Taylor Expansion, Finiteness and Strategies

We examine some recent methods introduced to extend Ehrhard and Regnier's result on Taylor expansion: infinite linear combinations of approximants of a lambda-term can be normalized while keeping all coefficients finite. The methods considered allow to extend this result to non-uniform calculi; we show that when focusing on precise reduction strategies, such as Call-By-Value, Call-By-Need, PCF or variants of Call-By-Push-Value, the extension of Ehrhard and Regnier's finiteness result can hold or not, depending on the structure of the original calculus.

In particular, we introduce a resource calculus for Call-By-Need, and show that the finiteness result about its Taylor expansion can be derived from our Call-By-Value considerations. We also introduce a resource calculus for a presentation of PCF with an explicit fixpoint construction, and show how it interferes with the finiteness result. We examine then Ehrhard and Guerrieri's Bang Calculus which enjoys some Call-By-Push-Value features in a slightly different presentation.