Resolution for hybrid logics

. We describe a resolution method and a procedure to transform formulae of some pure hybrid logics into their clausal form.


Introduction
Before applying the resolution method to the formulas of classical logic we transform them into clausal form. Well-known transformation methods of the formulas of classical logic are not suitable for neither modal nor hybrid logics. Transformation of formulas of hybrid logics needs a different approach. In [6,7] Mints et al. describe transformation of formulae into their clausal form for modal logics S4 and S5. A modal literal is defined as formula of the form l, 2l or 3l, where l is a propositional literal. A modal clause is a disjunction of modal literals. We prove that for every modal logic formula F there exist clauses D 1 , . . . , D n and a propositional literal l such that sequent ⊢ F is derivable in sequent calculus S4 (and, accordingly, S5) if and only if sequent 2D 1 , . . . , 2D n , l ⊢ is derivable. This transformation is the basis for the resolution calculus for modal logic S4 presented in [7]. F is a tautology if and only if an empty clause is derivable from the set {2D 1 , . . . , 2D n , l}. The paper [10] describes a procedure to transform formulae of hybrid logic H(@) over transitive and reflexive frames into their clausal form. This paper shows how we can transform formulas of hybrid logics H(@, ↓), H(@, ∃), H(@, E) into their clausal form. We also describe resolution method for the hybrid logics H(@, ↓), H(@, E). For more information about hybrid logic and its properties see [2,3,4,5].

Transformation
A literal of hybrid logic H(@) is a formula of the form l, 2l, 3l, @ i l (where l is a propositional variable, nominal or their negation; i -is nominal). In addition, ∀xl, ∃xl are also literals in the logic H(@, ∃), Al, El -in the logic H(@, E), ↓ x.l -in the logic H(@, ↓).
A clause is a formula of the form L, 2L, @ i L (where L is a disjunction of hybrid literals). In addition, a formula of the form @ i ∀xL is clause of logic H(@, ∃) and a formula @ i ↓ x.L is clause of logic H(@, ↓).
First of all, consider a formula of logic H(@). We transform a subformula of the form @ i G into clause @ i ¬G ∨ a (where a is a new propositional variable). We transform other subformulas similarly as described in [2,3,10]. If any subformula occur in the scope of modal operators 2, 3 we write @ z in the front. In the following we assume that the variable z is reserved, that is, z does not occur in the formulas under consideration.
Assume we want to know whether a sequent ⊢ F has a deduction in the sequent calculus H presented in [9,8]. We denote all the possible subformulas, with the exception of nominals, by the new propositional variables a, b, c, . . . . To the goal formula F assign letter a. Assume the formula F is in negation normal form, that is, the formula contains logical connectives only from the list ¬, ∨, ∧ and the negation symbol appears only in front of nominals. Recall that in this paper we transform only formulas of pure hybrid logic. Obtainable after transformation clauses contain propositional variables and nominals. Formula F has one of the forms: 1. G ∧ H, 2. G ∨ H, 3. 2H, 4. 3H, 5. @ i H. Suppose the variable b assigned to the formulas G, 2H, 3H, @ i H and variable c assigned to formula H.
In the first case formula is derivable if and only if ¬a, a ∨ ¬b ∨ ¬c ⊢ is derivable. In the second case if sequent ¬a, a ∨ ¬b, a ∨ ¬c ⊢ is derivable.
We add new rule to the calculus H and denote them by H ′ . In the third case ⊢ F is derivable in the calculus H iff ¬a, @ z (a ∨ 3¬b) ⊢ is derivable in calculus H ′ . Where u is a nominal. In addition, we can apply only the rule @ z to the formulas beginnig with @ z .
In the fourth case, if sequent ¬a, @ z (a ∨ 2¬b) ⊢ is derivable in calculus H ′ . In the fifth case, if sequent ¬a, a ∨ @ i ¬b) ⊢ is derivable.
We apply the transformation to subformulas b, c and its components as long as it possible. We will say that list of obtained clauses D 1 , . . . , D n corresponds to formula F . We get the following result. Theorem 1. For any formula F of logic H(@) a sequent ⊢ F is derivable in H iff a sequent D 1 , . . . , D n ⊢ corresponding to F is derivable in H ′ .
Let the letters e, c, f, d, b, a denote the subformula of, respectively ¬j, 3¬j, i ∨ j, The following list of clauses corresponds to formula F : ¬a, a∨¬b∨¬c, @ z (b∨3¬d), The case of logic H(@, E) is treated in a similar manner.