by

 (parser . prettyPrint . parser) = id  

I meant 

(prettyPrint . parser . prettyPrint) = id

for a valid input.

Simplifying, (parser ∷ String → something), and (prettyPrint ∷ something → String).

Therefore, (parser . prettyPrint . parser ∷ String → something) and (prettyPrint . parser . prettyPrint ∷ something → String).

Therefore, both criteria could only apply for (something ~ String). But as pretty printing adds quotation marks, not even that is true.

There are four formulations that might be applicable:

1. parser . prettyPrint ≍ id
   

2. prettyPrint . parser ≍ id -- ∷ String → String, useless here
   

3. prettyPrint . parser . prettyPrint ≍ prettyPrint

4. parser . prettyPrint . parser ≍ parser

5. Well, you could go beyond to (prettyPrint . parser . prettyPrint . parser ≍ prettyPrint . parser) etc…
   

I don't think 1 (or 2) follow from one of the last two. But 1 does imply them. So it is a stronger criterion than both, and therefore probably not the one to choose. Assuming the parser is internally consistent, 3 just says something about the internal consistency of the pretty printer, while 4 says something about the relationship of the pretty printer to the parser. Thus 4 looks like the best candidate for a criterion. Possibly with 3 as a secondary target.

Cheers,
MarLinn