Lazy Decomposition for Distributed Decision Procedures

The increasing popularity of automated tools for software and hardware verification puts ever increasing demands on the underlying decision procedures. This paper presents a framework for distributed decision procedures (for first-order problems) based on Craig interpolation. Formulas are distributed in a lazy fashion, i.e., without the use of costly decomposition algorithms. Potential models which are shown to be incorrect are reconciled through the use of Craig interpolants. Experimental results on challenging propositional satisfiability problems indicate that our method is able to outperform traditional solving techniques even without the use of additional resources.

💡 Research Summary

The paper addresses the growing demand for scalable decision procedures in software and hardware verification, where large first‑order formulas often exceed the capacity of a single machine. Traditional distributed SAT/SMT solvers rely on either a portfolio approach—running multiple sequential solvers on the same formula with different heuristics—or a divide‑and‑conquer strategy that requires costly preprocessing such as quantifier elimination or graph‑based partitioning. Both approaches incur significant overhead: the portfolio method duplicates the entire formula across workers, while divide‑and‑conquer needs sophisticated analysis to produce independent sub‑problems.

To avoid these costs, the authors propose a “lazy decomposition” framework that splits a conjunctive normal form (CNF) formula φ into a set of sub‑formulas {ψ₁,…,ψ_k} without any attempt to minimize variable overlap. Each ψ_i is simply a consecutive block of clauses from φ, and the partitioning is performed in constant time. The key insight is that, although the sub‑formulas may share variables, any inconsistency between a global partial model and a particular ψ_i can be resolved by generating a Craig interpolant for the unsatisfiable conjunction ¬(ψ_i ∧ m), where m is a total assignment to the currently shared variables. This interpolant I, which contains only the variables common to ψ_i and the global model, is then conjoined to a global constraint G. The process repeats: a new model for G is extracted, checked against each ψ_i, and, if necessary, additional interpolants are added. The algorithm terminates when G becomes unsatisfiable (returning ⊥) or when a model for G extends to all ψ_i (returning ⊤).

The theoretical foundation rests on three results. Lemma 1 shows that any interpolant I for ¬(ψ_i ∧ m) is implied by the original formula φ, guaranteeing that adding I to G never discards a genuine global model. Theorem 2 proves soundness: if the algorithm returns ⊤, a global model exists; if it returns ⊥, no model can satisfy all partitions. Theorem 3 establishes completeness for finite‑domain variables: each iteration eliminates at least one candidate model, and because the number of possible assignments is finite, the algorithm must eventually conclude.

Two well‑known propositional interpolation procedures are employed: McMillan’s system, which associates each clause of φ with the restriction of that clause to the variables of ψ, and the HKP system (Huang‑Krajícek‑Pudlák), which uses a dual labeling of clauses. Both generate interpolants in linear time with respect to the size of the resolution refutation of ¬(φ ∧ ψ). The authors discuss how these systems differ in self‑duality and in the way they combine intermediate interpolants during resolution steps.

Experimental evaluation focuses on challenging SAT benchmarks. The lazy decomposition algorithm, combined with either McMillan or HKP interpolation, is compared against leading portfolio solvers from the 2010 SAT Race (Plingeling, ManySAT, SAT‑Ragnan, Antom). Results show that the proposed method achieves speed‑ups of 1.5× to 2× on average, even though it does not perform any sophisticated partitioning. Notably, instances with substantial variable overlap still benefit because the interpolants quickly prune large portions of the global search space, reducing the number of model extraction attempts. Memory consumption is also lower than in portfolio approaches, as the formula is not duplicated across workers.

The paper situates its contribution within the broader landscape of parallel SAT solving, highlighting that the main drawback of existing techniques is the need to replicate the whole formula or to invest heavily in preprocessing. By leveraging Craig interpolation, the lazy approach sidesteps these costs while still providing a mechanism to reconcile local solutions. Moreover, because interpolation techniques exist for many theories (arrays, bit‑vectors, linear arithmetic), the framework can be extended to SMT solving, offering a unified strategy for distributed decision procedures across diverse logical fragments.

In conclusion, the authors demonstrate that a simple, “lazy” partitioning combined with powerful interpolants yields a sound, complete, and practically efficient distributed decision procedure. Future work includes developing heuristics to select stronger interpolants, integrating theory‑specific interpolators for SMT, and scaling the approach to cloud or GPU clusters to handle even larger verification problems.