Alexander Steen

Geoff Sutcliffe, Alexander Steen, Pascal Fontaine et al.

This paper describes the TPTP format for representing interpretations. It provides a background survey that helped ensure that the representation format is adequate for different types of interpretations: Tarskian, Herbrand, and Kripke interpretations. The needs of applications that use models are considered. The syntax and semantics of the format are expounded in detail, with multiple examples. Verification of models is discussed. Some tools that support processing the format are noted. The properties of interpretations represented in the format are discussed.

Alexander Steen, Geoff Sutcliffe, Christoph Benzmüller

This article 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 both typed first-order and higher-order logic in the TPTP language using an embedding approach, and solved using first-order resp. higher-order logic ATP systems and model finders. Additionally, the results from native modal logic ATP systems are considered, and compared with the results from the embedding approach. The findings are that the embedding process is reliable and successful when state-of-the-art ATP systems are used as backend reasoners, The first-order and higher-order embeddings perform similarly, native modal logic ATP systems have comparable performance to classical systems using the embedding for proving theorems, native modal logic ATP systems are outperformed by the embedding approach for disproving conjectures, and the embedding approach can cope with a wider range of modal logics than the native modal systems considered.

Alexander Steen

The logic embedding tool provides a procedural encoding for non-classical reasoning problems into classical higher-order logic. It is extensible and can support an increasing number of different non-classical logics as reasoning targets. When used as a pre-processor or library for higher-order theorem provers, the tool admits off-the-shelf automation for logics for which otherwise few to none provers are currently available.

Alexander Steen, David Fuenmayor

LegalRuleML is a comprehensive XML-based representation framework for modeling and exchanging normative rules. The TPTP input and output formats, on the other hand, are general-purpose standards for the interaction with automated reasoning systems. In this paper we provide a bridge between the two communities by (i) defining a logic-pluralistic normative reasoning language based on the TPTP format, (ii) providing a translation scheme between relevant fragments of LegalRuleML and this language, and (iii) proposing a flexible architecture for automated normative reasoning based on this translation. We exemplarily instantiate and demonstrate the approach with three different normative logics.

Alexander Steen, Geoff Sutcliffe

The TPTP World is the well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems. The TPTP World supports a range of classical logics, and since release v9.0.0 has supported non-classical logics. This paper provides a self-contained comprehensive overview of the TPTP World infrastructure for ATP in non-classical logics: the non-classical language extension, problems and solutions, and tool support. A detailed description of use of the infrastructure for quantified normal multi-modal logic is given.

Alexander Steen

Deontic logics are formalisms for reasoning over norms, obligations, permissions and prohibitions. Input/Output (I/O) Logics are a particular family of so-called norm-based deontic logics that formalize conditional norms outside of the underlying object logic language, where conditional norms do not carry a truth-value themselves. In this paper, an automation approach for I/O logics is presented that makes use of suitable reductions to (sequences of) propositional satisfiability problems. A prototypical implementation, named rio (reasoner for input/output logics), of the proposed procedures is presented and applied to illustrative examples.

Antoine Martina, Alexander Steen

An approach for encoding abstract dialectical frameworks and their semantics into classical higher-order logic is presented. Important properties and semantic relationships are formally encoded and proven using the proof assistant Isabelle/HOL. This approach allows for the computer-assisted analysis of abstract dialectical frameworks using automated and interactive reasoning tools within a uniform logic environment. Exemplary applications include the formal analysis and verification of meta-theoretical properties, and the generation of interpretations and extensions under specific semantic constraints.

Christoph Benzmüller, David Fuenmayor, Alexander Steen et al.

This paper reports on an exploration of Boolos' Curious Inference, using higher-order automated theorem provers (ATPs). Surprisingly, only suitable shorthand notations had to be provided by hand for ATPs to find a short proof. The higher-order lemmas required for constructing a short proof are automatically discovered by the ATPs. Given the observations and suggestions in this paper, full proof automation of Boolos' and related examples now seems to be within reach of higher-order ATPs.

Alexander Steen, David Fuenmayor, Tobias Gleißner et al.

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.

Alexander Steen, David Fuenmayor

We present an approach for representing abstract argumentation frameworks based on an encoding into classical higher-order logic. This provides a uniform framework for computer-assisted assessment of abstract argumentation frameworks using interactive and automated reasoning tools. This enables the formal analysis and verification of meta-theoretical properties as well as the flexible generation of extensions and labellings with respect to well-known argumentation semantics.

Alexander Steen, Christoph Benzmüller

Steen's (2018) Hintikka set properties for Church's type theory based on primitive equality are reduced to the Hintikka set properties of Brown (2007). Using this reduction, a model existence theorem for Steen's properties is derived.

Tomer Libal, Alexander Steen

A prototype for automated reasoning over legal texts, called NAI, is presented. As an input, NAI accepts formalized logical representations of such legal texts that can be created and curated using an integrated annotation interface. The prototype supports automated reasoning over the given text representation and multiple quality assurance procedures. The pragmatics of the NAI suite as well its feasibility in practical applications is studied on a fragment of the Smoking Prohibition (Children in Motor Vehicles) (Scotland) Act 2016 of the Scottish Parliament.

Alexander Steen, Christoph Benzmüller

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.

Alexander Steen, Christoph Benzmüller

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 many-sorted first-order logic and produces verifiable proof certificates. The prover is evaluated on heterogeneous benchmark sets.

Max Wisniewski, Alexander Steen, Christoph Benzmüller

LeoPARD supports the implementation of knowledge representation and reasoning tools for higher-order logic(s). It combines a sophisticated data structure layer (polymorphically typed λ-calculus with nameless spine notation, explicit substitutions, and perfect term sharing) with an ambitious multi-agent blackboard architecture (supporting prover parallelism at the term, clause, and search level). Further features of LeoPARD include a parser for all TPTP dialects, a command line interpreter, and generic means for the integration of external reasoners.