# Are there decidable non-trivial properties of a LBA's accepted language?

The halting problem and therefore the acceptance problem are decidable for LBAs, but are the infinite extensions of these problems decidable?

Given a LBA, can you decide whether there exists an input on which the LBA halts? Given a LBA, can you decide whether the LBA accepts a certain non-empty, non-finite language?

Given a Turing machine $$M$$, you can construct an LBA $$A$$ that given as input a word of length $$n$$, it simulates $$M$$ on the empty tape as long as $$M$$ uses up to $$n$$ space, and the moment $$M$$ exceeds space $$n$$, it enters an infinite loop. In contrast, if $$M$$ ever halts, then $$A$$ also halts. Thus $$A$$ halts on some input iff $$M$$ halts on the empty input.