Informal Description of a Non-Deterministic Turing Machine (NTM) for a Language

Question

I need help describing a Non-Deterministic Turing Machine (NTM) for the following language:

$L=\{(ww^c)^n | n>1$ and $w,w^c\in\{a,b\}^*\}$. If $w=w_1...w_k$ then $w^c=w_1^c...w_k^c$ where $w_i^c=b$ if $w_i=a$ and $w_i^c=a$ if $w_i=b$

How should I approach writing an informal description for this NTM? Any insight into the necessary non-deterministic steps would be really helpful!

Answer 1

Informal Description of a Non-Deterministic Turing Machine (NTM) for the Language:

The language is given by:

$$ L = \{ (wwc)^n \mid n > 1 \text{ and } w, wc \in \{a, b\}^* \} $$

Where:

• ( w ) is a string of symbols from ({a, b}),
• ( wc ) is a transformation of ( w ) such that:
  • If ( w_i = a ), then ( wc_i = b ),
  • If ( w_i = b ), then ( wc_i = a ),
• The string consists of repeated patterns of ( (wwc) ), where ( n > 1 ) (i.e., at least two repetitions).

Informal Description of the NTM:

1. Input: The NTM takes an input string of symbols from ({a, b}). The input string is expected to be in the form:

   $$ (wwc)^n $$

   where the machine must verify that the string consists of multiple repetitions of the pattern ( (wwc) ), with ( n > 1 ).

2. Step 1: Non-Deterministic Guessing of the Split for ( w ): The NTM will non-deterministically guess the position where the first occurrence of ( w ) ends and the second ( w ) begins. It guesses a boundary, dividing the string into potential segments of ( (wwc) ).

   • The machine chooses a point to mark the boundary between the first ( w ) and the second ( w ).
   • This non-deterministic choice is crucial, as it allows the NTM to try different positions for the first ( w ).

3. Step 2: Checking the Pattern ( ww ): After the NTM guesses where ( w ) ends, it will check if the second ( w ) matches the first ( w ).

   • The machine compares the first half of the string (which it believes to be ( w )) with the second part (which it believes to be another occurrence of ( w )).
   • It ensures that each pair of corresponding characters matches: ( w_1 = w_2, w_3 = w_4, \dots ).

4. Step 3: Checking the Transformation to ( wc ): The NTM now verifies that the third part of the string follows the transformation rules to form ( wc ) from ( w ). The transformation rule is as follows:

   • If ( w_i = a ), then ( wc_i = b ),
   • If ( w_i = b ), then ( wc_i = a ).

   This involves checking that for each character of ( w ), the corresponding character of ( wc ) satisfies the transformation condition.

5. Step 4: Repeating the Process for Multiple Repetitions: Since the string is in the form ( (wwc)^n ), where ( n > 1 ), the NTM needs to verify that the pattern repeats multiple times.

   • The NTM will non-deterministically guess the next boundary (the next position where a new ( (wwc) ) pattern starts).
   • It repeats Steps 2 and 3 for each new repetition, ensuring that:
     • The second ( w ) matches the first ( w ),
     • The corresponding characters in ( wc ) follow the transformation rules.

6. Step 5: Accept or Reject:

   • If the NTM successfully verifies all repetitions, it will accept the input string.
   • If any repetition fails (i.e., the second ( w ) doesn't match the first ( w ), or the transformation rules for ( wc ) are violated), the machine will reject the string.

Summary of Non-Deterministic Steps:

1. Non-deterministically guess the boundary between the first ( w ) and the second ( w ).
2. Verify the repetition: Ensure that the second ( w ) matches the first ( w ).
3. Check the transformation to ( wc ): For each character in ( w ), ensure the corresponding character in ( wc ) follows the transformation rules:
   • ( w_i = a \rightarrow wc_i = b ),
   • ( w_i = b \rightarrow wc_i = a ).
4. Repeat for subsequent ( (wwc) ) patterns, non-deterministically choosing new boundaries and checking the transformation and repetition rules.
5. Accept or reject based on whether all repetitions are valid.

Key Insight on Non-Determinism:

The most important non-deterministic step is in Step 1, where the machine guesses where each new repetition of ( (wwc) ) starts. This non-deterministic guess allows the NTM to explore multiple possible ways of dividing the string into repeated ( (wwc) ) patterns. For each guess, the NTM checks the corresponding characters for matching and transformation. If any part of the string doesn’t conform, the machine rejects the input.