The last post was about strings, so now something about languages. From the theoretical point of view a language is a set of words or more precisely sentences (strings). Possibly and usually an infinite set. Only sentences that are present in the set are part of the language, nothing else. Neat feature of languages is, that they interconnect two separate formal theories/models into one thing. Let me explain.

Language is a set. As in mathematical set. So anything that applies to sets applies to languages as well (like union, intersection, complement, Cartesian product). Members of any language are only strings. And there are some operations defined over strings (that is concatenation, reversal, sub-strings etc.). We can take advantage of this fact and abstract those, so they work with language as well. There is generally a lot you can do with a formal language either from the set or string point of view.

Definition 1. A formal language L over an alphabet Σ is a subset of Σ^*. Formally, L ⊆ Σ^*. The magical Σ^* denotes a set of all existing words and sentences over Σ. Sometimes it’s called an alphabet iteration.

Like I said earlier, all operations defined for sets work with languages as well. For example language union L₁ ∪ L₂ = {s; s ∈ L₁ ∨ s ∈ L₂} etc. More interesting are the string operations that can be abstracted so they work on languages. There is language concatenation:

Definition 2. Let L₁ be a language over Σ₁ and L₂ be a language over Σ₂. Concatenation of these two L = L₁⋅L2 = {xy; x ∈ L₁, y ∈ L₂} is a language over alphabet Σ = Σ₁ ∪ Σ₂.

Language concatenation as well as the string equivalent is not commutative operation. There is a couple of special cases:

• L ⋅ {ε} = L
• L ⋅ ∅ = ∅

The first, {ε} is a language with a single word — an empty string, in the second case we’re talking about an empty set. Concatenation can be looked at a sort of multiplication between the two languages (remember Cartesian product?). As a result of this thought, we define language power.

Definition 3. Let L be a language over Σ, then Lⁿ denotes n-power of language and

• L⁰= {ε}
• Lⁿ⁺¹=Lⁿ⋅L

With language power defined, we can advance to the last operation called language iteration. It’s basically an abstraction of language power.

Definition 4. L⁺ denotes positive iteration of language L defined accordingly

• L⁺ = L ∪ L²∪ … ∪ L^∞

Definition 5. L^* denotes iteration of language L defined accordingly

• L^* = {ε} ∪ L ∪ L²∪ … ∪ L^∞

Summary

In this post were introduced some of the most basic and most common operations that are defined on languages. The most important thing to remember is, that language is a set of sentences (or words) and the operations that we define originate from these two areas.

We have some set operations like union, intersection, complement as for generic sets. And we have abstractions for string operations like concatenation (notice, that it’s somewhat similar to cartesian product), language power and iteration.

Anyway, these couple of operations is nowhere near to the complete list of all existing functions for languages! After all, if you dive into some algebraic structures a little you can easily define your own :-).