Interpretation of co-NPCompleteness?

Question

Given a Problem $A$ that has an answer $true$ if and only if both conditions $1$ and $2$ are $false$, for some conditions 1 and 2.

Whether condition $2$ is $true$ can be tested with certainty in deterministic polynomial time. Thus, we can say that answer to an instance of $A$ is $false$ in polynomial time (in case condition $2$ holds). If condition $2$ is $false$, we still need to test condition $1$.

Checking if condition $1$ is $true$ is an NP-complete problem.

Can we thus conclude that the original problem $A$ with both conditions is coNP-complete?

Answer 1

As far as I understand your problem statement is as following:

Given a Problem $A$ that has an answer $true$ if and only if both (some) conditions 1 or 2 are $false$.

We have a decision problem $A(w) = \overline{B(w)} \wedge \overline{C(w)}$, where $B$ and $C$ correspond to the conditions 1 and 2 respectively.

Whether condition 2 is $true$ can be tested with certainty in polynomial time.

$C \in P$.

Checking if condition 1 is $true$ on $A$ is an NPComplete problem (if we consider all possible instances of $A$).

$B \in NP$ and $B$ is $NP$-hard.

(Please, correct me if I misunderstood the problem statement)

So, let's rewrite the statement using the language notation: $L_B \in NP$, $L_C \in P$, and $L_A = \overline{L}_B \cap \overline{L}_C$.

By definition $L_A \in coNP$ if $\overline{L}_A \in NP$. But $\overline{L}_A = L_B \cup L_C$ (De Morgan law). Since we are given that $L_B \in NP$ and $L_C \in P$, $L_B \cup L_C$ is in $NP$ as well. So, $L_A$ is in $coNP$.

Now, take $\overline{L_C} = \emptyset$ which is clearly is in $P$, and so is $L_C = \Sigma^*$. For any $NP$-complete language $L_B$, $L_B \cup L_C = \Sigma^*$. But $\Sigma^*$ is not $coNP$-complete.

Answer 2

Let $C_1$ and $C_2$ be languages representing the conditions $1$ and $2$, respectively. Then, $C_1 \in \text{NP-complete}$ and $C_2 \in \text{PTIME}$, and by writing down formally what you mentioned, we get $A = \overline{C_1} \cap \overline{C_2}$.

Now you can use the following claim:

Claim: A language $L$ is NP-complete iff $\overline{L}$ is coNP-complete.

So to understand whether $A$ is coNP-complete or not, you can check whether its complement is NP-complete. In your case, $\overline{A} = C_1 \cup C_2$, and since $C_2$ can be any problem in $PTIME$, then $\overline{A}$ can be a "very simple language". I leave the details to you.

Note: the important thing here is the above claim. To understand coNP-completeness, it is sufficient to understand NP-completeness.