Unification in the normal modal logic Alt1

The unification problem in a logical system L can be defined in the following way: given a formula φ(x1, . . . , xα), determine whether there exists formulas ψ1, . . ., ψα such that φ(ψ1, . . . , ψα) is in L. The research on unification for modal logics was originally motivated by the admissibility problem for rules of inference: given a rule of inference φ1(x1, . . . , xα), . . . , φm(x1, . . . , xα)/ψ(x1, . . . , xα), determine whether for all formulas χ1, . . ., χα, if φ1(χ1, . . . , χα), . . ., φm(χ1, . . . , χα) are in L then ψ(χ1, . . . , χα) is in L [1]. Within the context of description logics, the main motivation for investigating the unification problem was to propose new reasoning services in the maintenance of knowledge bases like, for example, the elimination of redundancies in the descriptions of concepts [2]. Combining algebraic and model-theoretic methods, Rybakov [7] demonstrated that the admissibility problem and the unification problem in intuitionistic propositional logic and modal logic S4 are decidable. Later on, Ghilardi [4], proving that intuitionistic propositional logic has a finitary unification type, yielded a new solution of the admissibility problem, seeing that determining whether a given rule of inference preserves validity in intuitionistic propositional logic is equivalent to checking whether the finitely many maximal unifiers of its premises are unifiers of its conclusion. These results incited researchers to determine whether there exists finitely many admissible rules of inference of intuitionistic propositional logic and modal logic S4 so that the remaining admissible rules of inference would be derivable from them [5]. With respect to the issue of computational complexity, the admissibility problem and the unification problem were mostly unexplored before the work of Jerábek [6] who established the coNEXPTIME-completeness of the admissibility problem for several intuitionistic and modal logics extending K4 such as S4 and GL, in contrast with the satisfiability problem for these logics which is usually PSPACE-complete and in contrast with the unification problem for modal logics contained in K4 which is undecidable if one considers a language with the universal modality [8]. One may ask whether the situation is getting better if the language is restricted in one way or another. Recently, the admissibility problem in the negation-implication fragment of intuition-