https://orbilu.uni.lu/bitstream/10993/37443/1/article-preprint-RG.pdf

Universal (meta-)logical reasoning: Recent successes

The automated theorem prover Leo-III for classical higher-order logic with Henkin semantics and choice is presented. Leo-III is based on extensional higher-order paramodulation and accepts every common TPTP dialect (FOF, TFF, THF), including their recent extensions to rank-1 polymorphism (TF1, TH1). In addition, the prover natively supports almost every normal higher-order modal logic. Leo-III cooperates with first-order reasoning tools using translations to (polymorphic) many-sorted first-order logic and produces verifiable proof certificates. The prover is evaluated on heterogeneous benchmark sets.

The Higher-Order Prover Leo-III

Designing normative theories for ethical and legal reasoning: LogiKEy framework, methodology, and tool support

Automation of classical higher-order logic faces various theoretical and practical challenges. On a theoretical level, powerful calculi for effective equality reasoning from first-order theorem proving cannot be lifted to the higher-order domain in a simple manner. Practically, implementations of higher-order reasoning systems have to incorporate procedures that often have high time complexity or are not decidable in general. In my dissertation, both the theoretical and the practical challenges of designing an effective higher-order reasoning system are studied. The resulting system, the automated theorem prover Leo-III, is one of the most effective and versatile systems, in terms of supported logical formalisms, to date.

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Extensional Paramodulation for Higher-Order Logic and its Effective Implementation Leo-III

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling e.g. proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.

Extensional Higher-Order Paramodulation in Leo-III

We present a procedure for algorithmically embedding problems formulated in higherorder modal logic into classical higher-order logic. The procedure was implemented as a stand-alone tool and can be used as a preprocessor for turning TPTP THF-compliant theorem provers into provers for various modal logics. The choice of the concrete modal logic is thereby specified within the problem as a meta-logical statement. This specification format as well as the underlying semantics parameters are discussed, and the implementation and the operation of the tool are outlined. By combining our tool with one or more THF-compliant theorem provers we accomplish the most widely applicable modal logic theorem prover available to date, i.e. no other available prover covers more variants of propositional and quantified modal logics. Despite this generality, our approach remains competitive, at least for quantified modal logics, as our experiments demonstrate.

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Theorem Provers For Every Normal Modal Logic

Modal logics have numerous applications in computational linguistics, artificial intelligence, rule-based reasoning, and, in general, alethic, deontic and epistemic contexts. Higher-order quantified modal logics additionally incorporate the expressiveness of higher-order formalisms and thereby provide a quite general reasoning framework. By exploiting this expressiveness, the Modal Embedding Tool (MET) allows to automatically encode higher-order modal logic problems into equivalent problems of classical logic, enabling the use of a broad variety of established reasoning tools. In this system description, the functionality and usage of MET as well as a suitable input syntax for flexible reasoning with modalities are presented.

The MET: The Art of Flexible Reasoning with Modalities

The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.

Knowledge Augmented Machine Learning with Applications in Autonomous Driving: A Survey

Non-classical logics are used in a wide spectrum of disciplines, including artificial intelligence, computer science, mathematics, and philosophy. The de-facto standard infrastructure for automated theorem proving, the TPTP World, currently supports only classical logics. Similar standards for non-classical logic reasoning do not exist (yet). This hampers practical development of reasoning systems, and limits their interoperability and application. This paper describes the latest extension of the TPTP World, which provides languages and infrastructure for reasoning in non-classical logics. The extensions integrate seamlessly with the existing TPTP World.

Automated Reasoning in Non-classical Logics in the TPTP World

This paper describes an evaluation of Automated Theorem Proving (ATP) systems on problems taken from the QMLTP library of first-order modal logic problems. Principally, the problems are translated to higher-order logic in the TPTP language using an embedding approach, and solved using higher-order logic ATP systems. Additionally, the results from native modal logic ATP systems are considered, and compared with those from the embedding approach. The findings are that the embedding process is reliable and successful, the choice of backend ATP system can significantly impact the performance of the embedding approach, native modal logic ATP systems outperform the embedding approach, and the embedding approach can cope with a wider range modal logics than the native modal systems considered.

Tobias Gleißner

Papers

Theorem Provers For Every Normal Modal Logic

The MET: The Art of Flexible Reasoning with Modalities

Knowledge Augmented Machine Learning with Applications in Autonomous Driving: A Survey

Automated Reasoning in Non-classical Logics in the TPTP World

Solving QMLTP Problems by Translation to Higher-order Logic