude that can be represented in this model. As shown in Section 2.3, G¨ardenfors takes a slightly different view, he considers the probability measure itself to be a representation of the belief state. In contrast, we would rather take the sentences or possible worlds as the model of an individual’s epistemic state and the probability measures as a means of expressing a richer epistemic attitude. Actually, from the previous discussion, we 1
Ontologies will be discussed in Chapter 6
53
agree that a successful unique change operator relies on some means of ranking, be it an epistemic entrenchment ordering, or a systems of sphere, or some other ranking mechanisms. This suggests the ranking among sentences (or possible worlds) is an inseparable element of an epistemological theory as far as belief change is concerned. In fact, we are inclined to treat the ranking which expresses the epistemic importance with respect to the new information as an integral part of an individual’s epistemic attitude. A probability space (S, X , P r) consists of a set S (the sample space), a set X of subset of S whose elements are called measurable sets and a probability measure P r : X → [0, 1], satisfying the following properties (known as the Kolmogorov axioms for probability): P1. 0 ≤ P r(X) ≤ 1 for all X ∈ X P2. P r(S) = 1 P3. P r(∪∞ i=1 Xi ) =
P∞
i=1
P r(Xi ), if the Xi s are pairwise disjoint members of X .
(Countable Additivity) Definition 4.2.1. A subset Y of X is said to be a basis of X if the members of Y are non-empty and disjoint, and if X consists precisely of countable unions of members of Y. The probability of every measurable set can be computed from the probability of every set in its basis by using P3. It is not hard to see that if X is finite then it has a unique basis, which consists precisely of the minimal elements of X . The above definition is defined on sets, to extend probability theory for logical sentences, Stalnaker(1970)[101] provides this language based definition on probability functions:
54
Definition 4.2.2. A (language-based) probability function is a function P from sentences in L into real numbers that meets the following six conditions for all sentences α,β, and γ: (I) 0 ≤ P r(α) ≤ 1 (II) P r(α) = P r(α ∧ α) (Reflexivity) (III) P r(α ∧ β) = P r(β ∧ α) (Commutativity) (IV) P r(α ∧ (β ∧ γ)) = P r((α ∧ β) ∧ γ) (Associativity) (V) P r(α) + P r(¬α) = 1 (VI) P r(α) = P r(α ∧ β) + P r(α ∧ ¬β)
Using the language-based Definition 4.2.2, a logic for L can be defined by following the two definitions below. Definition 4.2.3. A sentence α is logically valid iff P r(α) = 1 for all probability functions that satisfy conditions (i)-(vi). Definition 4.2.4. Two sentences α and β are logically disjoint iff ¬(α∧β) is logically valid. From Definition 4.2.4, a theorem equivalent to P3 (Countable Additivity) can be established for probabilities of sentences, Theorem 4.2.1. If α and β are logically disjoint, then for any probability function P r, P r(α ∨ β) = P r(α) + P r(β). As we can see from above, a probability measure can be defined on not only a set of worlds but also on a set of sentences. The probability measures thus defined are capable of representing partial beliefs 2 . In other words, in probabilistic settings, sentences can be believed with certain degrees of belief but not necessarily always as certain. 2
Partial beliefs refer to the sentences that are believed at a certain degree within a unit interval [0,1] but are not certain.
55
The rationality of belief change modelled by probability functions is governed by the coherence restriction. Intuitively, coherence requires that the new assignment of probabilities of the sentences in face of new information forms a standard probability function. Belief change in a probabilistic framework is captured by defining probability update functions. A probability update function is defined as a partial function from probability functions to probability functions, that is, if τ is a probability update function and P r is a probability function, then τ (P r) is the probability function that arises as a result of updating P r in the light of the new information encoded by τ . Traditionally, the standard way to update a probability function P r is to use the well-know Bayesian Conditioning using the conditional probability function P r(·|B), where
P (A|B) =
P r(A ∩ B) P r(B)
Moreover, a sequence of probability updates can be combined by composition. The composition operator ◦ is associative, but is not in general commutative; the order of updating matters. The result of updating by τ1 , then τ2 , and then τ3 can be given by the update function τ1 ◦ τ2 ◦ τ3 . The following proposition holds for Bayesian Conditioning, cond, given that B, C ∈ X , condB ◦ condC = condB∩C = condC ◦ condB For B ∈ X , condB is defined to be the probability update function such that condB (P r) = P r(·|B) if P r is a probability function on X when P r(B) > 0, and undefined otherwise3 . 3
The partiality of the update function allows it to be undefined if the new information that it
56
4.3
Belief Functions
A belief function is a function that assigns to every subset of a given set S a number between 0 and 1. The idea of a belief function was introduced by Dempster [16, 17] and put forward as a framework for reasoning about uncertainty by Shafer [98]. It has an attractive mathematical theory, many intuitively appealing properties, and is widely used as a standard tool in expert system applications. For the purpose of the work presented in this thesis, we are particularly interested in the so-called DempsterShafer’s rule of combination, which is an important technique for combining and merging knowledge bases. Here, we will briefly introduce the semantic underpinning of belief functions and their close relationship with probability theory and possibility theory. Formally, a belief function can be defined as a function satisfying the axioms below. These axioms can be viewed as a weakening of the Kolmogorov axioms that characterise probability functions. A belief function Bel on Ω is a function Bel : 2Ω → [0, 1] satisfying: (B0) Bel(∅) = 0 (B1) Bel(A) ≥ 0 (B2) Bel(Ω) = 1 (B3) Bel(A1 ∪ . . . ∪ Ak ) ≥
P
I⊆{1,...,k},I6=∅ (−1)
|I|+1
Bel(∩i∈I Ai ).
A more intuitive way of formulating belief functions is by defining so called mass functions. A mass function is a function m : 2s → [0, 1] such that encodes is incompatible with the probability to be updated.
57
(M1) m(∅) = 0 (M2)
P
A⊆Ω
m(A) = 1
m(A) is interpreted as the weight of evidence for A that has not already been assigned to some proper subset of A. Actually, a mass function and a belief function are interdefinable. Theorem 4.3.1 ([98]). 1. If m is a mass function on Ω,then the function Bel : 2Ω → [0, 1] defined by P Bel(A) = B⊆A m(B) is a belief function. 2. If Bel is a belief function on 2Ω and Ω is finite, then there is a unique mass P function m on 2Ω such that Bel(A) = B⊆A m(B) for every subset A of Ω.
In order to combine two or more independent 4 pieces of evidence, DempsterShafer’s rule of combination is widely used and shows many appealing features in real world applications. With the restriction of confining ourselves to finite sets S, the rule of combination can be described as follows. If m1 and m2 are mass functions with the same domain 2Ω , the combination of m1 ⊕ m2 is the mass function m(A) below when there exists a pair B1 , B2 such that
B1 ∩ B2 6= ∅ and m1 (B1 )m2 (B2 ) > 0 (Combination Condition) then P
4
m1 (B1 )m2 (B2 ) {B1 ,B2 |B1 ∩B2 6=∅} m1 (B1 )m2 (B2 )
{B1 ,B2 |B1 ∩B2 =A}
m(A) = P
Independence here is taken to be an intuitive, primitive notion. The probabilistic definition of independence - namely, that A and B are independent if P rA ∩ B = P r(A) × P r(B) - is a consequence of the intuitive notion, but not considered as sufficient or complete definition over it.[53]
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If there is no such pair B1 , B2 that meets the combination condition, then m1 ⊕m2 is left undefined, and the corresponding belief functions Bel1 and Bel2 are said to be not combinable. Using the rule of combination, Shafter shows that the update of a belief function in the face of new information can be captured by combining the existing belief function Bel with a single support function LearnB (A) which puts all the weights on the new evidence B; i.e. m(B) = 1 and m(A) = 0 if A 6= B. Then, we have,
B
Learn (A) =
(
1 if A ⊇ B
0 otherwise The new Bel after observing B is then Bel ⊕ LearnB . Actually the result of ∗
the combination is equivalent to what Shafter defined as conditional belief (Bel(·||B) which is analogous to Bayesian’s conditional probability (P r(·|B)).
Bel(A||B) =
¯ − Bel(B) ¯ Bel(A ∪ B) ¯ 1 − Bel(B)
Based on his two views of beliefs, when taking new information as a generalized probability (c.f. an evidence). Halpern proposed another version of conditional belief function. For those who are interested, please refer to Theorem 3.4 in his paper[53].
4.4
Possibility Theory
Possibility theory[23] is a new form of information theory which is related to, but independent of, both fuzzy sets and probability theory. Technically, a possibility distribution is a normal fuzzy set (at least one membership grade equals 1). For example, all fuzzy numbers5 are possibility distributions. However, possibility theory 5
Fuzzy sets are a further development of the mathematical concept of a set. Conventional sets have an either-or criterion for membership, whereas fuzzy sets adopt a grade of membership[114].
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can also be derived without reference to fuzzy sets. The rules of possibility theory are similar to probability theory, but use either MAX/MIN or MAX/TIMES calculus, rather than the PLUS/TIMES calculus of probability theory. In possibility theories, an epistemic state can be represented both semantically and syntactically. Most appealingly, the semantic ordering on possible worlds can be nicely translated into syntactic ordering on sentences. The semantic representation is captured by a possibility distribution π, which is a mapping from the set of possible worlds (Ω) to the unit interval [0, 1]. π(ω) represents the degree of compatibility of the possible world ω with [K], the set of possible worlds in which all sentences in K are simultaneously true. According to G¨ardenfors[41], “[K] is the largest set of possible worlds that is compatible with the individual’s convictions.” In particular, π(ω) = 0 means that the interpretation ω is impossible, and π(ω) = 1 means that nothing prevents ω from being the real world. When π(ω) > π(ω 0 ), ω is a preferred candidate to ω 0 for being the real state of the world. A possibility distribution is said to be normal if ∃ω ∈ Ω, such that π(ω) = 1 A possibility distribution π induces two different syntactical ways of ranking sentences in a language, namely, the possibility measure and the necessity measure. • A possibility measure evaluates the extent to which φ is consistent with the available knowledge expressed by π [115]. It is defined by:
Π(φ) = max{π(ω) : ω |= φ} Therefore, in fuzzy sets, the transition from membership to non-membership is gradual rather than abrupt.
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and it satisfies:
∀φ ∀ψ Π(φ ∨ ψ) = max(Π(φ), Π(ψ)) • A necessity measure evaluates to what extent φ is entailed by the available knowledge expressed by π . It is defined by:
N (φ) = inf {1 − π(ω) : ω |= ¬φ} = 1 − Π(¬φ) and it satisfies:
∀φ ∀ψ N (φ ∧ ψ) = min(N (φ), N (ψ)) The duality relation N (φ) = 1−Π(¬φ) is an extension of the entailment in classical logic. Traditionally, a sentence is entailed from a set of classical sentences if and only if its negation is not consistent with this set. Therefore, before being entailed by the available beliefs, the sentence concerned should be consistent with them first, which means N (φ) > 0 implies Π(φ) = 1. It is interesting to note the similarity between probability theory and possibility theory in their views of belief changes. Similar to probability theory’s traditional treatment of new information using conditional probability, in possibilistic settings, revision is defined by a conditional possibility measure Π(·|A). Back in 1978, Hisdal[58] first introduced the set function Π(·|A) through the following equality
∀B ∈ Ω, B ∩ A 6= ∅, Π(A ∩ B) = min(Π(B|A), Π(A))
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note here A = [α] is the set of models for new information α. However since the above equation may have more than one solution in terms of Π(B|A), Dubois and Prade[24] proposed the least specific solution, that is, when A ∩ B = ∅, Π(B|A) = 0. When A ∩ B 6= ∅
Π(B|A) =
(
if Π(A ∩ B) = Π(A) > 0
1
Π(A ∩ B) otherwise
The conditional necessity function is defined by duality,
¯ N (B|A) = 1 − Π(B|A) The possibility distribution π underlying the conditional possibility measure Π(·|A) is defined by π(·|A). If ω 6∈ A, π(ω|A) = 0; if ω ∈ A π(ω|A) =
(
1
if π(ω) = Π(A)
π(ω) if π(ω) < Π(A)
Benferhat has a more general view of belief revision in terms of fusion[4]. According to him, belief revision can be viewed as a fusion process where the new information has priority over the existing accumulated information in a prioritised knowledge base. Basically, fusion differs from revision as a symmetric operation since it does not necessarily distinguish the new information from the established ones. Benferhat’s belief revision is interpreted as building a new possiblistic knowledge base B⊕ with a distribution π⊕ obtained by aggregating π{φ,δ} and πB . He analyses and compares different fusion operators in paper [4], thus we can define different revision operators based on them. With the epistemic meaning defined above for both possibility measures and necessity measures, in the possiblistic framework, the syntactical representation of an
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epistemic state can be augmented to (φ, δ). It is important to note that, the certainty degree of the sentence is explicitly specified in a possbilistic knowledge base as compared to a traditional knowledge base, where only the sentence is explicit. Formally, a possiblistic knowledge base is made up of a finite set of weighted sentences, B = {(φi , δi ) : i = 1, n} where δi is understood as a lower bound on the degree of necessity N (φi ). Moreover, the syntactic representation can be translated to a corresponding semantic one by defining possibility distribution π out of δ guided by the intuition below[2]: • For a possibilistic knowledge base only consists of one sentence (φ, δ), then each interpretation ω which satisfies (i.e. consistent with) φ, π(ω) = 1; for each interpretation which falsifies φ will have a possibility degree π(ω) such that the higher δ is, the lower π(ω) is, a simple selection is 1 − δ. Therefore, the possibility distribution associated with B = {(φ, δ)} is: ∀ω ∈ Ω, π{φ,δ} (ω) =
(
1
if ω ∈ [φ]
1−δ
otherwise
• For a general possibilistic knowledge base B = {(φi , δi )}, the intuition is all the interpretations satisfying all the beliefs in B will have a highest possibility, 1; and the other interpretations will be ranked w.r.t. the highest belief that they falsify[23], ∀ω ∈ Ω, πB (ω) =
(
1
if ∀(φi , δi ) ∈ B, ω ∈ [φ]
1 − max{δi : (φi , δi ) ∈ B and ω 6∈ [φi ]} otherwise
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It is interesting to note that, the idea of a normalised possibilistic knowledge base coincide with definition of a knowledge system used by Williams [108, 111] in transmutations. It also turns out Williams’ Partial Entrenchment Rankings are equivalent to the Necessity Orderings [24]. The connection can be furthered through Peppas and Williams’ contribution [95] on a condition (ES) which governs the equivalence of an epistemic entrenchment ordering, a system of sphere and a nice pre-order. The condition (ES) is described in Section 3.3.3. Furthermore, in Dubois and Prade’s [24], one can find thorough discussions on how possibility theory in belief change can be related to the AGM paradigm and even a broader spectrum of research in this area in the last two decades, such as Spohn’s ordinal conditional functions [100] and potential surprise [109] etc.
4.5
Probability as Special Belief Functions
The major distinction between a probability function and a belief function is that a probability function assigns a number between 0 and 1 to some but not necessarily all of the subsets of a belief base, which are often called measurable sets in probability theory. In contrast, a belief function assigns a number to every subset of a set. There are two standard ways of extending a probability function P r so that it is defined on all subsets. One is to define the inner measure P r∗ introduced by P r, which is understood to be the best approximation to the probability P r from below. The other one is to define the outer measure P r ∗ , which is considered as the best approximation to P r from above. As it is often claimed pure statistical probabilities sometimes are difficult or even impossible to obtain in real world applications, the interval [P r ∗ , P r∗ ] is often considered as that the best range that real probability P r may fall into. The
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inner measure and outer measure induced by P r are defined for an arbitrary subset A ⊆ S[52].
P r∗ (A) = sup{P r(X) |X ⊆ A and X ∈ X } P r∗ (A) = inf {P r(X) |X ⊇ A and X ∈ X } For a finite sample space or if the number of measurable sets in the sample space is finite, the inner measure of A is just the measure of the largest measurable set contained in A, while the outer measure of A is the measure of the smallest measurable set containing A. For any set A, we have P r∗ ≤ P r∗ and if A is measurable, then P r∗ (A) = P r(A) = P r ∗ (A). It is easy to show that P r∗ is a belief function in that it satisfies the three axioms characterising belief functions and P r ∗ is the corresponding plausibility function. The connection between probability, belief and inner measure can be easily seen if using the mass functions in finite spaces. If P r is a probability function defined on a set X of measurable subsets of a finite set S, and Y is a basis of X , let m be the mass function such that
m(A) =
(
P r(A) if A ∈ Y 0
otherwise
and let Bel be the belief function corresponding to m. Then according to the definition of P r∗ , we can easily obtain Bel(A) = P r∗ (A) for all A ⊆ S. In fact, for all the measurable sets, Bel(A) = P r(A) and more generally, Bel(A) is equal to the inner measure on arbitrary subsets. It is important to notice that the mass function m has the property that its focal elements, i.e., those sets that have positive mass, are disjoint. Moreover, if the focal elements are not only disjoint but are also singletons,
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a belief function turns out to be a special type of probability function, in which every element in the sample space is measurable. Therefore, we have, Theorem 4.5.1 ([53]). A belief function is a discrete probability function if only if its focal elements are disjoint singletons.
