uni_notes/00 Inbox/29593940 - Formal Languages.md

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2026-04-08 10:20	29593850 - Automationtheory	languages		lecture	🔴	university

📌 Summary

[!abstract] Definition and example for formal languages

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📝 Content

A formal language of the alphabet Sigma is a subset L of Sigma^*.

[!EXAMPLE] For alphabet Sigma = {a, b}, let L_1 be the set of all string starting with b, followed by an arbitrary number of $a$'s, and ending with b: L_1 = {b a^n b | n in NN_0} Then b b in L_1, b a b in L_1, etc.

More in

Language Operations

Concatenation of languages

For languages X subset Sigma^*_X over alphabet Sigma_X and Y subset Sigma^*_Y over alphabet Sigma_Y, their concatenation is

X circle.small Y = X Y = {x y bar x in X and y in Y}

The concatenation of X and Y thus contains all string combinations where the prefix is a string from X and the suffix is a string from Y.

[!CONVENTION] Concatenation has a higher precedence than set operations (union, inter).

Exponentiation of languages

The n^"th" power of the language L subset.eq Sigma^* over alphabet Sigma is

• L^0 := { epsilon }
• L^n := L^(n-1) L if n > 0

Finite representation of languages

Goal: Represent a language using finite information

Using set notation

S = {a^n b^m bar n, m >= 0} = {epsilon, a, b, "aa", "ab", ...}

This is very inefficient.

Using regular expressions

A regular expression r over an alphabet Sigma is defined recursively:

• emptyset, epsilon and each a in Sigma are regular expression, which represent the Languages L(emptyset) = emptyset, L(epsilon) = {epsilon} and L(a) = {a}
• If r and s are regular expressions then
  • (r+s)