Geometry of interaction

In proof theory, 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[clarification needed] 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[1] of Lamping's algorithm[2] 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.[3] A bounded version of GoI dubbed the Geometry of Synthesis has been used to compile higher-order programming languages directly into static circuits.[4]

## References

1. ↑ Gonthier, G.; Abadi, M. N.; Lévy, J. J. (1992). "The geometry of optimal lambda reduction". Proceedings of the 19th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '92. pp. 15. doi:10.1145/143165.143172. ISBN 0897914538.
2. ↑ Lamping, J. (1990). "An algorithm for optimal lambda calculus reduction". Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '90. pp. 16–30. doi:10.1145/96709.96711. ISBN 0897913434.
3. ↑ Mackie, I. (1995). "The geometry of interaction machine". Proceedings of the 22nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '95. pp. 198–208. doi:10.1145/199448.199483. ISBN 0897916921.
4. ↑ Dan R. Ghica. Function Interface Models for Hardware Compilation. MEMOCODE 2011. [1]

## Further reading

* GoI tutorial given at Siena 07 by Laurent Regnier, in the Linear Logic workshop, [2]
* Geometry of Interaction in nLab

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