I've struggled in the closure properties of the general class of languages because I couldn't use any automata concept and grammars.

In specific, I'm interested in dependency of operations.

(The operation $A$ is dependent to other operations $\Sigma$ when every class of languages is closed under some operations $\Sigma$, then it must be closed under an operation $A$.)

So there is some questions about this.

  1. When I solve the problem about the closure properties of general class of languages(not CFL, REG, or else which is directly related to some grammar systems, machines and automata.), what concepts could I use?

For example, when I solve about the problem about full-trio I just use the definitions of morphism, inverse morphism, intersection by regular languages but this is too weak and hard to find suitable morphisms and regular languages. So, I could use the finite state transducer(this is guaranteed by Nivat's theorem).

  1. Is there any good textbooks for this(closure properties of general class of languages)? I read Hopcroft; Ullman, "Formal Languages and Their relation to Automata", especially Chapter 9. I think this is a good reference to study but I felt this is not enough to understand the theory fully.

And there is another question which is not really related to the original questions, so you don't need to answer it.

Several references study about only trio, full-trio, semi-AFL, AFL or some examples. Why most of the references only deal with these FLs(Family of languages)?

  • $\begingroup$ The set of all languages is closed under all operations on languages. Perhaps you're interested in recursive languages? $\endgroup$ – Yuval Filmus May 9 '18 at 12:28
  • $\begingroup$ @YuvalFilmus I think I used an ambigious words, the general class of language. I intend to express any family of languages, not the specific families like family of all languages, CFL, REG, or else. In the case, the classification of families by closure properties are meaningful. $\endgroup$ – ChoMedit May 9 '18 at 12:55

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