Stratification in Logics of Definitions

Proof systems for logics with recursive definitions typically impose a strict syntactic stratification on the body of a definition to ensure cut elimination and consistency of the logics, i.e., by forbidding any negative occurrences of the predicate being defined. Often such a restriction is too strong, as there are cases where such negative occurrences do not lead to inconsistency. Several logical frameworks based on logics of definitions have been used to mechanise reasoning about properties of operational semantics and type systems. However, some of the uses of these frameworks actually go beyond what is justified by their logical foundations, as they admit definitions which are not strictly stratified, e.g., in the formalisation of logical-relation type of arguments in typed λ-calculi. We consider here a more general notion of stratification, which allows one to admit some definitions that are not strictly stratified. We outline a novel technique to prove consistency and a partial cut elimination result, showing that every derivation can be transformed into a certain head normal form, by simulating its cut reductions in an infinitary proof system. We demonstrate this technique for a specific logic, but it can be extended to other richer logics.