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The famous PPAD class revolves around the End-Of-The-Line problem. Basically, it states that you are given two polynomial depth circuits, $P$ and $Q$, which act as "possible previous" and "possible next" functions, each with an exponentially sized domain. And the task is to find an input that satisfies $P(Q(x)) \neq Q(P(x)) \neq x$.

But I raise the question: How can these circuits be represented on a Turing machine?

To answer this question, I think two things must be proven.

  1. Arithmetic circuits have a polynomial time and space translation to a Turing machine language
  2. Polynomial depth circuits can compute any choice of $P$ and $Q$

My question is mostly about $1$, but I don't really see how $2$ can be true either. Do we need these to translate this problem to a Turing machine context?

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A circuit is not represented "on" a Turing machine, rather a representation of the circuit is provided as input. It is straightforward to represent a circuit as a bit-string. For instance, you can list each gate of the circuit, and which gates' outputs are connected to which gates' inputs.

I can't make sense of the things that you suggest must be proven. I suspect there is some misunderstanding or faulty premise.

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  • $\begingroup$ I think my second suggestion is faulty. My first one is asking for an algorithm for what you describe as "straightforward [method] to represent a circuit as a bit-string." (and how to evaluate such a bit-string as though it was a circuit) $\endgroup$
    – Loic Stoic
    Mar 17, 2023 at 9:30

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