38 research outputs found
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
The relational model is injective for Multiplicative Exponential Linear Logic
We prove a completeness result for Multiplicative Exponential Linear Logic
(MELL): we show that the relational model is injective for MELL proof-nets,
i.e. the equality between MELL proof-nets in the relational model is exactly
axiomatized by cut-elimination.Comment: 33 page
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
Taylor expansion in linear logic is invertible
Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded
into a differential net, which is its Taylor expansion. We prove that two
different MELL proof-nets have two different Taylor expansions. As a corollary,
we prove a completeness result for MELL: We show that the relational model is
injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the
relational model is exactly axiomatized by cut-elimination
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
Proof-Net as Graph, Taylor Expansion as Pullback
We introduce a new graphical representation for multiplicative and exponential linear logic proof-structures, based only on standard labelled oriented graphs and standard notions of graph theory. The inductive structure of boxes is handled by means of a box-tree. Our proof-structures are canonical and allows for an elegant definition of their Taylor expansion by means of pullbacks
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
On the discriminating power of tests in resource lambda-calculus
Since its discovery, differential linear logic (DLL) inspired numerous
domains. In denotational semantics, categorical models of DLL are now commune,
and the simplest one is Rel, the category of sets and relations. In proof
theory this naturally gave birth to differential proof nets that are full and
complete for DLL. In turn, these tools can naturally be translated to their
intuitionistic counterpart. By taking the co-Kleisly category associated to the
! comonad, Rel becomes MRel, a model of the \Lcalcul that contains a notion of
differentiation. Proof nets can be used naturally to extend the \Lcalcul into
the lambda calculus with resources, a calculus that contains notions of
linearity and differentiations. Of course MRel is a model of the \Lcalcul with
resources, and it has been proved adequate, but is it fully abstract? That was
a strong conjecture of Bucciarelli, Carraro, Ehrhard and Manzonetto. However,
in this paper we exhibit a counter-example. Moreover, to give more intuition on
the essence of the counter-example and to look for more generality, we will use
an extension of the resource \Lcalcul also introduced by Bucciarelli et al for
which \Minf is fully abstract, the tests