Exploring Properties of Normal Multi- modal Logics in Simple Type Theory with Leo-II

There are two well investigated approaches to automate reasoning in modal logics: the direct approach and the translational approach. The direct approach [6, 7, 14, 27] develops specific calculi and tools for the task; the translational approach [29, 30] transforms modal logic formulas into firstorder logic and applies standard first-order tools. Embeddings of modal logics into higher-order logic, however, have not yet been widely studied, although multimodal logic can be regarded as a natural fragment of simple type theory. Gallin [15] appears to mention the idea first. He presents an embedding of modal logic into a 2-sorted type theory. This idea is picked up by Gamut [16] and a related embedding has recently been studied by Hardt and Smolka [17]. Carpenter [12] proposes to use lifted connectives, an idea that is also underlying the embeddings presented by Merz [26], Brown [11], Harrison [18, Chap. 20], and Kaminski and Smolka [22]. In this paper we pick up and extend the embedding of multimodal logics in simple type theory as studied by Brown [11]. The starting point is a characterization of multimodal logic formulas as particular λ-terms in simple type theory. A distinctive characteristic of the encoding is that the definiens of the R operator λ-abstracts over the accessibility relation R. We illustrate that this supports the formulation of meta properties of encoded multimodal logics such as the correspondence between certain axioms and properties of the accessibility relation R. We show that some of these meta properties can even be efficiently automated within our higher-order theorem prover Leo-II [9] via cooperation with the first-order automated theorem prover E [32]. We also discuss some challenges to higher-order reasoning implied by this application direction. Moreover, we extend the