187790-3-20P Show that the single-tape TMs th... [FREE SOLUTION]

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Introduction to Theory of Computation

Michael Sipser

Computer Science

3rd Edition 路 458 pages

ISBN: 9781133187790

Chapter 0: 187790-3-20P (page 1)

Show that the single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.

Short Answer

Expert verified

Answer

It can be shown that the single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.

Step by step solution

Explain Turing Machine

Turing machine is the computational model that has an unlimited memory. It has a tape that can read and write symbols as input.

Show that the single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.

Consider the single-tape Turing Machine S that only reads symbols and cannot write on the portion of the tape that has an input string.

Consider the two events that happen in S, in-event and out-event. In the in-event, tape head moves to the input portion. In the out-event, that tape head moves to the portion on the right of the cell x.

Consider the state, $q_{1}$ in S , on entering the non-input portion. The following are the two possibilities.

Case 1:never enters the non-input portion

If accepts X , assign $q_{x}$ = $q_{a c c e p t}$

If S rejects, $q_{x} = q_{r e j e c t}$

For any that belongs to Q (set of inputs), such that $f_{x} (q) = q' .$

Thus, the S is in the state that undergoes in-event and then out-event will change S is q'

Case 2: S never enters the non-input portion again,

If S accepts x in input portion, assign $f_{x} (q) = q_{a c c e p t}$
If S rejects, $f_{x} (q) = q_{r e j e c t}$
Consider two strings, x and y and if x=y for all q , then $f_{x} (q) = f_{y} (q)$ .
Thus, S accepts xz if ana only if it accepts yz

Thus, the number of indistinguishable strings are finite. By theorem 鈥淢yhill-Nerode theorem鈥�, the language recognized by is regular.

Therefore, It has been shown that the single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.