Decision Problems for Non-regular Languages

We begin with studying closure properties of language classes. Informally, a closure property of a languages class C states that a certain type of transformation applied to languages in C always yields languages that belong to C as well. Examples for such transformations are: (i) Homomorphisms α : X∗ → Y ∗. They transform a language L ⊆ X∗ into the language α(L) = {α(w) | w ∈ L}. (ii) Inverse homomorphisms. If α : X∗ → Y ∗ is a homomorphism, then, inversely applied to L ⊆ Y ∗, it yields α−1(L) = {w ∈ X∗ | α(w) ∈ L}. (iii) Intersection with regular sets. For a regular language R ⊆ X∗, this transformation turns a language L ⊆ X∗ into L ∩R. A language is a subset of X∗ for some finite alphabet X. A language class is a collection of languages that contains at least one non-empty language. A language class C is said to be a full trio if it is closed homomorphisms, inverse homomorphisms, and intersection with regular languages. If instead of arbitrary homomorphisms, we only require closure under non-erasing homomorphisms (i.e. α(x) 6= ε for all x ∈ X), then we have a trio. Examples of full trios are: • the regular languages, • the context-free languages, and • the recursively enumerable languages (as we will see later). The context-sensitive languages constitute a trio, but not a full trio.