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I have a reduction and can't understand if it is Karp reduction. I want understand if my reduction is Karp reduction as well as understand how to determine which one is so.

Let I have formula $\Phi = (l_{1,1}\lor l_{1,2}\lor...\lor l_{1,k_1})\land(l_{2,1}\lor...\lor l_{2,k_2})\land...\land(l_{n,1}\lor...\lor l_{n,k_n})$

And I want to check if it is minimal CNF (decision variant of CNF minimization, one that requires to minimize amount of literal occurences). Determine if formula has an equivalent with $m$ or lower amount of literal occurences. This problem is $\Sigma^P_2$-complete. However, I have reduction:

$$Min(\Phi)=m\ge(length(\Phi)-(\overline{S(\Phi\land\overline {l_{1,1}}\land l_{1,2}\land...\land l_{1,k_1})}+\overline{S(\Phi\land l_{1,1}\land \overline{l_{1,2}}\land...\land l_{1,k_1})}+...+\overline{S(\Phi\land l_{1,1}\land l_{1,2}\land...\land\overline{l_{1,k_1}})}+\overline{S(\Phi\land\overline {l_{2,1}}\land...\land l_{2,k_2})}+...+\overline{S(\Phi\land l_{2,1}\land...\land\overline{l_{2,k_2}})}+...+\overline{S(\Phi\land\overline {l_{n,1}}\land...\land l_{n,k_n})}+...+\overline{S(\Phi\land l_{n,1}\land...\land\overline{l_{n,k_n}})}))$$

Here I use $S$ to denote $SAT$-problem.

$length(\Phi)$ counts amount of literal occurences in formula $\Phi$.

If this reduction is Karp reduction, then $\Delta^P_2 = \Sigma^P_2$ and $PH$ collapses.

But is this reduction Karp reduction, how can I determine it (general method is preferred)?

Okay, it seems, using $P^{NP}$ machine I can use $+,-,<,>,=$ operations (do not know about $\cdot,/$) for any data machine has at some step (after getting answer from oracle). If so, then I'm surprised that no one found such simple algorithm for CNF minimization.

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  • $\begingroup$ A Karp reduction from $A$ to $B$ is a polytime function $f$ such that $x \in A$ iff $f(x) \in B$. $\endgroup$ – Yuval Filmus Jun 29 '17 at 10:46
  • $\begingroup$ @YuvalFilmus, okay, in my case, $A$ is minimization problem, $B$ is a problem that answers $m\ge$ (length - number of UNSAT answers). Then, $x$ is $\Phi$. What here is $f(x)$? $\endgroup$ – rus9384 Jun 29 '17 at 10:50
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    $\begingroup$ That's for you to say. Your reduction is a Karp reduction if it can be implemented using a function $f$ which can be computed in polynomial time. I suggest looking at examples of NP-hardness proofs, which also feature the same type of reduction. $\endgroup$ – Yuval Filmus Jun 29 '17 at 10:51
  • $\begingroup$ @YuvalFilmus, I know, for example, that I can't invert the answer on NTM. Thus, reduction $NSAT(\Phi)=SAT(\overline \Phi)$ is not Karp reduction. However, here I use arithmetical operation $+$. Is it restricted for DTM (under Karp reductions) or not? $\endgroup$ – rus9384 Jun 29 '17 at 11:00
  • $\begingroup$ If your reduction collapses PH, then it is safe to say that it isn't a Karp reduction. So you shouldn't be able to express it as a function $f$ satisfying the constraints above. $\endgroup$ – Yuval Filmus Jun 29 '17 at 14:56
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Your reduction repeatedly calls a SAT oracle. Karp reductions only make one oracle call, and furthermore they must immediately return whatever the oracle returns (so it's like a "tail call"). So your reduction isn't a Karp reduction.

How to know if a reduction is a Karp reduction? Use the definition. A Karp reduction from a language $A$ to a language $B$ is a polytime function $f$ such that for all $x$ we have $x \in A$ iff $f(x) \in B$. If your reduction cannot be put in this form, it is not a Karp reduction.

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  • $\begingroup$ So, I can't use use $NP$ oracle in $P^{NP}$ machine multiple times to show that problem is in $P^{NP}$? $\endgroup$ – rus9384 Jun 29 '17 at 17:23
  • $\begingroup$ Definitely, but this doesn't constitute a Karp reduction. Rather, it is an oracle reduction (also known as a Turing reduction). $\endgroup$ – Yuval Filmus Jun 29 '17 at 17:41
  • $\begingroup$ I know that it's Turing reduction (I think, every reduction is Turing reduction). But it can't prove equality between $P^{NP}$ and $NP^{NP}$, right? However, as the paper says, prime CNF is already in $P^{NP}$ and thus I only need to show that prime CNF can be minimized on $P^{NP}$ or weaker machine. $\endgroup$ – rus9384 Jun 29 '17 at 17:59
  • $\begingroup$ Karp reductions are weaker than Turing reductions. In particular, there is a Turing reduction between the halting problem and its complement, but there is no such Karp reduction (even if you only require $f$ to be computable). $\endgroup$ – Yuval Filmus Jun 29 '17 at 18:16

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