Integrating the splitting rule into a saturation-based theorem prover may be highly beneficial for solving certain classes of fist-order problems. The use of splitting in the context of saturation-based theorem proving based on explicit case analysis (as implemented in SPASS) employs backtracking which is difficult to implement as it affects design of the whole system. Here we present a "cheap" and efficient technique for implementing splitting that does not use backtracking.

In this paper we show how to extend clausal temporal resolution to the ground eventuality fragment of monodic first-order temporal logic, which has recently been introduced by Hodkinson, Wolter and Zakharyaschev. While a finite Hilbert-like axiomatization of complete monodic first order temporal logic was developed by Wolter and Zakharyaschev, we propose a temporal resolutionbased proof system which reduces the satisfiability problem for ground eventuality monodic first-order temporal formulae to the satisfiability problem for formulae of classical first-order logic.

Until recently, First-Order Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a finitely axiomatisable fragment, termed the monodic fragment, has led to improved understanding of FOTL. Yet, in order to utilise these theoretical advances, it is important to have appropriate proof techniques for this monodic fragment. In this paper, we modify and extend the clausal temporal resolution technique, originally developed for propositional temporal logics, to enable its use in such monodic fragments. We develop a specific normal form for monodic formulae in FOTL, and provide a complete resolution calculus for formulae in this form. Not only is this clausal resolution technique useful as a practical proof technique for certain monodic classes, but the use of this approach provides us with increased understanding of the monodic fragment. In particular, we here show how several features of monodic FOTL can be established as corollaries of the completeness result for the clausal temporal resolution method. These include definitions of new decidable monodic classes, simplification of existing monodic classes by reductions, and completeness of clausal temporal resolution in the case of

We consider instantiation-based theorem proving whereby instances of clauses are generated by certain inferences, and where inconsistency is detected by propositional tests. We give a model construction proof of completeness by which restrictive inference systems as well as admissible simplification techniques can be justified. Another contribution of the paper are novel inference systems that allow one to also employ decision procedures for first-order fragments more complex than propositional logic. The decision procedure provides for an approximative consistency test, and the instance generation inference system is a means of successively refining the approximation.

This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur

Irrelevant clauses in resolution problems increase the search space, making it hard to find proofs in a reasonable time. Simple relevance filtering methods, based on counting function symbols in clauses, improve the success rate for a variety of automatic theorem provers and with various initial settings. We have designed these techniques as part of a project to link automatic theorem provers to the interactive theorem prover Isabelle. They should be applicable to other situations where the resolution problems are produced mechanically and where completeness is less important than achieving a high success rate with limited processor time. 1

Abstract. Resolution-based calculi are among the most widely used calculi for theorem proving in first-order logic. Numerous refinements of resolution are nowadays available, such as e.g. basic superposition, a calculus highly optimized for theorem proving with equality. However, even such an advanced calculus does not restrict inferences enough to obtain decision procedures for complex logics, such as SHIQ. In this paper, we present a new decomposition inference rule, which can be combined with any resolution-based calculus compatible with the standard notion of redundancy. We combine decomposition with basic superposition to obtain three new decision procedures: (i) for the description logic SHIQ, (ii) for the description logic ALCHIQb, and (iii) for answering conjunctive queries over SHIQ knowledge bases. The first two procedures are worst-case optimal and, based on the vast experience in building efficient theorem provers, we expect them to be suitable for practical usage. 1

We present TeMP---the first experimental system for testing validity of monodic temporal logic formulae. The prover implements fine-grained temporal resolution. The core operations required by the procedure are performed by an efficient resolution-based prover for classical first-order logic.

. This paper presents a translation-based resolution decision procedure for the multi-modal logic K (m) (\; [; ^) dened over families of relations closed under intersection, union and converse. The relations may satisfy certain additional frame properties. Dierent from previous resolution decision procedures which are based on ordering renements our procedure is based on a selection renement, the derivations of which correspond to derivations of tableaux or sequent proof systems. This procedure has the advantage that it can be used both as a satisability checker and a model builder. We show that tableaux and sequent-style proof systems can be polynomially simulated with our procedure. Furthermore, the nite model property follows for a number of extended modal logics. Keywords: modal logic, automated theorem proving, resolution decision procedures, tableaux proof systems, satisability testing, model generation, simulation, relative proof complexity, relative search space complex...

Abstract. We present a resolution-based decision procedure for the description logic SHOIQ—the logic underlying the Semantic Web ontology language OWL-DL. Our procedure is goal-oriented, and it naturally extends a similar procedure for SHIQ, which has proven itself in practice. Applying existing techniques for deriving saturation-based decision procedures to SHOIQ is not straightforward due to nominals, number restrictions, and inverse roles—a combination known to cause termination problems. We overcome this difficulty by using the basic superposition calculus, extended with custom simplification rules. 1