4.6
Possibility as Special Belief Functions
We can show via the following proof6 that the necessity measure is actually a special Dempster-Shafer’s belief function when the focal elements are nested singletons. Given two sentences φ and ψ ordered by “qualitative necessity relation” (≥c ) defined by the necessity measures. Assume the certainty degree of φ and ψ are N (φ) and N (ψ) respectively, with the help of the following proposition (proposition 2 in [24] and it is also proved in [40]:
if φ ` ψ then ψ ≥c φ We know that φ ` φ ∨ ψ and ψ ` φ ∨ ψ so we obtain:
φ ∨ ψ ≥c φ and φ ∨ ψ ≥c ψ therefore,
N (φ ∨ ψ) ≥ max(N (φ), N (ψ)) Recall the characteristic inequality of belief function, the degrees of belief on possible worlds[φ], [ψ], for φ and ψ respectively follows the inequality below, 6 Thanks to J´erˆome Lang for his valuable discussions on probability, possibility and belief functions and the proof presented here is mostly attribute to him.
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Bel([φ] ∪ [ψ]) ≥ Bel([φ]) + Bel([ψ]) − Bel([φ] ∩ [ψ]) Let us assume 1. The focal elements for defining BelN are nested singletons. Using the mass distribution function m on two focal elements A1 and A2 , the assumption is easily seen as equivalent to:
BelN (A1 ) ≥ BelN (A2 ) iff A1 ⊇ A2 2. Either [φ] ⊇ [ψ] or [φ] ⊇ [ψ] i.e. [φ] and [ψ] are nested. and we can get
BelN ([φ] ∪ [ψ]) ≥ max(BelN ([φ]), BelN ([ψ])) We can easily observe the match between N (φ) and BelN ([φ]), i.e. N (φ) = BelN ([φ]). This can be formulated into the theorem below similar to that of belief function and probability: Theorem 4.6.1. A belief function is a necessity function if only if its focal elements are nested singletons.
4.7
Summary
As shown in the beginning of Chapter 2, belief revision is an essential capability for an agent to exhibit intelligence and to be both proactive and reactive. Therefore, investigating issues that arise in the field of belief revision with a focus on implementing the proposed techniques for real world applications is of crucial importance.
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In the last two decades, belief revision became an increasingly important research area as it generated valuable discussions between disciplines that have traditionally ignored each other, such as philosophy, logic, computer science and mathematics. One of the major contributions to this field is the AGM paradigm introduced in Chapter 3, famous for its sets of rationality postulates and the simplicity of the framework. The AGM postulates circumscribe three types of belief change operators, expansion, contraction and revision. Belief states are represented by deductively closed sets of sentences, called belief sets. Operations of change take the form of either adding or removing specified sentences. When we contract a deductively closed set K of sentences by a nontautologous sentence α, the outcome Kα− is a deductively closed set of sentences that does not contain α. There are two operations that incorporate new information, namely, expansion Kα+ and revision Kα∗ . The outcome, Kα+ , which results from expanding K by α is simply Cn(K ∪ {α}). Clearly, expansion is a monotonic operation, and if ¬α ∈ K, then Kα+ is inconsistent. In contrast to an expansion, the outcome of revising K by α is a deductively closed set that contains α, which is consistent if only if α is consistent. Hence, in revision, but not expansion, old beliefs are sacrificed to achieve consistency. Based on the AGM postulates, expansion is a unique monotonic operation, but not revision and contraction. Two major ordering techniques, epistemic entrenchment and system of spheres are proposed as extra logical information devices for constructing certain classes of revision and contraction operators. Alongside or even before AGM’s syntactical qualitative approaches designed to develop a deeper understanding of belief revision under the umbrella of epistemological theories, there are quite a few quantitative and qualitative mathematical frameworks
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proposed for tackling belief change issues as well. These include the famous Bayesian probabilities, Dempster-Shafer’s belief function and Dubois-Prade’s possibility theories. Quite surprisingly, these all fit into the epistemological framework described by G¨ardenfors nicely. As shown by the brief presentation of these works in this chapter, although they are different works carried out separately by different researchers, they are very closely related. In [24], Dubios and Prade show that a qualitative necessity ordering is almost exactly the same as AGM’s epistemic entrenchment orderings. Furthermore, a necessity measure in possibility theory is actually a special case of a belief function, and a belief function can be used to define a probability function if it meets the disjoint singleton criteria presented in Section 4.5. The comparative analysis in the thesis provides a solid theoretical foundation for the development of a multi-agent belief revision system that might be operating in an heterogenous environment where agents can adopt their own revision mechanisms. In this thesis, we follow the literature in using the term ‘revision’ with three different senses. In its wide meaning, belief revision refers to the general problem of changing belief states; a narrower meaning refers to inserting an arbitrary piece of information into a belief system; still narrower is the sense just introduced; i.e. where the new piece of information is inconsistent with the original belief system.
Chapter 5 Belief Revision in a Multi-Agent Environment “... there are seven main components to a BDI agent: 1. a set of current beliefs, 2. a belief revision function (brf), which takes a perceptual input and the agent’s current beliefs, and on the basis of these, determines a new set of beliefs. 3. ...” — Michael Wooldridge (1999) in [113]
5.1
Brief Introduction to Multi-Agent Systems
Agent technologies and multi-agent systems have been identified as the mainstream software engineering[63] for large distributed domains, such as ecommerce/ebusiness, distributed sensor networks, air traffic control[77], resource allocation for Internet access[80] etc. The major focus of a multi-agent system is to create a system that interconnects separately developed agents such that the group of agents is able to function beyond the capabilities of any singular agent by itself. 69
70
A classical real-world example is a system inside which a personal travel agent will negotiate with other software agents who represent the interests of different shareholders, such as hotel agents, airline agents, and etc. This is a truly multi-agent problem which involves the inter-operation of separately developed and self-interest agents which provide a service beyond the capability of any single agent. In the process, most or all gain financially. Although there is no clear boundary between them, a multi-agent system is different from that of Parallel AI and Distributed Problem Solving. Nwana and Ndumu[89] point out that in agent research and practice, very often, people tend to confuse: agents and objects; distributed computing and agent-based computing; object-oriented systems, expert systems and agent-based systems. Before we can start discussing issues about belief revision in an multi-agent environment, we will have to first clarify these easy-to-confuse concepts. Woodridge[113] illustrates one of the major differences between objects and agents with an interesting slogan: “Agents do it for money. Objects do it for free.”. One other well-recognised difference is that agents have internal control on the running threads and most importantly have the ability to refuse the request, while objects are normally designed to be passive and methods are evoked by external events. As to object-oriented computing, distributed computing etc, according to Nwana and Ndumn[89], they do not in themselves offer solutions to multi-agent systems for the following reasons: • Distributed computing modules are designed to be passive/cooperative and homogeneous[25].
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• The communications between distributed computing modules are usually lowlevel while multi-agent systems require high-level messages. • Multi-agent system applications require a cooperative-knowledge level (i.e. ontology level) while object-oriented systems, expert systems etc typically operate at the symbol and knowledge level[87] at most. In fact, parallel AI and/or distributed problem solving techniques are more appropriate in the following areas, where multi-agent systems are not really needed: • Solutions are needed to enhance modularity (which reduces complexity), speed (due to parallelism), reliability (due to redundancy), efficiency and flexibility. • Problems are too large for a centralised single ‘agent’ to perform due to resource limitations or the risk of having one centralised system. • Solutions are used mainly to address problems requiring heterogeneous reasoning. Areas that multi-agent solutions really excel at include: • Problems requiring the interconnecting and inter-operation of multiple, autonomous, ‘self-interested’ existing legacy systems, e.g. expert systems or decision support systems. For example, when solutions need to be drawn from distributed autonomous experts, e.g. in health care provisioning or electricity distribution.1 • Problems that are inherently distributed, e.g. geographically, as in distributed sensor networks or air-traffic control. 1
Possible citation of Laurence’s paper on manager gathers information form individuals.
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• Problems whose solution require the collation and fusion of information, knowledge data from distributed, autonomous and ‘selfish’ information sources, e.g. British Telecommunication’s personal travel assistance prototype application[86], such systems also go under the name of cooperative information systems. Clearly, multi-agent systems are considered as a solution to complex real-world problems which in general involve multiple, autonomous, self-interested entities, require unsupervised interaction between the entities, especially require the collation and fusion of information from the distributed entities. Based on the above understanding of multi-agent systems, the term “multi-agent belief revision” may carry a two-fold meaning. Firstly, it implies a multi-agent solution for complex belief revision problems, where multiple heterogenous belief bases are involved and the conclusion can not be drawn without a consistent overview. Secondly, it can also be seen as an emergent behaviour that occurs in a multi-agent environment. Belief revision is essential for an agent to exhibit intelligent behaviour. In a multi-agent system - a society populated with intelligent agents, the issue of revising shared /common belief naturally arises which does not present in a single agent situation. This not only requires communication, negotiation, and cooperation among agents, but due to the autonomy of each individual (possibly selfish) agent, it exhibits behaviours much more complex than that of group decision making, which we will discuss under the title of “heterogeneity” in Section 5.4. Bearing in mind the global picture of where multi-agent systems reside in the family of distributed AI, let’s have a close look at the multi-agent environment and describe some important assumptions that define the boundaries of the system scope
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that we are to consider herein. We consider a general decentralised environment, where both control and data are logically and often geographically distributed. In addition, agents could come from different designers or different vendors, whose protocols or mechanisms are not foreseen. Each agent makes its own choices as to whether it will talk to another agent and at what time. Each agent has local memory, no memory needs to be shared by all agents. Each agent typically has several distinct knowledge bases, specifically, for example, some for domain knowledge, some for social knowledge. Furthermore, we assume that each agent has special expertise in specific areas. The expertise overlaps, but is not identical. An agent’s expertise and services are the only data open to other agents. Expertise is graded, e.g. by professional institutions. Lastly, we assume the communication between agents is always successful. Every agent can communicate with every other agent by sending and receiving messages. Any information sent out by agent x will arrive at agent y precisely with no distortion. For our special interest, those messages are in a formal form, we will discuss a reasonable form in the following chapters. We also assume the existence of low-level protocols (peer-peer, broadcast, multi-cast) to ensure effective and flexible communication. Agents could choose between connection oriented and connectionless to suit their needs. In addition to answering another agent’s queries, an agent could also offer information and services to other agents based on its own internal decision. We also assume the use of FIPA2 agent platform for information discovery. For example, we assume the existence of a directory facilitator for yellow page services and an agent name server for white page services. 2
http://www.fipa.org
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Under the above assumptions, each agent has an incomplete, local view of the environment. No agent is assumed to have sufficient information to solve all global problems and agents do not typically have full access to others’ internal states. However, the agent expertise and services are available to other agents through service descriptions and registrations with the directory facilitator. Viewing multi-agent systems as societies of heterogenous autonomous agents that satisfy the above assumptions, we formulate the following criteria for evaluating the existing work on multi-agent belief revision. Q1: Is there a super agent that has global access to all information? Q2: Are the agents heterogenous? For example, do they use different revision techniques, do they have different levels of expressing epistemic attitudes? Q3: Are the agents cooperative and trustworthy? Q4: Can agents choose what, who and when to communicate? In this chapter, first we review some existing frameworks for multi-agent belief revision by asking the above four questions. Then we present a picture of the field by classifying them according to whether the revision techniques are developed for a single agent environment or a multiple agent environment. Lastly, we identify different types of heterogeneities in order to address the weakness of the reviewed frameworks. Furthermore, the investigation for further understanding the heterogeneity issues helps define our research issues.
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5.2
Various Frameworks for Multi-Agent Belief Revision
5.2.1
Mutual Belief Revision of Van der Meyden
Mutual Belief Revision is a nested process during which an agent must revise not only its own beliefs about the world, but also its beliefs about other agents’ beliefs about the world and moreover about other agents’ beliefs about its own beliefs, and so on. A multi-agent version of perfect introspection logic K45n is employed to model the so-called mutual belief : Definition 5.2.1. A set of agents mutually believe ϕ iff each of them believes ϕ, and each of them believes that each of them believes ϕ, and so on, ad infinitum[31]. The theory of mutual belief revision ends up with a unique operator which satisfies the following four assumptions: (1) Agents have perfect introspection3 , but maybe inconsistent. (2) Agents revise their beliefs synchronously, in response to an event whose occurrence is a common knowledge. (3) The world is static; (4) Each agent’s revision method is a common knowledge. Assumption (3) simplifies the problem so that there is no need to consider the change of the environment and confines the problem to revision, not update. As noted earlier, it is this basic assumption that differentiates revision from update. But the other three assumptions ((1),(2) and (4)) limit the proposed theory’s capability of handling heterogeneities in multi-agent environment. By assuming perfect 3
Perfect introspection assumes a considerable degree of self-knowledge, [37] has detail.
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introspection, the system requires agents to know exactly what they believe and do not believe. On one hand, this is a quite rigid requirement and in the real world, agents only tend to know something after performing a search. On the other hand, perfect introspection only applies to ideal agents. However, a real system may have incompetent or stupid agents, who do not have perfect introspection, namely, they may not know what they believe and what they don’t or they may have wrong introspection about what they believed. Assuming perfect introspection limits the capability of dealing with heterogenous agents. Assumption (2) confines this theory to simple communication, namely, broadcasting. Assumption (2) and (4) together mean that not only the revision method is known to others but also the whole revision process and the input event that will trigger everyone’s revision simultaneously. Clearly mutual revision systems fail the test of being truly multi-agent solutions because assumptions (1),(2) and (4). Furthermore, the infinite recursive definition of mutual belief makes it almost impossible to implement. The idea of mutual revision is further illustrated in [85] by the “scientists in conference” example. In the example, all the agents work faithfully, broadcasting what they know and what they do not know. Considering a distributed knowledge base system, where each agent represents a local knowledge base. Based on the above four assumptions, the agent should store its own knowledge about the world. Meanwhile, it has full access to all other remote agents’ knowledge bases that belong to other agents. Therefore, every agent knows what the others know and what they do not know. On the receipt of new information, at first the agent revises its own beliefs, then because the revision method is common knowledge, it could readily and successfully predict what all the other agents will do with their beliefs.
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In fact, mutual belief revision can be thought of as a set of single belief revision processes carried out uniformly in a parallel but decentralised manner.
5.2.2
MABR of Kfir-dahav and Tennenholtz
In contrast to Van der Meyden, Kfir-dahav and Tennenholtz[68] approach MABR in the context of heterogeneous systems. The Private Domains (P Di ) and the Shared Domain (SD) of the agent knowledge base are defined in order to capture a general setting where each agent has private beliefs as well as beliefs shared with other agents. Under such a knowledge structure, each agent may have its own perspective of the world but needs to coordinate (ie agree on) its belief on shared elements. The shared domain also defines the communication language for the agents. One important question in this scenario is how do we manage and maintain the knowledge in P Di and SD? By definition[68], SD is just the intersection of each agent’s KB. Although it is not explicitly stated, the authors assume that agent i is aware of (knows) its own knowledge in P Di and the existence of SD. Since an agent does not know other agents’ P D, what happens if after several sequences of revision, the intersection of the P Di s is not empty? For example, consider the situation in which two agents(A1 , A2 ) engage in revision, Φi stands for the KB of Ai , i=1,2.
Φ1 = {γ, φ, ψ} Φ2 = {α, β, φ} According to the definition,
78
SD = {φ}; P D1 = {ψ, γ}; P D2 = {α, β} If A2 receives a piece of new information, say, α→ψ, then by Modus Ponens, ψ should be in P D2 . Thus,
P D1 ∩ P D2 = {ψ} 6= ∅ A2 will only expand P D2 but not SD providing that nobody says that ψ is also believed by A1 . The authors did not tell us what to do with this ψ. If ψ∈SD, it seems to us, they implicitly assumed that the existence of a super agent who knows all the agents’ private knowledge and the society’s shared knowledge, so that the intersection of private knowledge can always be upgraded to the shared domain. If ψ6∈SD, then the definition of SD is violated.
5.2.3
MSBR of Dragoni et al.
Recognising that agents may join a network with low degrees of competence or noncooperative intentions, Dragoni et al. [20] states that the reliability of the source affects the credibility of the information and vice-versa. Neglecting the “priority to the incoming information principle” is thus proposed and implemented by considering the information pair hinf ormation, sourcei rather than just hinf ormationi. During an AGM revision, priority is given to the incoming information. For instance, the new information will be accepted using an expansion if it is consistent with the current belief sets. In the sense of transmutation[108], the new information is allowed to come with a certain rank and to be accepted at this prescribed level. In this case although you don’t necessarily totally accept the new information, you do
79
need to respect the incoming rank. There is no explicit rational step to change the rank in these frameworks. While the non-priority of or in the extreme case, neglecting incoming information is thus a two step procedure, first, revise the rank according to the reliability of the informant and then incorporate the information with the new rank. Therefore, by evaluating the source reliability, the receiver agent has the flexibility of deciding whether to take the impinging information into account or not.
5.2.4
DBR of Dragoni et al.
In Distributed Truth Maintenance Systems[59], all the agents are both individually and mutually consistent with all other agents with whom they have exchanged knowledge. While in DBR[20][21], the Liberal Belief Revision Policy is adopted, that is, to let all the agents stand by their own beliefs based on their own view of the evidence. Therefore, the local consistency is considered as a prerequisite, but the global consistency is only considered as an end point which is eventually reached through some selection strategies. Every node (agent) in the DBR system is able to carry out MSBR as well as communicate with each other. As to the knowledge structure, local knowledge is distinguished from global knowledge. By using certain voting functions[20], the local knowledge can be selected to become global knowledge. Compared to Kfir-dahav and Tennenholtz’s terminology, Dragoni et al.’s local and global knowledge could be seen as the counterparts of the knowledge in P Di and SD, respectively. But a subtle improvement is made by Dragoni et al. in that certain voting functions are employed to generate global knowledge rather than simply taking the intersection of P Di s. Actually, private or local knowledge as defined here are not private in the real sense, because it is implicitly assumed
80
there is some super agent, at least a human developer, exists in the system to supervise the knowledge upgrade 4 . Private knowledge should be confidential, ie, invisible to all the outsiders. To classify agent knowledge in this way is still too simple to some extent. For example, in an agent society populated with multiple agents, some agents may wish to form small groups or teams to accomplish goals. Therefore, shared knowledge (or team knowledge) rather than global knowledge is sometimes required.
