In this talk, we introduce the syntax and semantics of Differential Linear Logic. We explain how its rules relate to Linear Logic's rules, and give informal intuitions in terms of functions and distributions. We show how its cut-elimination rules are a reflection of basic calculus rules. We also review Differential Lambda-calculus, with matching intuitions. At the end of the talk, we briefly review two recent development about Differential Linear Logic, in terms of Laplace transformation and co-promotion.[-]

Many models of (differential) linear logic and lambda-calculus can be regarded as a quantitative enrichment of the relational semantics of linear logic. This talk presents an introduction to these models, taking a simple but flexible approach. Relations can be enriched with coefficients drawn from any complete semiring — a structure which allows multiplication and infinite summation of quantities — and in each case we obtain a soundness result with respect to a quantitative operational semantics for a functional language with recursion, nondeterminism and quantitative effects. Examples include models that track the number of possible reduction paths, the length of the shortest reduction path, or the probability of termination of a program.[-]

Semistructured data is an umbrella term encompassing data models which are not logically organized in tables (i.e., the relational data model) but rather in hierarchical structures using markers such as tags to separate semantic elements and data fields in a ‘self-describing' way. In this lecture we survey some of the multiple connections between formal language theory and semi-structured data, in particular concerning the XML format. We will cover ranked and unranked tree automata, and its connections to Monadic Second Order logic, First Order logic, and XPath. The aim is to take a glimpse at the landscape of closure properties, algorithms and expressiveness results for these formalisms.[-]

Semistructured data is an umbrella term encompassing data models which are not logically organized in tables (i.e., the relational data model) but rather in hierarchical structures using markers such as tags to separate semantic elements and data fields in a ‘self-describing' way. In this lecture we survey some of the multiple connections between formal language theory and semi-structured data, in particular concerning the XML format. We will cover ranked and unranked tree automata, and its connections to Monadic Second Order logic, First Order logic, and XPath. The aim is to take a glimpse at the landscape of closure properties, algorithms and expressiveness results for these formalisms.[-]

In this programmatic talk, we will sketch both a conceptual and formal framework for reasoning about the notion of algorithm. This framework will arise from the analysis we will make of the relationships existing between the notion of algorithm and other similar (but still different) notions, like that of computation and that of program. We will first show that the Turing-Church thesis concerning effective computability is not sufficient to capture the notion of algorithm, as it identifies programs which are intensionally different. We will then show the limits of the existing models of computation in capturing some basic construction processes that we are willing to call algorithmic. In order to solve this problem, we propose a formalisation of the notion of model of computation on the base of which we claim that the notion of algorithm could eventually be analyzed. This approach centered around the dynamics of program execution, reconciles the more mechanical view of computation (such as formalized by Turing machines and automata) with the logical view - as it in particular stems from a generalization of Jean-Yves Girard's Geometry of Interaction programme.[-]

Gentzen's sequent calculus can be given additional structure via focusing. We illustrate how that additional structure can be used to construct synthetic inference rules. In particular, bipolar formulas (a slight generalization of geometric formulas) can be converted to such synthetic rules once polarity is assigned to the atomic formulas and some logical connectives. Since there is some flexibility in the assignment of polarity, a given formula might yield several different synthetic rules. It is also the case that cut-elimination immediately holds for these new inference rules. Such conversion of bipolar axioms to inference rules can be done in classical and intuitionistic logics.
This talk is based in part on a paper in the Annals of Pure and Applied Logic co-authored with Sonia Marin, Elaine Pimentel, and Marco Volpe (2022).[-]