Defining causal isomorphism

I pre­vi­ously posted this ques­tion in an­other dis­cus­sion, but it didn’t get any replies so, since I now have enough karma, I’ve de­cided to make it my first “ar­ti­cle”.

This brings up some­thing that has been on my mind for a long time. What are the nec­es­sary and suffi­cient con­di­tions for two com­pu­ta­tions to be (homeo?)mor­phic? This could mean a lot of things, but speci­fi­cally I’d like to cap­ture the no­tion of be­ing able to con­tain a con­scious­ness, so what I’m ask­ing is, what we would have to prove in or­der to say pro­gram A con­tains a con­scious­ness --> pro­gram B con­tains a con­scious­ness. “poin­t­wise” iso­mor­phism, if you’re say­ing what I think, seems too strict. On the other hand, al­low­ing any in­vert­ible func­tion to be a _mor­phism doesn’t seem strict enough. For one thing we can put any re­versible com­pu­ta­tion in 1-1 cor­re­spon­dence with a pro­gram that merely stores a copy of the ini­tial state of the first pro­gram and ticks off the nat­u­ral num­bers. Restrict­ing our func­tions by, say, re­source com­plex­ity, also seems to lead to both similar and un­re­lated is­sues...

Any tak­ers?