On Minimal Unsatisfiability and Time-Space Trade-offs for k-DNF Resolution

In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordstrom 2009] introduced the concept of minimally unsatisfiable sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formulas can have at most O((mk)^(k+1)) variables. They also gave an example of such sets with Omega(mk^2) variables. In this paper we significantly improve the lower bound to Omega(m)^k, which almost matches the upper bound above. Furthermore, we show that this implies that the analysis of their technique for proving time-space separations and trade-offs for k-DNF resolution is almost tight. This means that although it is possible, or even plausible, that stronger results than in [Ben-Sasson and Nordstrom 2009] should hold, a fundamentally different approach would be needed to obtain such results.

💡 Research Summary

The paper addresses a fundamental question in proof complexity: how many distinct variables can appear in a minimally unsatisfiable set of k‑DNF formulas that contains m formulas? A set is minimally unsatisfiable if it is unsatisfiable, yet weakening any single term (i.e., removing a literal from a conjunctive term) makes the whole set satisfiable. For the case k = 1 (CNF), Tarsi’s classic lemma shows that at most m − 1 variables are needed, and this bound is tight. Ben‑Sasson and Nordström (2009) extended the notion to k‑DNF, proving an upper bound of O((mk)^{k+1}) variables and providing a construction with Ω(m·k²) variables. The large gap between these bounds left an open problem.

The authors close this gap almost completely by constructing, for any fixed k ≥ 2, minimally unsatisfiable k‑DNF sets with Θ(m^{k}) variables. Their construction proceeds in two stages. First, they design a “weight‑constraint” k‑DNF family W_m(𝑥) that enforces the condition |𝑥| ≤ 1 (the Hamming weight of a vector of auxiliary variables is at most one). This family uses O(m) formulas and introduces auxiliary variables z_j and w_j to encode a sequential “first true” condition across blocks of variables. Crucially, if any term in W_m is weakened, the system becomes satisfiable with a vector of weight at least two, establishing minimal unsatisfiability.

Second, they replicate this weight‑constraint gadget for k − 1 independent blocks, each indexed by a (k − 1)-tuple (i₁,…,i_{k‑1}) ranging over