Question

3. Use Mathematical Induction on n to prove that if the TM (above) is started with a blank tape, after 10 n + 4 steps the machine will be in state 3 with the tape reading: ...0(0111)"011100.... That is, although there are three states with halting instructions, show why none of those instructions is actually encountered, and formulate this into a proof that this machine does not halt when started with a blank tape.

Answer 1

Check that after 5 steps were in the state 4, and the tape has 01110.. Where the red cell is the current location of the read/ write head. Check that after another 4 steps we're in state of 0, and the tape has 0111000 then check that the next 5 steps proceed just like the first 5,and we end up in state 2 with a tape 011101110. So far then we seen that after 10n+4 steps we end up in state 2 with a tape 0(0111) ^n 01110.. If n=0 or n=1.

The induction step is to show that if( 10 n+1) +4 steps take you to step 2 with a tape

0(0111) ^n+1 01110

Check that if you are on stage 2 and have tape

01110 and go 10 steps you end up in state 2 with a tape

011101110 write down the 10 combination of state and tape. You will find that the read/ write head neveroves far enough to the left to interfere with the (0111) ^n 0 part, so able to carry it along.