5.2.5
Summary
To summarise, mutual belief revision is only capable of revising knowledge, but not graded belief. Kfir-dahav and Tennenholtz’s MABR and Dragoni et al.’s DBR have overcome this by discussing revision in the broad sense of knowledge system transmutation. The social behavior that might affect the information credibility has been discussed by Dragoni et al. in the context of MSBR. This is an advance over other studies based on reliable, faithful and mutually trustworthy communication. The knowledge structures proposed by Kfir-dahav and Tennenholtz, and Dragoni et al. initiate an effort in classifying knowledge. However, they are not rich enough to eliminate some ambiguity or to offer some essential flexibility. It will be shown in Section 5.4 that finding a feasible way of classifying knowledge is an important step towards modelling social heterogeneity occurred in knowledge bases. None of the schemes discussed above address the issues that might arise from 4
Knowledge upgrade is one phase of knowledge migration, which also encompasses the dual phase of upgrade, ie knowledge degrade. Upgrade refers to the process that local knowledge is selected to become global knowledge, while degrade is the opposite process. Knowledge migration will be fully discussed in Chapter 9.
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multiple revision strategies, which would be highly desirable in a heterogenous revision system. Although MABR and DBR’s communication mode is not restricted to broadcast as mutual belief revision does, the underlying heterogeneity issues which might inhibit efficient and reliable communication has not been addressed in any depth. Heterogeneities appear in various forms in multi-agent systems. For the purpose of research with regard to belief revision in a multi-agent environment, the heterogeneities pertain to the domain need to be discussed and dealt with.
5.3
Belief Revision in Multi-Agent Systems - Concepts and Terminologies
As we can see from above, a variety of notations have been adopted by researchers investigating belief revision of multi-agent systems. A good understanding of the relationships between these approaches is essential before carrying out any further research. To clarify the terminologies and associated concepts for belief revision of multi-agent systems, let us revisit the definition of “agent”. Generally, an agent implies a problem solving entity that both perceives and acts upon the environment in which it is situated, applying its individual knowledge, skills and other resources to accomplish high-level goals. By employing various algorithms and processes, agents are capable of taking actions to achieve their individual goals or interacting with other agents to achieve mutual goals. According to whether belief revision is involved in individual goals or mutual goals, previous research efforts in belief revision of multi-agent system can be classified into two categories, ie, Belief Revision using information from Multiple Sources(MSBR) and Multi-Agent Belief Revision (MABR). On one hand, belief revision could be considered as part of the agent’s skills to
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maintain the consistency of its own epistemic state. In this case, an individual belief revision process is carried out in a multi-agent environment, where the new information may come from multiple sources and may be in conflict. Belief revision in this sense is called MSBR by Dragoni et al.[18][22][19]. Cantwell[11] tries to resolve conflicting information by ordering the information sources on the basis of their trustworthiness. This could be viewed as a rational way of generating the new information credibility based on the source reliability using the terms of MSBR. Benferhat et al.[2] investigate revision of information from multiple sources in the face of uncertainty as data fusion, using possibilistic logic; Liberatore and Schaef[74] treat the MSBR process as intelligent merging of knowledge bases, which they call Arbitration. On the other hand, belief revision could also be used to achieve a society’s or team’s mutual belief goals(e.g. reaching consensus before carrying out a plan). In this setting, more than one agent takes part in the process. In order to pursue a mutual goal, the agents involved need to communicate, cooperate, coordinate and negotiate with one another. A MABR system is a MAS whose mutual goal achievement involves belief revision. As we stated before, it carries a two-fold meaning: Firstly, it can be seen as a multi-agent solution for complex belief revision problems, where multiple heterogenous belief bases are involved and the conclusion can not be drawn without a consistent overview or consensus among multiple agents. Secondly, it can also be seen as a behaviour that emerged as a result of the interaction among multiple belief revision agents. Since a multi-agent system is actually an intelligent distributed system, an alternative name for MABR could be intelligent Distributed Belief Revision(DBR). MABR
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is the terminology adopted by Kfir-dahav and Tennenholtz[68]. Dragoni et al. prefer DBR[21] based on the comparison with Distributed Truth Maintenance(DTM). Van der Meyden’s semantical theory of BR in synchronous MAS, namely, Mutual Belief Revision, also falls into the MABR category. MSBR studies individual agent revision behaviours, ie, when an agent receives information from multiple agents towards whom it has social opinions. MABR investigates the overall BR behaviour of agent teams or a society. MSBR is one of the essential components of MABR. As shown in the previous chapter, the AGM paradigm[40] has been widely accepted as a standard framework for belief revision. But its major focus is in prescribing revision behaviours of an ideal single agent. The belief revision process is more complex in a multiple agent scenario. Besides the Principle of Minimal Change, there exist other requisites due to the sophisticated agent interactions. Therefore, the AGM framework is not rich enough to prescribe a satisfactory revision operator for MABR. In the following chapters, we develop a general framework based on an ontology to capture the necessary heterogenous properties so as to accommodate the sophisticated agent interactions. As a result, belief revision can be thought of in a narrower sense, which encompasses all previous work in AGM. It can be considered in a wider sense, taking into account MASs, from the viewpoint of a single agent or a society of agents. An agent is capable of carrying out Individual Belief Revision(IBR), while an agent society or team is capable of MABR, an agent is capable of participating in MABR is referred to as an MABR agent. IBR in a single agent environment(Single Belief Revision, SBR) could be achieved using classical belief revision techniques satisfying AGM
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Belief Revision
Individual Belief Revision (IBR)
Belief Revision in Single Agent Environment (SBR)
Multi-Agent Belief Revision (MABR) (Intelligent Distributed Belief Revision)
Individual Belief Revision in MASs Multiple-Source Belief Revision (MSBR)
Figure 5.1: Belief Revision Hierarchy postulates. IBR in a multiple agent environment is MSBR, ie, a single agent will have to process information coming from more than one source. After obtaining the new credibility of the new information on evaluating the multiple sources using some techniques(e.g.[11][19]), MSBR degenerates to SBR. We can classify the types of BR using the hierarchy in Figure 5.1.
5.4
Heterogeneities in Multi-Agent Systems
Much of the conceptual power of the MAS paradigm arises from the flexibility and sophistication of the agent interactions and organisations. As the basic skill for both individual agents and agent societies to maintain consistency, the belief revision processes and results will heavily depend on the way agents communicate, cooperate, coordinate and negotiate with one another. Heterogeneity is one of the basic origins of such flexibility but it also gives rise to complexity. Taking the opportunities as well as solving the difficulties could lead to a more dynamic and versatile belief revision perspective. Therefore, it is necessary to identify various types of heterogeneities and study how they might affect belief revision behaviours in multi-agent systems. Heterogeneities in a multi-agent system may take many forms, ranging from the hardware and software platform that each agent is based on, to the organisation schema that relates individual agents socially to others forming teams and societies,
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to the basic knowledge representation structure and reasoning strategy that makes the agent intelligent, to the problem domain that an agent specialises in. We focus on the issues raised by heterogeneities in the knowledge systems, which includes the last two cases. Following is a brief classification according to the source (i.e. level) of these knowledge system heterogeneities. • Social Character Heterogeneity(social-level): Within a multi-agent system, an agent is socially situated in a particular environment with its existence known to other agents. As a problem solving entity, an agent is also defined as a software entity that acts on behalf of the user to accomplish certain tasks assigned by the user. Therefore, just as a human could behave in a benevolent or malicious way, there is no reason to forbid agents from possessing such characters. In the context of modelling trust[65], “free will” has been defined to describe the mental decision process of judging between benevolent and malicious behaviour. Agents possessing free will of this type are designated as passionate. However, an agent, such as algorithms, protocols, software and hardware which could hardly be characterised as having a free will are classified as rational. On the other hand, due to technical or other possible reasons (e.g. hardware quality), the agent could either be competent or incompetent. Combining these characteristics, agents could be roughly classified into four categories: competent rational, competent passionate, incompetent rational and incompetent passionate agent. • Semantic or Logical Heterogeneity(meta-level): Borrowing terms from cooperative information systems(CIS)[60], semantic heterogeneity results when different conceptualisations and different database schemas are used to represent
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the same or overlapping data which is replicated in two or more databases. A simple real world example could be the various grading techniques in an educational system such as percentage or letter grade[54]. A generalisation of such heterogeneity also occurs in the agent knowledge bases. For example, the knowledge could be represented using different logics, in favour of an agent’s problem solving capability with respect to certain problem domain. • Syntactic Heterogeneity(content-level): This is a domain specific heterogeneity which arises from the fact that in many cases the same letters or words are used to represent different concepts or objects by different agents and vice versa. This happens during the process of building knowledge into autonomous agents to enhance their intelligence. Natural language is commonly transformed into a simple logical form. Developers have the freedom to name things as they wish. For example, some agents may choose the letter “a” for describing an apple, while some other agents might prefer the whole word “apple” or something else. This is also a common phenomenon in all the emerging research areas, different terminologies have been used to describe the same concept, or vice versa. While a research area matures, general terms and frameworks will be proposed to serve as a specification. It can be seen from the previous study in MABR, researchers have progressively put more accent on the heterogeneity that exists in agent knowledge bases. In the early 1990’s, Fagin et al.[31] semantically defined mutual belief and common knowledge. Van der Meyden extended this modal logic approach into MABR in 1994. Malheiro[81] in the same year defined private and shared belief to model the belief revision process in a distributed truth maintenance system. Similarly, Kfir-dahav and
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Tennenholtz claimed in 1996 that their work[68] is more amenable to solve the heterogeneity problem than Van der Meyden’s by stating that agent knowledge could fall into private and shared domains. Recently, other researchers[78][102], following Fagin et al.’s approach, defined various concepts such as team knowledge and shared knowledge. Dragoni et al. also distinguishes global knowledge from local knowledge[20]. Actually, defining shared, common and private knowledge paved the way for modelling the social character heterogeneity. In other words, the diversity of different kinds of knowledge is the reflection of social heterogeneity in agent knowledge bases. Private knowledge is needed by passionate agents to keep confidential information. Such privacy enables the possibility of malicious behaviour which is sometimes needed for an agent to maximise its own or group utility in a competitive environment. On the other hand, common or shared knowledge is essential to establish cooperationoriented communication and commitment. For example, incompetent agents could carry out teamwork by sharing knowledge so as to achieve high-level goals which could not be accomplished by any individual. Understanding the similarities and distinctions among these conceptualisations of agent knowledge poses a serious challenge when trying to integrate systems based on different knowledge structures. Semantic and syntactic heterogeneity have not yet been studied in the context of MABR. The existence of a variety of belief revision strategies is an example of semantic heterogeneity that exists in MABR. For belief revision within the AGM paradigm, many revision schemes have emerged during the past decade as shown in the previous chapter, such as numerical revision using probabilistic[93] or possibilistic logic[24], sentence-based revision using various transmutation[108] and so on. Special needs for
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communication and inter-operability arise, because a variety of ranking mechanisms 5 might be used when employing various revision strategies. How do agents communicate with each other in terms of the information credibility? In other words, how does an agent incorporate new information from another ranking system? Communication and inter-operability difficulties are associated with syntactic heterogeneity too. How could the information reliability be guaranteed during the information passing process in a system with such low-level heterogeneity that same string means something different?
5
Ranging from ordinal natural numbers(ie {0, 1, 2, ..., ∞}) to unit interval (ie [0, 1]).
Chapter 6 Ontology - Tackling the Heterogeneity Issues “Ontologies are nothing but names with standard meanings. In a world of data exchange names are incredibly important, because you and I cannot exchange information about a thing unless we agree on the name for the thing.”
— R.V. Guha (Editor for W3C’s standard for RDF)
6.1
Introduction
It is widely accepted that an ontology is an efficient approach that can be used to tackle heterogenous issues involved in the generalisation of low-level heterogenous data to relatively high-level concepts for the purposes of communication, system inter-operation and software reuse. Recent years have seen a promising surge in both research and practice of ontologies. Nowadays, ontology is no longer just a concept of the Artificial Intelligence laboratories where researchers recognise its importance, and hope other people - such as domain experts or knowledge engineers - will take
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up the tedious work of creating ready-to-use ontologies. In fact, more and more realworld implementations of ontologies, especially, ontologies for the World Wide Web have proved successful and are now heavily relied upon by e-commerce and e-business applications. Ontologies on the Web range from large taxonomies (subclasses-superclasses) categorising web sites (such as Yahoo! and Google’s Web Directories) to categorisations of products for sale and their features (such as on Amazon.com). These are the socalled first generation ontologies, which ground today’s machine to human web. In this machine to human web, the machine outputs search results, humans then have to interpret and filter the often large retrieved repository for relevant information. The next generation ontologies, will facilitate the revolution of the world wide web into Berners-Lee’s vision of semantic web[7]1 , which makes a huge amount of information available in machine-readable format. This will not only free human-beings from the information overflow but also enable the automatic online B2B trading. With the machine-readable information, large numbers of intelligent agents will be employed to do the knowledge gathering, reasoning, economic optimising or even bidding on human’s behalf. At the time of research, there are several large, general ontologies that are freely available, some examples are Cyc2 , WordNet 3 and World Fact Book4 . 1
The semantic web will enable machines to comprehend semantic documents and data, not human speech and writing. 2 http://www.cyc.com/cyc-2-1/cover.html 3 http://www.cogsci.princeton.edu/ wn 4 http://www-ksl-svc.stanford.edu:5915/doc/wfb
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6.2
Classifying Ontologies
Uschold [104] pointed out, ontologies may vary along three dimensions, namely, purpose, formality and the subject matter. In general, an ontology defines a common vocabulary for parties who need to share information in a domain. The information sharing may occur among people or among software agents and facilitating communication between people or between organisations, and inter-operation between heterogenous systems. The ontology used for such purposes serves as a translator between systems that “speak” different languages. For example, a human translator translating between English and Chinese has to understand both languages, and has to know the mapping between the two languages. Essentially, the two languages provide two different sets of vocabulary that describe the same perception of the world, or the domain of discourse. This is one of the dominant usages of ontology. Other possible usages [88, 104] are mainly for system engineering purposes, such as analysing the domain knowledge, making the domain assumptions explicit, separating the domain knowledge from the operational knowledge and thus enabling the reuse of domain knowledge. If categorised by its subject matter, a Domain ontology is for special subjects such as medicine, finance and etc. Upper model or top-level ontology[51] is general world knowledge, such as the WorldFact Book we mentioned in the Introduction Section 6.1. A task, method or problem solving ontology is for the subject of problem solving. For example, the AGM postulates prescribe a method ontology that specifies a set of revision functions. A representation ontology is for the subject of a knowledge representation language. For example, ontology languages such as CycL, KIF, DAML+OIL, XML, and RDF etc are themselves representation ontologies for
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representing knowledge and capturing relations among the defined concepts. Ontologies can also be classified according to their formality. An ontology is highly informal if expressed loosely in natural language; structured informal if in a restricted and structured form of natural language; Semi-formal if in an artificial formally defined language such as Ontolingua and Knowledge Interchange Format(KIF)(links in [47]) and etc; rigorously formal if in meticulously defined terms with formal semantics, theorems and proofs of such properties as soundness and completeness. Furthermore, the semi-formal and rigorously formal ontology languages can be categorised based on the underpinning logical paradigm. For example, the syntax of both CycL and KIF are derived from First-Order Predicate Logics. They both provide an important extension which allows reification of formulas as terms used in other formulas. Therefore, CycL and KIF allow meta-level statements. Ontolingua and Frame Logic integrate frames concepts (i.e. classes) into predicate logic. Their central modelling primitives are classes with certain properties (i.e. attributes), as compared to KIF and CycL, whose central primitives are predicates. On the other hand, recently, there is much interest in ontology languages based on Description Logics[9], such as CLASSIC and DAML. A distinguishing feature of Description Logics is that classes (usually called concepts) can be defined intentionally using descriptions. The descriptions are used to specify the properties that objects must satisfy in order to belong to the concept. Moreover, the language used to express the descriptions allows the construction of composite descriptions. Such features makes description logic based ontology languages good candidates for web service descriptions. For our purposes, as will be shown in the later sections, a first-order logic based ontology that is able to support identification of concepts, attributes and describing
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relations between concepts would suffice. In summary, ontologies are important for both human beings and software agents who are involved in knowledge sharing. Actually we may not be aware of an important fact that, we, human beings are using ontologies in every aspect of our life and developing ontologies since the ancient civilisation began. Our everyday languages can be seen as a form of informal ontology. It is a matter of fact that it is scientific discovery that builds and continuously adds to our general world knowledge, i.e. our informal top ontology. It is often the case that ontologies that we use are not in formal languages. This is due to the fact that as intelligent human beings, we understand natural languages and are able to reasoning using them. However, the existence of explicit machine interpretable ontologies is crucial to facilitate communication between heterogenous agents, as well as a means of accessing and referring to them (such as an access protocol or a naming space)[29]. In an open environment, agents are developed by different programmers and organisations, and designed around various ontologies, either explicit or implicit. Assuming agents interact with an open system sharing intrinsically the same ontology is therefore not realistic. This is also one of the main driving forces for current research activities in the Semantic Web[7] and service description languages[84]. An ontology based communication envisaged by FIPA is illustrated in Figure 6.1[29]. In a more general and ambitious setting, FIPA’s ontology service specification XC00086D[29], suggests that one or more Ontology Agents should be implemented in an open environment to facilitate the following:
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Ontology
Ontology Query
Ontology Query
ACL Communication = Agent A
Agent B Ontology-Based Communication
Figure 6.1: Ontology-Based Communication Model • querying the definition of terms • selecting the shared ontology • checking the equivalence of two ontologies • locate a particular ontology • translating a term
6.3
Representing Ontologies
Now that we have discussed what an ontology can do, along with the dimensions on which it may differ, we can now have a closer look at ontologies themselves. What is an ontology about? What should it look like and what should actually appear in the representation of an ontology? According to Guarino[49], philosophically, a piece of reality is often captured by a set of informal rules that constrain its structure, which is referred to as a conceptualisation. It involves the underlying model of the domain in terms of objects, attributes and relations. A conceptualisation may be implicit, e.g. exist only in someone’s mind, or embodied in a piece of software. An explicit account or representation of some part
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of a conceptualisation[48] is usually called an ontology. In other words, ontologies are explicit specifications of the terms in the domain and relations among them[46]. In addition to conceptualisation, there are two other important features of an explicit ontology: • Vocabulary: this involves assigning symbols or terms to refer to those objects, attributes and relations. • Axiomatisation: this involves encoding rules and constrains which capture significant aspects of the domain model. In Figure 6.2, we illustrate that two ontologies may be based on different conceptualisations (Person Ontology 1 v.s. 3), they may be based on the same conceptualisation but use different vocabularies (Person Ontology 1 v.s. 2), they may also be differ in how much they attempt to axiomatise in the ontologies (Person Ontology 2 v.s. 4). In other words, ontolgies are not necessarily absolute but could be context sensitive. It is important to note that the trees in Figure 6.2 only describe one essential relation between concepts, namely, the “IsA” relation. Other relations do not appear in Figure 6.2, for example, FatherOf(man, children). Essentially, an ontology consists of a vocabulary of non-logical symbols, definitions of symbols (maybe empty for the primitive symbols) and use the logical formulation for non-primitives. Axioms constraining the interpretation of primitive symbols (which maybe empty or outside the logical formulation). In Ontolingua and Frame Logic[32], for the purpose of creating an ontology[88], the central frame-based modelling primitives needed are defined and realised. In Ontolingua, an ontology is defined as a formal explicit description of concepts in a
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Person
Person
Male
Female
Man
Woman
Father
Mother
Dad
Mum
(1) Person Ontology 1
(2) Person Ontology 2
Person
Anglo-Saxon
Asian
Latino
...
