Complete Axiomatizations of Finite Syntactic Epistemic States

An agent who bases his actions upon explicit logical formulae has at any given point in time a finite set of formulae he has computed. Closure or consistency conditions on this set cannot in general be assumed – reasoning takes time and real agents frequently have contradictory beliefs. This paper discusses a formal model of knowledge as explicitly computed sets of formulae. It is assumed that agents represent their knowledge syntactically, and that they can only know finitely many formulae at a given time. In order to express interesting properties of such finite syntactic epistemic states, we extend the standard epistemic language with an operator expressing that an agent knows at most a particular finite set of formulae, and investigate axiomatization of the resulting logic. This syntactic operator has also been studied elsewhere without the assumption about finite epistemic states [5]. A strongly complete logic is impossible, and the main results are non-trivial characterizations of the theories for which we can get completeness. The paper presents a part of a general abstract theory of resource bounded agents. Interesting results, e.g., complex algebraic conditions for completeness, are obtained from very simple assumptions, i.e., epistemic states as arbitrary finite sets and operators for knowing at least and at most.