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I saw that the injectivity "derives directly from the theorem", but i can't see how it's happen, any explanation?

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    $\begingroup$ I'm not sure what exactly do they mean, but isn't it true that the resulting circuit can only be constructed from a unique nondeterministic TM? $\endgroup$
    – user114966
    Jan 25, 2021 at 18:21
  • $\begingroup$ @Dmitry Thank you, I think it's not about TM, maybe the answer to this question is: to each CNF can be added one or more clauses that make it possible map each unique domain to unique codomain. but I'm not sure $\endgroup$ Jan 25, 2021 at 21:09
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    $\begingroup$ Can you edit your question to provide more context and background? I'm not sure what your question is asking, and I don't know where you read that quote or the context or what it is referring to, so it's hard to know how to answer. $\endgroup$
    – D.W.
    Jan 26, 2021 at 5:04

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Your question is a bit unclear, but let me hazard a guess. Let $L$ be an NP language. According to the Cook–Levin theorem, there is a polynomial time reduction $\phi_L$ from $L$ to SAT, that is, $\phi_L$ is a polynomial time function that satisfies $x \in L$ iff $\phi_L(x) \in \mathrm{SAT}$. You are asking why $\phi_L$ is injective.

Roughly speaking, $\phi_L(x)$ constructs a formula which describes the operation of a nondeterministic Turing machine for $L$ executing on $x$. The input $x$ is hardcoded in this description, and in particular, we can extract $x$ from $\phi_L(x)$. This implies that $\phi_L$ is injective.

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