Convert Regular Expression to Turing Machine

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Turing Machines are the most powerful machines that can accept any language which is acceptable by less powerful machines like Finite State Machine, Pushdown Automata, Linear Bounded Automata, etc.

In this chapter, we will see how to convert a regular expression into a Turing machine. We will start with the basics of Turing machines and regular expressions, and then see step-by-step process of the conversion with examples for a better understanding.

Turing Machines and Regular Expressions

A Turing machine is a theoretical computing device that can simulate any computer algorithm.

• It consists of a tape (which serves as memory),
• a head (which reads and writes symbols on the tape), and
• a set of states that guide its operations based on the symbols it reads.

On the other hand, a regular expression is a pattern that specifies a set of strings within a formal language. Regular expressions are often used to describe search patterns in text.

The process of converting a regular expression into a Turing machine involves several steps. This conversion helps us understand how simple patterns described by regular expressions can be executed by a Turing machine.

Steps to Convert a Regular Expression into a Turing Machine

The conversion process from a regular expression to a Turing machine involves five key steps. We will discuss each step in detail and provide an example to illustrate the process.

Convert the Regular Expression to a Finite Automata

The first step in converting a regular expression to a Turing machine is to transform the regular expression into a finite automata. A finite automata is a simple machine used to recognize patterns. It consists of states, transitions between those states, and accepts or rejects input strings based on these transitions.

For instance, consider the regular expression (a + b)* (aa + bb) (a + b)*. This expression describes strings made of a combination of as and bs. To convert this into a finite automata:

• Start by building loops that handle (a + b)*, meaning any combination of as and bs.
• Then, create transitions for (aa + bb).
• Finally, include another loop for (a + b)* at the end.

Modify Initial and Final States to Intermediate States

After converting the regular expression to a finite automata, the next step is to modify the initial and final states of this automata. These states will become intermediate states in the Turing machine.

This modification is necessary because we will introduce a new initial state and a new final state in the Turing machine. The original initial and final states from the finite automata will now serve as transitional or intermediate states.

Insert a New Initial State and a New Final State

The third step is to introduce a new initial state and a new final state in the Turing machine.

The new initial state will have a transition that begins with a blank symbol. The blank symbol is typically represented as B, , or another placeholder symbol, and it indicates an empty space on the tape. This transition will lead to what was previously the initial state of the finite automata.

Similarly, a new final state will be introduced, with transitions that end with a blank symbol, indicating that the machine has finished processing the input.

These new states help the Turing machine begin and end the computation in a controlled manner.

Modify Transitions to Include Tape Operations

The fourth step is to convert the transitions of the finite automata into a form suitable for a Turing machine. In a Turing machine, each transition involves not just changing states, but also reading and writing symbols on the tape, and moving the tape head either left or right.

For example, if a finite automata transition reads an input a, we modify this to a Turing machine transition that reads a, writes a back on the tape, and then moves the head to the right. This is represented as (input, input, R) where R stands for the right movement of the tape head. Similarly, if the input is b, the transition will be (b, b, R).

By following this method, every transition in the finite automata is adjusted to perform the necessary tape operations in the Turing machine.

Finalize the Turing Machine

The last step is to complete the conversion by ensuring all transitions are properly defined. The Turing machine should now be capable of accepting the same language that was described by the original regular expression.

Conclusion

Converting a regular expression to a Turing machine is a systematic process that converts simple pattern recognition with complex computation. By following the five steps we discussed above with examples, we can transform any regular expression into a Turing machine that accepts the same language.