I'm trying to understand the Cook-Levin theorem proof, as it attempts to create a polynomial-time reduction from any NP problem to SAT (as presented in the book by Michael Sipser).Most requirements are absolutely clear, but I don't understand why the $\phi_{cell}$ formula is required. Of course, a tableau with improper values, such that fails $\phi_{cell}$, is broken and therefore will not translate correctly to the original NP problem on the other side of the reduction. We can easily break a tablaue by putting a $q_{accept}$ somewhere where it shouldn't, and the machine's language will change.But I seem to miss something basic. A reduction says that if A reduces to B, then for each $w\in\Sigma^* $, $w\in A$ iff $f(w)\in B$.The $f(w)$ part may be the broken tableau, but can we really call it $f(w)$? After all, such broken tableaus are never in the image of $f$.So, why do we care about them?
Cook-Levin theorem proof's requirement of $\phi_{cell}$
complexity theory;formal languages