How can I understand if reduction is Karp reduction? 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 $+,-,&lt;,&gt;,=$ 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.
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.