Dynamic epistemic logic (DEL) is a logical framework dealing with knowledge and information change. Typically, DEL focuses on situations involving multiple agents and studies how their knowledge changes when events occur. These events can change factual properties of the actual world (they are called ontic events): for example a red card is painted in blue. They can also bring about changes of knowledge without changing factual properties of the world (they are called epistemic events): for example, a card is revealed publicly (or privately) to be red. Originally, DEL focused on epistemic events. Only some of the basic ideas are present in this entry of the original DEL framework; more details about DEL in general can be found in the references.
Due to the nature of its object of study and its abstract approach, DEL is related and has applications to numerous research areas, such as computer science (artificial intelligence), philosophy (formal epistemology), economics (game theory) and cognitive science. In computer science, DEL is for example very much related to multi-agent systems, which are systems where multiple intelligent agents interact and exchange information.
As a combination of dynamic logic and epistemic logic, dynamic epistemic logic is a young field of research. It really started in 1989 with Plaza's logic of public announcement. Independently, Gerbrandy and Groeneveld proposed a system dealing moreover with private announcement and that was inspired by the work of Veltman. Another system was proposed by van Ditmarsch whose main inspiration was the Cluedo game. But the most influential and original system was the system proposed by Baltag, Moss and Solecki. This system can deal with all the types of situations studied in the works above and its underlying methodology is conceptually grounded. This entry will present some of its basic ideas.
Formally, DEL extends ordinary epistemic logic by the inclusion of event models to describe actions, and a product update operator that defines how epistemic models are updated as the consequence of executing actions described through event models. Epistemic logic will first be recalled. Then, actions and events will enter into the picture and we will introduce the DEL framework.A distinction is sometimes made between events and actions, an action being a specific type of event performed by an agent.
Epistemic logic
Epistemic logic is a modal logic dealing with the notions of knowledge and belief. As a logic, it is concerned with understanding the process of reasoning about knowledge and belief: which principles relating the notions of knowledge and belief are intuitively plausible? Like epistemology, it stems from the Greek word or ‘episteme’ meaning knowledge. Epistemology is nevertheless more concerned with analyzing the very nature and scope of knowledge, addressing questions such as “What is the definition of knowledge?” or “How is knowledge acquired?”. In fact, epistemic logic grew out of epistemology in the Middle Ages thanks to the efforts of Burley and Ockham. The formal work, based on modal logic, that inaugurated contemporary research into epistemic logic dates back only to 1962 and is due to Hintikka. It then sparked in the 1960s discussions about the principles of knowledge and belief and many axioms for these notions were proposed and discussed. For example, the interaction axioms and are often considered to be intuitive principles: if an agent Knows then (s)he also Believes , or if an agent Believes , then (s)he Knows that (s)he Believes . More recently, these kinds of philosophical theories were taken up by researchers in economics, artificial intelligence and theoretical computer science where reasoning about knowledge is a central topic. Due to the new setting in which epistemic logic was used, new perspectives and new features such as computability issues were then added to the research agenda of epistemic logic.
Syntax
In the sequel, is a finite set whose elements are called agents and is a set of propositional letters.
The epistemic language is an extension of the basic multi-modal language of modal logic with a common knowledge operator and a distributed knowledge operator . Formally, the epistemic language is defined inductively by the following grammar in BNF:
where , and . The basic epistemic language is the language without the common knowledge and distributed knowledge operators. The formula is an abbreviation for (for a given ), is an abbreviation for , is an abbreviation for and an abbreviation for .
Group notions: general, common and distributed knowledge.
In a multi-agent setting there are three important epistemic concepts: general knowledge, distributed knowledge and common knowledge. The notion of common knowledge was first studied by Lewis in the context of conventions. It was then applied to distributed systems and to game theory, where it allows to express that the rationality of the players, the rules of the game and the set of players are commonly known.
General knowledge.
General knowledge of means that everybody in the group of agents knows that . Formally, this corresponds to the following formula:
Common knowledge.
Common knowledge of means that everybody knows but also that everybody knows that everybody knows , that everybody knows that everybody knows that everybody knows , and so on ad infinitum. Formally, this corresponds to the following formula
As we do not allow infinite conjunction the notion of common knowledge will have to be introduced as a primitive in our language.
