[For the entire collection see Zbl 0741.00073.]
A formal analysis of reasoning about knowledge is presented in the paper. A Kripke model for a (two person) knowledge situation is defined. It consists of a state space \(W\) and two equivalence relations \(\equiv_ 1,\equiv_ 2\). Intuitively, \(s\equiv_ i t\) means that states \(s,t\) are indistinguishable to an individual \(i\), where \(i=1,2\). The crucial concept is that of a dialogue system (DS), which is a mapping \(f: W\times N^ +\to\text{``no''}\cup W\). The mapping \(f\) “generates” responses in the dialogue. An argument from \(N^ +\) represents the stage of the dialogue. The values of \(f\) are, intuitively, answers in a dialogue (“no” means “I don’t know”, \(f(s,n)=t\) means: “the current situation is \(t\)”) and they depend only on the equivalence relations and on the previous answers in the dialogue. Interesting dialogue systems are sound (\(f(s,n)=s\), if \(f(s,n)\neq\text{``no''}\)) and optimal (any other sound \(g\) gives a positive answer later, i.e. if \(n,m\) are the minimal stages such that \(f(s,n)=s\) and \(g(s,m)=s\), then \(n\leq m\) for all \(s\)).
A simple game is used as an introductory example. The game is analyzed in the formal terms of the paper. It is shown that a natural DS provides the optimal strategy for the game. Special attention is devoted to infinite dialogues. It is shown that sometimes knowledge can only be acquired in an infinite dialogue. The probabilistic case is studied, too. If justified risk is considered, the situation improves dramatically.