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International audience; We present a formalization of proof-nets (and more generally, proofstructures) for the multiplicative-exponential fragment of linear logic, with a novel treatment of boxes: instead of integrating boxes into to the graphical structure (e.g., by adding explicit auxiliary doors, plus a mapping from boxed nodes to the main door, and ad hoc conditions on the nesting of boxes), we fix a graph morphism from the underlying graph of the proof-structure to the tree of boxes given by the nesting order. This approach allows to apply tools and notions from the theory of algebraic graph transformations, and obtain very synthetic presentations of sophisticated operations: for instance, each element of the Taylor expansion of a proof-structure is obtained by a pullback along a morphism representing a thick subtree of the tree of boxes. A treatment of cut elimination in this framework currently under development.

This special issue is devoted to some aspects of the new ideas that recently arose from the work of Thomas Ehrhard on the models of linear logic (LL) and of the λ-calculus. In some sense, the very origin of these ideas dates back to the introduction of LL in the 80s by Jean-Yves Girard. An obvious remark is that LL yielded a first logical quantitative account of the use of resources: the logical distinction between linear and non-linear formulas through the introduction of the exponential connectives. As explicitly mentioned by Girard in his first paper on the subject, the quantitative approach, to which he refers as ‘quantitative semantics,’ had a crucial influence on the birth of LL. And even though, at that time, it was given up for lack of ‘any logical justification’ (quoting the author), it contained rough versions of many concepts that were better understood, precisely introduced and developed much later, like differentiation and Taylor expansion for proofs. Around 2003, and thanks to the developments of LL and of the whole research area between logic and theoretical computer science, Ehrhard could come back to these fundamental intuitions and introduce the structure of finiteness space, allowing to reformulate this quantitative approach in a standard algebraic setting. The interpretation of LL in the category Fin of finiteness spaces and finitary relations suggested to Ehrhard and Regnier the differential extensions of LL and of the simply typed λ-calculus: Differential Linear Logic (DiLL) and the differential λ-calculus. The theory of LL proof-nets could be straightforwardly extended to DiLL, and a very natural notion of Taylor expansion of a proof-net (and of a λ-term) was introduced: an element of the Taylor expansion of the proof-net/term α is itself a (differential) proof-net/term and an approximation of α.

International audience; A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a (infinite) 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.

International audience; We show that every connected Multiplicative Exponential Linear Logic (MELL) proof-structure (with or without cuts) is uniquely determined by a well-chosen element of its Taylor expansion: the one obtained by taking two copies of the content of each box. As a consequence, the relational model is injective with respect to connected MELL proof-structures.

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.

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.

Proponiamo una rassegna di alcuni aspetti degli sviluppi recenti della teoria della dimostrazione e dei suoi legami con l'informatica.

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.

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.

Process calculi based on logic, such as πDILL and CP, provide a foundation for deadlock-free concurrent programming. However, in previous work, there is a mismatch between the rules for constructing proofs and the term constructors of the π-calculus: the fundamental operator for parallel composition does not correspond to any rule of linear logic. Kokke et al. [12] introduced Hypersequent Classical Processes (HCP), which addresses this mismatch using hypersequents (collections of sequents) to register parallelism in the typing judgements. However, the step from CP to HCP is a big one. As of yet, HCP does not have reduction semantics, and the addition of delayed actions means that CP processes interpreted as HCP processes do not behave as they would in CP. We introduce HCP − , a variant of HCP with reduction semantics and without delayed actions. We prove progress, preservation, and termination, and show that HCP − supports the same communication protocols as CP.

International audience; We introduce a new graphical representation for multiplica-tive 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.

L'opera si propone come testo di riferimento per acquisire una solida preparazione specialistica nella Logica, presentando in maniera rigorosa ed innovativa argomenti tradizionalmente affrontati nei corsi universitari di secondo livello. Questo secondo volume, che completa l'opera, presenta le basi della teoria della ricorsività, l'aritmetica di Peano ed i teoremi di incompletezza, gli assiomi della teoria assiomatica degli insiemi di Zermelo-Fraenkel e la teoria degli ordinali e dei cardinali che ne deriva.

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" subsystems), 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.

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.

International audience; The 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.

We address the question of injectivity of coherent semantics of linear logic proof-nets. Starting from Girard's 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.

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.

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"., 36 pages

We give a semantic characterization of bounded complexity proofs. We introduce the notion of obsessional clique in the relational model of linear logic and show that restricting the morphisms of the category RscrEscrLscr to obsessional cliques yields models of ELL and SLL. Conversely, we prove that these models are relatively complete: an LL proof whose interpretation is an obsessional clique is always an ELL/SLL proof. These results are achieved by introducing a system of ELL/SLL untyped proof-nets, which is both correct and complete with respect to elementary/polynomial time complexity

To attack the problem of ``computing with the additives'', we introduce a notion of sliced proof-net for the polarized fragment of linear logic. We prove that this notion yields computational objects, sequentializable in the absence of cuts. We then show how the injectivity property of denotational semantics guarantees the ``canonicity'' of sliced proof-nets, and prove injectivity for the fragment of polarized linear logic corresponding to the simply typed lambda-calculus with pairing.

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.