Lorenzo Tortora de Falco

Polarized and focalized linear and classical proofs

Abstract We give the precise correspondence between polarized linear logic and polarized classical logic. The properties of focalization and reversion of linear proofs are at the heart of our analysis: we show that the tq-protocol of normalization for the classical systems LK pol η and LK pol η , ρ perfectly fits normalization of polarized proof-nets. Some more semantical considerations allow us to recover LC as a refinement of multiplicative LK pol η .

Obsessional experiments for linear logic proof-nets

We address the question of injectivity of coherent semantics of linear logic proof-nets. Starting from Girards definition of experiment, we introduce the key-notion of ‘injective obsessional experiment’, which allows us to give a positive answer to our question for certain fragments of linear logic, and to build counter-examples to the injectivity of coherent semantics in the general case.

Strong normalization property for second order linear logic

The paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girards original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girards proof-nets, and we apply to sps Gandys 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 Girards reducibility candidates.

Additives of linear logic and normalization: part I: A (restricted) Church--Rosser property

We define a generalized cut-elimination procedure for proof-nets of fall linear logic (without constants), for which we prove a (restricted) Church-Rosser property.

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

Abstract 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”.

The additive multiboxes

Abstract We introduce the new notion of additive “multibox” for linear logic proof-nets. Thanks to this notion, we define a cut-elimination procedure which associates with every proof-net of multiplicative and additive linear logic a unique cut-free one.

Denotational Semantics for Polarized (But-non-constrainted) LK by Means of the Additives

Following the idea that a classical proof is a superimposition of ‘constructive’ proofs, we use the linear connective “&” to define an embedding Pa of polarized but non-constrained LK in linear logic, which induces a denotational semantics. A classical cut-elimination step is now a cut-elimination step in one of the ‘constructive slices’ superimposed by the classical proof.

A semantic account of strong normalization in Linear Logic

Abstract We prove that given two cut-free nets of linear logic, by means of their relational interpretations one can determine: 1) whether or not the net obtained by cutting the two nets is strongly normalizable, 2) (in case it is strongly normalizable) the maximum length of the reduction sequences starting from that net. As a by-product of our semantic approach, we obtain a new proof of the conservation theorem for Multiplicative Exponential Linear Logic ( MELL ) which does not rely on confluence; this yields an alternative proof of strong normalization for MELL .

Strong Normalization for All-Style LK

We prove strong normalization of tq-reduction for all standard versions of sequent calculus for classical and intuitionistic (second and first order) logic and give a perspicuous argument for the completeness of the focusing restriction on sequent derivations.

Generalized Standardization lemma for the additives

Abstract This is a preliminary report, in which we prove the so-called standardization lemma for the additives, in presence of a generalization (the natural one) of the commutative elementary reduction step. The technical tools developped to achieve this result suggest some remarks on the non confluence of full linear logic.