A Linear Algebra Formulation for Boolean Satisfiability Testing

In the article \The State of SAT", the authors asked whether a procedure dramatically different from DPLL can be found for handling unsatisfiable instances. This study proposes a new linear programming approach to address this issue efficiently. Our experiments showed that the new method works for many unsatisfiable instances. However, we must concede that this method should be incomplete; otherwise, it will imply P=co-NP.

💡 Research Summary

The paper proposes a novel, non‑DPLL‑based framework for detecting unsatisfiable (UNSAT) instances of Boolean formulas by translating them into a linear‑algebraic setting. The authors first reduce any given 3‑CNF formula to a 1‑in‑3‑UNSAT instance, where each clause must have exactly one true literal. This reduction yields a set of linear equations of the form X_i + X_j + X_k = 1 together with Boolean constraints X_u ∈ {0,1}, which are enforced by adding quadratic constraints X_u = X_u².

The core of the method consists of three stages: Linear Transformation (LT), Quadratic Propagation (QP), and Linearization (ReL). In LT the original formula is expressed as the linear system described above. QP expands this system by multiplying equations pairwise (inner quadratic propagation) and by multiplying each equation with the quadratic constraints (constraint quadratic propagation), producing a larger set of quadratic equations that preserve the set of Boolean solutions (BoS). ReL then replaces every quadratic monomial X_i X_j with a fresh variable Z_{ij}, converting the quadratic system into a purely linear one.

The authors prove Theorem 2.04, establishing the equivalence of three statements: (1) the original linear system has a Boolean solution, (2) the quadratic system has a Boolean solution, and (3) the linearized system ReL(Z) has a Boolean solution. Proposition 2.05 provides a sufficient condition for unsatisfiability: if ReL(Z) is inconsistent over the reals ℝ or the integers ℤ, then the original formula is definitely UNSAT. This condition is not necessary, so the method may return “Unknown” on many satisfiable instances.

Algorithm 1 implements the pipeline. It first builds the LT system and checks its consistency over ℝⁿ and ℤⁿ. If inconsistent, it immediately reports UNSAT. If consistent and the solution is unique, it verifies whether the solution is Boolean, yielding a definitive SAT or UNSAT answer. When multiple solutions exist, the algorithm proceeds to QP and ReL, finally testing the consistency of ReL(Z). A consistent ReL(Z) leads to an “Unknown” result; an inconsistent one yields UNSAT.

Experimental evaluation uses randomly generated 3‑CNF formulas with varying numbers of variables (V) and clauses (C). The results indicate that the method efficiently identifies many UNSAT instances, especially when the clause‑to‑variable ratio exceeds roughly 0.9. However, it performs poorly on SAT instances, frequently returning “Unknown”. No comparison with state‑of‑the‑art CDCL solvers (e.g., MiniSat, Glucose) is provided, and the scalability concerns are evident: QP can generate O(m²) quadratic equations, and ReL introduces O(n²) new variables, leading to large integer linear programs whose feasibility checking is itself NP‑hard.

The paper’s contributions are the formal reduction from SAT/UNSAT to a Boolean‑solution‑preserving linear system, the introduction of quadratic propagation and linearization as a means to obtain sufficient UNSAT certificates, and an empirical demonstration of the approach’s ability to detect many UNSAT cases. Nonetheless, the work has several limitations. The exponential blow‑up in the number of constraints undermines practical applicability; the reliance on solving integer/real linear programs offers no theoretical advantage over existing methods; the completeness gap (many SAT cases remain undecided) limits its utility as a general SAT solver; and the lack of rigorous proofs (many arguments are hand‑waved) and insufficient positioning relative to prior algebraic proof systems (e.g., Polynomial Calculus, Cutting‑Planes, Sum‑of‑Squares) weaken its novelty claim.

Future research directions suggested include developing compression techniques to reduce the size of the quadratic system, tailoring the linearization to special formula classes (Horn, XOR, cardinality constraints), integrating the linear‑algebraic test as a preprocessing or pruning step within CDCL solvers, and establishing tighter theoretical connections with existing algebraic proof frameworks. With such enhancements, the linear‑algebraic perspective could become a valuable complementary tool in the SAT/UNSAT toolbox, but in its current form the proposed method remains more of a theoretical curiosity than a competitive alternative to mature SAT solving technologies.