Algorithmic Definability and Completeness in Modal Logic

Abstract

One of the nice features of modal languages is that sometimes they can talk about abstract properties of the corresponding semantic structures. For instance the truth of the modal formula $\square p\Rightarrow p$ in the Kripke frame (W,R) is equivalent to the reflexivity of the relation R. Using a terminology from modal logic [13], we say that the condition of reflexivity – (∀x)(xRx), is a first-order equivalent of the modal formula $\square p\Rightarrow p$, or, that the formula $\square p\Rightarrow p$ is first-order definable by the condition (∀x)(xRx). More over, adding the formula $\square p\Rightarrow p$ to the axioms of the minimal modal logic K we obtain a complete logic with respect to the class of reflexive frames and the completeness proof can be done by the well known in modal logic canonical method (such formulas are called canonical). Let us note that definability and completeness are some of the good features in the applications of modal logic, and hence it is important to have algorithmic methods for establishing such properties. In our talk we will describe several algorithmic approaches to this problem.