(3) Person Ontology 3
Person
Man
Dad
Woman
Grandpa
Mum
(4) Person Ontology 4
Figure 6.2: Person Ontology
Grandma
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domain of discourse (class(sometimes called concepts)), properties of each concept describing various features and attributes of the concept (slots (sometimes called roles or properties), and restrictions on slots (facets (sometimes called role restrictions)). An ontology together with a set of individual instances of classes constitute a knowledge base. Take a person ontology for example, described below: • Concept PERSON has two attributes name and address. This implies the domains of name and address are the elements of PERSON. The range of name and address are concepts of NAME and ADDRESS. • Concepts MAN and WOMAN are defined as subclasses of PERSON. This implies that all attributes defined for PERSON are also applicable for MAN and WOMAN. • Concept FATHER is a subclass of MAN that implements a FatherOf relation. The domain of a FatherOf relation is Father, the range of the relation is the elements of CHILDREN. • Similarly, concept MOTHER is a subclass of WOMAN that implements a MotherOf relation. The domain of a MotherOf relation is MOTHER, the range of the elements of CHILDREN. • Concept CHILDREN is a class that has attribute children. The range of children a non-empty set of the elements of PERSON. The above ontology can be represented using the syntax of Ontolingua as: //Concepts class PERSON(?NAME, ?ADDRESS)\\
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class class class class class class class
NAME(?class)\\ ADDRESS(?class)\\ MAN(?class)\\ WOMAN(?class)\\ FATHER(?class)\\ MOTHER(?class)\\ CHILDREN(?class)\\
//attributes are instances of corresponding concepts relation instance-of(?name, ?NAME)\\ relation instance-of{?address, ?ADDRESS)\\ //MAN, WOMAN are subclasses of PERSON relation subclass-of(?MAN, ?PERSON)\\ relation subclass-of(?WOMAN,?PERSON)\\ //FATHER, MOTHER are subclasses of MAN and WOMAN, respectively relation subclass-of(?FATHER, ?MAN)\\ relation subclass-of(?MOTHER, ?WOMAN)\\ //Attributes are inherited relation domain(?name, ?PERSON)\\ relation domain(?address, ?PERSON)\\ relation domain(?name, ?MAN)\\ relation domain(?address, ?MAN)\\ relation domain(?name, ?WOMAN)\\ relation domain(?address, ?WOMAN)\\ //FatherOf and MotherOf are one-to-many-realtions relation one-to-many-relation(?FatherOf)\\ relation one-to-many-relation(?MotherOf)\\ relation FatherOf(?FATHER, ?CHILDREN)\\ relation MotherOf(?MOTHER, ?CHILDREN)\\ //The value of children instances in the FatherOf //relation is of type PERSON relation instance-of(?children ?PERSON)\\ relation value-type(?children, ?FatherOf, ?PERSON)\\ relation value-type(?children, ?MotherOf, ?PERSON)\\ //Possible Functions: function value-cardinality(?children,?FatherOf):-> ?n\\ function value-cardinality(?children,?MotherOf):-> ?n\\
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This person ontology can be further implemented in JAVA to achieve agent communication at an ontological level. For the purpose of demonstrating how an ontology is implemented in a programming language, part of the ontology code adapted from JADE2.4 user guide[30] is shown below: public class PeopleOntology extends FullOntology { // The name of this ontology. public static final String ONTOLOGY_NAME = "PEOPLE_ONTOLOGY"; // Concepts, i.e., objects public static final String public static final String public static final String
of the world. PERSON = "PERSON";\\ MAN = "MAN";\\ WOMAN = "WOMAN";\\
// Slots of concepts, i.e., attributes of objects. public static final String NAME = "NAME";\\ public static final String ADDRESS = "ADDRESS";\\ // Predicates public static final String FATHER_OF = "FATHER_OF"; \\ public static final String MOTHER_OF = "MOTHER_OF";\\ // Roles in predicates, i.e., names public static final String FATHER = public static final String MOTHER = public static final String CHILDREN
of arguments for predicates "FATHER";\\ "MOTHER"; = "CHILDREN";\\
private static PeopleOntology theInstance = new PeopleOntology(); public static PeopleOntology getInstance() { return theInstance; } public PeopleOntology(Ontology base) { super(ONTOLOGY_NAME, ACLOntology.getInstance()); // Add definitions of schemata here. ...... } }
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6.4
Ontologies for Multi-Agent Belief Revision
Three levels of heterogeneities were identified in Section 5.4, namely, social character heterogeneity at the social level, semantic heterogeneity at the meta level and syntactical at the content level. While most ontology languages serve at the content level, the syntactic heterogeneity is highly application dependent and requires knowledge acquisition in the domain (e.g. Internet resource allocation, travel industry). The domain specific ontology can be implemented using the syntax available in JADE ontology support. The JADE ontology base class diagram is shown in Appendix C. The following two sections will not discuss the syntactic heterogeneities as they are domain dependent. However, we discuss the social heterogeneities and semantic heterogeneities pertaining to belief revision frameworks for multiple agents.
6.4.1
Tackling the Social Character Heterogeneity
We define a heterogenous society as a multi-agent system that is populated with both rational and passionate agents, who might be competent or incompetent in a certain domain. To enable the inter-operability within and between heterogenous societies, ontologies are needed for the purpose of supporting communication by translating between different modelling methods and paradigms for instance. The social heterogeneities presented in multi-agent belief revision frameworks are typically reflected by the modelling method of shared belief and the modelling method of trust. As shown in the discussion in Chapter 5, existing frameworks have different models to represent shared belief and private belief. To complicate the matter even further,
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there is no consensus as to how private belief can be protected from shared belief. In other words, the knowledge structures are heterogenous. Therefore, to have agents built around different knowledge structures, an ontological description of a knowledge structure is needed. The knowledge structure has to be general enough to represent each structure considered and sophisticated enough to separate shared belief from private belief. According to the classification dimensions introduced in the previous sections, here we need a computer-executable semi-formal ontology that addresses heterogeneities in the domain of structuring knowledge. Another type of heterogeneity involves the way of modelling trust. In a heterogenous society, agents can range from trustworthy or untrustworthy. Due to the fact that agents are built by different vendors and represent different users’ interest, we can not assume agents are all competent and always sincere. Agents can be unfaithful in order to maximise their utility. Agents may also be incompetent due to programming defects. Therefore, it is crucial to understand how trust can be interpreted in a multi-agent belief revision domain, how it affects the sharing of information and how an agent can survive and build up their knowledge about others in an untrustworthy environment. An ontological description of the trust issue on information sources (as information passing and sharing are the main activities concerned in belief revision involving multiple agents) are needed to describe the model of domain belief and the necessary format of social belief. Such an ontology will enable the representation of an agent’s insincere behaviour as an information source. For example, delivering information that it does not believe in. The ontology also facilitates reasoning about other agents’ competency and thus minimises the possibility of being defrauded by
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untrustworthy agents.
6.4.2
Tackling the Semantic Heterogeneity - Translating among various ranking systems
In general, there are two major ranking systems employed by the major belief revision techniques discussed in chapters 3 and 4. Namely, the use of ordinals and the use of the unit interval [0,1]. These two ranking systems are not equally expressive. In most cases, the unit interval is considered richer than the ordinals. Therefore, information may be lost when one translates ranks from a rich ranking system like [0,1] to an ordinal one. Benferhat et al.[5] shows that it is possible to translate between Spohn’s kuppa function (ordinals) to the necessity ordering of possibility theory (unit interval), namely,
µ = e−σ
(6.1)
where 0 < µ < 1 and σ is an ordinal. This equation works for translating between infinite orderings. In the case of translating between two finite orderings, one possible transformation from [b1 , b2 ] to [0,1] can be done by a conditioning function, which maps the highest ordinal to 1 and the lowest ordinal to 0.
µ=
o − b1 b2 − b 1
(6.2)
Numerically, it is realisable to translate between a unit interval to a ordinal. However, it is not always feasible to translate between two ranking systems or even within one ranking system that based on different revision scheme.
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In fact, even if the same ranking mechanism is adopted, the ordering can take on different meanings. For example, in AGM’s epistemic entrenchment settings, the ordering describes the underlying importance of each sentence in face of change. In other words, the relative importance may need to be altered when the epistemic input is different. Furthermore, in Williams transmutations, the ordinal used for partial entrenchment ranking can reflect the ordering of preference or the different level of necessities or capturing the degree of suprise[109]. Similarly, the unit interval can take on meanings such as the degree of belief(Theory of Evidence), statistical probabilities (Bayesian Theory) and possibilities (Possibility Theory) in different mathematical revision paradigms. Determining the appropriate translation may require substantial communication between agents to clarify the intended meaning of their ranking and the underlying revision scheme. According to the classification criteria above, on the semantic heterogeneity of ranking systems, a problem solving ontology is needed to transform one ranking system to another.
6.5
Discussion
Ontologies are not panacea, they can be both constructive and destructive. Information distortion could occur during the generalisation process of ontology design. Many criteria exist to prescribe a good ontology[46] and more are still under investigation. Essentially, the key is to thoroughly understand the problem domain. To design an ontological knowledge structure and model trust, both philosophical and psychological investigations into the nature of shared/common, private knowledge
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and trust issues are necessary. Trust issues are investigated in chapters 7 and 8. Basic components of evaluating trustworthiness of information sources are identified, together with the way of representing malicious and incompetent behaviour of information sources. This not only provides the essential data needed for fusion and conflict resolution but also prepares the ontological components of an agent’s social beliefs and domain beliefs as shown in Section 10.4.1. In Chapter 9, a general knowledge structure, which encompasses private, shared and accessible knowledge, is proposed. In the knowledge migration phase, a proper revision operator on a shared domain poses a great challenge due to the fact that belief revision in this case is a mixed process composed of traditional belief revision, communication and other interactions. Sophisticated decision making techniques are required of such a belief revision operator, because the revision process could branch according to communication mode and social intention. Therefore in certain circumstances, the revision result might not be easily determined.
Chapter 7 Trust Evaluation “.... trust is a social good to be protected just as much as the air we breathe or the water we drink. When it is damaged, the community as a whole suffers; and when it is destroyed, societies falter and collapse.” — Sissela Bok (1978) in [8]
7.1
Introduction
A dominant approach to trust evaluation in the agent literature stems from viewing agents as software modelled on humans. In this approach agents are ascribed mental states (beliefs, desires etc), have personalities (benevolent, malicious etc) and are able to make decisions and act autonomously (i.e. without human intervention). The most promising attribute of agents is that, in addition to the ability to stand alone, make decisions and perform on their own as individual entities, they can interact with each other, establish (cooperative or competitive) relationships, thus forming so-called agent societies. Some researchers go even further by exploring the idea of a digital city1 , which hosts citizen agents under the governance of police agents and 1
Digital City: http://www.digitalcity.gr.jp/virtualcommunity/tourguide.html
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military agents, and so forth. In such virtual agent societies, analogous to physical human societies, it is necessary to consider the possibilities of fraud, being cheated, and being deceived during information exchanges. This raises the issue and fuels the growing interest in trust. The interest in a formal theory of trust for the purpose of agent research can be traced back to the early 90’s[62]. Furthermore, there has been a resurgence in recent years inspired by The First International Workshop on Deception, Fraud and Trust in Agent Societies in 1998. Despite the broad use of the term “trust” in network security, cryptography and e-commerce, it is rarely generally defined in the agent literature. One reason for this is that the notion itself is highly context-sensitive and can be given a variety of readings; one can define it for specific applications but not easily for the general case. Elofson[26] attempted to give a composite definition of trust after studying several perspectives on the meanings of trust: Definition 7.1.1. Trust is the outcome of observations leading to the belief that the actions of another may be relied upon, without explicit guarantee, to achieve a goal in a risky situation. From our vantage point, this definition is a good definition because it emphasises the belief aspects of trust, the risk involved in trust and the close relationship between trust and delegation. However, there is an important element missing, one that we are particularly interested in, that is the “reliance” on the information an agent conveyed2 . It is important to clarify the phrase trust of an information source. This is different 2
On the other hand, communication can be thought of as a special form of action performed by the trustee agent, which is perceived by the trustor agent. One could argue Elofson definition covers the special case of defining trust on information sources.
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from trust of another agent’s capability and responsibility for carrying out certain delegated tasks. It is about the credibility of the information delivered by the source agent. For example, consider the following questions: 1. Shall I trust the informant, is it possible that he is lying to me? 2. Is he competent in the subject matter, how can I know that he is not telling me something stupid? Thus, an agent’s trustworthiness as an information source is evaluated using the credibility of the information conveyed. In this and the following chapter, we focus on how an agent can obtain and evaluate the trustworthiness of an information source. Two essential constituents belief are identified in Section 7.2, namely, competency and sincerity. In Section 7.3, we propose the corresponding evaluation procedures. By evaluating sincerity and competency separately, and in addition to the information’s pedigree, we are able to analyse and model the process of passing-on information in Section 8.1. Using an example, Section 8.2 demonstrates an application of evaluating trustworthiness during the process of resolving conflicting information. The work presented in these two chapters is incorporated into a social belief revision agent in Chapter 10 which is capable of revising both its social knowledge base and its domain knowledge base (i.e. revising information from multiple sources). In addition, the belief revision agent is capable of interacting with other agents to carry out multi-agent belief revision.
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7.2
Essential Constituent Beliefs for Trusting Information Sources
Castelfranchi and Falcon[13] pointed out that the degree of trust is a function of the subjective certainty of the pertinent beliefs, namely, the cognitive constituents that are essential to establish trust with another agent. These include competence belief, disposition belief, dependence belief, fulfillment belief, willingness belief, persistence belief and self-confidence belief. The component beliefs are identified and evaluated to help the decision making in delegating certain tasks to a trustee, not for the purpose of determining the reliability of a trustee as an information source. However, it does shed light on how to decompose the problem of evaluating trustworthiness into tractable components. Taking a similar approach, in this section, we identify the essential constituent beliefs for trusting an information source. The communicative act considered here for exchanging information is most appropriately described as an inform act, using performatives defined in FIPA ACL3 . Provided that the communication is successful, the result of the inform act is that the receiver successfully receives the information the sender intended to send it to. In other words, when an agent receives some information from another agent, the action of informing has already been performed by the trustee (i.e. the sender). Therefore, the evaluation on disposition, willingness, fulfillment and persistence belief is no longer necessary, as these are beliefs regarding the extent to which the delegated action (i.e. informing in this case) is to be carried out. This is the major difference between trust on delegation and trust of information sources, when trust of information sources is considered as a special case of trust on delegation. What remains is 3
FIPA: Foundation of Intelligent Physical Agent: http://www.fipa.org, ACL: Agent Communication Language.
