Ground Stratification for a Logic of Definitions with Induction

Abstract

The logic underlying the Abella proof assistant includes mechanisms for interpreting atomic predicates through fixed point definitions that can additionally be treated inductively or co-inductively. However, the original formulation of the logic includes a strict stratification condition on definitions that is too restrictive for some applications such as those that use a logical relations based approach to semantic equivalence. Tiu has shown how this restriction can be eased by utilizing a weaker notion referred to as ground stratification. Tiu's results were limited to a version of the logic that does not treat inductive definitions. We show here that they can be extended to cover such definitions. While our results are obtained by using techniques that have been previously deployed in related ways in this context, their use is sensitive to the particular way in which we generalize the logic. In particular, although ground stratification may be used with arbitrary fixed-point definitions, we show that weakening stratification to this form for inductive definitions leads to inconsistency. The particular generalization we describe accords well with the way logical relations are used in practice. Our results are also a intermediate step to building a more flexible form for definitions into the full logic underlying Abella, which additionally includes co-induction, generic quantification, and a mechanism referred to as nominal abstraction for analyzing occurrences of objects in terms that are governed by generic quantifiers.