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Hyper tableaux with equality
(2007)

In most theorem proving applications, a proper treatment of equational theories or equality is mandatory. In this paper we show how to integrate a modern treatment of equality in the hyper tableau calculus. It is based on splitting of positive clauses and an adapted version of the superposition inference rule, where equations used for paramodulation are drawn (only) from a set of positive unit clauses, the candidate model. The calculus also features a generic, semantically justified simplification rule which covers many redundancy elimination techniques known from superposition theorem proving. Our main results are soundness and completeness, but we briefly describe the implementation, too.

The E-KRHyper system is a model generator and theorem prover for first-order logic with equality. It implements the new E-hyper tableau calculus, which integrates a superposition-based handling of equality into the hyper tableau calculus. E-KRHyper extends our previous KRHyper system, which has been used in a number of applications in the field of knowledge representation. In contrast to most first order theorem provers, it supports features important for such applications, for example queries with predicate extensions as answers, handling of large sets of uniformly structured input facts, arithmetic evaluation and stratified negation as failure. It is our goal to extend the range of application possibilities of KRHyper by adding equality reasoning.

This dissertation investigates the usage of theorem provers in automated question answering (QA). QA systems attempt to compute correct answers for questions phrased in a natural language. Commonly they utilize a multitude of methods from computational linguistics and knowledge representation to process the questions and to obtain the answers from extensive knowledge bases. These methods are often syntax-based, and they cannot derive implicit knowledge. Automated theorem provers (ATP) on the other hand can compute logical derivations with millions of inference steps. By integrating a prover into a QA system this reasoning strength could be harnessed to deduce new knowledge from the facts in the knowledge base and thereby improve the QA capabilities. This involves challenges in that the contrary approaches of QA and automated reasoning must be combined: QA methods normally aim for speed and robustness to obtain useful results even from incomplete of faulty data, whereas ATP systems employ logical calculi to derive unambiguous and rigorous proofs. The latter approach is difficult to reconcile with the quantity and the quality of the knowledge bases in QA. The dissertation describes modifications to ATP systems in order to overcome these obstacles. The central example is the theorem prover E-KRHyper which was developed by the author at the Universität Koblenz-Landau. As part of the research work for this dissertation E-KRHyper was embedded into a framework of components for natural language processing, information retrieval and knowledge representation, together forming the QA system LogAnswer.
Also presented are additional extensions to the prover implementation and the underlying calculi which go beyond enhancing the reasoning strength of QA systems by giving access to external knowledge sources like web services. These allow the prover to fill gaps in the knowledge during the derivation, or to use external ontologies in other ways, for example for abductive reasoning. While the modifications and extensions detailed in the dissertation are a direct result of adapting an ATP system to QA, some of them can be useful for automated reasoning in general. Evaluation results from experiments and competition participations demonstrate the effectiveness of the methods under discussion.