Скачать или смотреть Turing Machine as a Comparator: Compare two numbers a &b | a = b | (Design & Simulation) Explanation

Welcome to Smart But Clever! In this session, we treat the Turing Machine as a comparator—a transducer that reads two binary numbers and outputs whether a greater than b, a=b, or a less than b. If you’re preparing for GATE/UGC NET or strengthening Theory of Computation, this builds practical intuition for TM-based decision/transduction tasks.

We formalize inputs as x#y (two binary strings separated by ‘#’) and define the output convention as a single symbol written on the tape: ‘greater than’, ‘=’, or ‘less than’. Design idea:

Normalize/skip leading zeros on both sides.

Length test by synchronous crossing-off: repeatedly mark one bit from x and one from y. If one side finishes first, decide by length (longer ⇒ larger).

If lengths are equal, perform a left-to-right lexicographic compare using position-wise marking: read the next unmarked bit of x, shuttle across ‘#’ to the corresponding bit of y, and decide at the first mismatch (1 vs 0 ⇒ a is greater than b, 0 vs 1 ⇒ a is less than b); if no mismatch to the end, a=b.
We discuss state/transition patterns, a clean hand simulation, edge cases (all zeros, different lengths, leading zeros), and a brief complexity note (single-tape implementations are typically O(n²) due to back-and-forth scans). You’ll also get shortcut tricks: use an end-marker, encode outcomes in terminal states, and a compact cleanup to leave only the relation symbol.

What you’ll learn:

TM as a transducer for comparison (input/output conventions)
x#y format, leading-zero handling, and delimiter logic
Length-first vs lexicographic comparison strategy on a single tape
Reusable state diagram/transition templates for comparators
Worked simulations and edge cases
Correctness invariants and quick time/space notes
High-yield exam tips for GATE/UGC NET and interviews

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Prerequisites: Basic TM model (tape alphabet, blank, L/R moves), comfort reading TM state diagrams.