Interpolation in modal logics

Abstract

Since Craig's landmark result on interpolation for classical predicate logic, proved as the main technical lemma in [14], interpolation is considered one of the centra! concepts in pure logic. Various interpolation properties find their applications in computer science and have many deep purely logical consequences. We focus on two propositional versions of Craig interpolation property: Craig Interpolation Property: for every provable implication (A -+ B) there is an interpolant I containing only only common variables of A and B such that both implications (A -+ I) and (I-+ B) are provable. Craig interpolation, although it seems rather technical, is a deep logical property. It is dosely related to expressive power of a logic - as such it entails Beth's definability property, or forces functional completeness. It is also related to Robinson's joint consistency of two theories that agree on the common language. Craig interpolation has an important algebraic counterpart - it entails amalgamation or superamalgamation property of appropriate algebraic structures. In case of modal provability logics, Craig interpolation entails fixed point theorem. There are other interpolation properties, defined w.r.t. a consequence relation rather then w.r.t. a provable implication. In presence of deduction theorem the two...