# 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 $a^nb^n$? Jul 10, 2018 at 9:41
• Are working tapes available? Jul 11, 2018 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.)