Resolution-based methods for modal logics

Abstract

In this paper we give an overview of resolution methods for extended propositional modal logics. We adopt the standard translation approach and consider different resolution refinements which provide decision procedures for the resulting clause sets. Our procedures are based on ordered resolution and selection-based resolution. The logics that we cover are multi-modal logics defined over relations closed under intersection, union, converse and possibly complementation.

A new methodology for developing deduction methods

“…The translation is similar to the one used in [46,48]; other structural translations have been used in [10,25,26,28], for instance.…”

“…In this paper we discuss and extend an approach of developing tableaux calculi for modal logics that has been suggested and followed in our previous work [10,19,25,26,28,46,48]. Although resolution calculi apparently operate considerably differently from tableau calculi, we have shown that it is possible to linearly simulate many forms of modal logic or description logic tableau calculi with standard techniques of first-order resolution theorem proving.…”

“…These simulation results show that it is possible to use first-order resolution in a way that it closely simulates modal logic and description logic tableau procedures. The close connection exhibited in these papers between tableau and a certain instance of resolution is exploited in [10] in order to develop a tableau calculus for a logic that has not been considered before. The logic considered was the dynamic modal logic K (m) (∧, ∨, ).…”

“…The logic corresponds to the description logic A L C in which conjunction, disjunction and converse of roles are allowed. In [10] we have shown how a tableau calculus can essentially be 'read off' from the clausal form of the translation of formulae in K (m) (∧, ∨, ). In [46,48] we use resolution methods to develop a new translation mapping (called the axiomatic translation) of traditional style modal logics.…”

Rewriting Conjunctive Queries over Description Logic Knowledge Bases

“…The key to achieving this is to extend the clause set in Table 2 to cover the new constructors, and to prove that Lemma 2 still holds. We believe we can do this along the lines of [17,16]; the main technical difficulty is to precisely describe the interaction between the new types of clause and clauses of type Q1. Once this is done, however, the rest of this paper (i.e., the material in the following two sections), should hold with only minor changes.…”