Defining causal isomorphism

badtheatre

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?

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?

Two points of relevance that I see are:

If we care about the nature of morphisms of computations only because of some computations being people, the question is fundamentally what our concept of people refers to, and if it can refer to anything at all.

If we view isomorphic as a kind of extension of our naïve view of equals, we can ask what the appropriate generalisation is when we discover that equals does not correspond to reality and we need a new ontology as in the linked paper.

Actually, I started thinking about computations containing people (in this context) because I was interested in the idea of one computation simulating another, not the other way around. Specifically, I started thinking about this while reading Scott Aaronson's review of Stephen Wolfram's book. In it, he makes a claim something like: the rule 110 cellular automata hasn't been proved to be turing complete because the simulation has an exponential slowdown. I'm not sure if the claim was that strong, but definitely it was claimed later by others that turing co... (read more)

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Defining causal isomorphism

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Defining causal isomorphism

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