Just a technical point that hints at a significant philosophical idea: The asymptote cannot reach 100% of program behavior in any finitary way. That would solve the halting problem. The x-axis should go off to infinity. Also, it's not a smooth progression. There are huge jumps in expressivity involved here. Going from Java-style types to Hindley-Milner to full System F are all massive jumps in expressivity. There are…
> The asymptote cannot reach 100% of program behavior in any finitary way. That would solve the halting problem. There are languages that enforce termination. They only accept programs that can be shown to terminate through syntactic reasoning (e.g., when processing lists, you only recurse on the tail), or where you can prove termination by other means. Coq is like this, as is Isabelle, as is F* , as are others. They…
There are two ways to see type systems. In the first way you construct terms along with their types, this is called Church style. In the second way, the terms exist before their types and you use types to describe their behavior, this is called Curry style. In particular take System F. In Church style the terms of System F come with their types. In Curry style we see System F types as a way to describe the behavior of untyped lambda terms.
I used to think Church style was more important but lately I've been more partial to Curry style. Programs exist before you type them, type systems tell you how they behave. They also tell you how to construct programs but this is subordinate to the more fundamental descriptive capacity.