I have just begun reading about Epistemic Logic from Reasoning About Knowledge, Fagin et. al. $\mathcal K_i$ is defined as the possibility relation between two worlds, and the author says that for the most part - one would want $\mathcal K_i$ to be an equivalence relation. I do not understand the author's intuition.
Here is some setup to give you background - there are $n$ agents, $\{1,2,...,n\}$. As usual, $\Phi$ denotes a set of propositional variables, and $S$ is a non-empty set of states/possible worlds for the Kripke structure. $\pi$ is a valuation, which associates each state in $S$ with a map as follows: $$\pi(s): \Phi\to\{T,F\}$$ Then, the author says the following about $\mathcal K_i$:
The binary relation $\mathcal K_i$ is intended to capture the possibility relation according to agent $i$: $(s, t) \in \mathcal K_i$ if agent $i$ considers world $t$ possible, given his information in world $s$. We think of $\mathcal K_i$ as a possibility relation, since it defines what worlds agent $i$ considers possible in any given world. Throughout most of the book (in particular, in this chapter), we further require that $\mathcal K_i$ be an equivalence relation on $S$. We take $\mathcal K_i$ to be an equivalence relation since we want to capture the intuition that agent $i$ considers $t$ possible in world $s$ if in both $s$ and $t$ agent $i$ has the same information about the world, that is, the two worlds are indistinguishable to the agent. Making $\mathcal K_i$ an equivalence relation seems natural and turns out to be the appropriate choice for many applications.
I find it difficult to understand what intuition the authors are trying to capture with such a definition. Could someone help me better understand the motivation and reason behind such an assumption? Thanks a lot!
There are several motivations behind taking possibility relations to be equivalence relations. Some are logical, some have to do with applications of epistemic logic.
From a logical point of view, using equivalence models (i.e. modal models with equivalence relations) allows reducing modal models to a very simple form, at least in the monomodal case (just one possibility relation): you can equate such models with sets of classical propositional models (sets of atoms). Correspondingly, the satisfiability problem for formulas in equivalence models is extremely simple: it is NP-complete and so does not add any complexity to the case of classical propositional logic.
From a pragmatic point of view using equivalence models is a natural way of formalizing distributed systems, which is a core area of applying epistemic logic to computer science. Fagin et al. provide such applications in the very book you are citing.