Halting on a (possibly one-way) write-once tape Consider Turing machines (with the Busy Beaver "design specifications")
that must write a 1 whenever they read a 1 (i.e., "write once").

${}$1. $\:$ Is the halting problem decidable for these machines?



If those specifications are modified to use a [one-way infinite] tape instead,

${}$2. $\;$ Is the halting problem decidable for the resulting type of machines if they

a. $\;$ stay where they are

b. $\;$ halt

whenever they try to move off the end of the tape?
Ad 1. Write-once TMs are equivalent to ordinary TMs (I think this is an exercise in Sipser, but I don't have the text at hand to confirm) so $\rm woBB$ is uncomputable.

Ad 2. No. The usual zipper encoding also works for write-once TMs, so write-once TMs that never move past the end of the one-way infinite tape are equivalent those with two-way infinite tape. Therefore $\rm owwoBB$ is uncomputable, and the choice of behaviour when moving past the end of the tape doesn't affect this.