Dynamic epistemic logic

Typically, DEL focuses on situations involving multiple agents and studies how their knowledge changes when events occur.

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.

[1] Independently, Gerbrandy and Groeneveld[2] proposed a system dealing moreover with private announcement and that was inspired by the work of Veltman.

[3] Another system was proposed by van Ditmarsch whose main inspiration was the Cluedo game.

[5][6] This system can deal with all the types of situations studied in the works above and its underlying methodology is conceptually grounded.

In fact, epistemic logic grew out of epistemology in the Middle Ages thanks to the efforts of Burley and Ockham.

More recently, these kinds of philosophical theories were taken up by researchers in economics,[11] artificial intelligence and theoretical computer science[12] where reasoning about knowledge is a central topic.

As we do not allow infinite conjunction the notion of common knowledge will have to be introduced as a primitive in our language.

In other words, he wanted to know what kind of knowledge is needed so that everybody feels safe to drive on the right.

is a function specifying which propositional facts (such as ‘Ann has the red card’) are true in each of these worlds.

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.

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.

So, we are now going to define some particular classes of epistemic models that all add some extra constraints on the accessibility relations

[16] Axiom 4 is nevertheless widely accepted by computer scientists (but also by many philosophers, including Plato, Aristotle, Saint Augustine, Spinoza and Schopenhauer, as Hintikka recalls ).

Most philosophers (including Hintikka) have attacked this axiom, since numerous examples from everyday life seem to invalidate it.

[17] In general, axiom 5 is invalidated when the agent has mistaken beliefs, which can be due for example to misperceptions, lies or other forms of deception.

Axiom B states that it cannot be the case that the agent considers it possible that she knows a false proposition (that is,

They can be characterized in terms of intuitive interaction axioms relating knowledge and beliefs.

[19] The Hilbert proof system K for the basic modal logic is defined by the following axioms and inference rules: for all

–models is the class of epistemic models whose accessibility relations satisfy the properties listed above defined by the axioms of

, if we restrict to finite nesting, then the satisfiability problem is NP-complete for all the modal logics considered.

If we then further restrict the language to having only finitely many primitive propositions, the complexity goes down to linear in time in all cases.

Dynamic Epistemic Logic (DEL) is a logical framework for modeling epistemic situations involving several agents, and changes that occur to these situations as a result of incoming information or more generally incoming action.

: this public announcement and correlative update constitute the dynamic part.

However, epistemic events can be much more complex than simple public announcement, including hiding information for some of the agents, cheating, lying, bluffing, etc.

We will first focus on public announcements to get an intuition of the main underlying ideas of DEL.

Other significant puzzles include the sum and product puzzle, the Monty Hall dilemma, the Russian cards problem, the two envelopes problem, Moore's paradox, the hangman paradox, etc.

changes the current epistemic model like in the figure below.The proof system

In this last statement, we see at work an interesting feature of the update process: a formula is not necessarily true after being announced.

The insight of the DEL approach is that one can describe how an event is perceived by the agents in a very similar way.