page_title: Dynamic epistemic logic
section_title: Public Events
hierarchical_section_title: Dynamic epistemic logic / Adding Dynamics / Public Events
caption_attribution_description: English: DEL: Muddy children
context_page_description: Dynamic epistemic logic 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: for example a red card is painted in blue. They can also bring about changes of knowledge without changing factual properties of the world: for example a card is revealed publicly 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, philosophy, economics 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.
context_section_description: In this section, we assume that all events are public. We start by giving a concrete example where DEL can be used, to better understand what is going on. This example is called the muddy children puzzle. Then, we will present a formalization of this puzzle in a logic called Public Announcement Logic (PAL). The muddy children puzzle is one of the most well known puzzles that played a role in the development 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.
  Muddy Children Example:
  We have two children, A and B, both dirty. A can see B but not himself, and B can see A but not herself. Let  be the proposition stating that A is dirty, and  be the proposition stating that B is dirty.
  We represent the initial situation by the pointed epistemic model  represented below, where relations between worlds are equivalence relations. States  intuitively represent possible worlds, a proposition (for example ) satisfiable at one of these worlds intuitively means that in the corresponding possible world, the intuitive interpretation of  (A is dirty) is true. The links between worlds labelled by agents (A or B) intuitively express a notion of indistinguishability for the agent at stake between two possible worlds. For example, the link between  and  labelled by A intuitively means that A can not distinguish the possible world  from  and vice versa. Indeed, A cannot see himself, so he cannot distinguish between a world where he is dirty and one where he is not dirty. However, he can distinguish between worlds where B is dirty or not because he can see B. With this intuitive interpretation we are brought to assume that our relations between worlds are equivalence relations.
  Now, suppose that their father comes and announces that at least one is dirty (formally, ). Then we update the model and this yields the pointed epistemic model represented below. What we actually do is suppressing the worlds where the content of the announcement is not fulfilled. In our case this is the world where  and  are true. This suppression is what we call the update. We then get the model depicted below. As a result of the announcement, both A and B do know that at least one of them is dirty. We can read this from the epistemic model.
  Now suppose there is a second (and final) announcement that says that neither knows they are dirty (an announcement can express facts about the situation as well as epistemic facts about the knowledge held by the agents). We then update similarly the model by suppressing the worlds which do not satisfy the content of the announcement, or equivalently by keeping the worlds which do satisfy the announcement. This update process thus yields the pointed epistemic model represented below. By interpreting this model, we get that A and B both know that they are dirty, which seems to contradict the content of the announcement. However, if we assume that A and B are both perfect reasoners and that this is common knowledge among them, then this inference makes perfect sense.
  Public announcement logic (PAL):
  We present the syntax and semantic of Public Announcement Logic (PAL), which combines features of epistemic logic and propositional dynamic logic.
  We define the language  inductively by the following grammar in BNF:
  where .
  The language  is interpreted over epistemic models. The truth conditions for the connectives of the epistemic language are the same as in epistemic logic (see above). The truth condition for the new dynamic action modality  is defined as follows:
  where  with
  ,
   for all  and
  .
  
  The formula  intuitively means that after a truthful announcement of ,  holds. A public announcement of a proposition  changes the current epistemic model like in the figure below.
  The proof system  defined below is sound and strongly complete for  w.r.t. the class of all pointed epistemic models.
  The axioms Red 1 - Red 4 are called reduction axioms because they allow to reduce any formula of  to a provably 
caption: Initial situation: pointed epistemic model 
    
      
        
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