Dynamic epistemic logic

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. We only present in this entry some of the basic ideas of the original DEL framework; more details about DEL in general can be found in the references. 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. We only present in this entry some of the basic ideas 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. We will present in this entry 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. 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 ϵ π ι σ τ η μ η {displaystyle epsilon pi iota sigma au eta mu eta } 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 K p → B p {displaystyle Kp ightarrow Bp} and B p → K B p {displaystyle Bp ightarrow KBp} are often considered to be intuitive principles: if an agent Knows p {displaystyle p} then (s)he also Believes p {displaystyle p} , or if an agent Believes p {displaystyle p} , then (s)he Knows that (s)he Believes p {displaystyle p} . 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. In the sequel, A G T S = { 1 , … , n } {displaystyle AGTS={1,ldots ,n}} is a finite set whose elements are called agents and P R O P {displaystyle PROP} 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 C A {displaystyle C_{A}} and a distributed knowledge operator D A {displaystyle D_{A}} . Formally, the epistemic language L EL C {displaystyle {mathcal {L}}_{ extsf {EL}}^{C}} is defined inductively by the following grammar in BNF: L EL C : ϕ     ::=     p   ∣   ¬ ϕ   ∣   ( ϕ ∧ ϕ )   ∣   K j ϕ   ∣   C A ϕ   ∣   D A ϕ {displaystyle {mathcal {L}}_{ extsf {EL}}^{C}:phi ~~::=~~p~mid ~ eg phi ~mid ~(phi land phi )~mid ~K_{j}phi ~mid ~C_{A}phi ~mid ~D_{A}phi } where p ∈ P R O P {displaystyle pin PROP} , j ∈ A G T S {displaystyle jin {AGTS}} and A ⊆ A G T S {displaystyle Asubseteq {AGTS}} . The basic epistemic language L E L {displaystyle {mathcal {L}}_{EL}} is the language L E L C {displaystyle {mathcal {L}}_{EL}^{C}} without the common knowledge and distributed knowledge operators. The formula ⊥ {displaystyle ot } is an abbreviation for ¬ p ∧ p {displaystyle eg pland p} (for a given p ∈ P R O P {displaystyle pin PROP} ), ⟨ K j ⟩ ϕ {displaystyle langle K_{j} angle phi } is an abbreviation for ¬ K j ¬ ϕ {displaystyle eg K_{j} eg phi } , E A ϕ {displaystyle E_{A}phi } is an abbreviation for ⋀ j ∈ A K j ϕ {displaystyle igwedge limits _{jin A}K_{j}phi } and C ϕ {displaystyle Cphi } an abbreviation for C A G T S ϕ {displaystyle C_{AGTS}phi } .