There is actually a pretty large choice of OS for ARM-based PDAs:
Windows Mobile, Symbian, Palm OS, Linux, netBSD, iPhone OS X, RISC OS, ...
RISC OS is not currently used on any PDAs, but it can easily be adapted to do so (it has been used on a touchscreen tablet PC, which isn't far from a modern PDA).
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Re: Operating system
There is actually a pretty large choice of OS for ARM-based PDAs:
Windows Mobile, Symbian, Palm OS, Linux, netBSD, iPhone OS X, RISC OS, ...
RISC OS is not currently used on any PDAs, but it can easily be adapted to do so (it has been used on a touchscreen tablet PC, which isn't far from a modern PDA).