Drawing Sound Conclusions from Unsound Premises

TL;DR

This paper investigates the stability of logical consequence under the removal of formulas from sets of boolean formulas, reducing the problem to the consequence problem in infinite-valued extL_ logic, and proving its coNP-completeness.

Contribution

It introduces a quadratic reduction from a stability problem in propositional logic to the consequence problem in infinite-valued extL_ logic, establishing its computational complexity.

Findings

The consequence problem in infinite-valued extL_ logic is coNP-complete.

The stability of logical consequence under formula removal can be reduced to extL_ consequence.

The paper provides a self-contained proof of the complexity result.

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.