Turing machines review problems

1. Draw the state diagram for a Turing machine that increments a binary number. Assume that the input tape contains at least one non-blank symbol. For example, if the binary representation of 4 is initially on the tape

   ..b100b..
   

   then the output should be the binary representation of 5,

   ..b101b..

   or if the initial tape contains the binary representation of 11

   ..b1011b..

   then the output should be the binary representation of 12,

   ..b1100b..

   or if the initial tape contains the binary representation of 7

   ..b111b..

   then the output tape should be the binary representation of 8,

   ..b1000b..

2. Suppose we try to construct a Turing machine to solve a particular problem, but we are not successful. Does this mean that no Turing machine exists that can solve that problem? Explain and justify your answer.

3. Does the fact that the Halting Problem is not computable mean that we can never tell if a program we have written is going to halt? Explain.