# Does Turing machine move left on particular input?

We know that RE language is the collection of unrestricted grammar which is known as type-0 grammar that's why emptiness, finiteness of every RE languages is undecidable. My question is how I check decidability "the Turing machine makes move left or not" on particular input string. I have found some internet contents but very difficult to understand. I want to understand just intuition which is brief, not the concrete proof.

• – D.W.
Nov 8, 2021 at 4:03

What would happen if we never move left? Either we halt at some time, or we always see the input $$\sqcup$$ (blank symbol) and choose to move right. But there is a finite number of states, and by the pigeonhole principle after a long time we will get stuck in a loop in those states.
Checking if we get stuck in a loop in the states, is not hard when the input is always $$\sqcup$$ - since we have to consider only the states.