Drawing Sound Conclusions from Unsound Premises

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.

💡 Research Summary

The paper introduces the “Stable Consequence” problem, a natural generalisation of the classical propositional entailment question. Given finite families of Boolean formulas Φ₁,…,Φ_k and integers e₁,…,e_k with 0 ≤ e_i < |Φ_i|, one may delete arbitrarily e_i formulas from each Φ_i. The task is to decide whether, for every possible choice of deletions, the remaining conjunction is unsatisfiable – i.e., whether the target formula ω (or, in the special case of the abstract, ¬ω) still follows “stably” despite the removal of up to e_i premises from each group. This problem subsumes the decision version of MAX‑SAT and captures reasoning under unreliable premises.

To analyse its computational complexity, the authors construct a polynomial‑time reduction to the consequence problem in infinite‑valued Łukasiewicz logic (L∞). The core of the reduction is a syntactic transformation, denoted by the “‡‑operator”, that maps any Boolean formula ψ into an L∞‑formula ψ‡. First ψ is put into negation normal form; then each occurrence of a variable X is replaced by the L∞‑term ¬X ∨ (X ⊕ X), while each occurrence of ¬X is replaced by X ∨ ¬(X ⊙ X). The resulting ψ‡ evaluates to 1 exactly when the original Boolean valuation assigns X the truth value 1, and evaluates to the special value e/(e+1) when the Boolean valuation assigns 0, where e ≥ 2 is a parameter that will later be chosen as max(2, e₁,…,e_k). This correspondence is proved in Proposition 4.3.

Using this translation, each set Φ_i is turned into a single L∞‑conjunction of its transformed formulas. Additional L∞ constraints of the form (X_{e_t} ↔ ¬X_t) ∨ (X_t ↔ ¬e_t·X_t) (for t = 1,…,k) force every variable to take one of the two admissible real values 1/(e+1) or e/(e+1). The reduction then builds two L∞‑formulas:

θ = ∧_{t=1}^k