Exploiting Dynamically Propositional Logic Structures in SAT

The 32-bit hwb (hwb-n32 for short) problem is from equivalence checking that arises in combining two circuits computing the hidden weighted bit function. Since 2002, it remains still unsolvable in every SAT competition. This paper focuses on solving problems such as hwb-n32. Generally speaking, modern solvers can detect only XOR, AND, OR and ITE gates. Other non-clausal formulas (propositional logic structures) cannot be detected. To solve the hwb-n32 problem, we extract dynamically some special propositional logic structures, and then use a variant of DPLL-based solvers to solve the subproblem simplified by the extracted structure information. Using the dynamic extraction technique, we solved efficiently the hwb-n32 problem, even some of which were solved within 3000 seconds.

💡 Research Summary

The paper tackles the long‑standing challenge of solving the 32‑bit hidden weighted bit (hwb‑n32) SAT instances, which have remained unsolved in every SAT competition since 2002. The authors observe that modern DPLL/CDCL solvers dynamically recognize only a limited set of gate‑like structures (XOR, AND, OR, ITE). However, many hard instances, especially those originating from equivalence checking of circuits implementing the hidden weighted bit function, contain richer propositional structures that are not captured by these primitives.

To address this gap, the authors define seven logical inference rules (denoted (1)–(7)) that capture specific relationships among variables appearing in the CNF encoding of the hwb problem. Each rule has the form α ⊢ β, where α is a set of clauses (often including easily detectable XOR constraints) and β is a non‑CNF logical consequence (a combination of AND, OR, XOR, etc.). Although β is not present in the original CNF, it becomes a valid consequence whenever the premises α are satisfied during search.

The solving methodology proceeds in three stages:

1. Problem Splitting and Look‑ahead – The original formula F is recursively split into sub‑problems using a simple look‑ahead heuristic. DecisionVar1 extracts variables that appear in the premise of rule (1); DecisionVar2 computes a heuristic score H(x) = occCNF(x)·occCNF(¬x) for each candidate variable, where occCNF counts occurrences outside XOR constraints.

2. Dynamic Extraction of Logical Consequences – When the search reaches a decision level greater than the number of extracted premise variables (|P|+4), the current sub‑problem is simplified (equivalent literal detection and backward subsumption). Then the algorithm scans the simplified formula for any of the seven premises. Whenever a premise is found, the corresponding β is generated and encoded into CNF using standard Tseitin‑style transformations. The set of all such β’s is denoted G.

3. CDCL Solving of Enriched Sub‑problem – The enriched formula F ∪ CNF(G) is handed to a CDCL engine. The authors choose MXC 0.99 as the underlying solver because it has shown the best performance on hwb instances in the SAT‑2009 competition. MXC then performs conflict‑driven learning, unit propagation, and restarts on the enriched problem.

The implementation, named LogicSAT, is essentially MXC augmented with the dynamic extraction module. The authors also experimented with building the same module on top of PrecoSAT, but observed significantly slower runtimes (e.g., solving hwb‑n32‑02 took 11 205 s versus < 7 300 s with MXC).

Experimental Evaluation – Benchmarks include several families of hwb instances (hwb‑n2, hwb‑n3, hwb‑n32), as well as unrelated SAT/UNSAT instances from the SAT‑2002 competition (lisa21_99, lisa19_99, pb‑sat‑40, pb‑unsat‑40). Each solver was given a 10 000‑second timeout per instance on a 2.40 GHz Intel Core 2 Quad Q6600 with 2 GB RAM running Mint Linux. Results (Table 1) show:

• LogicSAT consistently outperforms MXC on all hwb‑n2 and hwb‑n3 instances, often by a factor of 2–5.
• Both PrecoSAT and MXC fail to solve any hwb‑n32 instance within the timeout, whereas LogicSAT solves all three, with runtimes ranging from ~2 800 s to ~7 300 s.
• On some non‑hwb benchmarks (e.g., lisa19_99, pb‑sat), PrecoSAT remains faster, reflecting that the specialized logical rules are not beneficial there.
• Approximately 90 % of generated sub‑problems are simplified by applying the logical rules, and the CDCL engine is typically invoked at decision levels 8–12.

Conclusions and Future Work – The authors demonstrate that dynamically extracting propositional logic structures beyond the traditional gate primitives can dramatically reduce the search space for certain hard SAT problems. While the current rule set is limited to the hwb domain, the approach suggests a promising direction: extending the rule library, automating rule discovery, and integrating dynamic extraction with other CDCL solvers. The paper positions this hybrid static‑dynamic structural reasoning as a viable path toward solving SAT instances that have resisted all previous competition‑level solvers.