When a subroutine (such as Rototurret) says trash 1 program, does it have to be an installed program?

For example, if a runner has a personal workshop or Djinn out and cards on them and they encounter a Rototurret and can't bypass the subroutine can you trash a program that is hosted on those cards?

You can only trash installed cards.

Wondering why you would not just trash Djinn. Any hosted programs would be lost when Djinn gets trashed anyway.

Programs hosted on Djinn are installed and are valid targets. Programs hosted on Personal Workshop are not installed and are not valid targets. But yeah, target the Djinn.

Djinn was just used as an example, as there are a lot of cards that have similar abilities. Many of the Caissa cards have wording that says that they are hosted but not installed as well. Could you use the trash 1 program subroutine on these as well?

I was asking more as to these outside cases, rather than Djinn itself.

Caissa have to be installed prior to hosting. Almost every instance of hosting in the game requires an installation, either prior to hosting (such as Caissa) or as part of the same action (such as Parasite, or anything on Djinn or Dinosaurus). Personal Workshop is actually unusual in this sense, I believe it's currently the only card which hosts another card without that card being installed.