Before defining the language with this new operator, we are going to give an example introduced by Lewis that illustrates the difference between the notions of general knowledge and common knowledge. Lewis wanted to know what kind of knowledge is needed so that the statement : “every driver must drive on the right” be a convention among a group of agents. In other words, he wanted to know what kind of knowledge is needed so that everybody feels safe to drive on the right. Suppose there are only two agents and . Then everybody knowing (formally ) is not enough. Indeed, it might still be possible that the agent considers possible that the agent does not know (formally ). In that case the agent will not feel safe to drive on the right because he might consider that the agent , not knowing , could drive on the left. To avoid this problem, we could then assume that everybody knows that everybody knows that (formally ). This is again not enough to ensure that everybody feels safe to drive on the right. Indeed, it might still be possible that agent considers possible that agent considers possible that agent does not know (formally ). In that case and from ’s point of view, considers possible that , not knowing , will drive on the left. So from ’s point of view, might drive on the left as well (by the same argument as above). So will not feel safe to drive on the right. Reasoning by induction, Lewis showed that for any , is not enough for the drivers to feel safe to drive on the right. In fact what we need is an infinite conjunction. In other words, we need common knowledge of : .
Distributed knowledge.
Distributed knowledge of means that if the agents pooled their knowledge altogether, they would know that holds. In other words, the knowledge of is distributed among the agents. The formula reads as ‘it is distributed knowledge among the set of agents that holds’.
Semantics
Epistemic logic is a modal logic. So, what we call an epistemic model is just a Kripke model as defined in modal logic. The set is a non-empty set whose elements are called possible worlds and the interpretation is a function specifying which propositional facts (such as ‘Ann has the red card’) are true in each of these worlds. The accessibility relations are binary relations for each agent ; they are intended to capture the uncertainty of each agent (about the actual world and about the other agents' uncertainty). Intuitively, we have when the world is compatible with agent ’s information in world or, in other words, when agent considers that world might correspond to the world (from this standpoint). We abusively write for and denotes the set of worlds .
Intuitively, a pointed epistemic model , where , represents from an external point of view how the actual world is perceived by the agents .
For every epistemic model , every and every , we define inductively by the following truth conditions:
iffiffiffiffiffiff
where is the transitive closure of : we have that if, and only if, there are and such that and for all , .
Despite the fact that the notion of common belief has to be introduced as a primitive in the language, we can notice that the definition of epistemic models does not have to be modified in order to give truth value to the common knowledge and distributed knowledge operators.
Card Example:
Players , and (standing for Ann, Bob and Claire) play a card game with three cards: a red one, a green one and a blue one. Each of them has a single card but they do not know the cards of the other players. Ann has the red card, Bob has the green card and Claire has the blue card. This example is depicted in the pointed epistemic model represented below. In this example, and . Each world is labelled by the propositional letters which are true in this world and corresponds to the actual world. There is an arrow indexed by agent from a possible world to a possible world when . Reflexive arrows are omitted, which means that for all and all , we have that .
stands for : " has the red card''
stand for: " has the blue card''
stands for: " has the green card''
and so on...
When accessibility relations are equivalence relations (like in this example) and we have that , we say that agent cannot distinguish world from world (or world is indistinguishable from world for agent ). So, for example, cannot distinguish the actual world from the possible world where has the blue card (), has the green card () and still has the red card ().
In particular, the following statements hold:
'All the agents know the color of their card'.
' knows that has either the blue or the green card and that has either the blue or the green card'.
'Everybody knows that has either the red, green or blue card and this is even common knowledge among all agents'.
Knowledge versus Belief We use the same notation for both knowledge and belief. Hence, depending on the context, will either read ‘the agent Knows that holds’ or ‘the agent Believes that holds’. A crucial difference is that, unlike knowledge, beliefs can be wrong: the axiom holds only for knowledge, but not necessarily for belief. This axiom called axiom T (for Truth) states that if the agent knows a proposition, then this proposition is true. It is often considered to be the hallmark of knowledge and it has not been subjected to any serious attack ever since its introduction in the Theaetetus by Plato.
The notion of knowledge might comply to some other constraints (or axioms) such as : if agent knows something, she knows that she knows it. These constraints might affect the nature of the accessibility relations which may then comply to some extra properties. So, we are now going to define some particular classes of epistemic models that all add some extra constraints on the accessibility relations . These constraints are matched by particular axioms for the knowledge operator . Below each property, we give the axiom which defines the class of epistemic frames that fulfill this property. ( stands for for any .)
+ Properties of accessibility relations and corresponding axiomsserialDtransitive4Euclidicity5reflexiveTsymmetricBconfluent.2weakly connected.3semi-Euclidean.3.2R1.4