ABSTRACT. Species of structures were first introduced by Joyal as a unified framework for the theory of generating series in enumerative combinatorics. In 1988, Girard introduced normal functors as a model of pure lambda-calculus where terms are interpreted as infinite series with sets as coefficients (which correspond to a special case of Joyal's species). Fiore presented a generalized definition that both encompasses Joyal's species and constitutes a model of differential linear logic.

Since species encode families of labelled structures, much work has been done to investigate their connection with algebraic data types. We want to explore an alternative viewpoint of seeing them as terms in an extension of lambda-calculus motivated by the relationship between species and differential models of linear logic and use the combinatorial intuition as a guide in the design of the syntax. The next step would be to study methods of resolution of differential equations in the setting of generalized species with the ultimate goal being to establish in linear logic a combinatorial interpretation of the obtained differential calculus.

ABSTRACT. Affine lambda-terms are lambda-terms in which each bound variable occurs at most once and linear lambda-terms are lambda-terms in which each bound variable occurs once and only once. In this paper we count the number of affine closed lambda-terms of size n, linear closed lambda-terms of size n, affine closed beta-normal forms of size n and linear closed beta-normal forms of size n, for several measures of the size of lambda-terms. From these formulas, we show how we can derive programs for generating all the terms of size n for each class. The foundation of all of this is a specific data structure, made of contexts in which one counts all the holes at each level of abstractions by lambda's.

ABSTRACT. Finding lower bounds in complexity theory has proven to be an extremely difficult task. Nowadays, only one research direction is commonly acknowledged to have the ability to solve current open problems, namely Mulmuley's Geometric Complexity Theory programme. Relying on heavy techniques from algebraic geometry, it stemmed from a first lower bound result for a variant of Parallel Random Access Machines.

We analyse this original proof, together with two other proofs of lower bounds from a semantics point of view, interpreting programs as graphings -- generalizations of dynamical systems. We show that, abstracted to a more general setting that exploits the classic notion of topological entropy, this three proofs share the same structure. This reformulation recentres the proofs around dynamical aspects, relegating the use of algebraic geometry to a model-driven choice rather than a key concept of the method.

ABSTRACT. Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly underdeveloped. In particular, very little is known about the computational behavior of such proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a proof system: cut elimination and focalization. The first one is only known to hold for restricted (purely additive) infinitary calculi, thanks to the work of Santocanale and Fortier; the second one has never been studied in infinitary systems. In this paper, we consider the infinitary proof system µMALL∞ for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results. We thus establish µMALL∞ as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.

ABSTRACT. In this talk we want to present the recent results of \cite{DK}. We construct several smooth classical denotational models of Linear Logic: they are smooth as non-linear proofs are interpreted as infinitely differentiable functions, and they feature an involutive linear negation. The starting point of this work consists in noticing that the multiplicative disjunction $\parr$ corresponds to the well-known Schwartz' epsilon product. Requiring its associativity then asks for a completeness notion, while the linear involutive negation is ensured by considering a good topology (the Arens topology) on the dual, ensuring that the linear involutive negation works as an orthogonality relation.

ABSTRACT. We propose to return to the construction carried precedently, where we claimed that Simply-typed approximations = intersection types derivations in the precise sense that we built a categorical equivalence between specific type systems (that encompass all well-known intersection type systems used to characterize normalization) and simply-typed approximations, that we realize as approximation functors, that arise from the translation of the language into linear logic. By studying these specific functors, we claim that their main feature is that they map the exponential of linear logic into what can reasonably be called a resource modality, corresponding either to linear, affine or cartesian intersection types. So, we present the story under the slightly different, and less syntactic, slogan: Intersection type system = multiplicative linear logic + resource modality These resource modalities are linked with well-know systems. In particular, generalized species of structure can be seen as a strictification of the Kleisli category of the linear resource modality. The study of the link between these different resource modality can shed a new light on the extensional collapse, and paving the way for a study of this collapse for dynamic semantics (such as the Geometry of Interaction and ordered combinatory algebras, used to account for forcing and realizability).

ABSTRACT. Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well- established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we seek to start a discussion of benchmarks for Girard’s linear logic and some of its variants. For some quick bootstrapping of the collection of problems, we use translations of the collection of Kleene’s intuitionistic theorems in the traditional monograph Introduction to Metamathematics. We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using linear logic based provers with focusing.

ABSTRACT. We introduce the notion of coherent graphs, and show how those can be used to define dynamic semantics for Multiplicative Linear Logic. The models thus obtained are finite with respect to several aspects (finite graphs, finite generation of types) and thus improve in this way previous constructions by Seiller. We also discuss how the added notion of coherence can also be used to introduce non-determinism.

ABSTRACT. [joint work with Klaas Pruiksma]

Session types were designed to express communication behavior between
concurrent processes.  They were later discovered to correspond to
propositions in linear logic, where sequent proofs describe processes,
and cut reduction is the basis for message-passing computation.  In
this talk we provide a new operational interpretation of the linear
sequent calculus in terms of shared memory, with some surprising
twists when compared to the message-passing semantics.

ABSTRACT. Semantic evaluation has proven to be a valuable tool to prove computational complexity results on monomorphic type systems. Can we apply it in presence of impredicative polymorphism? The linearity and stratification at work in light logics might make it possible.

ABSTRACT. We present a declarative and modular specification of an automated trading system (ATS) in the concurrent linear framework CLF. We implemented it in Celf, a CLF type checker which also supports executing CLF specifications. We outline the verification of a representative property of trading systems using generative grammars, an approach to reasoning about CLF specifications.