Bouncing Threads for Circular and Non-Wellfounded Proofs: Towards Compositionality with Circular Proofs

Abstract

Given that (co)inductive types are naturally modelled as fixed points, it is unsurprising that fixed-point logics are of interest in the study of programming languages, via the Curry-Howard (or proofs-as-programs) correspondence. This motivates investigations of the structural proof-theory of fixed-point logics and of their cut-elimination procedures. Among the various approaches to proofs in fixed-point logics, circular – or cyclic – proofs, are of interest in this regard but suffer from a number of limitations, most notably from a quite restricted use of cuts. Indeed, the validity condition which ensures soundness of non-wellfounded derivations and productivity of their cut-elimination prevents some computationally-relevant patterns of cuts. As a result, traditional circular proofs cannot serve as a basis for a theory of (co)recursive programming by lack of compositionality: there are not enough circular proofs and they compose badly. The present paper addresses some of these limitations by developing the circular and non-wellfounded proof-theory of multiplicative additive linear logic with fixed points () beyond the scope of the seminal works of Santocanale and Fortier and of Baelde et al. We define bouncing-validity: a new, generalized, validity criterion for , which takes axioms and cuts into account. We show soundness and cut elimination theorems for bouncing-valid non-wellfounded proofs: as a result, even though bouncing-validity proves the same sequents (or judgments) as before, we have many more valid proofs at our disposal. We illustrate the computational relevance of bouncing-validity on a number of examples. Finally, we study the decidability of the criterion in the circular case: we prove that it is undecidable in general but identify a hierarchy of decidable sub-criteria.