I will give an introduction to type refinement systems in the style of Frank Pfenning et al., discuss the phenomenon of how refinement typing makes some aspects of evaluation order visible in the type system, and give some general indications of how type refinement may be understood from a categorical perspective. For background reading, you can see my notes from OPLSS 2016 on “Principles of type refinement” (https://noamz.org/oplss16/refinements-notes.pdf).

Dans le but de factoriser et étendre les techniques d'Analyses Amorties de Ressource Automatisées, nous introduisons une machine abstraite comme représentation intermédiaire pour étendre les possibilités d'analyses aux langages linéaires et à sémantiques Call-By-Push-Value (CBPV), qui intègre à la fois l'appel par valeur et l'appel par nom.

La machine est une extension de la machine ILL_rho^eta de Curien et al.[1] avec des points fixes et une dépendance des types sur des paramètres de structures (taille, longueur, coût) qui sont contraints par une formule du premier ordre. Une phase de réécriture systématique, de et vers la machine, enrichit les programmes et définitions d'ADT avec des annotations de paramètres, de passage d'état et des primitives de coût. Elle implémente AARA comme un effet qui produit à la compilation une approximation sûre des manipulations de ressources à l'exécution.

Nous présentons l'exemple d'un miniML en appel par valeur des avec blocs monadiques et transformeurs de monades pour l'état mutable et les exceptions locales. L'analyse AARA demeure inchangée grâce aux implémentations canoniques des primitives monadiques dans CBPV. Une procédure de typage basée sur HM(X) a été implémentée et génère une contrainte globale sur les paramètres du programme entiers. Dans le cas classique où les coûts sont représentés par les polynômes en plusieurs variables, une procédure d'élimination des quantificateurs permet de se ramener à la programmation entière pour instancier des bornes explicites via un solveur externe.

Cet exposé reprend en grande partie le travail qui a été présenté à la conférence LOPSTR 2023 [2]

[1] doi.org/10.1145/2837614.2837652 [2] doi.org/10.1007/978-3-031-45784-5_5

Constructive modal logics are extensions of intuitionistic logic obtained by adding unary operators, called modalities, to the language of intuitionistic logic. These logics have generated increasing interest over the past decades, being investigated from both proof theory and type theory perspectives.

In this talk, we investigate the Curry-Howard correspondence for constructive modal logic CK in light of the gap between the proof equivalences enforced by the lambda calculi from the literature and by the recently defined winning strategies for this logic.

We present a game semantics for CK and a new lambda-calculus system for this logic, obtained by enriching the calculus from the literature with additional reduction rules. We then provide a typing system in the style of focused proof systems, allowing us to provide a unique proof for each term in normal form. Finally, we use this result to show a one-to-one correspondence between terms in normal form and winning innocent strategies for CK.

(Joint work with Matteo Acclavio and Federico Olimpieri).

Structural proof theory has been widely used in the study of term structures. In this talk, I will illustrate this tight connection between proofs and terms by presenting the focused proof system LJF as a framework for designing term structures. Since the proof theory of LJF does not pick a canonical polarization for atomic formulas, two different approaches to term representation arise. We will illustrate these two approaches by applying them to the encoding of untyped lambda terms. When atomic formulas are given the negative polarity, LJF proofs yield the usual tree-like representation of untyped lambda terms. When atomic formulas are given the positive polarity, LJF proofs yield a term structure in which sharing is possible via explicit substitution. In the second part of the talk, I will present the positive lambda calculus, a calculus with explicit substitution based on the positive polarization, and show its close relationship with Accattoli and Paolini's value substitution calculus.

In this work I will present a new feature proposed for the OCaml programming language, “constructor unboxing”, first suggested by Jeremy Yallop in March 2020. It enables a more efficient representation of certain sum types, but requires a static analysis to forbid certain unboxing requests that would be unsound.

To define this static analysis, one has to solve a problem of normalization of first-order definitions in presence of recursion. In the talk I hope to explain my current understanding of this halting problem, and present an algorithm to compute normal forms and reject (in finite time) non-normalizable definitions.

Is λ-calculus a reasonable machine according to Van Emde Boas' Invariance Thesis? By defining the useful evaluation strategy, B. Accattoli and U. Dal Lago were able to answer positively to this question. However, their characterization doesn't facilitate the use of inductive arguments to reason about it. And, up to this date, there are no quantitative models to capture its time complexity.

In this talk I'll discuss the first inductive characterization of the useful evaluation for open call-by-value. We achieve this by parameterizing the evaluation relation with respect to the information provided by the context in which the computation takes place. First we prove some properties of our notion of useful calculus. Then, we give a quantitative model for the useful strategy, using a non-idempotent intersection type system with tight rules. Therefore, typing a term t gives exact information about how many steps does the computation requires to compute the normal form of t.

Closure conversion is a program transformation at work in compilers for functional languages to turn inner functions into global ones, by building 'closures' pairing the transformed functions with the 'environment' of their free variables. Abstract machines rely on similar and yet different concepts of 'closures' and 'environments'.

In this talk, we analyze the relationship between the two approaches. We adopt a very simple lambda-calculus with tuples as source language and study abstract machines for both the source language and the target of closure conversion.

We overview three contributions. Firstly, a new simple proof technique for the correctness of closure conversion, inspired by abstract machines. Secondly, we show how the closure invariants of the target language allow us to design a new way of handling environments in abstract machines, not suffering the shortcomings of other styles. Thirdly, we analyze the machines from the point of view of sharing and time complexity, dissecting how different aspects of closure conversion impact on the cost of the execution.

Cut-reduction procedures for classical sequent calculus are notoriously non-deterministic and non-confluent, both in the original formulation by Gentzen and in later reformulations. It is natural to ask whether those instances of non-confluence are superficial in nature, i.e. whether syntactically distinct normal forms of the same derivation are in fact correlated in a non-trivial way, as is the case in the intuitionistic and linear versions of sequent calculus. A famous counter-example by Lafont purports to show that the answer is negative, that is, every interpretation of derivations in LK that is invariant under classical cut-elimination must be a trivial one that identifies at least all proofs of the same sequent. A long-standing open question has then been whether it could be possible to work around Lafont's example by natural and non-trivial adjustments of the calculus and/or of cut-reduction steps, without resorting to symmetry-breaking techniques like polarization or embeddings into intuitionistic or linear logic.

Working within the propositional fragment of the context-sharing formulation of LK — where parallel logical rules permute freely — we show that the graph constructed by tracing the history of atomic formula occurrences through axiom and cut rules is invariant under arbitrary rule permutations in cut-free proofs, thus providing a canonical representation of normal-form proofs.

We then introduce a refinement of the notion of axiom-induced graph that allows extending the invariance result to proofs with cuts, although at the cost of a strong assumption on the shape of derivations. Because cut-reduction in this formulation of LK can be implemented entirely by logical rule permutations plus a pair of local rewriting steps that preserve the refined axiom graphs, the result yields a non-trivial invariant of cut-reduction.

Our work introduces a quantitative version of the simple type assignment system, starting from a suitable restriction of non-idempotent intersection types. The key idea is to restrict multiset formation to uniform types, two types being uniform if they differ only in the cardinality of the multisets occurring in it. The resulting system has the same typability power as the simple type assignment system; thus, assigning types to terms supplies the very same qualitative information given by simple types, but at the same time provides some interesting quantitative information. It is well known that typability for simple types is equivalent to unification; we prove a similar result for the newly introduced system. More precisely, we show that typability is equivalent to a unification problem which is a non-trivial extension of the classical one: in addition to unification rules, our typing algorithm makes use of an operation that increases the cardinality of multisets whenever needed.