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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?

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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.

You can try to show that the second property is undecidable using a similar argument.

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