Here's the question:
Assume there exists a polynomial time machine $M$ that receives two formulas $\varphi_1,\varphi_2$ and satisfies the following:
- If $\varphi_1 \in \mathrm{SAT}$ and $\varphi_2 \notin \mathrm{SAT}$, then $M(\varphi_1,\varphi_2)=1$
- If $\varphi_1 \notin \mathrm{SAT}$ and $\varphi_2 \in \mathrm{SAT}$, then $M(\varphi_1,\varphi_2)=0$
Show that there also exists a polynomial time algorithm for SAT.
Note that we have no guarantee for the output of $M(\varphi_1,\varphi_2)$ whenever $\varphi_1,\varphi_2$ are both satisfiable (or when both are not).
Can I get any hint on how to start the solution?