I am currently on a frolic into the literature on Control Flow Analysis (CFA), and discuss what this is, for pure lambda calculus. A wonderful reference for this is this paper by Palsberg.

In this episode, I talk about what we should consider to be a measure function. Such functions can be used to show termination of some process or program, by assigning a measure to each program, and showing that as the program computes, the measure decreases in some well-founded ordering. But what should count as a measure function? The context for this is RTA Open Problem 19, on showing termination for the simply typed lambda calculus using a measure function.

Let's call this the start of season 7, because it seems about time for that.

To solve the problem raised in the last episode, I propose schematic affine recursion. We saw that affine lambda calculus (where lambda-bound variables are used at most once) plus structural recursion does not enforce termination, even if you restrict the recursor so that the function to be iterated is closed when you reduce ("closed at reduction"). You have to restrict it so that recursion terms are disallowed entirely unless the function to be iterated is closed ("closed at construction"). But this prevents higher-order functions like map, which need to repeat a computation involving a variable f to be mapped over the elements of a list. The solution is to allow schematic definition of terms, using schema variables ranging over closed terms.