Abstract
This thesis is about logical models of belief (and knowledge) representation and belief change. This means that we propose logical systems which are intended to represent how agents perceive a situation and reason about it, and how they update their beliefs about this situation when events occur. These agents can be machines, robots, human beings. . . but they are assumed to be somehow autonomous.
The way a fixed situation is perceived by agents can be represented by statements about the agents' beliefs: for example 'agent A believes that the door of the room is open' or 'agent A believes that her colleague is busy this afternoon'. 'Logical systems' means that agents can reason about the situation and their beliefs about it: if agent A believes that her colleague is busy this afternoon then agent A infers that he will not visit her this afternoon. We moreover often assume that our situations involve several agents which interact between each other. So these agents have beliefs about the situation (such as 'the door is open') but also about the other agents' beliefs: for example agent A might believe that agent B believes that the door is open. These kinds of beliefs are called higher-order beliefs. Epistemic logic [Hintikka, 1962; Fagin et al., 1995; Meyer and van der Hoek, 1995], the logic of belief and knowledge, can capture all these phenomena and will be our main starting point to model such fixed ('static') situations. Uncertainty can of course be expressed by beliefs and knowledge: for example agent A being uncertain whether her colleague is busy this afternoon can be expressed by 'agent A does not know whether her colleague is busy this afternoon'. But we sometimes need to enrich and refine the representation of uncertainty: for example, even if agent A does not know whether her colleague is busy this afternoon, she might consider it more probable that he is actually busy. So other logics have been developed to deal more adequately with the representation of uncertainty, such as probabilistic logic, fuzzy logic or possibilistic logic, and we will refer to some of them in this thesis (see [Halpern, 2003] for a survey on reasoning about uncertainty).
But things become more complex when we introduce events and change in the picture. Issues arise even if we assume that there is a single agent. Indeed, if the incoming information conveyed by the event is coherent with the agent's beliefs then the agent can just add it to her beliefs. But if the incoming information contradicts the agent's beliefs then the agent has somehow to revise her beliefs, and as it turns out there is no obvious way to decide what should be her resulting beliefs. Solving this problem was the goal of the logic-based belief revision theory developed by Alchourrón, Gärdenfors and Makinson (to which we will refer by the term AGM) [Alchourrón et al., 1985; Gärdenfors, 1988; Gärdenfors and Rott, 1995]. Their idea is to introduce 'rationality postulates' that specify which belief revision operations can be considered as being 'rational' or reasonable, and then to propose specific revision operations that fulfill these postulates. However, AGM does not consider situations where the agent might also have some uncertainty about the incoming information: for example agent A might be uncertain due to some noise whether her colleague told her that he would visit her on Tuesday or on Thursday. In this thesis we also investigate this kind of phenomenon. Things are even more complex in a multi-agent setting because the way agents update their beliefs depends not only on their beliefs about the event itself but also on their beliefs about the way the other agents perceived the event (and so about the other agents' beliefs about the event). For example, during a private announcement of a piece of information to agent A the beliefs of the other agents actually do not change because they believe nothing is actually happening; but during a public announcement all the agents' beliefs might change because they all believe that an announcement has been made. Such kind of subtleties have been dealt with in a field called dynamic epistemic logic (Gerbrandy and Groeneveld, 1997; Baltag et al., 1998; van Ditmarsch et al., 2007b]. The idea is to represent by an event model how the event is perceived by the agents and then to define a formal update mechanism that specifies how the agents update their beliefs according to this event model and their previous representaton of the situation. Finally, the issues concerning belief revision that we raised in the single agent case are still present in the multi-agent case.
So this thesis is more generally about information and information change. However, we will not deal with problems of how to store information in machines or how to actually communicate information. Such problems have been dealt with in information theory [Cover and Thomas, 1991] and Kolmogorov complexity theory [Li and Vitányi, 1993]. We will just assume that such mechanisms are already available and start our investigations from there.
