Question

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Given a Turing machine M, consider the following language LLEFT {(( Describe a Turing machine that decides this language. M), r)| M moves its head to the left at least once during its execution on input x}.

Answer 1

ANS:

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A turing machine, which moves its head at least once during its execution on input x, can be said as, machine takes input x, moves its head to left and can move futher left without even immediate x.

So we describe Turing machine:

Input alphabet $\sum =\left \{ x \right \}$

tape alphabet $\Gamma =\left \{ x,\Lambda \right \}$ where $\Lambda$ is blank symbol

$q_{s}$ is start state

The transition function $\delta$ is specified by following instruction:

1) $q_{s}\Lambda \rightarrow q_{s}\Lambda R$

2) $q_{s}x \rightarrow q_{0}x N$

3) $q_{0} x\rightarrow q_{1}xL$

4) $q_{1} x\rightarrow q_{0}xL$

5) $q_{2} x\rightarrow q_{0}xL$

6) $q_{1} \Lambda \rightarrow q_{2}\Lambda L$

7) $q_{2} \Lambda \rightarrow q_{1}\Lambda L$

Now lets take example, say input string is $\Lambda \Lambda \Lambda \Lambda x\Lambda$

a) on accpeting $\Lambda$ in initial state it will move forward till x is found and enters state $q_{0}}$ ( by transition function1 and 2)

b) in this state, on accepting x, it will move left and enters state $q_{1}}$ ( by transition function 3)

c) here, on accepting $\Lambda$, it will continue to move left and enters $q_{2}$ ( by transition function 6)

d) here, on accepting $\Lambda$, it will again go to state $q_{1}$ and so on. ( by transition function 7)

Thus the head will move atleast once when input x occurs.

If there were other input x during step b or c, it will go to state $q_{0}$ ( by transition function 4 and 5) and will ensure again atleast one left move.