arXiv subjects

Nathan Guermond

Publications and source records attributed to Nathan Guermond.

2 recordsLinked to original sources

arXiv2026

Towards Weak Stratification for Logics of Definitions

The logic of definitions is a family of logics for encoding and reasoning about judgments, which are atomic predicates specified by inference rules. A definition associates an atomic predicate with a logical formula, which may itself depend on the predicate being defined. This leads to an apparent circularity which can be resolved by interpreting definitions as monotone fixed-point operators on terms, and which is enforced by imposing a stratification condition on definitions. In many instances, it is useful to consider definitions in which the predicate being defined appears negatively in the body of its definition. In the logic $\mathcal G$, underlying the Abella proof assistant, this is not allowed due to the stratification condition. One such application violating this condition is that of defining logical relations, which is a technique commonly used to prove properties about programming languages. Tiu has shown how to relax this stratification condition to allow for a broader body of definitions including that needed for logical relations. However, he only showed how to extend a core fragment of $\mathcal G$ with the weakened stratification condition, resulting in a logic he called $\mathrm{LD}$. In this work we show that the weakened stratification condition is also compatible with generic (nabla) quantification and general induction. The eventual aim of this work is to justify an extension of the Abella proof assistant allowing for such definitions.

cs.LO↗

arXiv2025

Ground Stratification for a Logic of Definitions with Induction

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.

cs.LO↗