Theorem Proving in Dependently-Typed Higher-Order Logic

Abstract

Higher-order logic HOL offers a very simple syntax and semantics for representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a rich type system, but has rather substantial conceptual differences to HOL, as well as comparatively poor proof automation support.We introduce a dependently-typed extension DHOL of HOL that retains the style and conceptual framework of HOL. Moreover, we build a translation from DHOL to HOL and implement it as a preprocessor to a HOL theorem prover, thereby obtaining a theorem prover for DHOL.

Modelling Value-Oriented Legal Reasoning in LogiKEy

“…We note that our normative assignment here is widely in accordance with classifications in the AI and Law literature (Berman and Hafner [6], Bench-Capon [52]). LogiKEy takes the position that expressive logics such as classical HOL (or possibly beyond, see Rothgang et al [108]) are suited to serve as a universal meta-logic for knowledge representation and reasoning as motivated in Benzmüller [10] (regarding the choice of a meta-logic, we are thus in opposition to Quine [109], who advocated first-order logic for the task). This contrasts with the widespread view in the field of knowledge representation and reasoning in AI that decidability should be taken as a hard limiting criterion for the development of any logic tools and associated infrastructure.…”