I'd like to understand the modal logic $K$ better. Some modal logics appear as "modal companion" of some superintuitionistic logic. Is there some logic over the classical propositional language (i.e. without the modalities) which has $K$ as modal companion? Or is there some logic over the classical propositional language, which has frame semantics similar to intuitionistic logic, but over the class of all Kripke frames, like $K$? Does the Gödel-translation need to be adapted to achieve this?
With classical propositional language I mean the set of fornulae generated by countably many propositional variables, the constant $\bot$ and the binary operations $\land, \lor, \to$.
Update: I found the "basic logic" by Visser, which has $K4$ as modal companion. This logic does not satisfy the usual modus ponens, but only a weakened version of it. One can probably adapt the definitions to obtain an even weaker logic which has $K$ as modal companion.