Question

Let Lpair be the set of all encodings of Turing machines such that there is a pair of consecutive binary numbers that are both accepted by m. (For example, if a Turing machine m accepts both 100 and 101, then is in Lpair.)

a. Prove that Lpair is recognizable (R.E.) by giving a high-level description of a Turing machine that accepts Lpair.
b. Prove that Lpair is not decidable (not recursive).
A) For part a, describe the high-level functioning of a Turing machine that accepts Lpair.
B) For part b, provide a proof outline demonstrating that Lpair is not decidable.
C) Both A and B are correct.
D) Both A and B are incorrect.

Answer 1

Lpair is a set of Turing machine encodings that accept pairs of consecutive binary numbers. A Turing machine can recognize Lpair by enumerating and testing all pairs of binary numbers for acceptance. Lpair is undecidable because its decidability would imply a solution to the undecidable Halting problem.

Step-by-step explanation:

The question asks us to address two parts regarding the set Lpair, which contains all encodings of Turing machines that accept a pair of consecutive binary numbers. In part A, we are to describe a Turing machine that recognizes Lpair, providing a high-level description of its functioning. In part B, we are to provide a proof outline demonstrating that Lpair is not decidable.

Part A: Recognizability of Lpair

To prove that Lpair is recognizable, consider a Turing machine M that performs the following:

1. 
   
3. Iterate through all possible pairs of consecutive binary numbers.
4. 
   
6. Simulate the Turing machine encoded by the input on each pair.
7. 
   
9. If both numbers in the pair are accepted, accept the input encoding.
10. 
    
12. If one or both numbers are rejected, move to the next pair and repeat the process.

This Turing machine M will accept an encoding if and only if it corresponds to a Turing machine that accepts a pair of consecutive binary numbers. Therefore, Lpair is recognizable.

Part B: Undecidability of Lpair

To establish that Lpair is not decidable, we can use a reduction from the Halting problem, which is known to be undecidable. If Lpair were decidable, we could construct a decider for the Halting problem by creating a Turing machine M' that, for any input and Turing machine, constructs a machine that halts and accepts if it receives two consecutive binary numbers where the second number represents a halting computation of the original machine on the input. Since the Halting problem is undecidable, it follows that Lpair must also be undecidable.

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