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Deciding Regular Grammar Logics with Converse Through First-Order Logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF2 include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators.

A Resolution Decision Procedure for the Guarded Fragment

We give a resolution based decision procedure for the guarded fragment of [ANB96]. The relevance of the guarded fragment lies in the fact that many modal logics can be translated into it. In this way the guarded fragment acts as a framework explaining the nice properties of these modal logics. By constructing an effective decision procedure for the guarded fragment we define an effective procedure for deciding these modal logics.

Resolution in modal, description and hybrid logic

We provide a resolution-based proof procedure for modal, description and hybrid logic that improves on previous proposals in important ways. It avoids translations into large undecidable logics, and works directly on modal, description or hybrid logic formulas instead. In addition, by using the hybrid machinery it avoids the complexities of earlier propositional resolution-based methods for modal logic. It combines ideas from the method of prefixes used in tableaux, and resolution ideas in such a way that some of the heuristics and optimizations devised in either field are applicable.

Deciding the guarded fragments by resolution

The guarded fragment (GF) is a fragment of first-order logic that has been introduced for two main reasons: first, to explain the good computational and logical behaviour of propositional modal logics. Second, to serve as a breeding ground for well-behaved process logics. In this paper we give resolution-based decision procedures for the GF and for the loosely guarded fragment (LGF) (sometimes also called the pairwise guarded fragment). By constructing an implementable decision procedure for the GF and for the LGF, we obtain an effective procedure for deciding modal logics that can be embedded into these fragments. The procedures have been implemented in the theorem prover Bliksem.

A Resolution-Based Decision Procedure for the Two-Variable Fragment with Equality

The two-variable-fragment L? 2 of first order logic is the set of formulas that do not contain function symbols, that possibly contain equality, and that contain at most two variables. This paper shows how resolution theorem-proving techniques can be used to provide an algorithm for deciding whether any given formula in L? 2 is satisfiable. Previous resolution-based techniques could deal only with the equality-free subset L2 of the two-variable fragment.

Geometric resolution: a proof procedure based on finite model search

We present a proof procedure that is complete for first-order logic, but which can also be used when searching for finite models. The procedure uses a normal form which is based on geometric formulas. For this reason we call the procedure geometric resolution. We expect that the procedure can be used as an efficient proof search procedure for first-order logic. In addition, the procedure can be implemented in such a way that it is complete for finding finite models.

A Resolution Decision Procedure for the Guarded Fragment with Transitive Guards.

We show how well-known refinements of ordered resolution, in particular redundancy elimination and ordering constraints in combination with a selection function, can be used to obtain a decision procedure for the guarded fragment with transitive guards. Another contribution of the paper is a special scheme notation, that allows to describe saturation strategies and show their correctness in a concise form.

Splitting Through New Proposition Symbols

The splitting rule is a tableau-like rule, that is used in the resolution context. In case the search state contains a clause C1 V C2, which has no shared variables between C1 and C2, the prover splits the search state, and tries to refute C1 and C2 separately. Instead of splitting the state of the theorem prover, one can create a new proposition symbol α, and replace C1 V C2 by C1 V α and ¬α V C2. In the first clause a is the least preferred literal. In the second clause α is selected. In this way, nothing can be done with C2 as long as C1 has not been refuted. This way of splitting simulates search state splitting only partially, because a clause that inherits from C1 V α cannot subsume or simplify a clause that does not inherit from C1. With search state splitting, a clause that inherits from C1 can in principle subsume or simplify clauses that do not derive from C1. As a consequence, splitting through new symbols is less powerfull than search state splitting. In this paper, we present a solution for this problem.

Resolution Games and Non-Liftable Resolution Orderings

We prove the completeness of the combination of ordered resolution and factoring for a large class of non-liftable orderings, without the need for any additional rules, as for example saturation. This is possible because of a new proof method which avoids making use of the standard ordered lifting theorem. This new proof method is based on a new technique, which we call the resolution game.

Translation of resolution proofs into short first-order proofs without choice axioms

We present a way of transforming a resolution-style proof containing Skolemization into a natural deduction proof without Skolemization. The size of the proof increases only moderately (polynomially). This makes it possible to translate the output of a resolution theorem prover into a purely first-order proof that is moderate in size.