Periodic Reporting for period 1 - REGALIA (RELatinG quALIties and quantities by resource Approximation)
Berichtszeitraum: 2024-09-01 bis 2026-08-31
Qualitative type systems are a widespread technique exploited in the study of programming languages. From this one can obtain relevant information on the behaviours of programs, such as termination of evaluation. REGALIA aims to deepen our understanding of the relationship between these systems and quantitative ones, that are used to obtain information about complexity and resource consumption of the computation. In order to do so, we shall further develop the theory of resource approximation, by extending Girard's approximation theorems to proofs with cuts and by establishing a translation algorithm between qualitative systems and quantitative ones. We shall then exploit these results to define modular methods to study programming languages, alternative to Tait-Girard reducibility, that will offer quantitative interpretation of relevant qualitative systems in the context of both purely functional and effectful computation.
The host istitution will be the University of Bologna, the perfect environment for REGALIA, being a living center for research on implicit complexity and programming languages. The supervisor will be Ugo dal Lago, a leading expert in programming languages theory.
The host istitution will be the University of Bologna, the perfect environment for REGALIA, being a living center for research on implicit complexity and programming languages. The supervisor will be Ugo dal Lago, a leading expert in programming languages theory.
During the last 6 months we were able to achieve the main objective of the project, namely the extension of Girard's approximation theorems by the means of linearization. In order to do so, we defined the structural resource calculus, a new formalism in which strongly normalizing terms of the lambda-calculus can naturally be represented, and at the same time any type derivation can be internally rewritten to its linearization. The calculus is shown to be normalizing and confluent. Noticeably, every strongly normalizable lambda-term can be represented by a type derivation. This is the first example of a system where the linearization process takes place internally, while remaining purely finitary and rewrite-based. We submitted a paper to LICS 2025 about this result.
Known linearization procedures were either infinitary (like the Taylor expansion) or non-terminating (like Kfoury's original construction). We defined a finitary and terminating procedure, which paves the way for applications in the analysis of program behavior.