Conservative extensions in modal logic

Every normal modal logic L gives rise to the consequence relation ' |=L which holds if, and only if, is true in a world of an L-model whenever ' is true in that world. We consider the following al- gorithmic problem for L. Given two modal formulas '1 and '2, decide whether '1^'2 is a conservative extension of'1 in the sense that whenever '1 ^'2 |=L and does not contain propositional variables not occurring in '1, then '1 |=L. We first prove that the conservativeness problem is coNExpTime-hard for all modal logics of unbounded width (which have rooted frames with more than N successors of the root, for any N < !). Then we show that this problem is (i) coNExpTime-complete for S5 and K, (ii) in ExpSpace for S4 and (iii) ExpSpace-complete for GL.3 (the logic of finite strict linear orders). The proofs for S5 and K use the fact that these logics have uniform interpolants of exponential size.