The Structure of Rigid Frames of Restricted Depth

In this paper 1 we describe in precise the structure of all rigid modal rooted frames of depth 2. The interest to rigid frames came from the paper [5] where it has been showed that any given quasi-characterizing inference rule is self-admissible if and only if the frame of the algebra generated that rule is not rigid. The study of admissibility of inference rules was, in particular, motivated by H.Friedman problem [3] of determining the admissibility for inference rules in intuitionistic propositional logic IPC. The current state of the theory of admissible rules can be found in the monograph [6]. We note here only that the introduction of quasi-characteristic rules by Citkin [1], as a generalization of Jankov’s characterizing formulas [4], brought important results and a description of all quasi-characterizing rules admissible in IPC. As we noted above, it is proved that quasi-characterizing rule is self-admissible iff the frame of the algebra originated that rule is not rigid. Therefore in this paper we restrict our attention to describe the structure of all rigid frames of restricted depth. Since in whole the structure can be very complicated we will deal with only frames of depth 2. By a considerably simple reasoning we will be in a position to give very transparent description for all rigid frames of depth 2. And hence we are in a position to specify in precise self-admissible quasi-characteristic rules of algebras of depth 2.