A semantic account of strong normalization in Linear Logic

We prove that given two cut free nets of linear logic, by means of their relational interpretations one can: 1) first determine whether or not the net obtained by cutting the two nets is strongly normalizable 2) then (in case it is strongly normalizable) compute the maximal length of the reduction sequences starting from that net.Comment: 41 page

The relational model is injective for Multiplicative Exponential Linear Logic (without weakenings)

We show that for Multiplicative Exponential Linear Logic (without weakenings) the syntactical equivalence relation on proofs induced by cut-elimination coincides with the semantic equivalence relation on proofs induced by the multiset based relational model: one says that the interpretation in the model (or the semantics) is injective. We actually prove a stronger result: two cut-free proofs of the full multiplicative and exponential fragment of linear logic whose interpretations coincide in the multiset based relational model are the same "up to the connections between the doors of exponential boxes".Comment: 36 page

An Abstract Approach to Stratification in Linear Logic

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic

Glueability of Resource Proof-Structures: Inverting the Taylor Expansion

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

Relational type-checking for MELL proof-structures. Part 1: Multiplicatives

Relational semantics for linear logic is a form of non-idempotent intersection type system, from which several informations on the execution of a proof-structure can be recovered. An element of the relational interpretation of a proof-structure R with conclusion $\Gamma$ acts thus as a type (refining $\Gamma$) having R as an inhabitant. We are interested in the following type-checking question: given a proof-structure R, a list of formulae $\Gamma$, and a point x in the relational interpretation of $\Gamma$, is x in the interpretation of R? This question is decidable. We present here an algorithm that decides it in time linear in the size of R, if R is a proof-structure in the multiplicative fragment of linear logic. This algorithm can be extended to larger fragments of multiplicative-exponential linear logic containing $\lambda$-calculus

Injectivity of relational semantics for (connected) MELL proof-nets via Taylor expansion

International audienceWe show that: (1) the Taylor expansion of a cut-free MELL proof-structure R with atomic axioms is the (most informative part of the) relational semantics of R; (2) every (connected) MELL proof-net is uniquely determined by the element of order 2 of its Taylor expansion; (3) the relational semantics is injective for (connected) MELL proof-nets

Gluing resource proof-structures: inhabitation and inverting the Taylor expansion

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. As a consequence, we also prove semi-decidability of the type inhabitation problem for cut-free MELL proof-structures.Comment: arXiv admin note: substantial text overlap with arXiv:1910.0793

Strong normalization property for second order linear logic

AbstractThe paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girard’s original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girard’s proof-nets, and we apply to sps Gandy’s method to infer SN from weak normalization (WN). We prove a standardization theorem for sps: if WN without erasing steps holds for an sps, then it enjoys SN. A key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness, namely acyclicity slice by slice. We conclude by showing how standardization for sps allows to prove SN of LL, using as usual Girard’s reducibility candidates

A semantic measure of the execution time in Linear Logic

We give a semantic account of the execution time (i.e. the number of cut-elimination steps leading to the normal form) of an untyped MELL (proof-)net. We first prove that: 1) a net is head-normalizable (i.e. normalizable at depth 0) if and only if its interpretation in the multiset based relational semantics is not empty and 2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then define a size on every experiment of a net, and we precisely relate the number of cut-elimination steps of every stratified reduction sequence to the size of a particular experiment. Finally, we give a semantic measure of execution time: we prove that we can compute the number of cut-elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the (untyped) lambda-calculus

Gluing resource proof-structures: inhabitation and inverting the Taylor expansion

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