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the competence belief and the dependence belief, where dependence belief is closely related to sincerity belief defined in the list below. According to Demolombe[15], a sender agent’s trustworthiness on delivering correct information can be analysed on the basis of four elementary properties: 1. Sincerity: agent A is sincere with respect to agent B when he believes what he says to B. 2. Credibility: agent A is credible when what he believes is true in the world. 3. Co-operativity: agent A is co-operative with respect to agent B if he tells B what he believes. 4. Vigilance: agent A is vigilant with respect to p if, whenever p is true in the world, then A believes p. Not surprisingly, philosopher John Hardwig[56] pointed out that the reliability of a receiver’s belief depends on the reliability of the sender’s character, both his moral character and his epistemic character. The moral character is mainly the agent’s truthfulness, while the epistemic character involves his level of competence, conscientiousness and epistemic self-assessment. It is not hard to see the two essential pertinent beliefs that both Demolombe and Hardwig agree on here are: the sincerity and the competency. Furthermore, the discussion earlier in this section using Castelfranchi and Falcon’s approach confirms the same decomposition. Observation: The sincerity and the competency beliefs are the necessary constituents in the evaluation of a social cognitive agent’s trustworthiness as an
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information source. Although these two are not the only ones required, they are sufficient in the sense that most other possible constituents fall into these two categories. What about the cooperativity and vigilance proposed by Demolombe[15]? Should they be in the set of the component beliefs for trusting an information source? As a necessary condition for an agent to be credible, vigilance falls into the broader category of competency. Cooperation and trust have a strong reciprocal relationship with each other. That is, a trusting climate encourages cooperation, and a positive cooperative outcome in turn enhances the level of trust for all parties involved. Marsh[83] claims that cooperation can be classified according to the incentive as cooperation towards mutual rewards, cooperation based on communal relationships and cooperation that emerges from coordination. No matter what incentive makes an agent cooperative or not, the result of cooperativity in information exchange is a matter of whether the receiver (trustor) obtained some information from the sender (trustee) or not. Generally, agents who are not cooperative will either provide no information or wrong/misleading information. Whereas, agents who are cooperative will provide information truthfully. As a consequence cooperativity analysis for information exchange can be considered as an important factor that may affect the degree of sincerity. It then becomes part of the sincerity analysis but not necessarily an independent component belief in its own right. Assumption: By accepting the separation of trustworthiness into competence and truthfulness beliefs, from now on, we can assume the trustworthiness of information sources are scalable and comparable.
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non-sense agents honest trustworthy dishonest ignoramus agents experts
Figure 7.1: Four Types of Agents in terms of Sincerity and Competency Therefore, by interpreting trustworthiness in terms of agents’ competency and sincerity, we are able to model the following types of agents, which are depicted in Figure 7.1: • G1 : Trustworthy Agents (agents who are both truthful and credible) • G2 : Honest Ignoramus (agents who are honest but not credible) • G3 : Dishonest Expert (agents who are credible and not honest) • G4 : Nonsense Agents (agents who are neither truthful nor credible)
7.3
Trustworthiness and Degree of Beliefs
In the fields of data fusion[3], weighted knowledge base merging[76], multiple source conflict resolution [19] and multi-agent belief revision[68], the value of trustworthiness is always assumed to be given. In this section, based on the observation in Section 7.2 above, we present two ways of using trustworthiness. One is that the degree of trustworthiness t is used as a single value, the other is the degree of sincerity (s) and the degree of competency (c) used separately with the degree of beliefs δ.
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7.3.1
Degree of Trustworthiness
Given the value of an information source’s competency and sincerity, the total trust on an information source T can be computed using an evaluation function τ (s, c) : S × C → T , which maps a degree of sincerity s ∈ S and a degree of competency c ∈ C to the trustworthiness of an information source t ∈ T . s, c, t ∈ [0, 1], with 1 the most truthful, credible and trustworthy respectively, 0 the least. The greater the value of t, the more trust an agent has in an information source. 1, 0,
τ (s, c) = c,
s,
if s = c = 1 if s = c = 0 if s = 1 if c = 1
s ⊕ c, otherwise. and ∀y∃z, tx (y) ≤ tx (z) or tx (z) ≥ tx (y)
Subscripts and variables (e.g. x, y, z...) are used to indicate agents, i.e., trustee and trustor. For example, tx (y) is agent x’s faith in agent y as an information source. Similarly, we can have sx (y) and cx (y) for agent x’s evaluation of agent y’s sincerity and competency respectively. Three simple operators that can be used for ⊕ are: • τmulitply = s × c • τmin = min(s, c) • τssf = 1 − (1 − s)(1 − c) = s + c(1 − s) = c + s(1 − c)
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The multiplication τmulitply and the minimum operator τmin are straightforward, and need little explanation. The third function τssf is the so-called Bernoulli’s rule of combination[98], which is a special case of Dempster-shafer’s rule of combination on Simple Support Functions(SSF), where s and c are viewed as single support functions on an agent’s sincerity and competency respectively[98]. It can be easily shown that: τmultiply ≤ τmin ≤ τssf In our model an agent can choose different operators for different situations, or an agent can be designed with different characteristics in terms of how it evaluates others. Multiplication τmultiply tends to be cautious but pessimistic, Bernoulli’s combination shows optimistic behaviour, while minimum operator is mild and represents a compromise between the two.
7.3.2
Degree of Deception and Degree of Sincerity
Given two agents, x and y, sx (y) denotes x’s evaluation of y’s general 4 sincerity on information that he delivered, and 0 ≤ sx (y) ≤ 1. The intended interpretation of this value is to what extent y tells exactly what he believes in his own knowledge base. For example, y believes p at a degree δ, but he may decide to tell x that he believes p at a 0
degree δ . To allow agents to communicate both the content and the degree of beliefs, we assume the message takes the form: (p, δyx ), where p is the information, δyx is the degree of belief delivered along with the information p, whilst the real degree held by agent y is δy . To minimise the negative effect of being deceived, agent x’s task is 4
We use general here to indicate that the degree of sincerity sx (y) is not content sensitive. The content sensitive value will be represented using sx (y, O) as show in Section 7.3.3.
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to guess a value as close as possible to δy before making any changes that might be triggered by the information from y. Using this general message format, we are able to model different levels of anticipated deception. The degree of belief discussed in this paper is considered as a Bayesian’s probability measure, where δ(p) = 1 means that p is accepted, δ(p) = 0 means p is rejected or ¬p is accepted and undetermined if 0 < p < 1[40]. Particularly, δ(p) = δ(¬p) =
1 2
is considered as the way of expressing complete ignorance5 .
Therefore, in the extreme case, an agent can tell the opposite of what he believes (i.e. δy = 0 but δyx = 1 and vice versa) or tell something he is completely ignorant about (δy (p) = δy (¬p) = 12 , but 0 < δyx < 1). It is true in the real world, sometimes, truthful messages might be unbelievable, and sending them could be irrational. On the other hand, some lies might be believable, or at least could sway the receiver to reconsider or change its actions in favour of the sender. As 0 ≤ δ ≤ 1, there are boundaries as to what extent an agent can lie, and the degree of deception(γ) is thus defined:
γ=
|δyx − δy | max(δy , 1 − δy )
The sender agent’s decision about the value of γ is affected by various factors, but mainly (and intuitively) by two: the utility and the believability. In other words, there is a optimal value for γ which can maximise his utility and also sound true but not absurd. As this is not our major concern, we are not going to pursue this further here. If interested, please refer to Gmytrasiewicz’s[44] study on why and when it is 5
Bayesian theory is often criticised for its inappropriate representation of ignorance, but our definition on the degree of deception can be easily modified to fit into other definitions of degree of beliefs such as the one used in Dempster-Shafer’s belief functions.
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feasible for an agent to lie to others. The receiver agent x, on receiving a information couple (p, δyx ), will have to guess the value of δy and use the obtained δy0 along with the competency belief (c) for his further decision making. Assume agent x himself uses the above mechanism for γ to defraud others, then he is likely to make the assumption that y is using similar mechanisms for himself[25]. A reasonable guess can be achieved if we assume, degree of deception ≡ 1 - degree of sincerity (i.e. γ ≡ 1 − s) Substituting γ with 1−s in the above definition sentence for γ, which is the subjective guess of γ, we have δy0 =
δyx 1 and δy0 ≥ where s 6= 0 1 ± (1 − s) 2
(7.1)
δyx ± (1 − s) 1 and δy0 < where s 6= 0 1 ± (1 − s) 2
(7.2)
OR δy0 =
In the case where s = 1, δy0 = δy , which means x believes y is completely truthful in the information that y delivered. In the case of s = 0, agent x can just ignore the information and not make any changes to his knowledge base. The “cleaned” degree of belief δy0 can then be used together with the degree of competency c to obtain the degree of acceptance 6 δxy . To obtain this value, the combination operators proposed in Section 7.3.1 can be used. This will provide reliable input to belief change and knowledge base integration etc, which is illustrated by an example in Section 8.2. 6
Degree of acceptance is the degree of belief at which an receiver agent will incorporate the information into his own knowledge base.
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7.3.3
The Evaluation of Competency and Sincerity
Sections 7.3.1 and 7.3.2 describe two ways of using degrees of competency (C) and degrees of sincerity (S) provided an agent already has the values c and s. But how and where does an agent get these values? In the real world, we often have some general feelings about others’ trustworthiness, such as x generally trusts y to a degree, which is denoted by tx (y) in Section 7.3.1. But quite often, trust on a certain action is specific to task(s). Similarly, an information agent’s trustworthiness specifically depends on the domain area of the incoming information. For example, we might trust our father’s opinion on how to get our TV fixed if he is an an electronic engineer, but would not trust his opinion on the direction of our AI research. Therefore, a general static trust value is not sufficient and we need a context sensitive measure of trust. So we introduce a topic specific trust evaluation towards others in both competency and sincerity. Thus, we augment the notation with one more variable, the topic O. We denote cx (y, O) as x’s evaluation of y’s competency with respect to topic O. Similarly we have sx (y, O). According to whether there is any previous experience with a trustee, Marsh [83] suggests three possible situations that an agent might encounter in assessing other’s trustworthiness. They are adapted to suit our needs and listed below. 1. The agent is not known, in this or similar topics. 2. The agent is known, but not in this or similar topics. 3. The agent is known and trusted in this or similar topics. Inspired by Jonker and Trenur’s trust dynamics[64], the above three situations can be handled respectively as shown below:
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1. In the first case, without previous trust influencing experiences but with a given competency or sincerity threshold, cδ or sδ , the information source can be treated as either initially competitive cx (y, O) ≥ cδ (faithful sx (y, O) ≥ sδ ) or initially uncompetitive cx (y, O) < cδ (unfaithful sx (y, O) < sδ ). Practical strategies can be added to enable agents to choose different initial values in different situations. 2. If the agent is known in some other topics, the non-relevant previous experience can be used as a coefficient (0 ≤ α ≤ 1) to adjust the initial competency (sincerity) as assigned in the first case. 3. For the third case, the previous experiences can be used to calculate the current competency (sincerity) using a competency (sincerity) evaluation or update function. Such functions can be defined to suit various applications by following the properties and constraints proposed by Jonker and Trenur[64]. Although such functions are applicable for both competency and sincerity evaluation, one may find that an agent’s competency value is unlikely to change as dramatically as the sincerity value does. Therefore, it is reasonable to expect a slow dynamic in competency evaluation/update functions as compared to the ones for sincerity. The above procedures are applicable for information that contains a single topic. For information that contains more than one topic, we will have to obtain the competency value and the sincerity value for each individual topic first and then combine them using the same set of combination operators (i.e. τmultiply , τmin and τssf ) proposed for trustworthiness evaluation in Section 7.3.1.
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Using formal methods to encode the incoming information, we can calculate the topics for it[82]. Topics of a sentence p are defined as the minimum set of atoms occurring in p. For example, given φ = a ∨ b, φ0 = a ∧ b, ψ = (a → b), and ψ 0 = (a → ¬b), using uppercase letters A and B to denote the topic for proposition a and b respectively, we have
topic(φ) = topic(φ0 ) = topic(ψ) = topic(ψ 0 ) = {A, B} Consider the following example: Agent y told agent x that e-commerce does not necessarily involve computer science (i.e. a → ¬b). There are two topics: e-commerce (A) and computer science (B) involved in the information delivered by y. Given the values of cx (y, a) and cx (y, b), to calculate the combined competency of agent y on this information, we have
cx (y, {A, B}) = cx (y, A) ⊕ cx (y, B) The above function can easily be extended to calculate the combined competency (sincerity) for more than two topics.
Chapter 8 Information Pedigree “The foundation theory holds that some of one’s beliefs ‘depend on’ others for their current justification; these other beliefs may depend on still others, until one gets to foundational beliefs that do not depend on any further beliefs for their justification. ... On the other hand, according to the coherence theory, it is not true that one’s ongoing beliefs have or ought to have the sort of justificational structure required by the foundations theory. In this view ongoing beliefs do not usually require any justification. Justification is taken to be required only if one has a special reason to doubt a particular belief. Such a reason may consist in a conflicting belief or in the observation that one’s beliefs could be made more “coherent”, that is, more organised or simpler or less ad hoc, if the given belief were abandoned (and perhaps if certain other changes were made).”
— G. Harman (1986) in [57]
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8.1
Evaluating Trust on Passing-On Information
It is known that a large majority of knowledge and beliefs come from indirect experiences, namely, communication. In the real world, communication is not necessarily restricted to conversation with other agents. It also includes emails, publications, books, newspapers and TV channels and more recently the Web. We do not distinguish these special information channels from agents as information sources. Instead, they are considered as special types of information sources and an agent must evaluate their trustworthiness. Furthermore, the communicated information itself could come from the sender’s indirect experiences as well. In such cases, the sender serves as a mediator who passes on information from a source agent to a receiver agent. Literature related to the trust of information sources often only consider the reciprocal trust attitudes among agents. There are few studies on the trust of passing-on information. Even in the real world, people tend to ignore the importance of passingon information, which may result in the receiver’s inability to diagnose the sources of conflict. Consequently, this will affect the receiver’s evaluation of other agents, and as a result the quality of their acquired information. For example, in a Robot Soccer scenario1 , the perception of a robot is limited, the robot who is far away from the ball has to rely on communication to obtain the estimated position of the ball. Suppose Robot-1 is close enough to perceive the ball position, he sends information about the position of the ball to the closest robot, say Robot-2, and Robot-2 passes this information onto Robot-3. Typically, Robot-3 will consider the received information as Robot-2’s perception because the information 1
Thanks to James Brusey at RMIT University for the valuable discussion on Robot Soccer.
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Robot-4
Robot-3
ball
Robot-2
Robot-1
Figure 8.1: Robot Soccer Scenario source (Robot-1) is normally not mentioned. Suppose at the same time (the ball hasn’t been moved yet), Robot-3 got a message stating a different position message from Robot-4 who is also close enough, then Robot-3 has to judge which position is more accurate before taking action. In such situations where conflict occurs, it is crucial for an agent to identify the original sources that are responsible for the conflicting information. The need for this is intensified in highly risky situations.
8.1.1
Representing Information Pedigrees
One natural and sensible approach a proactive rational agent could take to further clarify the conflicting information sources is to interrogate the sender agents. By conflicting information sources we mean those agents whose information when considered together will spoil the consistency of the receiver agent’s knowledge base. To facilitate such interrogation, idealistically agents would retain an infinite reasonable depth of information pedigree, which records the origin of the information.
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Figure 8.2: Information Pedigrees However, this is not practical. We will show in this section, by allowing agent to describe passing-on information, a three level information pedigree would be sufficient in terms of keeping information about the sources. It is widely accepted that agents receive information from two major channels via their perceptions directly from the environment and via communication with other agents. Thus, on other agents’ query about the source of information an agent should have one of the following answers: 1. The information in doubt is my own observation. 2. The information in doubt is from agent-1 (maybe also from agent-2, agent-3, ... agent-n), the pedigree tree is thus enlarged. We depict different scenarios of information gathering using information pedigree diagrams in Figure 8.2, where agents are nodes and the directed arcs stands for information route.
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The trustworthiness assessment represented by Scenario 1 and 2 in the pedigree diagram concerns pairs formed by individual senders with the receiver. Scenario 1 illustrates that agent-2 tells agent-1 some of his observed information, whereas Scenario 2 represents the situation where agent-1 receives observational information from multiple agents. With the degree of trustworthiness (or degree of sincerity and degree of competency) prepared as in Section 7.3, Scenario 1 turns into a traditional belief change problem - how to maintain a consistent knowledge/belief base in face of new information. While Scenario 2 becomes a fusion (or knowledge base merging) problem - how to incorporate new information from multiple sources. Scenarios 3, 4 and 5 illustrate more complicated cases where senders pass on information from the original sources to a receiver. The general information format (p, δyx ) used in Section 7.3.2 is only suitable for the pair-wise communication shown in scenarios 1 and 2. Such representation is not sufficient for passing-on information, as the receiver agent also expects the sender agent to tell him some information about the original source. Therefore, we propose the following four tuple, Iyx : (p, z, δyx , δzyx ) where, x, y, z are agent IDs, p is the information, z is the original source of information p, δyx is agent y’s degree of belief of p that agent y told agent x, δzyx is agent z’s degree of belief of p that agent y told agent x. When z = y, that is, the sender is the original source, (p, z, δyx , δzyx ) is then equivalent to (p, δyx ). Using the above information format, a sender can send a receiver not only his own degree of belief on p but also pass on other agent’s degree of belief on p. Moreover, Scenarios 3, 4 and 5 can then be represented using information matrix similar to the example shown in Table 8.1 for Scenario 4,
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Information p ... p p p
Sender Source Sender’s DOB* 2 3 δ21 ... ... ... n 3 δn1 n ... δn1 n m δn1 ∗DOB - Degree Of Belief
Passing-on DOB* δ321 ... δ3n1 δ...n1 δmn1
Table 8.1: Agent-1’s Information Matrix for Scenario 4
8.1.2
Information Pedigree Transformation using Trust Evaluation
As we are talking about sincerity and competency, we have to consider the sender’s trustworthiness on both the information itself and passing-on information about the original source. Taking Scenario 3 for instance, on receiving (p, 3, δ21 , δ321 ), agent-1 faces the following uncertainties, 1) Agent-2 may or may not be faithful about his degree of belief on p, that is, δ2 ? = δ21 , where δ2 is agent-2’s true degree of belief on p and δ21 is what agent-2 told agent-1. For evaluation, degree of sincerity on topic P (i.e. s1 (2, P )) could be employed here. 2) Agent-2 may or may not be competent in evaluating p. For evaluation, degree of competency on topic P (i.e. c1 (2, P )) could be used here. 3) Agent-2 may or may not be faithful about what agent-3 told him, that is, δ32 ? = δ321 , where δ32 is the real value agent-2 received from agent-3 and δ321 is what agent-2 told agent-1 about δ32 . Using a similar approach as equation
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0 (7.1) and (7.2), we can estimate δ32 . The estimated value δ32 is thus a function
of δ321 and degree of sincerity s1 (2, 3) in the topic about agent-3. 4) Agent-2 may/may not be competent in passing information onto others, e.g. not certain whether the information is distorted during the process or not. This could be evaluated using the degree of competency c1 (2, 3) in the topic about agent-3. Uncertainty of types 1) and 2) can be handled using the standard pair-wise trustworthiness evaluation proposed in Section 7.3. For uncertainty types 3) and 4), the topic of the trustworthiness evaluation is no longer about p, but about another agent, which is why we have the notation s1 (2, 3) and c1 (2, 3) in the above list. The ultimate goal of sincerity analysis is to get a close estimation of the source agent’s degree of belief in p. In the case when agent-3 tells agent-1 directly about p, according to equation (7.1) and (7.2), we have, δ30 = f (δ31 , s1 (3, P ))
(8.1)
But in the case of Scenario 3 in Figure 8.2, agent-1 has to estimate the relationship between agent-2 and agent-3 and thus figure out the possibility that agent-3 lies to agent-2, which could be captured using the degree of sincerity s1 (32, P ). Therefore, 0 and s1 (32, P ) to substitute δ31 , s1 (3, P ) in the above formula 8.1, agent-1 is using δ32
able to estimate δ30 shown as follows, 0 δ30 = f (δ32 , s1 (32, P ))
To generalize the above process, we can use the following transformation shown in Figure 8.3.
