Monday 24 March 2025
The Geometry of Interaction program, a research effort that seeks to bridge the gap between proof theory and algebraic geometry, has taken another significant step forward. A new paper published on arXiv details how researchers have developed a model for multiplicative exponential linear logic (MELL), a language that allows for more complex calculations than its predecessors.
To understand this breakthrough, it’s necessary to first grasp the basics of proof theory and algebraic geometry. Proof theory is concerned with understanding the logical structures underlying mathematical proofs, while algebraic geometry deals with the geometric properties of spaces defined by polynomial equations. The Geometry of Interaction program combines these two fields in an attempt to develop a deeper understanding of how proofs work.
MELL, specifically, is a language that allows for exponential operations, which enable more complex calculations than its predecessors. However, this increased power comes at a cost: MELL’s semantics are much harder to understand and work with than those of earlier languages.
The new paper proposes a model for MELL based on the Hilbert scheme, a mathematical construct used to study the geometry of algebraic varieties. In essence, the authors use the Hilbert scheme to represent the proofs in MELL as geometric objects, allowing them to analyze the properties of these proofs in a more intuitive and visual way.
This approach has several benefits. For one, it provides a new way of understanding the behavior of exponential operations in MELL, which is crucial for developing practical applications of the language. Additionally, the model can be used to study the geometry of interaction between different components of a proof, shedding light on how these interactions affect the overall structure of the proof.
The authors also explore connections with other areas of mathematics, such as elimination theory and syzygies, which may seem unrelated at first glance. However, by bridging the gap between proof theory and algebraic geometry, researchers can gain new insights into these fields and develop novel techniques for solving problems in both areas.
While this breakthrough is significant, it’s just one step forward in a long journey of research. The Geometry of Interaction program has many more challenges to overcome before it can be applied practically. Nevertheless, the potential rewards are substantial: if successful, this work could lead to new insights into the nature of proof and computation itself, with far-reaching implications for fields such as computer science, mathematics, and philosophy.
The paper is available on arXiv and provides a detailed explanation of the model and its applications.
Cite this article: “Geometry of Interaction Program Takes Significant Step Forward with New Model for Multiplicative Exponential Linear Logic”, The Science Archive, 2025.
Proof Theory, Algebraic Geometry, Geometry Of Interaction, Mell, Hilbert Scheme, Exponential Operations, Logical Structures, Geometric Properties, Computation, Mathematics
Reference: William Troiani, Daniel Murfet, “Linear Logic and the Hilbert Scheme” (2025).
