Designing a Deterministic Turing Machine for the Language a^n b^n c^n

Welcome to Smart But Clever! In this session, we tackle a core Theory of Computation challenge: designing a single-tape deterministic Turing Machine for the language aⁿbⁿcⁿ (n ≥ 1). If you’re preparing for GATE/UGC NET or strengthening your automata theory foundation, this is a must-learn topic.

We begin by clarifying the language definition and why DFA and (single-stack) PDA are insufficient—since aⁿbⁿcⁿ is not context-free, we need Turing power (or at least two stacks). Then we walk through the mark-and-match strategy across three blocks: repeatedly mark the leftmost unmarked a → X, scan right to find the next b → Y, then the next c → Z, and return left to continue. The machine accepts when only X, Y, Z (and blanks) remain; it rejects on wrong order (e.g., b before finishing a’s), mismatched counts, or leftover unmarked symbols. You’ll see the state diagram, transition patterns, and a clean worked simulation (e.g., on aaabbbccc), plus a short note on time complexity (why single-tape runs in quadratic time) and common exam pitfalls.

What you’ll learn:

Why aⁿbⁿcⁿ is not context-free; PDA limits vs TM power

Three-phase mark-and-match (a→X, b→Y, c→Z) with clear scanning loops

Full state/transition pattern you can reuse in exams

Worked simulation and edge cases (wrong order, extra symbols, missing block)

Correctness intuition (invariants) and brief complexity note

High-yield tips for GATE/UGC NET and interviews

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Prerequisites: Basics of TM (tape alphabet, blank, L/R moves), reading state diagrams; helpful to know the aⁿbⁿ design.