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I want to formally show that a computational system $\mathcal M$ is computationally universal by showing it is computationally equivalent to some already known universal system, i.e. some UTM.

To show that system $\mathcal M$ is computationally equivalent I need to prove something along the lines of this following definition:

Definition(Computationally Equivalent Systems) Let $A, B, A_0 , B_0$ be alphabets and let $\mathcal M$ and $\mathcal N$ be computational systems with their computed functions being of the form $f_M : A^* \to B^*$ and $f_N : A_0^* \to B_0^*$. Then if there exist encoding functions $e_1 : A^* \to A_0^*$ and $e_2 : B^* \to B_0^*$ such that $e_2 \circ f_M = f_N \circ e_1$ holds we say $\mathcal M$ and $\mathcal N$ are computationally equivalent w.r.t. the encodings $e_1$ and $e_2$.

The problem is I can't find a useful definition for encodings. They obviously should be total and Turing computable (and possibly reversible) functions. However, they must not be too powerful since if the encodings partake in the actual computation any systems can be "equivalent" and that is not what we want.

Example: If you want to compare a machine that computes addition on a pair of numbers (in base 10) with a machine that computes addition on a pair of binary numbers those should be equivalent in their computation since only the representation they use for input and output changes. However, if you now consider an "encoding function" that it self performs the addition and returns the result, the first machine would be "equivalent" to a machine that simply performs the identity function. This is obviously not an intended result of the definitions above. In this example it is easy to see intuitively that the "encoding" was to "strong". But how can I describe that formally? How can I further restrict the computational power of encodings to be lower than the computation that I am studying?

The same problem arises when comparing machines regarding computational universality. How can I define encodings that are not able to perform universal computations themselves (so that they don't interfere in the results I am trying to prove), but that are still powerful enough to be useful for actual translations of complex representations of inputs and outputs?

My initial idea was to restrict the encodings to functions that can be computed by some kind of automata that is provably less powerful than a TM. For example a linear-bounded TM. However (if I am not mistaken) such a linear-bounded TM would already be to weak, since it can not compute functions like $\text{DecToBin}: \{0,1,2,...,9\}^* \to \{0,1\}^*$ since the length of the output grows exponentially compared to the input.

Edit: I am mistaken. A linear-bounded TM CAN compute functions like decimal to binary conversion, since their growth is linear and NOT exponential. So maybe functions computable by linear-bounded TMs would be a solution?

An other idea I think might be useful to incorporate is some semantic preservation property. I.e. if the if the words over the input and output alphabets are equipped with some notion of semantic we require that the encodings preserve this semantic.

However this does feel like "cheating" at feels not at all formally sound.

This leads me to the question: How can one formally restrict the computational power of encodings such that a formally sound study of universality is possible?

All proofs of universality I've read so far simply brush over the fact that some kind of encoding is happening. If you know of a work where this was explicitly discussed I would be very interested.

Thank you!

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Excellent question. I'm not aware of any formal way to define this, to my knowledge. Instead, as far as I'm aware, in cases where computational equivalence comes up in practice, usually this issue doesn't arise, as the encoding are so obviously weak compared to the notion of computational power we care about, that it's obvious that everything is fine.

There are some specific instances where we can provide a formal definition for what encodings are allowable. For instance, when proving NP-completeness, we require the reductions (which are essentially encodings, in the sense of the definition you gave) to run in polynomial time.

A similar issue arises in complexity theory, when we translate an informally stated problem into something that can be analyzed using complexity theory. We typically have to pick a representation of the inputs and outputs (a way that they are encoded in binary), to turn that into a precisely defined problem (specifically, a formal language). Complexity theory ultimately proves theorems about the resulting formal language. In principle, a particularly obtuse choice of representation might make a very hard problem easy. In practice, people use 'the obvious' representation, which is obviously so easy to implement efficiently that just don't bother about it. But from a pedantic or formalistic perspective, what we have actually proven theorems about are the formal languages, not the informal problem statements. We then treat those theorems as telling us something useful about the informal problem statements, making use of our recognition that the representation is obviously reasonable.

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  • $\begingroup$ You are right on point. This notion of "encoding are so obviously weak" feels like cheating to me and I am rather confused that I can't find a rigorous definition anywhere, since so many accepted results seem to be based on this "obvious" thing. However it might be that simply functions that can be computed by eg linear-bounded TM are sufficient (see my edit). I am yet to retry and see if I can get It working with this as a restriction. $\endgroup$
    – Yannik Eik
    Oct 11, 2023 at 12:34

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