Drawing Sound Conclusions from Unsound Premises

Abstract

Given sets $\Phi_1=\{\phi_{11},...,\phi_{1u(1)}\}, ...,\Phi_{z}=\{\phi_{z1},...,\phi_{zu(z)}\}$ of boolean formulas, a formula $\omega$ follows from the conjunction $\bigwedge\Phi_i= \bigwedge \phi_{ij}$ iff $\neg \omega\wedge \bigwedge_{i=1}^z \Phi_i$ is unsatisfiable. Now assume that, given integers $0\leq e_i < u(i)$, we must check if $\neg \omega\wedge \bigwedge_{i=1}^z \Phi'_i$ remains unsatisfiable, where $\Phi'_i\subseteq \Phi_i$ is obtained by deleting $\,\,e_{i}$ arbitrarily chosen formulas of $\Phi_i$, for each $i=1,...,z.$ Intuitively, does $\omega$ {\it stably} follow, after removing $e_i$ random formulas from each $\Phi_i$? We construct a quadratic reduction of this problem to the consequence problem in infinite-valued \luk\ logic \L$_\infty$. In this way we obtain a self-contained proof that the \L$_\infty$-consequence problem is coNP-complete.