Simple type theory is suited as framework for combining classical and non-classical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about combinations of logics. Combinations of modal logics and other logics are particularly relevant for multi-agent systems.

We report on the application of higher-order automated theorem proving in ontology reasoning. Concretely, we have integrated the Sigma knowledge engineering environment and the Suggested Upper-Level Ontology (SUMO) with the higher-order theorem prover LEO-II. The basis for this integration is a translation from SUMO’s SUO-KIF representations into the new typed higher-order form representation language TPTP THF. We illustrate the benefits of our integration with examples, report on experiments and analyze open challenges. 1

Abstract. Prominent logics, including quantified multimodal logics, can be elegantly embedded in simple type theory (classical higher-order logic). Furthermore, off-the-shelf reasoning systems for simple type type theory exist that can be uniformly employed for reasoning within and about embedded logics. In this paper we focus on reasoning about modal logics and exploit our framework for the automated verification of inclusion and equivalence relations between them. Related work has applied first-order automated theorem provers for the task. Our solution achieves significant improvements, most notably, with respect to elegance and simplicity of the problem encodings as well as with respect to automation performance. 1

Reasoning with embedded formulas is relevant for the SUMO ontology but there is limited automation support so far. We investigate whether higher-order automated theorem provers are applicable for the task. Moreover, we point to a challenge that we have revealed as part of our experiments: modal operators in SUMO are in conflict with Boolean extensionality. A solution is proposed.

Abstract. Sigma is an open source environment for the development of logical theories. It has been under development and regular release for nearly a decade, and has been the principal environment under which the open source Suggested Upper Merged Ontology (SUMO) has been created. We discuss its features and evolution, and explain why it is an appropriate environment for the development of expressive ontologies in first and higher order logic. 1

Abstract. Reasoning with embedded formulas is relevant for the SUMO ontology but there is limited automation support so far. We investigate whether higher-order automated theorem provers are applicable for the task. Moreover, we point to a challenge that we have revealed as part of our experiments: modal operators in SUMO are in conflict with Boolean extensionality. A solution is proposed. 1 EMBEDDED FORMULAS IN SUMO The open source Suggested Upper Merged Ontology 3 (SUMO) [9] (and similarly, proprietary Cyc [13]) contains a small but significant amount of higher-order representations. The approach taken in these systems to address higher-order challenges has been to employ specific translation ’tricks’, possibly in combination or in addition to some pre-processing techniques. Examples of such means are the quoting techniques for embedded formulas as employed in SUMO [11] and the heuristic-level modules in CYC [13]. Unfortunately, however, these solutions are strongly limited. The effect is that many desirable inferences are currently not supported, so that many relevant queries cannot be answered. This includes statements in which formulas are embedded as arguments of terms, for example, statements that employ epistemic operators such as believes or knows, temporal operators such as holdsDuring, and further operators such as disapproves or hasPurpose. While first-order automated theorem proving (FO-ATP) for SUMO has strongly improved recently [12], there is still only very limited support for reasoning with non-trivial embedded formulas; we give an example (free variables in premises are universal and those in the query are existential): Ex. 1 (Reasoning in temporal contexts.) What holds that holds at all times. Mary likes Bill. 4 During 2009 Sue liked whoever Mary liked. Is there a year in which Sue has liked somebody?

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Numerous classical and non-classical logics can be elegantly embedded in Church’s simple type theory, also known as classical higher-order logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with off-the-shelf higher-order automated theorem provers.