Studying and proposing logical models for belief change and belief representation has applications in several areas. First in artificial intelligence, where machines or robots need to have a formal representation of the surrounding world (which might involve other agents), and formal mechanisms to update this representation when they receive incoming information. Such formalisms are crucial if we want to design autonomous agents, able to act autonomously in the real world or in a virtual world (such as on the internet). Indeed, the representation of the surrounding world is essential for a robot in order to reason about the world, plan actions in order to achieve goals... and it must be able to update and revise its representation of the world itself in order to cope autonomously with unexpected events. Second in game theory (and consequently in economics), where we need to model games involving several agents (players) having beliefs about the game and about the other agents' beliefs (such as agent A believes that agent B has the ace of spade, or agent A believes that agent B believes that agent A has the ace of heart...), and how they update their representation of the game when events (such as showing privately a card or putting a card on the table) occur. Third in cognitive psychology, where we need to model as accurately as possible epistemic state of human agents and the dynamics of belief and knowledge in order to explain and describe cognitive processes.
The thesis is organized as follows. In Chapter 2, we first recall epistemic logic. Then we observe that representing an epistemic situation involving several agents depends very much on the modeling point of view one takes. For example, in a poker game the representation of the game will be different depending on whether the modeler is a poker player playing in the game or the card dealer who knows exactly what the players' cards are. In this thesis, we will carefully distinguish these different modeling approaches and the. different kinds of formalisms they give rise to. In fact, the interpretation of a formalism relies quite a lot on the nature of these modeling points of view. Classically, in epistemic logic, the models built are supposed to be correct and represent the situation from an external and objective point of view. We call this modeling approach the perfect external approach. In Chapter 2, we study the modeling point of view of a particular modeler-agent involved in the situation with other agents (and so having a possibly erroneous perception of the situation). We call this modeling approach the internal approach. We propose a logical formalism based on epistemic logic that this agent uses to represent 'for herself' the surrounding world. We then set some formal connections between the internal approach and the (perfect) external approach. Finally we axiomatize our logical formalism and show that the resulting logic is decidable.
In Chapter 3, we first recall dynamic epistemic logic as viewed by Baltag, Moss and Solecki (to which we will refer by the term BMS). Then we study in which case seriality of the accessibility relations of epistemic models is preserved during an update, first for the full updated model and then for generated submodels of the full updated model. Finally, observing that the BMS formalism follows the (perfect) external approach, we propose an internal version of it, just as we proposed an internal version of epistemic logic in Chapter 2.
In Chapter 4, we still follow the internal approach and study the particular case where the event is a private announcement. We first show, thanks to our study in Chapter 3, that in a multi-agent setting, expanding in the AGM style corresponds to performing a private announcement in the BMS style. This indicates that generalizing AGM belief revision theory to a multi-agent setting amounts to study private announcement. We then generalize the AGM representation theorems to the multi-agent case. Afterwards, in the spirit of the AGM approach, we go beyond the AGM postulates and investigate multi-agent rationality postulates specific to our multi-agent setting inspired from the fact that the kind of phenomenon we study is private announcement. Finally we provide an example of revision operation that we apply to a concrete example.
In Chapter 5, we follow the (perfect) external approach and enrich the BMS formalism with probabilities. This enables us to provide a fined-grained account of how human agents interpret events involving uncertainty and how they revise their beliefs. Afterwards, we review different principles for the notion of knowledge that have been proposed in the literature and show how some principles that we argue to be reasonable ones can all be captured in our rich and expressive formalism. Finally, we extend our general formalism to a multi-agent setting.
In Chapter 6, we still follow the (perfect) external approach and enrich our dynamic epistemic language with converse events. This language is interpreted on structures with accessibility relations for both beliefs and events, unlike the BMS formalism where events and beliefs are not on the same formal level. Then we propose principles relating events and beliefs and provide a complete characterization, which yields a new logic EDL. Finally, we show that BMS can be translated into our new logic EDL thanks to the converse operator: this device enables us to translate the structure of the event model directly within a particular axiomatization of EDL, without having to refer to a particular event model in the language (as done in BMS).
In Chapter 7 we summarize our results and give an overview of remaining technical issues and some desiderata for future directions of research.
Parts of this thesis are based on publication, but we emphasize that they have been entirely rewritten in order to make this thesis an integrated whole. Sections 4.2.2 and 4.3 of Chapter 4 are based on [Aucher, 2008]. Sections 5.2, 5.3 and 5.5 of Chapter 5 are based on [Aucher, 2007]. Chapter 6 is based on [Aucher and Herzig, 2007].