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​).

Prove that Lpair​ is recognizable (R.E.) by giving a high-level description of a Turing machine that accepts Lpair​.

Answer 1

To prove that Lpair is recognizable, we describe a Turing machine that systematically enumerates all binary strings, simulating the input machine on each and the subsequent incremented string. If it finds a pair of consecutive accepted strings, it halts and accepts, showing that Lpair is a recognizable set.

Step-by-step explanation:

The student's question involves proving that the set Lpair, which is the set of all encodings of Turing machines that accept a pair of consecutive binary numbers, is recognizable. To prove that Lpair is recognizable, we can describe a high-level Turing machine that accepts Lpair.

The procedure for the Turing machine would be as follows:

1. 
   
3. Start with the encoding of any Turing machine m and begin a systematic enumeration of all binary strings in ascending order.
4. 
   
6. For each string x, simulate m on x and x incremented by 1 (denote this as x+1).
7. 
   
9. If m accepts both x and x+1, halt and accept.
10. 
    
12. If not, continue the enumeration until a matching pair is found or indefinitely if no such pair exists.

Because the Turing machine halts and accepts when it finds a matching pair of consecutive binary numbers that are both accepted by m, Lpair is recognized by this Turing machine. This means that Lpair is a recognizable (recursively enumerable) set. Since recognizability does not require the machine to halt on all inputs (it may run indefinitely on inputs not in Lpair), the setup described effectively demonstrates that Lpair is recognizable.