Tableau with Holes: Clarifying NP-Completeness

In the context of defining $\mathcal{NP}$-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula $\psi$, defined as $\psi ={\psi }_{cell}\wedge {\psi }_{rest}$. The component ${\psi }_{cell}$ enforces the constraint that each cell in the tableau contains exactly one symbol, while ${\psi }_{rest}$ incorporates constraints governing the step-by-step behavior of N on w. Intuitively, ${\psi }_{rest}$ appears to pose a much greater challenge for satisfiability. This raises the question of whether the distinction between ${\psi }_{cell}$ being a 3cnf formula, rather than a cheap 2cnf formula, actually matters. We show that if, hypothetically, ${\psi }_{rest}$ can be succinctly represented as a Horn formula, then satisfying $\psi$ can be achieved efficiently in $K^{f(n,k)}$ steps, where N operates within $O\left(n^k\right)$ steps and both k and K are constants. Asymptotically, $f(n,k)\approx n^{\frac{2}{3}k}$. Our method has the potential for iterative application. Technically, we trim ${\psi }_{cell}$ down to a 2cnf–Horn formula, whose satisfiability allows for empty cells, or “holes,” in the tableau. This modified tableau represents exponentially many paths of N on w, rather than a single accepting path p. While a tableau with holes conceptualizes the satisfiability of ${\psi }_{trim}$—a trimmed-down version of $\psi$—it does not directly address the satisfiability of $\psi$. Therefore, we introduce an external user who efficiently employs backtracking to fill in specific holes, ultimately verifying the satisfiability of the original $\psi$.