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Figure 8.3: Information Pedigree Transformation
Figure 8.3 shows us that a 3-level pedigree tree can be transformed into a 2-level one. This will finally enable the use of belief change techniques, data fusion, merging or integration techniques to help the receiver agent maintain a consistent knowledge base. As one may notice, the information pedigree diagrams in Figure 8.2 are no more than 3 levels in depth. This is because as soon as the receiver is notified about another agent by the sender, the receiver agent can directly contact the third involved agent for further information requested. Adding in another level of pedigree will unnecessarily increase the complexity of the trustworthiness evaluation process, which requires an agent to estimate the relationship of too many other agents. Therefore, to keep no more than one level of information source is sensible and sufficient for an agent to maintain the origin of communicational information.
8.2
An Example in Resolving Conflicting Information
Consider the following example, John is interested in acquiring a property. After inspecting a potential property, he suspects that the dining room is a new extension and is not sure whether it is a legal extension. Then he inquired about this with the real estate agent, asked his
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Agent Lawyer: 1 Real Estate Agent: 2 John’s observation: 3 Friend A: 4 Friend B: 5
Information No building approval in the past 7 years. No building extension within the last 10 years. Dinning area seems to be newly extended Statements about the purchaser’s inspection. Statements suggest a possible illegal extension.
e → ¬l ¬e e s s → ¬l
Table 8.2: John’s House Inspection
Agent Information 1 2 3 4 5
e → ¬l ¬e e s s → ¬l
Degree Of Belief δyx 1 0.9 1 1 0.9
Sincerity Competency sx (y, P ) sx (y, P ) 1 1 0.4 0.8 1 0.5 1 0.8 1 0.6
Trustworthiness(×) sx (y, P ) 1 0.32 0.5 0.8 0.6
Table 8.3: John’s Social Knowledge
lawyer to read legal documents on the property, and also listened to two friends’ opinions. Following is a summary of what he has gleaned from different sources: In Table 8.2, we assume that John has the values for degree of sincerity (s) and degree of competency (c) in his social knowledge base already. Then we can either work out the degree of trustworthiness (t) based on the values of s and c, or use s and c separately with δyx as shown in Section 7.3. The trustworthiness values in Table 8.2 are obtained using operator τmultiply introduced in Section 7.3.1. This prepares the essential values that are required by Cantwell[11] in his process of resolving conflicting information. Using the notation of relative support, Cantwell[11] obtains the order of sentences
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based on the joint trustworthiness2 of the sources that have rejected the states. Taking the example illustrated in Table 8.2 for instance, the conflict resolving process suggested by Cantwell can be described using the following steps: 1. Construct all the possible worlds based on the information state: t1 : e → ¬l; t2 : ¬e; t3 : e; t4 : s; t5 : s → ¬l The possible worlds thus formed are {(e, l, s), (e, l, ¬s), (e, ¬l, s), (e, ¬l, ¬s), (¬e, l, s), (¬e, l, ¬s), (¬e, ¬l, s), (¬e, ¬l, ¬s)}. 2. Given each agent(witness)’s individual trustworthiness, find out the joint trustworthiness of agents who reject the world, for example, for world (e, l, s), since agent 1 and 2 reject e, agent 1 and 5 reject l, agent 5 reject s, the joint trustworthiness of rejecting world (e, l, s) is then t1 +t2 +t1 +t5 +t5 = 3.52. Similarly, by applying such procedure to each every possible world, we would thus associate a joint trustworthiness value to each world. 3. The worlds which have the largest joint trustworthiness value are the candidates to be rejected. The above method does not guarantee a unique solution. Actually, in a special case when all the witnesses have equivalent trustworthiness, it generates the maximal consistent set of worlds, which is equivalent to the result of theory extraction[107]. Moreover, the worlds that are apt to be rejected are often the ones that do not satisfy the proposition that no agent is against. This implies that things uttered by agents take priority, if there is no objection, the receiver agent should accept it without doubt. 2
In this case, the joint trustworthiness of a set of agents is just the simple addition of the individual trustworthiness of each agent.
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Alternatively, we can use equations (7.1) and (7.2) to work out the estimated degree of belief (δy0 , y = 1, 2, 3, 4, 5) and combine δy0 with c to obtain the degree y of acceptance (δjohn , y = 1, 2, 3, 4, 5) that the receiver agent (John) is willing to
consider. This will turn the above example into a standard belief revision problem, which could be easily solved using the existing belief revision techniques, such as Williams’ adjustment[108]. y 1 4 5 3 2 After calculating δjohn , we have δjohn > δjohn > δjohn > δjohn > δjohn , therefore,
1 δjohn (1) : e → ¬l 4 δjohn (0.8) : s 5 δjohn (0.54) : s → ¬l 3 δjohn (0.5) : e 2 δjohn (0.45) : ¬e
As the real estate agent’s claim has the lowest degree of acceptance, a rational agent will give up ¬e and continue to accept that the extension is illegal to keep a consistent knowledge base. In addition, with the help of information pedigree, the receiver agent can figure out the information that is in contradiction then ask the sender agents involved to further track down the information source. For example, after inquiring, agent-2 (Real Estate Agent) told the receiver agent, my information is from the city council, which is a definite trustworthy authority, the piece of information ¬e from agent-2 will then take a much higher priority in the above ordering. In this case John’s own observation e can be discarded, i.e. he was wrong about the existence of an extension.
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As shown by this example, the trust evaluation process proposed in here provides valuable insights into areas such as belief revision, conflict resolution, fusion and etc. In particular, the separation of sincerity and competency analysis offers a clarified view of the analysis required for evaluating the trustworthiness of an information source, which in turn enables the analysis of passing-on information and offers a more flexible way of handling conflicting information. With information pedigrees, we give agents another means of searching for more evidence so as to arrive at a unique sensible solution. Furthermore, the trust evaluation based on experience makes the framework applicable in iterative situations as well.
8.3
Discussion
Trust is often referred to as an adaption, a generalisation or as a means for the reduction of complexity[83]. Adaption (via evolution) and generalization enables agents to learn from experience in beneficial ways and provides agents with means of reasoning about and to make general assumptions about other agents or their environment for which they are ignorant. Therefore, trust evaluation is a valuable and essential survival skill for agents as they interact in an uncertain social and physical environment. In this chapter and the previous chapter, we investigated a procedure for evaluating an agent’s trustworthiness as an information source. By separating this process into competency analysis and sincerity analysis, we were able to deal with complicated cases of passing-on information, where the same information may reach the receiver agent through different routes. In order to keep information about the source agent,
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we introduced information pedigrees as a means to maintain the history of communicational information. Information sources’ trustworthiness analysis and computation described herein can be used to prepare agents to perform fusion, weighted knowledge base merging, and multiple source conflict resolution. The results from these two chapters serves as the basis for an agent-oriented implementation that supports the acquisition of information from multiple sources with different degree of trustworthiness. Chapter 10 details the design of the system which use the Java-based multi-agent programming paradigm JADE together with Belief Revision and Theory Extraction engines[107] from SATEN.
Chapter 9 Shared Knowledge Structure and Knowledge Migration “For certain, consumers must be assured that agents will not compromise private information and deviate beyond their constraints.” — Pattie Maes and Rober Guttman (MIT Media Labaoratory)
9.1
Introduction
It is well-known that e-commerce is one of the major application areas of agent technologies. From a consumer buying behaviour (CBB) [79] perspective, agents can play important roles in three primary CBB stages: product brokering, merchant brokering, and negotiation as to what to buy, who to buy it from and how to determine the terms of the transaction respectively. In particular, there already exists agent systems that can negotiate and make agreements in domains such consumer-to-consumer (eBay’s Auction Web), business-to-business (FairMarket) and stock markets (E-Trade and OptiMark Technologies). In order to trust and conduct trading using these systems, the agent owners must be assured that the agent will not compromise private information and deviate beyond its design goals. 132
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One of the major drawbacks of the frameworks we visited in Chapter 5 is that there is no clear indication as to how the agent could explicitly separate owner’s sensitive private information from the information that needs to be shared with other agents. This is a context dependent decision. In the case when information is exchanged mainly via communication, a simple solution could be to simply flag every piece of information with an extra property - level of privacy. However, some problems may still arise. First, hard-coding a privacy property could lead to maintenance problems when the circumstance changes or the trading partner changes. Secondly, this also poses problems when the involved parties can actually query each other’s databases. Especially, considering that the two major goals of e-commerce is inter-operability and automation, more and more information services provide direct database access. Therefore, to have a dynamic knowledge structure is important for the purposes of protecting private information from shared information and allowing different level of sharing with regards to different agents in different circumstances. In addition, the proposed sophisticated knowledge structure serves as a upper-level ontology that addresses the social heterogeneities that may existed in an open system, which is lacking in the current frameworks. In most of the frameworks presented in Chapter 5, agents knowledge bases are considered to be accessible by all agents and every agent provides faithful information. Therefore, with the informal ontology proposed herein we are able to discuss shared knowledge and private knowledge in different frameworks. As we will also show in Section 9.5, the process of multi-agent belief revision under the shared knowledge structure becomes the process of knowledge migration.
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9.2
Shared Knowledge Structure
The accessible knowledge(KAcc ) is defined as a subset of an agent’s knowledge base (KB), which is open to anonymous viewers or certain viewers specified by the agent itself. KAcc can exist in various circumstances, for example, a web site that provides personalised information to personal assistance agents, or a buyer agent who can dynamically change preferences according to the merchant it interacts with. Figure 9.1 illustrates three agents’ belief bases, with their subsets classified according to dynamic accessibility.
KC R11
1
1 Kpri(1)
1 K3-Acc(1,23)
Agent1
K2-Acc(1,3)
Agent2
K2-Acc(2,1) Kpri(2)
R22
K3-Acc(2,13)
R12 K2-Acc(1,2)
K2-Acc(2,3) R21 R31
R32
R13 Kpri(3)
R23
K3-Acc(3,12) K2-Acc(3,2) K2-Acc(3,1)
Agent3 R33
Figure 9.1: Accessibility to Knowledge Bases From agenti ’s point of view, the agent acquaintance base is populated by the agents that it interacts with. Let us assume that there are n agents in the society
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including agenti . We denote the KB of agenti to be K(i). It is composed of maccessible knowledge and m-semiprivate knowledge as defined below: 1. m-Accessible Knowledge: Accessible knowledge for agentj , denoted by KAcc (i, j), represents the set of sentences that agenti determines to open to agentj . Since agenti can fully access its own knowledge base, when i = j, KAcc (i, i) is actually just K(i). To avoid any confusion that might arise, K(i) is adopted. Therefore, in the following sections, KAcc (i, j) only stands for the cases when j 6= i. As the subset of KAcc (i, j) may also be accessible to agents other than i and j, it is necessary to define strictly 2-accessible knowledge (K2−Acc (i, j)) and accessible knowledge for multiple agents (KAcc (i, j · · · k)). Following is the definition: K2−Acc (i, j) = KAcc (i, j) ∩ (∪(· · · , KAcc (i, k), · · ·)) where k 6= i and k 6= j. Let l be the cardinality of the set {KAcc (i, j), · · · , KAcc (i, k), · · ·}, the right hand side of the set operation results in a strictly 2−1 2-Accessible knowledge set iff l = Cn−1 = n − 1, where Csr is the binomial
coefficients,
s r
=
s! , (s−r)!r!
n is the population of the society.
KAcc (i, j · · · k) = ∩(KAcc (i, j), · · · , KAcc (i, k)) where m is the cardinality of the set {i, j, · · · , k}, 1 < m ≤ n. m is called the degree of accessibility, KAcc (i, j · · · k) is the accessible knowledge for m agents. Similar to the definition of K2−Acc (i, j), the m-accessible knowledge1 Km−Acc (i, j · · · k) could be defined based on KAcc (i, j · · · k) 1
m − accessible means strictly accessed by m agents. While the knowledge accessible by m agents means this knowledge is accessible by any subset of the m agents.
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KB n-semiprivate knowledge Kn-Acc(i,j...n) Ks(ij...n)
n-Accessible Knowledge
n-shared knowledge
K2-Acc(i,j)
Ks(ij) 2-shared knowledge for agent i and j
......
Kn-sp(i,j...n)
K2-Acc(i,n)
maximally n-1 2-Accessible Knowlege
Kpri(i) private knowledge
Figure 9.2: Shared Knowledge Structure Km−Acc (i, j · · · k) = KAcc (i, j · · · k) ∩ (∪(· · · , KAcc (i, p · · · q), · · ·)) KAcc (i, p · · · q) has the same degree of accessibility as KAcc (i, j · · · k), but set {j, · · · , k} has no intersection with set {p, · · · , q}. Let l be the cardinality of the set {KAcc (i, j · · · k), · · · , KAcc (i, p · · · q), · · ·}, the right hand side of the set m−1 operation results in a strictly m-Accessible knowledge set iff l = Cn−1 , n is the
population of the society. The bottom up m-accessible knowledge classification results in a tree structure, which is displayed in Figure 9.2, where m = 1, · · · , n. 2. m-Semiprivate/Private Knowledge: As the complement of Km−Acc (i, j · · · k), the m-semiprivate knowledge Km−sp (i, j · · · k) is private towards agentj , · · · , agentk , which is accessible by agenti itself and agents other than j, · · · , k. The following relation holds: ∪(· · · Km−Acc (i, j · · · k) · · ·) ∩ Km−sp (i, j · · · k) = K(m+1)−sp (i, j · · · k + 1) |
{z
m−1 Cn−1
}
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When m = 1, the knowledge is only accessible by agent i itself and not by any one else, it becomes the private knowledge of agent i, denoted by Kpri (i). 3. KB of agenti : K(i) is the knowledge base of agent i. Within this KB, the agent can assign various level of accessibility to various agents in the society and reserve the rest of the KB as its private knowledge Kpri (i). K(i) = ∪(Kpri (i), K2−Acc (i, j), · · · , Km−Acc (i, j · · · k), · · ·, Kn−Acc (i, j · · · n)) |
{z
2−1 Cn−1
}|
{z
} |
m−1 Cn−1
= ∪(Kpri (i), KAcc (i, j), · · · , KAcc (i, k) · · · , KAcc (i, n)) {z
|
{z 1
}
}
n−1
where j 6= i and j = 1, · · · n for an n agent society.
