Restricted Unification in the DL $$\mathcal {FL}_0$$

Unification types in Euclidean modal logics

“…As for their unification types, it is only known that extensions of K5 which contain K4 are of unification type 1 [13,27]. Indeed, results about unification types in modal logics are scarce 3 : Alt 1 ⊕ n ⊥ is of type 1 when n≥2 [10]; K⊕ n ⊥ is of type ω when n≥2 [10]; Alt 1 is of type 0 [12]; extensions of S4.3 are of unification type 1 [18]; transitive modal logics like K4 and S4 are of unification type ω [21,24]; K4.2 + is of type 1 [22,25]; K is of type 0 [26]; extensions of K4D1 are of unification type 1 [27]. And one observes that, while trying to determine the unification types of modal logics, little, if anything, from standard tools and techniques such as canonical models and filtrations is helpful 4 .…”

“…Fourthly, we prove in (Proposition 28) Section 6 that if L contains K5 and L is global 6 then for all substitutions σ, every formula L-unified by σ is implied in K by an L-projective formula based on the variables of the given formula and having σ as one of its L-unifiers. 2 Reciprocally, since a formula ϕ is L-unifiable if and only if the associated rule of inference ϕ ⊥ is not L-admissible, we can turn any algorithm deciding L-admissibility into an algorithm deciding L-unifiability. 3 For instance, the types of KD, KT, KB, KDB and KTB are not known.…”

“…2 Reciprocally, since a formula ϕ is L-unifiable if and only if the associated rule of inference ϕ ⊥ is not L-admissible, we can turn any algorithm deciding L-admissibility into an algorithm deciding L-unifiability. 3 For instance, the types of KD, KT, KB, KDB and KTB are not known. Indeed, the modal logics KD, KT, KB, KDB and KTB have been proved to be of type 0 within the context of unification with parameters [6,7,9,31].…”