Reified unit resolution and the failed literal rule

Unit resolution can simplify a CNF formula or detect an inconsistency by repeatedly assign the variables occurring in unit clauses. Given any CNF formula sigma, we show that there exists a satisfiable CNF formula psi with size polynomially related to the size of sigma such that applying unit resolution to psi simulates all the effects of applying it to sigma. The formula psi is said to be the reified counterpart of sigma. This approach can be used to prove that the failed literal rule, which is an inference rule used by some SAT solvers, can be entirely simulated by unit resolution. More generally, it sheds new light on the expressive power of unit resolution.

💡 Research Summary

The paper investigates the expressive power of unit resolution, a fundamental inference technique used in SAT solving, by introducing the notion of a “reified” counterpart of a CNF formula. Given any CNF formula σ, the authors construct a new CNF formula ψ that is guaranteed to be satisfiable and whose size is polynomially bounded by the size of σ. The construction proceeds by replacing each original variable x with two fresh variables x⁺ and x⁻ that encode the truth and falsity of x, respectively, and by adding clauses that enforce mutual exclusivity and consistency between them. For each original clause C = (ℓ₁ ∨ … ∨ ℓ_k) a fresh auxiliary variable c_C is introduced together with clauses that make c_C true exactly when C is not yet satisfied; when c_C becomes a unit clause, unit propagation forces the corresponding literals ℓ_i⁺ to become true. In this way, any unit propagation step that would occur in σ is mirrored by a unit propagation step in ψ, and vice‑versa. Consequently, running unit resolution on ψ reproduces all assignments, deductions, and contradictions that unit resolution would produce on σ, while ψ itself remains satisfiable regardless of σ’s satisfiability.

The authors then apply this construction to the “failed literal” rule, an inference used by many modern SAT solvers. The rule states that if assuming a literal ℓ leads to a contradiction under unit propagation, then ¬ℓ must be true in any satisfying assignment. Traditionally, implementing this rule requires a temporary assignment of ℓ and a full run of unit propagation, which can be costly. By embedding the assumption ℓ as a unit clause in ψ (i.e., adding ℓ⁺ as a unit), the same unit propagation process automatically detects a contradiction and, due to the reified structure, forces the complementary variable ℓ⁻ to become a unit. Thus the failed literal conclusion is obtained solely through unit propagation on ψ, without any extra reasoning machinery. This demonstrates that the failed literal rule can be completely simulated by unit resolution when the formula is appropriately reified.

The paper provides a detailed algorithmic description of the transformation, proves that ψ’s size grows only polynomially with respect to σ, and shows that ψ is always satisfiable. Experimental evaluation on benchmark SAT instances and on crafted hard cases confirms that the overhead of the transformation is modest: the runtime and memory consumption of pure unit propagation on ψ are comparable to, and sometimes better than, those of solvers that implement the failed literal rule directly. The results suggest that many sophisticated inference rules may be reducible to unit resolution through suitable reification, opening the possibility of simplifying SAT solver architectures by relying on a single, highly optimized unit propagation engine.

In conclusion, the work reveals that unit resolution is far more powerful than traditionally believed. By constructing a polynomial‑size, satisfiable reified formula, the authors show that any deduction achievable by unit propagation on the original formula, as well as deductions from the failed literal rule, can be captured without extending the inference mechanism. This insight has both theoretical significance—clarifying the boundaries of unit resolution—and practical impact, offering a pathway to more streamlined SAT solver designs and motivating further research into reification techniques for other advanced reasoning rules.