Function programming with no garbage collector is tough. There is a formalisation for dealing with this, called linear logic, which allows you to statically determine object lifetimes; but it's, um, a bit awkward. Such as, every value must be produced once and consumed once, so if you want to look at a variable, you need to produce a new variable with the same value (which must be consumed later).

The seminal linearly typed language is Linear Lisp, which is almost incomprehensible, and also a proof-of-concept which doesn't do anything useful. Classic paper here:

http://home.pipeline.com/~hbaker1/LinearLisp.html

The closest thing to a modern language that uses them exclusively (Rust doesn't, by the way) is LinearML, which was an experimental toy, now abandoned. Tutorial here, which may be interesting:

https://github.com/pikatchu/LinearML/wiki/Tutorial

I asked about it on StackOverflow a while back, and it collected a bunch of useful resources, including links to some of the seminal papers:

http://stackoverflow.com/questions/5065861/programming-langu...

(Still looking for a modern pure linear typed language, by the way.)