A complete transformation system for polymorphic higher-order unification

Polymorphic higher-order unification is a method for unifying terms in the poly\-mor\-phi\-cally typed $\lambda$-calculus, that is, given a set of pairs of terms $S_0 = \{s_1 = t_2,\ldots,s_n = t_n\}$, called a unification problem, finding a substitution $\sigma$ such that $\sigma(s_i)$ and $\sigma(t_i)$ are equivalent under the conversion rules of the calculus for all $i$, $1\leq i\leq n$. I present the method as a transformation system, i.e.\ as a set of schematic rules $U \Rightarrow U'$ such that any unification problem $\delta({U})$ can be transformed into $\delta({U'})$ where $\delta$ is an instantiation of the meta-level variables in $U$ and $U'$. By successive use of transformation rules one possibly obtains a solved unification problem with obvious unifier. I show that the transformation system is correct and complete, i.e.\if $\delta({U}) \Rightarrow \delta({U'})$ is an instance of a transformation rule, then the set of all unifiers of $\delta({U'})$ is a subset of the set of all unifiers of $\delta({U})$ and if $\cal U$ is the set of all unification problems that can be obtained from successive applications of transformation rules from an unification problem $U$, then the union of the set of all unifiers of all unification problems in $\cal U$ is the set of all unifiers of $U$. The transformation rules presented here are essentially different from those in Gallier and Snyder (1989) or Nipkow (1990). The correctness and completeness proofs are in lines with those of Gallier and Snyder (1989)

Theorem Proving for Pointwise Metric Temporal Logic Over the Naturals via Translations

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A Resolution Prover for Coalition Logic

We present a prototype tool for automated reasoning for Coalition Logic, a non-normal modal logic that can be used for reasoning about cooperative agency. The theorem prover CLProver is based on recent work on a resolution-based calculus for Coalition Logic that operates on coalition problems, a normal form for Coalition Logic. We provide an overview of coalition problems and of the resolution-based calculus for Coalition Logic. We then give details of the implementation of CLProver and present the results for a comparison with an existing tableau-based solver

Hyperresolution for guarded formulae

AbstractThis paper investigates the use of hyperresolution as a decision procedure and model builder for guarded formulae. In general, hyperresolution is not a decision procedure for the entire guarded fragment. However we show that there are natural fragments of the guarded fragment which can be decided by hyperresolution. In particular, we prove decidability of hyperresolution with or without splitting for the fragment GF1− and point out several ways of extending this fragment without losing decidability. As hyperresolution is closely related to various tableaux methods the present work is also relevant for tableaux methods. We compare our approach to hypertableaux, and mention the relationship to other clausal classes which are decidable by hyperresolution

Motel user manual

MOTEL is a logic-based knowledge representation languages of the KL-ONE family. It contains as a kernel the KRIS language which is a decidable sublanguage of first-order predicate logic. Whereas KRIS is a single-agent knowledge representation system, i.e. KRIS is only able to represent general world knowledge or the knowledge of one agent about the world, MOTEL is a multi-agent knowledge representation system. The MOTEL language allows modal contexts and modal concept forming operators which allow to represent and reason about the believes and wishes of multiple agents. Furthermore it is possible to represent defaults and stereotypes. Beside the basic resoning facilities for consistency checking, classification, and realization, MOTEL provides an abductive inference mechanism. Furthermore it is able to give explanations for its inferences

Resolution-based decision procedures for subclasses of first-order logic

This thesis studies decidable fragments of first-order logic which are relevant to the field of nonclassical logic and knowledge representation. We show that refinements of resolution based on suitable liftable orderings provide decision procedures for the subclasses E+, K, and DK of first-order logic. By the use of semantics-based translation methods we can embed the description logic ALB and extensions of the basic modal logic K into fragments of first-order logic. We describe various decision procedures based on ordering refinements and selection functions for these fragments and show that a polynomial simulation of tableaux-based decision procedures for these logics is possible. In the final part of the thesis we develop a benchmark suite and perform an empirical analysis of various modal theorem provers.Diese Arbeit untersucht entscheidbare Fragmente der Logik erster Stufe, die mit nicht-klassischen Logiken und Wissensrepräsentationsformalismen im Zusammenhang stehen. Wir zeigen, daß Entscheidungsverfahren für die Teilklassen E+, K, und DK der Logik erster Stufe unter Verwendung von Resolution eingeschränkt durch geeignete liftbare Ordnungen realisiert werden können. Durch Anwendung von semantikbasierten Übersetzungsverfahren lassen sich die Beschreibungslogik ALB und Erweiterungen der Basismodallogik K in Teilklassen der Logik erster Stufe einbetten. Wir stellen eine Reihe von Entscheidungsverfahren auf der Basis von Resolution eingeschränkt durch liftbare Ordnungen und Selektionsfunktionen für diese Logiken vor und zeigen, daß eine polynomielle Simulation von tableaux-basierten Entscheidungsverfahren für diese Logiken möglich ist. Im abschließenden Teil der Arbeit führen wir eine empirische Untersuchung der Performanz verschiedener modallogischer Theorembeweiser durch

Modal Resolution: Proofs, Layers and Refinements

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof in order to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur in order to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements