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A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing (and deciding in the finite case) those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures. We also prove semi-decidability of the type inhabitation problem for cut-free MELL proof-structures.

Non-wellfounded and circular proofs have been recognised over the past decade as a valuable tool to study logics expressing (co)inductive properties, e.g. μ-calculi. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. While the cut-free fragment of circular proofs is satisfactory, cuts are poorly treated and the non-canonicity of sequent proofs becomes a major issue in the non-wellfounded setting. The present paper develops for (multiplicative linear logic with fixed points) the theory of infinets – proof-nets for non-wellfounded proofs. Our structures handles infinitely many cuts therefore solving a crucial shortcoming of the previous work [19]. We characterise correctness, define a more complete cut-reduction system and proving a cut-elimination theorem. To that end, we also provide an alternate cut reduction for non-wellfounded sequent calculus.

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures.

Starting from an exact correspondence between linear approximations and non-idempotent intersection types, we develop a general framework for building systems of intersection types characterizing normalization properties. We show how this construction, which uses in a fundamental way Melliès and Zeilberger's ``type systems as functors'' viewpoint, allows us to recover equivalent versions of every well known intersection type system (including Coppo and Dezani's original system, as well as its non-idempotent variants independently introduced by Gardner and de Carvalho). We also show how new systems of intersection types may be built almost automatically in this way.