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A decision problem is in NP if a candidate "yes" can be verified in polynomial time.

Are there decision problems whose "yes" is verified with an algorithm that is P-complete (reducible in logspace)? Are those problems in NP?

If so, can the "yes" be verified in logspace?

I'm told "for every NP language we can find a verifier that uses only logspace."

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Every language in NP has a logspace verifier.

Indeed, let $L$ be any language in NP. Therefore, there is a polynomial time machine $T$ and a polynomial $p$ such that $x \in L$ iff some string $y$ of size at most $p(|x|)$ makes $T(x,y)$ accept.

While the witness $y$ may be hard to check, we can make help a logspace machine verify it by including the transcript of the accepting computation. We construct another machine $T'$ which accepts a new witness $z$. The witness $z$ consists of a sequence of configurations of the machine $T$, starting with an initial configuration encoding $x$ and $y$, and ending at an accepting configuration. Since each configuration has polynomial length, a logspace machine can verify that $z$ indeed consists of a valid sequence of configurations.

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  • $\begingroup$ When T is a verifier we can design it so that it only accepts valid certificates y. But when T (or its sequence of states) is part of a witness z, it can be anything and T' then needs to check that T is not some random but valid sequence that accepts a bogus certificate y. Can T' be designed in that way? $\endgroup$ – Albert Hendriks May 9 '20 at 16:17
  • $\begingroup$ The machine $T$ is fixed ahead of time. It is hard-coded inside $T’$. $\endgroup$ – Yuval Filmus May 9 '20 at 16:24

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