Concept

Geometry of interaction

The Geometry of Interaction (GoI) was introduced by Jean-Yves Girard shortly after his work on linear logic. In linear logic, proofs can be seen as various kinds of networks as opposed to the flat tree structures of sequent calculus. To distinguish the real proof nets from all the possible networks, Girard devised a criterion involving trips in the network. Trips can in fact be seen as some kind of operator acting on the proof. Drawing from this observation, Girard described directly this operator from the proof and has given a formula, the so-called execution formula, encoding the process of cut elimination at the level of operators. One of the first significant applications of GoI was a better analysis of Lamping's algorithm for optimal reduction for the lambda calculus. GoI had a strong influence on game semantics for linear logic and PCF. GoI has been applied to deep compiler optimisation for lambda calculi. A bounded version of GoI dubbed the Geometry of Synthesis has been used to compile higher-order programming languages directly into static circuits.

Official source

https://en.wikipedia.org/wiki/Geometry_of_interaction

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Linear logic

Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction.

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