# A Turing Machine that Doesn't Move to the left [duplicate]

My question is if the following statement is true or false:

Does every turing-recognized $B$ language has a turing machine $M$ that recognizes $B$ and fullfiles the following statement:

For every word $w$ the belongs to $B$, $w \in B$, the writing-reading head never moves to the left?

Notice that the condition is required only for words in the language. if $w \notin B$ then the head can move to the left as usual.

The statemet seems correct to me. I think I can simulate a regular $TM$ with this "special" $TM$, while saying that every time the head moves left, it will stay put. Couldn't prove it, though.

• Can you construct a TM like that that accepts $a^nb^n$? – ratchet freak Jul 10 '18 at 9:41
• Are working tapes available? – Andrej Bauer Jul 11 '18 at 8:00

This makes it into a finite state machine, which is significantly weaker than a Turing machine. (As ratchet freak points out in the comments, a finite state machine can't recognize the language $\{ 0^n 1^n | n \in \mathbb{N} \}$, while a Turing machine can.)