Applying the definitions above to the other agent’s knowledge bases, K(i) can be further classified: 1. Mutually Accessible Knowledge: It is the set of sentences that agents i, j, · · · , k all believe but may not know whether everyone knows that everyone believes those sentences and may not know to what degree. 2-mutually accessible knowledge KAccM utual (ij) is a special case when both agents believe and know each other knows that each of them believes, which is defined as: KAccM utual (ij) = ∩(Kacc (i, j), Kacc (j, i)) The ideal m-mutually Accessible Knowledge defined in the similar way is not realisable. Since the knowledge is classified from agenti ’s viewpoint, it is impossible for i to know about the knowledge classification of other agents. KAccM utualIdeal (ij · · · k) = ∩(KAcc (i, j · · · k), · · · , KAcc (j, i · · · k), KAcc (k, i · · · j) |
{z
m! m−1 =m Cm
}
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where, m is the cardinality of set {i, j, · · · , k}. Therefore, only super agents who can access every agent’s KB can do the above evaluation. For agent i, it could only reason about the possible m-mutually accessible knowledge at a coarse level: KAccM utualP ossi (ij · · · k) = ∩(KAccM utual (ij), · · · , KAccM utual (ik)) ⊇ KAccM utualIdeal (ij · · · k) but KAccM utual (ij) = KAccM utualP ossi (ij) = KAccM utualIdeal (ij) 2. m-Shared Belief: Ks (ij) represent the 2-shared knowledge of agent i, and j. It describe a package of knowledge that agent i and j both believe and both of them know that each other believes it to an agreed degree. Ks (ij) is defined as: Ks (ij) = (∗)KAccM utual (ij) where ∗ stands for the various kinds of operator that can generate shared knowledge from the mutually accessible knowledge. One simple candidate is to simply take the mutually accessible knowledge as shared knowledge and derive a unanimous ranking on it. Alternatively it could be used to initiate conversations among the agents to select an agreed subset. m-shared belief is defined as Ks (ij · · · k) = (∗) ∩ (Ksij , · · · , Ksik ) where ∗ has the same meaning as defined above. The discussion for possible m-mutually accessible knowledge suggests that ∗ in this case should enable
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agenti to send out queries to other agents. This is to determine the accessibility of KAccM utualP ossi (ij · · · k) from the other agents viewpoint. If all the agents faithfully answer queries whenever required, KAccM utualIdeal (ij · · · k) would be the final result of such queries. 3. Common Belief: Kc is the n-shared beliefs, which is a special case of shared belief when the set of sentences is believed by the whole society with an agreed degree(i.e. n-shared knowledge), and everyone in the society knows that everyone believes it at a certain degree and · · · and everyone knows that · · · that everyone believes it to a certain degree. The mutually accessible knowledge is stored separately in each agent’s local knowledge base, the rank of the same sentence in KAccM utual (ij · · · k) does not necessarily have to be the same according to individual agent’s judgment. But when the agents decide to share some knowledge, it is necessary for them to agree on the rank of the shared knowledge. Thus the rest of each agent’s knowledge base needs to be revised to accommodate the agreed degree. By the way, shared/common belief could be extracted from each agent’s knowledge base and stored in a common location if everyone agrees to do so. This would be beneficial in terms of saving space and it would enhance the robustness of the dynamics of shared information.
9.3
Relationship with Other Knowledge Structures
Essentially, two types of knowledge structures for modelling common/shared knowledge exist in the literature, (I) The labelled tree of mutual belief by Van de Meyden[85] based on possible
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worlds semantics of modelling knowledge and belief in MASs[31]; (II) The knowledge in a Shared Domain (SD) and a Private Domain (PD) by Kfir-dahav and Tennenholtz [68]. In the case of mutual belief [85], the accessibility relations are defined on the possible worlds according to the agents current beliefs. A labelled tree formally represents this, which is a mixture of domain and social knowledge. Each agent is able to reason uniformly because the whole tree structure is visible to everyone. It is inflexible to some extent because it does not support private knowledge. Tidhar et al.[102] define team knowledge as “the team knows only what is known to every subteam”. Such team knowledge is actually the intersection of subteams’ knowledge base, which is similar as SD defined in [68]. It is shown in Chapter 5 such a classification is not capable of modelling truely private knowledge. It implicitly presumes a super agent who can access all the agents’ KBs, draw the conclusions from the shared knowledge and impose that on each individual. Imposing team knowledge top down from a team leader is only one way of achieving team knowledge. The other way is to derive consensus (via. negotiation or argumentation etc.) from the team members. The latter way is not supported by the structures in [68][102]. Intuitively, the agents knowing the same thing does not necessarily imply either the awareness of, or the sharing of intentions.
While agents sharing some
knowledge means the shared knowledge must be known to each agent involved, i.e. Shared
=⇒ 6⇐=
Known. Previous research reported in the literature fails to capture such
intuition. However our share knowledge structure is successful in achieving it by authorising agents to organise their own knowledge bases. The accessibility as defined in Section 9.2 is totally determined by an individual agent’s personal stance. It is
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considered as an attribute of a certain subset of the agent KB regardless of how the agent is going to reason on it. Since there is no need to reveal all of its KB, true privacy is maintained when cooperating. The operator (*) is suggested to derive shared belief from a rational set of mutually accessible knowledge without the aid of super agents. However, this knowledge structure does not preclude the possibility of the existence of a super agent. This could be done by simply forcing all the other agents to register their platform to give full accessibility to the super agent. The super agent, on the other hand, would allow only limited or no accessibility to others. The proposed shared knowledge structure is capable of simulating previous structures by rearranging the accessibility relations. For example, if every agent opens its domain knowledge base to all the others, the n-possible mutually accessible knowledge becomes n-ideal mutually accessible knowledge. The n-shared knowledge achieved is just the knowledge in SD. The complement is left for P D. If every agent fully opens both its social knowledge base and domain knowledge base, the accessibility relation becomes transitive. Therefore, shared knowledge structure is able to capture the semantics of mutual belief[85], and in this case, Kpri (i) is empty.
9.4
Inconsistency Principle
In this framework, the consistency of an individual’s knowledge base with the common/shared beliefs is our primary concern. This is achieved using standard single agent belief revision techniques. So local consistency is still a prerequisite of each agent, but the global consistency of the society is not required either during or after the process of belief revision. Hence, inconsistency across the knowledge bases of the society is permitted. This is called inconsistency principle. This is different from the
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Liberal Belief Revision Policy[20] where the final distributed belief revision goal is to achieve global consistency. Observation: If sentence p is inconsistently believed in the agent society, then neither p ∈ Kc nor ¬p ∈ Kc This observation also holds for the shared belief bases. That is, among the agents who share the belief, neither it or its negation are derivable from the shared belief base. It becomes one of the key postulates that should be satisfied during the multiagent belief revision process. Therefore, as long as a sentence is not derivable from the common belief base, it could be inconsistently believed across the society or across several groups; similarly, as long as a sentence is not in a shared belief base of a certain group, it could be held inconsistently at least across the agents in this group or across several subgroups. The major goal of belief revision based on our proposed shared knowledge structure and the inconsistency principle is to maintain the consistency of K(i) with K c , and with shared belief base Ks (ij · · · k). Common/shared beliefs are distinct from mutually accessible knowledge by the fact that the rank in Kc /Ks (ij · · · k) should be respected, while different ranks could be assigned to the same mutually accessible knowledge by different agents. Therefore, maintaining the consistency of K(i) with Kc means aligning the rank of common beliefs in every agents knowledge base when new information is accepted by the society. That is, the revision process to revise K(i) by a set of sentences with fixed ranks (in Kc /Ks (ij · · · k)) and also the new information, while the rank of new information and sentences in K(i) are changeable, those in Kc are not.Revision regarding the shared knowledge Ks (ij · · · k) uses a similar process but within a smaller groups of agents.
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Communication and negotiation might be necessary to reach a consensus on ranking of Kc /Ks (ij · · · k). Consequently, belief dynamics considered here are more sophisticated than is required for single agent revision and could be extended to knowledge migration in terms of knowledge grade, which is discussed in the next section.
9.5
Knowledge Grade and Knowledge Migration
Based on the shared knowledge structure, an agent’s knowledge base could be ordered according to the degree to which beliefs are shared: Common B (n, · · · , 2)-Shared B (n, · · · , 2)-AccNotShared B Pri B HIGH . Compared to the core of a belief (sentence, p), we call the rest elements for an ideal belief as the axillaries of a belief. All combinations of a core sentence with any number of the axillary elements are considered to be valid beliefs in the framework. For example, < p, δ >, < p, T opic >, < p, Source, T opic > are considered as valid domain beliefs.
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Similarly, an ideal piece of social belief is either a 3-tuple < AID, T rustworthiness, T opic > or a 4-tuple < AID, Sincerity, Competency, T opic > depending on whether the trustworthiness is a simple subjective estimation or a composite value calculated from the component beliefs of trust. It is also true that any combination of the core with any other axillaries (< AID, T rustworthiness >, < AID, T opic > or < AID, Sincerity, Competency, T opic >) would make valid social beliefs. 0..*
DomainBelief -sentence +addComponent() +removeComponet() +getChildren()
1 Degree -degree +getValue()
Source
Topic
AccessibleBy
-source
-topic
-AccessibleBy
+getValue()
+getValue()
+getValue()
CompositeDomainBelief
Figure 10.4: UML: Class Diagram of Domain Beliefs Figure 10.4 shows that through inheritance and composition we are able represent to various kind of domain beliefs as well as social beliefs. When implementing the above fields for a domain belief base and social belief base, we would like not only to fulfil the flexibility requirement for representing various types of domain and social beliefs, but also to make sure further development does not necessarily interfere with the current implementation. For example, the social belief tuple < AID, Sincerity, Competency, T opic > may not be an exhaustive list of all the possible information a social belief need to encapsulate. An agent may want to keep a relationship field as well such as “friend, competitor, normal” and so on.
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AID «uses» JADE Agent Platform SocialBelief
0..*
-AID +addComponent() +removeComponent() +getChildren()
1 CompositeSocialBelief
0..* Trustworthiness
Topic
-trustworthiness
-topic
+getValue()
+getValue()
+getValue() +addComponent() +removeComponent() 1
Sincerity
Competency
-sincerity
-competency
+getValue()
+getValue()
CompositeTrustworthiness
Figure 10.5: UML: Class Diagram of Social Beliefs Therefore, the composite pattern[39] is used to compose a composite belief from the leaves (e.g. δ, T opic and ect) as shown by the UML diagram in both Figure 10.4 and Figure 10.5.
10.4.2
Belief Revision Services - Use Case Study
Refering back to the ontological description of the Section 5.3 and the discussion in previous sections, the belief revision agent herein described can provide three types of belief revision services: SBR, MSBR and MABR. These services are described using use case diagrams illustrating the functionalities that the 3-layered belief revision engine consists of. The use cases described here focus on the external behaviour of the system but
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Layer:SBR
DegreeOfBelief * TheoryExtraction ***
*
Actor
«uses»
«uses»
Transmutation
Fusion
* *
Figure 10.6: UML Use Case: Single Belief Revision not how things are actually performed inside the system, which will be described instead using the class diagrams. Single Belief Revision There are four main use cases for a single belief revision, which are shown in the use case diagram Figure 10.8. There are basically four types of use cases in the single belief revision layer. The DegreeOfBelief use case models a scenario where the agent answers queries of the degree of believing in a sentence. The TheoryExtration performs the request of extracting a consistent theory from the given belief base. The Transmutation and Fusion use cases perform revising a belief base in face of new information and fusing multiple belief bases respectively. An example description of the DegreeOfBelief use case is shown below. Objective: Query the degree of belief δ of a sentence p
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Initiated: When an actor sends a query message in ACL with QUERY performative Message Flow: Actor to send a QUERY message and the sentence. The communication between the actor and the use case should follow FIPA “request-like”4 interaction protocols. Value returned to actor: One of the following: • Refuse(reason) • Not-Understood • Failure(reason) • Inform(result) If the request is successfully processed, the actor should receive a value that indicates the degree of belief for the sentence. Belief Revision using information from Multiple Sources Figure 10.7 illustrates the use cases for the MSBR layer. Two use cases GeneralTrust and TopicSpecificTrust are for query agents to inquire about the level of trustworthiness of a particular acquaintance, either general or specific to a topic. Meanwhile, there is a use case in this layer to allow dynamic update of social beliefs. The core use case in this layer is the MSBR use case. It uses the agent trustworthiness to compute a new degree of acceptance. The the degree of acceptance is used as important input for fusion in the SBR layer. 4
The FIPA request like protocols include FIPA-Request, FIPA-query, FIPA-propose, FIPARequest-When, FIPA-recruiting, FIPA-brokering, FIPAsubscribe and etc.
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Layer:MSBR DegreeOfTrust «extends»
«extends»
* *
GeneralTrust **
TopicSpecificTrust
SocialBeliefUpdate
* UpdateActor
«uses» *
«uses» SocialInfoQueryActor
Layer:SBR
«uses» MSBR
1..* 1
Fusion MSBRActor
Figure 10.7: UML Use Case: Belief Revision using Information from Multiple Sources Multi-Agent Belief Revision The MABR layer will contain use cases such as updating the accessibility of the domain beliefs, query about the shared beliefs, and core functionality of knowledge migration. This is shown in Figure 10.8. Imagine agent-x shares φ with agent-y, it also believes that φ → ψ, but this is not shared with agent-y. Instead it shares the logical consequence ψ with agent-z. Therefore, when agent-y send a request to remove φ, agent-x has to communicate with agent-z as well to make a decision on whether to remove ψ or degrade that to a private knowledge. A typical MABR process (i.e. the knowledge migration) involves: 1. Keep a copy of the original belief base K
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Layer:MABR Accessbility «uses»
Layer:MSBR *
SharedBelief *
PerceptionActor
MSBR «uses»
Layer:SBR 1..*
KnowledgeMigration 1
Transmutation «uses»
MABRActor
Figure 10.8: UML Use Case: Multi-Agent Belief Revision 2. Carry out a SBR or a MSBR and obtain a resulting K 0 3. Compare K and K 0 , and work out all the affected partners. 4. Inform all the involved partners about the possible changes 5. Based on the decision making rules (e.g. compare agent-y and agent-z’s trustworthiness) and partners feedback to choose between upgrade or degrade the belief and all the logical consequences Therefore, the core use case knowledge migration will use the functionality provided in either the MSBR layer or the SBR layer or both.
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10.4.3
Layered Belief Revision Engine
As we can see from the above use case study, each layer of the engine has a core use case. Moreover, to accomplish the core functionality, the higher layer will need functionalities provided by its lower layer(s). Therefore, we can treat the layer’s functionality as sub-capability of the higher layer’s capability. This provide a modularised design that meets the requirements of agent-oriented programming[43, 91]. It also facilitates the possibility of plug-in compatible capabilities in order to incorporate specific revision techniques. In JADE, each functionality/service provided by an agent should be implemented as one or more behaviours. A scheduler, internal to the base Agent class and hidden to the programmer, automatically manages the scheduling of behaviours. User defined behaviours should extend the base class behaviours and added to the agent class using Agent.addBehaviour(). The functionalities of each layer is therefore wrapped into behaviours and then a JADE belief revision agent can use it by simply calling Agent.addBehaviour(). The layered belief revision engine can be implemented using the composite pattern as shown in Figure 10.9. The UML class diagram is closely related to the use case diagrams for each layer, with most of the use cases implemented in classes. Furthermore, the MABR process is implemented as a composite of all the functionalities it can provide, with MSBR process as a component function that supports the overall functionality. Similarly, the MSBR process consists of functionalities such as updating social beliefs, querying the degree of trust, and obtaining the acceptance level of new beliefs based on social beliefs, then feeding the new degree of beliefs into the SBR layer to achieve MSBR.
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0..* «interface» CompositeMABR
1 «interface» CompositeMSBR
0..*
«interface» SharedBelief
«interface» KnowledgeMigration
«interface» MABR
1 «interface» MSBR
«interface» SBR
«interface» DegreeOfTrust
«interface» SocialBeliefUpdate
Figure 10.9: The Composite Pattern used for Belief Revision SBR is basically a reformulation of the existing SATEN functionalities, as shown in the UML class diagram in Figure 10.10. It shows the hierarchy of a single belief revision process and at the same time leave spaces for possible incorporation of other revision schemes such as Probability, Possibility and Belief Functions.
10.5
Agent Communication Channel - JADE Agent Wrapper for Belief Revision Services
As shown in Section 10.3 and Appendix D, a JADE agent platform automatically registers an agent with the AMS when the agent is created and thus the agent will gain access the Message Transport Services. It provides base classes such as DFService for a JADE agent to use the service provided by the directory facilitator. Moreover, it also extracts the common structure of FIPA communication protocol as shown in Figure 10.11 below. There are basically two roles that an agent can perform when involved in a
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Behaviour
JADE Agent Platform «interface» SBR
«interface» TheoryExtraction
Adjustment
«uses»
«interface» Possibility
Maxi-Adjustment
«uses»
Transmutation-Adjust
«interface» Probability
«uses»
DMA
«uses»
Fusion-Adjust Transmutation-MA
«interface» BeliefFunctions
Fusion-MA
«uses»
Transmutation-DMA
«uses»
Fusion-DMA
Figure 10.10: Single Belief Revision communicative act, namely, an initiator and a responder. In order to make sure the communication between two agents is meaningful and rational for both parties, JADE introduces the concept of rational effects of communicative acts[30]. With the achieve-rational-effect functionalities provided by two classes AchieveREInitiator and AchieveREResponder, an agent can be involved in any FIPA request like conversations. The initiator sends a message (i.e. performs a communicative act, as shown in the white box). The responder can then reply by sending a not-understood, or a refuse to achieve the rational effect of the communicative act, or also an agree message to communicate the agreement to perform (possibly in the future) the communicative act, as shown in the first row of shaded boxes. The responder then performs the action and, finally, must respond with an inform of the result of the
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Communicative Act
Not-Understood
Refuse Reason
Agree
Failure
Inform Done(Action)
Inform (iota x (result action)x)
Figure 10.11: Homogenous Structure of Agent Communication Protocol[30] action (eventually just that the action has been done) or with a failure if anything went wrong. JADE has extended the protocol to make optional the transmission of the agree message. If performing the action takes a short time, sending the agree message will create an ineffective overhead. In such cases, the agree to perform the communicative act is subsumed by the reception of the next message in the protocol. Figure 10.12 shows how the JADE communication functionalities can be used together with the belief revision behaviours by a BRInitiator agent. The BRInitiator agent extends the JADE base class Agent and adds the AchieveREInitiator behaviour into its behaviour base. It also adds the DFService from the JADE library to its behaviour base to be able to register its service to a directory facilitator. Finally, it adds to its behaviour base BRBehaviour as its major service. The BRBehaviour class in Figure 10.12 is a fictional class which is going to be replaced by the actual class that extends JADE behaviour and implements one of the interfaces SBR, MSBR and MABR which forms the hierarchy shown in Figure
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10.9. Depends the services, the BRBehaviour class will interact with either the DomainBeliefBase or SocialBeliefBase or both. The two types of belief bases are aggregation of the DomainBeliefs and the SocialBeliefs illustrated in Figure 10.4 and Figure 10.5 respectively. Communication Wrapper Agent
AchieveREInitiator
DFService
Behaviour
JADE Agent Platform
BRInitiator
«interface» CompositeMABR
«uses»
SocialBeliefBase «uses»
«uses»
-socialBeliefs
BRBehaviour «uses»
DomainBeliefBase -domainBeliefs
Figure 10.12: Agent Communication Channel - JADE Agent Wrapper
10.6
Summary
We illustrated in this chapter using a software engineering approach of how to design and implement the multi-agent belief revision framework and the concepts involved. The investigated concepts in the former chapters, such as various revision schemes, trust, information pedigree, shared beliefs and ontologies, eventually come together within a flexible and general framework. The framework is adaptable for incorporating new belief revision schemes and viable for a heterogenous multi-agent environment. In particular, as the layering approach is adopted, a belief revision agent designed under such a framework is capable of providing services at various levels. It can model single
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belief revision behaviour, revision behaviour for information from multiple sources, as well as cooperative revision in multi-agent situation where revision on shared or common knowledge is desired. This provides a tangible architecture of a belief revision service provider for an dynamic, heterogenous, autonomous open environment.
