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Turing Machine Structure

Algorithmics (HESS)

About these notes

Area Of Study

Computer science: past and present

Unit

Unit 4: Principles of algorithmics

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Turing Machine Structure

Algorithmics (HESS)
01 May 2026

Characteristics of a Turing Machine

A Turing machine is a theoretical model of computation introduced by Alan Turing in 1936. It provides a precise mathematical definition of algorithm and computation.

Components

Component Description
Infinite tape One-dimensional tape divided into cells, each holding one symbol. Extends infinitely in both directions.
Read/write head Positioned over one cell; can read and write a symbol.
State register Holds the current state from a finite set \(Q\).
Transition function \(\delta: Q \times \Gamma \rightarrow Q \times \Gamma \times \{L, R\}\) — given current state and tape symbol, outputs new state, symbol to write, and direction to move.
Initial state A designated start state \(q_0\).
Halting states \(q_{\text{accept}}\) and \(q_{\text{reject}}\) end computation.

Formal Definition

\[M = (Q,\; \Sigma,\; \Gamma,\; \delta,\; q_0,\; q_{\text{accept}},\; q_{\text{reject}})\]

Where \(\Sigma\) is the input alphabet and \(\Gamma \supseteq \Sigma\) is the tape alphabet (includes blank \(\sqcup\)).

Operation

  1. Input is written on the tape; head starts at the leftmost symbol.
  2. At each step: read current symbol, look up \(\delta\), write symbol, move head, change state.
  3. Halt when entering \(q_{\text{accept}}\) (accept) or \(q_{\text{reject}}\) (reject), or loop forever.

Example Transition Table

A machine to accept inputs with an even number of 0s:

State Read Write Move New State
\(q_0\) (even) 0 0 R \(q_1\)
\(q_0\) \(\sqcup\) \(\sqcup\) — \(q_{\text{accept}}\)
\(q_1\) (odd) 0 0 R \(q_0\)
\(q_1\) \(\sqcup\) \(\sqcup\) — \(q_{\text{reject}}\)

Key Properties

  • Universality: A Universal Turing Machine simulates any other Turing machine by taking the description of that machine and its input as its own input — analogous to a general-purpose computer running any program.
  • Church-Turing Thesis: Any function computable by any effective procedure can be computed by a Turing machine.
  • Determinism: A deterministic Turing machine follows exactly one transition per step. Non-deterministic TMs are theoretically equivalent in power.

Why It Matters

The Turing machine model allows precise definitions of:
- Decidable problems — a TM always halts with the correct answer
- Semi-decidable problems — a TM halts on YES instances but may loop on NO
- Undecidable problems — no TM exists that always halts correctly
- Time and space complexity — measured in terms of steps and tape cells used

KEY TAKEAWAY: The Turing machine is the theoretical basis for what can and cannot be computed. It is not a physical device but a mathematical abstraction.

EXAM TIP: VCAA may ask you to describe the components of a Turing machine and explain what the transition function does. Focus on: infinite tape, read/write head, finite states, transition function, halting states.

COMMON MISTAKE: The tape is conceptually infinite — a Turing machine is a theoretical model, not constrained by physical memory. Real computers are strictly less powerful than a Turing machine.

VCAA FOCUS: Describe the general structure (tape, head, states, transition function). Give an example of a simple computation. Understand the concept of a Universal Turing Machine and why it matters.

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