Call by value in the lambda calculus

Question

I\'m working my way through Types and Programming Languages, and Pierce, for the call by value reduction strategy, gives the example of the term id (id (λz. id z)).

Answer 1

Short answer: yes. You can never reduce inside a lambda-term you can only reduce term outside, starting by right.

The set of evaluation contexts in lambda-calculus by value is defined as follow:

E = [ ] | (λ.t)E | Et

E is what you can value..

For example in lambda calculus by name the evaluation context is :

E = [ ] | Et | fE

as you can reduce an application even if a term is not a value. For example (λx.x)(z λx.x) is stuck in call by value but in call by name it reduce to (z λx.x), which is a normal form.
In the context grammar f is a normal form (in call by name) defined as:

f = λx.t | L  
L = x | L f

You can see another definition of contexts at chapter 19.5.3 of the Pierce.

Answer 2

Is the answer that 'outermost' and 'innermost' only refers to lambda abstractions? So for a term t in λz. t, t can't be reduced, but in a redex s t, t is reduced to a value v if this is possible, and then s v is reduced?

Yes, that's exactly right.