Introduction Link to heading

Turing Machines are a fundamental concept in computer science, playing a crucial role in the theory of computation. Named after the British mathematician and logician Alan Turing, these abstract machines help us understand the limits of what can be computed. This post will explore the history, definition, and significance of Turing Machines, complete with examples and diagrams to elucidate their operation.

Alan Turing

Historical Background Link to heading

Alan Turing introduced the concept of the Turing Machine in his 1936 paper, “On Computable Numbers, with an Application to the Entscheidungsproblem”1. His work addressed the decision problem (Entscheidungsproblem), a challenge posed by mathematicians to determine whether a given mathematical statement is provable from axioms using a finite set of rules.

Turing’s abstract machine provided a mathematical model of computation that could simulate the logic of any computer algorithm. This model laid the groundwork for modern computing and theoretical computer science.

Definition and Components of a Turing Machine Link to heading

A Turing Machine consists of the following components:

  1. Tape: An infinite strip divided into cells, each holding a symbol from a finite alphabet.
  2. Head: A read/write head that moves left or right along the tape, reading and writing symbols.
  3. State Register: Holds the state of the Turing Machine, selected from a finite set of states.
  4. Transition Function: Determines the machine’s next action based on the current state and the symbol being read.

Formally, a Turing Machine can be represented as a 7-tuple (Q, Σ, Γ, δ, q_0, q_accept, q_reject), where:

  • Q is a finite set of states.
  • Σ is the input alphabet (excluding the blank symbol).
  • Γ is the tape alphabet (including the blank symbol).
  • δ is the transition function: Q × Γ → Q × Γ × {L, R}.
  • q_0 is the initial state.
  • q_accept is the accept state.
  • q_reject is the reject state.

Example of a Turing Machine Link to heading

Let’s consider a simple Turing Machine that determines whether a binary string contains an even number of 1s.

Components Link to heading

  • States: Q = {q_0, q_1, q_accept}

  • Input Alphabet: Σ = {0, 1}

  • Tape Alphabet: Γ = {0, 1, _}

  • Transition Function: Defined as follows:

    δ(q_0, 0) = (q_0, 0, R)
δ(q_0, 1) = (q_1, 1, R)
δ(q_1, 0) = (q_1, 0, R)
δ(q_1, 1) = (q_0, 1, R)
δ(q_0, _) = (q_accept, _, N)
δ(q_1, _) = (q_reject, _, N)

Operation Link to heading

  1. The machine starts in state q_0 at the leftmost symbol of the tape.
  2. It reads the current symbol and transitions according to the transition function.
  3. It moves the read/write head left (L) or right (R), or halts (N) when it reaches the accept or reject state.

Example Execution Link to heading

Consider the input tape containing the binary string 1101_:

  1. Initial State: q_0, Tape: 1101_, Head: at the first symbol 1.
  2. Step 1: (q_0, 1) → (q_1, 1, R), Tape: 1101_, Head: at the second symbol 1.
  3. Step 2: (q_1, 1) → (q_0, 1, R), Tape: 1101_, Head: at the third symbol 0.
  4. Step 3: (q_0, 0) → (q_0, 0, R), Tape: 1101_, Head: at the fourth symbol 1.
  5. Step 4: (q_0, 1) → (q_1, 1, R), Tape: 1101_, Head: at the blank symbol _.
  6. Step 5: (q_1, _) → (q_reject, _, N), Halts.

The machine halts in the reject state, indicating that the input does not have an even number of 1s.

Significance and Applications Link to heading

Turing Machines are not just a theoretical construct; they have practical implications in various fields of computer science.

Computability Theory Link to heading

Turing Machines help define the limits of what can be computed. The notion of Turing-completeness is used to describe systems that can simulate a Turing Machine, thus capable of performing any computation that can be algorithmically defined.

Complexity Theory Link to heading

Turing Machines are used to analyze the complexity of algorithms. By understanding the resources (time and space) required by a Turing Machine to solve a problem, we can categorize problems into complexity classes such as P, NP, and NP-complete.

Automata Theory Link to heading

Turing Machines extend the concept of finite automata and pushdown automata, providing a more powerful model for recognizing languages. The Chomsky hierarchy classifies languages based on the computational power required to recognize them, with Turing Machines recognizing recursively enumerable languages.

Advanced Concepts Link to heading

Universal Turing Machine Link to heading

A Universal Turing Machine (UTM) is a Turing Machine that can simulate any other Turing Machine. Alan Turing introduced the concept of the UTM, illustrating that a single machine can be programmed to perform any computation given the appropriate input and instructions.

Non-deterministic Turing Machine Link to heading

A Non-deterministic Turing Machine (NDTM) is a variant where multiple transitions are possible from a given state and symbol. NDTMs are used in theoretical discussions about computational complexity and are central to the P vs NP problem.

Conclusion Link to heading

Turing Machines are a cornerstone of theoretical computer science, providing a framework to understand computation’s fundamental limits and capabilities. Their influence extends across various domains, from algorithm analysis to language recognition. By studying Turing Machines, we gain insight into the principles that underlie all modern computing systems.

References Link to heading

Turing Machine Diagram

Further Reading Link to heading

  1. Turing, A. M. (1936). “On Computable Numbers, with an Application to the Entscheidungsproblem”. Proceedings of the London Mathematical Society, 2(42), 230-265. Link ↩︎