Alphabet, Strings, and Languages

 

🔠 Alphabet, Strings, and Languages

📘 1. Alphabet (Σ)

Definition :

An alphabet is a finite, non-empty set of symbols. It is usually denoted by the Greek letter Σ (sigma).

Examples:

  • Σ = {0, 1} → binary alphabet

  • Σ = {a, b} → common for regular languages

  • Σ = {0, 1, 2, ..., 9} → decimal digits

  • Σ = set of all ASCII characters → for programming languages

📌 Note: The alphabet must be finite. 

Teaching Tip:

Think of the alphabet as the set of building blocks—like letters in a language or keys on a keyboard.

📄 2. String (or Word)

Definition:

A string (or word) over an alphabet Σ is a finite sequence of symbols chosen from Σ.

  • If Σ = {a, b}, then:

    • a, ab, bba, aaabb are all strings over Σ.

    • The string xyz is not a string over Σ.

Special Case – Empty String (ε):

  • The empty string, denoted ε, is the string with zero symbols.

  • It is not the same as a string containing a space or "null".

  • Every alphabet has ε as a valid string.

Operations on Strings:

  • Concatenation: Combining two strings
    If x = ab and y = ba, then xy = abba

  • Length: Denoted |w| for a string w
    If w = aba, then |w| = 3, and |ε| = 0

🌐 3. Language

Definition:
A language is a set of strings formed from an alphabet Σ.

  • So:
    A language is a subset of Σ*
    (Where Σ* is the set of all possible finite-length strings over Σ)

Examples:

  • Σ = {0, 1}

    • L = {ε, 0, 1, 00, 11} → A finite language

    • L = {w ∈ Σ* | w has even number of 0s} → An infinite language

  • Σ = {a, b}

    • L = {a^nb^n | n ≥ 0} → L = {ε, ab, aabb, aaabbb, ...}

Common Notations:

  • Σ*: Set of all strings over Σ (including ε)

  • Σ⁺: Set of all non-empty strings over Σ
    (i.e., Σ⁺ = Σ* − {ε})

🔁 4. Key Theoretical Properties 

  • Languages are the input/output for automata.

  • We classify languages based on complexity and recognizability:

    • Regular languages (finite automata)

    • Context-free languages (pushdown automata)

    • Recursively enumerable (Turing machines)

  • Every string belongs to exactly one of: a language or its complement (Σ* − L)

🎓 Summary:

Term       Description      Example (Σ = {a, b})
    Alphabet                        Finite set of symbol            {a, b}
    String                        Finite sequence from Σ             abba, ε
    Σ*                        All strings over Σ            ε, a, b, aa, ab...
    Language                        Subset of Σ*            {ab, ba, aab}


Understanding alphabet, strings, and languages is essential before moving to:

  • Regular expressions and finite automata

  • Grammar derivations

  • Decidability and Turing machines

They form the foundation of all theoretical models of computation.

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