# What are the technical reasons for which the empty string is not allowed to be accepted by a Turing machine?

Below are the excerpts from the automata text by Peter Linz.

Definition 9.3

Let $$M = (Q,\Sigma,\Gamma,\delta,q_0,\square,F)$$ be a Turing machine. Then the language accepted by $$M$$ is

$$L(M) = \{ w \in \Sigma^+ : q_0w \vdash ^* x_1q_fx_2 \text{ for some } q_f \in F,x_1,x_2 \in \Gamma^*\}$$

From the above definition it is clear that the $$\Sigma^+$$ in it does not allow the empty string to be accepted by the Turing machine.

Later in the example:

Example 9.6

For $$\Sigma = \{0,1\}$$, design a Turing machine that accepts the language denoted by the regular expression $$00^*$$.

Solution :

Though the diagram is not given, I give it to simplify the situation. Then Linz adds,

Note that the Turing machine also halts in a final state if started in state $$q_0$$ on a blank. We could interpret this as acceptance of $$\lambda$$ (empty string), but for technical reasons the empty string is not included in Definition 9.3.

I have read the automata text by Ullman, but there was no such restrictions imposed on Turing machines. Using the concepts I have learnt from Ullman text, I could solve the problem as:

this no longer has the problem of acceptance of $$\lambda$$.

So what might be the technical reason or it is just a discretion of the author or reduction of the number of states?

• There is no real reason that prevents a Turing machine from accepting the empty word. I suspect that "technical reason" is used in the sense of "technical convenience", i.e., some later proof or discussion in the book becomes easier/less tedious to argue about if there is no need to handle the empty word. Since I haven't read that book I can't say with certainty that this is what's going on, nor point you to a proof/argument where this technical simplification is used. – Steven Apr 4 at 17:12
• @Steven I see. Ok. – Abhishek Ghosh Apr 4 at 17:42
• Their own exercise 7g doesn't agree with their definition unless they expect the answer to be that no such machine exists. – plop Apr 4 at 17:48
• @plop 7g ? I do not get it... – Abhishek Ghosh Apr 4 at 18:45