Turing machines are simple abstract computational devices first described by Alan Turing in 1936–7. They are intended to help investigate the extent and limitations of what may be calculated.

A Turing machine is a theoretical computing device that can read, write, and erase symbols on an infinite tape. It operates using a set of rules to determine its actions based on the current state and the symbol it's reading.

In this blog, we will learn about the Turing Machine in TOC in detail. Let's start with the definition of Turning Machine.

What is Turing Machine in TOC?

In theoretical computer science, a Turing machine is a hypothetical computing machine that can read, write, and erase symbols on a tape-based on a set of rules. It serves as a model for understanding the limits of computation and the concept of algorithmic computation. In simpler terms, it is a machine that can perform a series of tasks based on specific rules and input data.

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Basic Model of Turing machine

A Turing machine(TM) is a computer model that reads and writes to an infinite tape to perform computations. The tape is divided into cells into which input is given. It contains a reading head that reads the input tape.

A state register is used to store the state of the Turing machine. The cell in the tape is replaced with another symbol, its internal state is changed, and it travels from one cell to the right or left after reading an input symbol.

The input string is accepted if the TM reaches the end state; else, it is denied.

Formal definition of Turing machine

A TM is defined as a 7-tuple (Q, X, ∑, δ, q0, B, F) where −

Q represents a finite set of states

X is a tape alphabet

∑ is the input alphabet

δ is the transition function; δ : Q × X → Q × X × {Left_shift, Right_shift}.

q0 is the initial state

B represents the blank symbol

F is the set of final states

The following representation can be used to model the turning machine.

The input tape has an infinite number of cells, each containing one input symbol, allowing the input string to be taped. The empty tape is filled with blank characters as shown below:

The tape head and the finite control are responsible for reading the current input symbol. The tape head moves from left to right.

It consists of a finite set of states that a machine must pass through.

External symbols are a finite set of symbols used to build the logic of a Turing machine.

Language accepted by Turing machine

All languages, even if they are recursively enumerable, the Turing machine accepts them all. The terms recursive enumerable refer to the same set of rules being repeated an infinite number of times. Computable functions, such as addition, multiplication, subtraction, division, power function, and many others, are accepted by the TM.

Example

Q: Construct a Turing machine that accepts the language of aba over ∑ = {a, b}.

Solution:

We'll assume the string 'aba' is placed on the input tape as follows:

IMAGE

The tape head will read out the sequence up to the Δ characters. Now, we'll see how this Turing machine works for the string 'aba.'

We'll see how this Turing machine works for the string 'aba' now.

The initial state is q0, and the head points to an as follows:

2. The move will be δ(q0, a) = δ(q1, A, R), implying that it will proceed to state q1, with a replaced by A and the head moving to the right as follows:

3. The move will be δ(q1, b) = δ(q2, B, R), implying that it will proceed to state q2, with b replaced by B and the head moving to the right as follows:

4. The move will be δ(q2, a) = δ(q3, A, R), meaning it will proceed to state q3, replace a with A, and move the head to the right as follows:

5. The move δ(q3, Δ) = (q4, Δ, S) indicates that it will go to state q4, the HALT state, which is always an accepted state for any TM.

Turing machines are used to model and analyze algorithms and computation and serve as the basis for the design of modern computers. They also have applications in artificial intelligence, cryptography, and complexity theory.

What are the advantages of Turing machine in TOC?

The are many advantages to Turing Machines in TOC. The main advantage among them is that Turing Machines can read any language and compute the model functions for which the algorithm is possible.

What is the Turing machine TOC?

It is a theoretical model of computation that manipulates symbols on an infinite tape according to a set of rules.

What are the 7 components of a Turing machine?

The 7 components of a Turing machine are:

Finite set of states (Q).

Tape alphabet (Σ).

Blank symbol (B).

Transition function (δ).

Start state (q₀).

Accepting states (F).

Read/write head.

What is the Turing machine in theoretical computer science?

A foundational concept used to define the limits of mechanical computation. It can theoretically solve any problem solvable by a computer algorithm.

What are the types of Turing machine?

There are various classifications, including:

Deterministic vs. Nondeterministic

Single-tape vs. Multi-tape

Universal Turing machines (powerful enough to simulate any other Turing machine).

Conclusion

In this article, we learned about the Turing Machine in TOC.

We learned that a Turing Machine (TM) is a mathematical model consisting of an infinitely long tape separated into cells into which input is delivered. It has a reading head that reads the input tape. A state register is used to store the state of the Turing machine.

A Turing machine accepts all languages, even if they are recursively enumerable.