Linear Bounded Automata (LBA)

Linear Bounded Automata (LBA) are a special class of Turing Machines that operate within restricted computational resources. Specifically, an LBA uses a tape of finite length that is proportional to the size of the input. LBAs are used to recognize context-sensitive languages, making them an important model in computational complexity and formal language theory.

Definition of Linear Bounded Automata

An LBA is a nondeterministic Turing Machine with the following restrictions:

  • The tape is bounded to a length proportional to the input size. Specifically, the tape length is at most a constant multiple of the input size.
  • The read-write head cannot move beyond the portion of the tape containing the input or the cells immediately adjacent to it.

Formally, an LBA is defined as a 7-tuple, just like a Turing Machine:

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(Q, Σ, Γ, δ, q₀, qᴀ, qʀ)

Where:

  • Q: A finite set of states.
  • Σ: The input alphabet (does not include the blank symbol).
  • Γ: The tape alphabet, which includes Σ and additional symbols for computation.
  • δ: The transition function, which maps the current state and tape symbol to a new state, a symbol to write, and a direction (Left, Right, or Stay).
  • q₀: The start state.
  • qᴀ: The accept state.
  • qʀ: The reject state.

The key difference between an LBA and a standard Turing Machine is the restricted tape length, which limits the computational resources available to the machine.

Languages Recognized by LBAs

LBAs recognise the class of context-sensitive languages (CSLs), which are defined by context-sensitive grammars (CSGs). CSLs are more expressive than context-free languages but less expressive than recursively enumerable languages.

Examples of Context-Sensitive Languages

  • Balanced Powers of a and b: L = {aⁿbⁿcⁿ | n ≥ 1}. An LBA can verify that the number of a‘s, b‘s, and c‘s are equal.
  • Palindrome with Constraints: Palindromes where the middle character must satisfy certain conditions.

LBAs cannot recognize all recursively enumerable languages, but they are more powerful than pushdown automata.

Characteristics of LBAs

  • Bounded Tape: The tape size is limited to k × n, where k is a constant and n is the input length.
  • Deterministic and Nondeterministic LBAs: Both deterministic and nondeterministic LBAs are equivalent in power.
  • Computational Complexity: LBAs operate within polynomial space, making them significant in complexity theory.

Example of an LBA

Consider the language L = {aⁿbⁿcⁿ | n ≥ 1}. An LBA can be constructed to recognize this language by:

  1. Marking the first a, matching it with the first b, and then matching it with the first c.
  2. Repeating the process for subsequent a‘s, b‘s, and c‘s.
  3. Accepting if all symbols are matched and rejecting if there is a mismatch or leftover symbols.

The bounded tape ensures the LBA only uses space proportional to the input size, unlike a standard Turing Machine which uses an infinite tape.

Key Takeaways

  • Linear Bounded Automata (LBA): LBAs are Turing Machines with a tape length restricted to be proportional to the input size.
  • Context-Sensitive Languages: LBAs recognize context-sensitive languages, which are more expressive than context-free languages.
  • Computational Complexity: LBAs operate within polynomial space and are important for understanding space-bounded computation.
  • Applications: LBAs provide insights into languages that require constraints, such as matching symbols in balanced patterns.