Gödel’s system T revisited

The linear lambda calculus, where variables are restricted to occur in terms exactly once, has a very weak expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with natural numbers and a restricted iterator: only closed linear functions can be iterated. We show properties of this linear version of Gödel’s Tusing a closed reduction strategy, and study the class of functions that can be represented. Surprisingly, this linear calculus offers a huge increase in expressive power over previous linear versions of T, which are ‘closed at construction’ rather than ‘closed at reduction’. We show that a linear Twith closed reduction is as powerful as T.