Chapter 11 Discussion, Conclusions and Implications “Mind maps are tools which help you think and learn ...” — From http://www.maps.jcu.edu.au/netshare/learn/mindmap/index.html
has belief revision behaviours Intelligent Agents
is implemented in
need
extend JADE Agent Class
Domain Beliefs and Social Beliefs
trust forms
JADE BRAgent
specifies
Belief Revision
Multi-Agent Systems
implemented
Muti-Agent Belief Revision
Ontologies for Belief Revisioin
has heterogenieties need ontologies
requires shared belief and private belief
shared knoweldge structure
SBR v.s. MSBR v.s MABR
Layered Belief Revision Engine
ontologies JADE Agent Framework
implemented in
JADE Basic Ontologies
Figure 11.1: Mind Map of Multi-Agent Belief Revision
11.1
Discussion and Conclusion
Agent technology has gained tremendous attention from both the research community and industry. The Internet and the World Wide Web provide a large-scale truly distributed, dynamic, open and heterogenous test-bed for agent and multi-agent theories 168
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and techniques. Agent technology is an innovative programming paradigm that is different to the object-oriented approach. It is particularly powerful and useful in modelling and tackling problems inherent in large distributed systems, such as air traffic control, distributed manufacturing system, Health Care, e-commerce and e-business, and etc. To meet the surging interest from the industry and to explore the agent technology, many agent programming languages, APIs, SDKs and frameworks have been developed. Some treat agents as normal software entities that can perform normal functions and processes with the common capability of communicating with others, i.e. sending and receiving messages. Some focus on implementing intelligent agents based on human beings’ practical reasoning, thus bestowing mental attributes to agents, such as belief, intension, desire and goals. Agents built around this paradigm are often referred to as BDI agents. The deliberation power of BDI agents makes it particularly of interest to us, as these agents exhibit rational behaviour such as persistence in achieving goals, social awareness and robustness in terms of recovering from failure. Often, belief revision is considered as one of the essential functions that a BDI agent should possess so that it can maintain a consistent belief base. However, most of the agent development toolkits or frameworks either do not support the representation of agent internal states such as beliefs, or belief set structures but use simple database insertion and deletion to represent the update of beliefs. In fact, maintaining a consistent agent belief set is crucial for agents to exhibit rational behaviour. The web based belief revision system - SATEN provides a nice tool for human researchers to test out consistency of theories but the process is not automated,
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meaning it can not communicate with other software programs or agents to further its potential as a belief revision service or component. Therefore, there is the need of re-programming SATEN into a belief revision agent, which provides belief revision services to other types of agents. The design process carried out in Chapter 10 facilitates the wrapping of SATEN into a service oriented belief revision agent with social awareness. On the other hand, the research in the area of belief revision mainly focus on maintaining beliefs kept with one single agent. Seldom are there studies into belief revision issues that are presented in a multi-agent environment. We investigated belief revision techniques for individual agents as well as the frameworks that address belief revision issues in a multi-agent environment. The research into individual belief revision shows us a diverse range of revision schemes, some operate on theories, some on theory bases, some abide by the AGM postulates, some fall into the traditional Bayesian probability paradigm, some follow a more general way of representing beliefs, such as possibility theory and belief functions. It is interesting to note that all these revision schemes are somewhat related, basically they can all be described using the essential elements of an epistemological theory, that is, epistemic state, epistemic input, epistemic attitude and epistemic changes. The epistemic changes are all regulated by certain rationality criteria, such as minimal change or coherence requirement. Furthermore, we showed that both probabilities and possibilities are special cases of belief functions and can be represented by belief functions when certain conditions are met. An important outcome of the research into the available multi-agent belief revision frameworks is an ontological classification of the various terminologies adopted in
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the literature, where there is no consensus as yet. We classified belief revision into three major categories, namely, simple belief revision (SBR), belief revision using information from multiple agents (MSBR) and multi-agent belief revision(MABR). This provided a sound basis that aided our investigation into the domain. Another outcome of the research is the identification of a major drawback of the current frameworks. That is, they are not capable of dealing with heterogeneity issues. It is not feasible therefore to use these frameworks in open systems, where agents are designed by different vendors and representing interests of different uses. This motivates our research into classifying various types of heterogeneities that might exist and affect the belief revision process in a multi-agent environment. We identified three common types of heterogeneities: social heterogeneity, semantic heterogeneity and syntactical heterogeneity. Ontologies are an effective way of handling heterogeneities and thus come into play. Basic promises as to what ontologies can do have been investigated and it further evokes the necessary research into two areas: the trust issues on information sources and the issues of representing shared beliefs without compromising private beliefs. We analysed the differences between trust in an action and trust in a piece of information. Two constituents of evaluating an information source’s trustworthiness were thus identified, namely, the belief of sincerity and the belief of competency. We also proposed several ways of generating general or topic specific trustworthiness from the degree of sincerity and degree of competency. The obtained trustworthiness serves as crucial input, and is often required by fusion and weighted knowledge base merging when considering revision on information received from multiple sources. Our investigation into the feasibility of keeping information about the source shows that it
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is advisable for an agent to keep a pedigree of at least 3 levels of depth. This enables the agent to analyse and keep track of passing-on information, whose credibility can be wrongly computed in some real application domains, such as robot soccer. Overall, the research into trust issues not only provided a clear domain description but also formulated the necessary components of agent beliefs, i.e. domain beliefs and social beliefs. The implementation details are provided in Chapter 10. After clarifying the trust issues, we proposed a shared knowledge structure that is capable of representing both shared knowledge as well as private knowledge. This allows us to leverage our research from a MSBR level to a MABR level. At the MABR level supported by shared knowledge structure, agents can dynamically manage their shared knowledge with other agents and participate in group decision making processes in maintaining the consistency of the shared beliefs. We define a knowledge grade for a belief according to the level of accessibility. MABR can therefore be achieved through knowledge migration where a piece of information can be upgraded and degraded. The investigated concepts, such as various revision schemes, trust, information pedigree, shared beliefs and ontologies, eventually come together within a flexible and general framework. The framework is adaptable for incorporating new belief revision schemes and viable for a heterogenous multi-agent environment. In particular, as the layering approach is adopted, a belief revision agent designed under such a framework is capable of providing services at various levels. It can model single belief revision behaviour, revision behaviour for information from multiple sources, as well as cooperative revision in multi-agent situation where revision on shared or common knowledge is desired. This provides a tangible architecture of a belief revision service
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provider for an dynamic, heterogenous, autonomous open environment.
11.2
Limitations and Implications for Future Research
The belief revision agent we described is service oriented, the agent instances may operate on a request agent’s knowledge base. This may raise security concerns. Intelligent agents are considered to be autonomous, thus able to making decisions without human interventions. Therefore, if the infrastructure does not enforce the security and privacy rules that it should abide by, a belief revision agent may not be stopped from trading some of the trustor’s private beliefs to increase its own utilities. Therefore, in some case, a subsumption structure would be better, where the belief revision agent works as an internal process of a manager agent. The manager agent can initiate a belief revision agent and delegate revision tasks to it. The revision agent could function as a passive process of the manager agent and have no external access directly, meaning all inter-agent communication should be conducted through the manager agent. Alternatively, such belief revision functionalities (SBR, MSBR and MABR) can be implemented into capabilities, which can then be kept internal to the agent itself. Further research into an infrastructure that addresses the security and privacy concerns is certainly of interest and would have practical implications to real applications. Although JADE is used as the agent platform and the belief revision agent extends a JADE base agent, the framework does not preclude the possibility of using other agent development environment and infrastructure to achieve more sophisticated agent behaviour. For example, JACK - a BDI agent programming environment
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- focuses more on the agent internal behaviour as compared to JADE agent, which focuses more on the external behaviour such as communication and etc. JACK agents exhibit goal-directed behaviour and are more suitable to be described as intelligent agents. JACK provides a more robust environment for modelling agent internal states such as belief sets, which simplify the implementation of social beliefs and domain beliefs. However, JADE agents excel at ontology description, supporting FIPA compliant features for agent management and communication. With the newly developed FIPA-JACK1 and further work into it, it is possible to combine the strength of two agent development environments to have a better agent wrapper for belief revision services. On the MABR level, group decision making processes are often needed to achieve the possible consensus on the shared beliefs. This coincides with the research into knowledge merging, such as arbitration merging[74] and majority merging[75]. Further research into the connection between knowledge migration and different merging operators would be exceedingly useful. In particular, domains such as cooperative information systems, distributed databases, resolving conflicts among multiple agents, would be appealing application areas for such research. It is also possible for our multi-agent belief revision design to serve as a flexible test-bed for various merging operators.
1
http://www.cs.rmit.edu.au/agents/protocols/
Appendix A Agent Environment Properties • Accessible vs inaccessible. An accessible environment is one in which the agent can obtain complete, accurate, up-to-date information about the environment’s state. Most moderately complex environments(includeing, for example, the everyday physical world and the Internet) are in accessbile. The more accessible an envirom=nment is, the simpler it is to build agents to operate in it. • Deterministic vs non-deterministic. As we have already metnioned, a deterministic environment is one in which any action has a single guaranteed effect - there is no uncertainty about the state that will result from performing an action. The pysical world can to all intents and purposes be regarded as non-deterministic. Non-deterministic environments present greater problems for the agent designer. • Episodic vs non-episodic. In an episodic environment, the performance of an agent is dependent on a number of discrete episodes, which no link between the performance of an agent in different scenarios. An example of an episodic environment would be a mail 175
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sorting system. Episodic environments are simpler from the agent developers’ perspective because the agent can decide what action to perform based only on the current episode - it need not reason about the interactions between this and furture episode. • Static vs dynamic. A static environment is one that can be assumed to remain unchanged except by the performance of actions by the agent. A dynamic environment is one that has other processes operating on it, and which hence changes in ways beyond the agent’s control. The physical world is a highly dynamic environment. • Discrete vs continuous. An environment is discrete if there are a fixed, finite number of actions and percepts in it. A chess game is an example of discrete environment, and taxi drivinng is an example of a continous one. As Russell and Norvig observe[96], if an environment is sufficiently complex, then the fact that it is actually deterministic is not much help: to all intents and purposes, it may as well be non-deterministic. The most complex general class of environments are those that are inaccessible, non-deterministic, non-episodic, dynamic, and continuous.
Appendix B Logical Foundations In order to formalise the above examples and introduce belief revision theory formally, this section will present the logical notations and terminologies that we are going to use throughout the thesis. Facts (e.g. It is raining) are expressed as sentences1 in some formal language L (e.g. propositional logic or first order logic). Assume the language L is closed under the logical connectives (i.e. boolean operators): ¬ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), and ↔ (if and only if). Letters p, q, r .... are used for atomic sentence and letters A, B, C ... for arbitrary sentences. They represent the sentences in L, thus the facts that an agent has in its knowledge/belief set or base. Each sentence has a meaning (semantics) - either t or f - relative to the meaning of the propositional symbols it contains. For example, sentence A ∨ B’s meaning is relative to the t and f of both A and B. An interpretation, or truth assignment, for a set of sentences is a function from its set of propositional symbols to {t, f }. 1
closed formulae.
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An interpretation satisfies a sentence if the sentence evaluates to t under the interpretation. In order words, a sentence is satisfiable if there exist a truth assignment that can make the forumla true. A set S of sentences entails A if every interpretation that satisfies all elements of S, also satisfies A. This is denoted S ` A. It is usual to write A ` B instead of {A} ` B, where {A} is a one-element set with the only element A. A set S of sentences is valid or tautology if every interpretation for S satisfies every formula in S. In other words, no matter what truth values that are assigned to S, every sentence in S evaluates to t. This is denoted ` S. In propositional logic, a valid sentence is also called a tautology, which is denoted similiarly ` A. A set S of sentences satisfiable (or consistent) if there is some interpretation for S that satisfies every sentence in S. A set S of sentences is unsatisfiable (or inconsistent) if it is not satisfiable, ie, there is no interpretation for S that satisfies every sentence in S. An inconsistent sentence A is denoted 6` A. Formulae A and B are equivalent, A ' B, provided that A ` B and B ` A. For example:
A → B ' ¬A ∨ B ¬A ' A → f A belief base is a set of sentences in L. A belief set is a set of sentences in L which satisfies the integrity constraint: (I) If K logically entails B, then B ∈ K. (Integrity Constraint)
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In other words, a belief set includes both explicit beliefs(ie, sentences) plus all of the implicit beliefs which can be derived from them.
Appendix C JADE Implementation of Ontologies V2.4 The package jade.onto.basic includes a set of classes that are commonly part of every ontology, such as Action, TruePredicate, FalsePredicate, ResultPredicate, The BasicOntology can be joined to any user-defined ontology as described in section 3.6. Notice that the Action class should be used to represent actions. It has a couple of methods to set/get the AID of the actor (i.e. the agent who should perform the action) and the action itself (e.g. Register/Deregister/Modify).
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ContentElement
GenericAction
CommunicationAct
E.g. Inform, Request
AgentAction
E.g. Buy, Sell
Term
Proposition
Predicate
E.g. FatherOf, SonOf
ActionPredicate
E.g. Done
HigerOrderPredicate
E.g. Believe, Intend
Figure C.1: Ontology Elements Represented as JADE Classes[30]
Concept
E.g. Person, Address
Appendix D FIPA Agent Platform Components The agent management reference model consists of the following logical components, each representing a capability set. These can be combined in physical implementations of Agent Platforms (AP): • An Agent is the fundamental actor on an AP which combines one or more service capabilities into a unified and integrated execution model that may include access to external software, human users and communications facilities. An agent must have at least one owner, for example, based on organisational affiliation or human user ownership, and an agent may support several notions of identity. An Agent Identifier (AID) labels an agent so that it may be distinguished unambiguously within the Agent Universe. An agent may be registered at a number of transport addresses at which it can be contacted and it may have certain resource brokering capabilities for accessing software. • A Directory Facilitator (DF) is a mandatory component of an AP. The DF provides yellow pages services to other agents. Agents may register their services with the DF or query the DF to find out what services are offered by other agents. Multiple DFs may exist within an AP and may be federated. 182
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• An Agent Management System (AMS) is a mandatory component of an AP. The AMS exerts supervisory control over access to and use of the agent platform. Only one AMS will exist in a single agent platform. The AMS maintains a directory of AIDs which contain transport addresses (amongst other things) for agents registered with the AP. The AMS offers white pages services to other agents. Each agent must register with an AMS in order to get a valid AID. • An Message Transport Service (MTS) is the default communication method between agents on different APs. • An Agent Platform (AP) provides the physical infrastructure in which agents can be deployed. The AP consists of the machine(s), operating system, agent support software, FIPA agent management components (DF, AMS and MTS) and agents. The internal design of an AP is an issue for agent system developers and is not a subject of standardisation within FIPA. APs and the agents which are native to those APs, either by creation directly within or migration to the AP, may use any proprietary method of inter-communication. It should be noted that the concept of an AP does not mean that all agents resident on an AP have to be co-located on the same host computer. FIPA envisages a variety of different APs from single processes containing lightweight agent threads, to fully distributed APs built around proprietary or open middleware standards. • Software describes all non-agent, executable collections of instructions accessible through an agent. Agents may access software, for example, to add new
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services, acquire new communications protocols, acquire new security protocols/algorithms, acquire new negotiation protocols, access tools which support migration, etc.
Appendix E Publications 1. Wei Liu and Mary-Anne Williams, Trustworthiness of information sources and information pedigrees, Intelligent Agents VIII (Agent Theories, Architectures and Languages), John-Jules Ch. Meyer, Milind Tambe (Eds.), Springer (Hot Topics in LNAI), 2002. 2. Wei Liu and Mary-Anne Williams, A framework for multi-agent belief revision, Journal of Studia Logia, edited by David Makinson, 2001. 3. Wei Liu and Mary-Anne Williams, A framework for multi-agent belief revision(part ii : A layered model and shared knowledge structure), in the Proceedings of the 8th International Workshop on Non-Monotonic Reasoning (NMR2000) (Colorado, USA), 2000. 4. Wei Liu and Mary-Anne Williams, A framework for multi-agent belief revision (part i: The role of ontology), in the Proceedings of 12th Australian Joint Conference on Artificial Intelligence (Sydney, Australia) (Norman Foo, ed.), Lecture Notes in Artificial Intelligence, Springer-Verlag, 1999.
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