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Appears in collection : Differential $\lambda$-Calculus and Differential Linear Logic, 20 Years Later / $ \lambda $-calcul différentiel et logique linéaire différentielle, 20 ans après

Many models of (differential) linear logic and lambda-calculus can be regarded as a quantitative enrichment of the relational semantics of linear logic. This talk presents an introduction to these models, taking a simple but flexible approach. Relations can be enriched with coefficients drawn from any complete semiring — a structure which allows multiplication and infinite summation of quantities — and in each case we obtain a soundness result with respect to a quantitative operational semantics for a functional language with recursion, nondeterminism and quantitative effects. Examples include models that track the number of possible reduction paths, the length of the shortest reduction path, or the probability of termination of a program.

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  • EHRHARD, Thomas. Finiteness spaces. Mathematical Structures in Computer Science, 2005, vol. 15, no 4, p. 615-646. - https://doi.org/10.1017/S0960129504004645
  • EHRHARD, Thomas, TASSON, Christine, et PAGANI, Michele. Probabilistic coherence spaces are fully abstract for probabilistic PCF. In : Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. 2014. p. 309-320. - https://doi.org/10.1145/2535838.2535865
  • LAMARCHE, François. Quantitative domains and infinitary algebras. Theoretical computer science, 1992, vol. 94, no 1, p. 37-62. - https://doi.org/10.1016/0304-3975(92)90323-8
  • LAIRD, Jim, MANZONETTO, Giulio, MCCUSKER, Guy, et al. Weighted relational models of typed lambda-calculi. In : 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, 2013. p. 301-310. - https://doi.org/10.1109/LICS.2013.36

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