Abstract
This note is a brief survey and a discussion of recent ideas and open problems in the understanding of an important aspect of Type Theory: how terms may depend on types. This problem is at the core of the distinction between “ad hoc” and proper polymorphism and inspired the large amount of work on “parametricity”.
Contents. 1 Types; 2 Parametricity; 3 Genericity; 4 Axiom C and Dinatural Transformations; 5 Axiom C and the Isomorphisms of Types; 6. Types as Parameters; 7. True type dependency or “ad hoc” polymorphism.
The presentation is directly endebted to joint work and many stimulating discussions with Giuseppe Castagna, Giorgio Ghelli, Roberto DiCosmo, Simone Martini, Kathleen Milsted, Sergei Soloviev. With Simone, in particular, I discussed of the dinatural interpretation (§.4), while Sergei and Roberto pointed out to me the connection to the isomorphisms of types in §.5.
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Longo, G. (1993). Types as parameters. In: Gaudel, M.C., Jouannaud, J.P. (eds) TAPSOFT'93: Theory and Practice of Software Development. CAAP 1993. Lecture Notes in Computer Science, vol 668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56610-4_96
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