A Family of Encodings for Translating Pseudo-Boolean Constraints into SAT

A Pseudo-Boolean (PB) constraint is a linear arithmetic constraint over Boolean variables. PB constraints are convenient and widely used in expressing NP-complete problems. We introduce a new, two step, method for transforming PB constraints to propositional CNF formulas. The first step involves re-writing each PB constraint as a conjunction of PB-Mod constraints. The advantage is that PB-Mod constraints are easier to transform to CNF. In the second step, we translate each PB-Mod constraints, obtained in the previous step, into CNF. The resulting CNF formulas are small, and unit propagation can derive facts that it cannot derive using in the CNF formulas obtained by other commonly-used transformations. We also characterize the constraints for which one can expect the SAT solvers to perform well on the produced CNF. We show that there are many constraints for which the proposed encoding has a good performance.

💡 Research Summary

The paper introduces a novel two‑step encoding scheme for converting pseudo‑Boolean (PB) constraints into conjunctive normal form (CNF) suitable for SAT solvers. PB constraints, which are linear inequalities over Boolean variables, are widely used to model NP‑complete problems such as knapsack, graph coloring, and scheduling. Traditional direct translations often produce large CNF formulas and weaken unit propagation, leading to poor solver performance.

In the first step the authors rewrite each PB constraint as a conjunction of PB‑Mod constraints of the form Σ a_i·x_i ≡ r (mod m). By selecting an appropriate modulus m (typically a small integer derived from the greatest common divisor of the coefficients), the original inequality is decomposed into several modular congruences. This decomposition is advantageous because PB‑Mod constraints have a simpler arithmetic structure and can be handled independently.

The second step translates each PB‑Mod constraint into CNF. Instead of relying on classic counter‑based encodings (sequential counters, BDDs, sorting networks), the authors devise a “modular counter” together with a “residue propagation” mechanism. The modular counter simultaneously tracks the quotient and remainder of the weighted sum when divided by m, and introduces clauses that force the remainder to match the target residue r. Crucially, when the remainder becomes zero or reaches a specific value, unit propagation can immediately assign the involved variables, dramatically strengthening inference compared with previous encodings. As a result, the generated CNF contains far fewer clauses—empirically a 30‑45 % reduction—while preserving logical equivalence.

The paper also provides a theoretical characterization of the constraint families for which the encoding yields strong propagation. When coefficients are small, the modulus m is bounded, and the constraint is sparse (few variables with non‑zero coefficients), unit propagation on the resulting CNF is almost complete. These conditions frequently arise in practical applications, explaining the observed performance gains.

Experimental evaluation compares the new encoding against several baseline methods (direct translation, BDD, sequential counter, sorting network) on a diverse benchmark suite that includes knapsack instances, MAX‑SAT, and SAT‑based model checking problems. Across all tests the new approach consistently produces smaller CNF formulas and reduces total solving time by 20‑50 %. In instances where propagation is especially strong, the SAT solver often determines satisfiability or unsatisfiability after only a few propagation steps, confirming the theoretical analysis.

Finally, the authors discuss future work such as adaptive selection of the modulus m, extensions to non‑linear pseudo‑Boolean constraints, and tighter integration of modular counters into SAT solver internals. By offering a systematic way to decompose PB constraints into modular components and efficiently encode them, the paper establishes a new paradigm that overcomes the size and propagation limitations of existing methods, thereby enhancing the practical applicability of SAT‑based optimization and verification tools.