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1. Generating Extended Resolution Proofs with a BDD-Based SAT Solver

   https://doi.org/10.1145/3595295  

   Bryant, Randal E.; Heule, Marijn J. (October 2023, ACM Transactions on Computational Logic)

   In 2006, Biere, Jussila, and Sinz made the key observation that the underlying logic behind algorithms for constructing Reduced, Ordered Binary Decision Diagrams (BDDs) can be encoded as steps in a proof in the extended resolution logical framework. Through this, a BDD-based Boolean satisfiability (SAT) solver can generate a checkable proof of unsatisfiability. Such a proof indicates that the formula is truly unsatisfiable without requiring the user to trust the BDD package or the SAT solver built on top of it. We extend their work to enable arbitrary existential quantification of the formula variables, a critical capability for BDD-based SAT solvers. We demonstrate the utility of this approach by applying a BDD-based solver, implemented by extending an existing BDD package, to several challenging Boolean satisfiability problems. Our results demonstrate scaling for parity formulas as well as the Urquhart, mutilated chessboard, and pigeonhole problems far beyond that of other proof-generating SAT solvers. 

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2. Clausal Proofs for Pseudo-Boolean Reasoning

   https://doi.org/10.1007/978-3-030-99524-9_25  

   Bryant, Randal E.; Heule, Marijn J. (January 2022, International Conference on Tools and Algorithms for the Construction and Analysis of Systems)

   Fisman, D.; Rosu, G. (Ed.)

   When augmented with a Pseudo-Boolean (PB) solver, a Boolean satisfiability (SAT) solver can apply apply powerful reasoning methods to determine when a set of parity or cardinality constraints, extracted from the clauses of the input formula, has no solution. By converting the intermediate constraints generated by the PB solver into ordered binary decision diagrams (BDDs), a proof-generating, BDD-based SAT solver can then produce a clausal proof that the input formula is unsatisfiable. Working together, the two solvers can generate proofs of unsatisfiability for problems that are intractable for other proof-generating SAT solvers. The PB solver can, at times, detect that the proof can exploit modular arithmetic to give smaller BDD representations and therefore shorter proofs. 

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3. Generating Extended Resolution Proofs with a BDD-Based SAT Solver

   https://doi.org/10.1007/978-3-030-72016-2_5  

   Bryant, Randal E.; Heule, Marijn J. (January 2021, Tools and Algorithms for the Construction and Analysis of Systems)

   Groote, J. F.; Larsen, K. G. (Ed.)

   In 2006, Biere, Jussila, and Sinz made the key observation that the underlying logic behind algorithms for constructing Reduced, Ordered Binary Decision Diagrams (BDDs) can be encoded as steps in a proof in the extended resolution logical framework. Through this, a BDD-based Boolean satisfiability (SAT) solver can generate a checkable proof of unsatisfiability. Such proofs indicate that the formula is truly unsatisfiable without requiring the user to trust the BDD package or the SAT solver built on top of it. We extend their work to enable arbitrary existential quantification of the formula variables, a critical capability for BDD-based SAT solvers. We demonstrate the utility of this approach by applying a prototype solver to obtain polynomially sized proofs on benchmarks for the mutilated chessboard and pigeonhole problems—ones that are very challenging for search-based SAT solvers. 

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4. Dual Proof Generation for Quantified Boolean Formulas with a BDD-based Solver

   https://doi.org/10.1007/978-3-030-79876-5_25  

   Bryant, R. E.; Heule, M. J. (January 2021, Conference on Automated Deduction)

   Platzer, A.; Sutcliffe, G. (Ed.)

   Existing proof-generating quantified Boolean formula (QBF) solvers must construct a different type of proof depending on whether the formula is false (refutation) or true (satisfaction). We show that a QBF solver based on ordered binary decision diagrams (BDDs) can emit a single dual proof as it operates, supporting either outcome. This form consists of a sequence of equivalence-preserving clause addition and deletion steps in an extended resolution framework. For a false formula, the proof terminates with the empty clause, indicating conflict. For a true one, it terminates with all clauses deleted, indicating tautology. Both the length of the proof and the time required to check it are proportional to the total number of BDD operations performed. We evaluate our solver using a scalable benchmark based on a two-player tiling game. 

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5. Certified Knowledge Compilation with Application to Formally Verified Model Counting

   https://doi.org/10.1613/jair.1.15958  

   Bryant, Randal E; Nawrocki, Wojciech; Avigad, Jeremy; Heule, Marijn_J H (January 2025, Journal of Artificial Intelligence Research)

   Computing many useful properties of Boolean formulas, such as their weighted or unweighted model count, is intractable on general representations. It can become tractable when formulas are expressed in a special form, such as the decision decomposable negation normal form (decision-DNNF). Knowledge compilation is the process of converting a formula into such a form. Unfortunately existing knowledge compilers provide no guarantee that their output correctly represents the original formula, and therefore they cannot validate a model count, or any other computed value.We present Partitioned-Operation Graphs (POGs), a form that can encode all of the representations used by existing knowledge compilers. We have designed CPOG, a framework that can express proofs of equivalence between a POG and a Boolean formula in conjunctive normal form (CNF).We have developed a program that generates POG representations from decision-DNNF graphs produced by the state-of-the-art knowledge compiler D4, as well as checkable CPOG proofs certifying that the output POGs are equivalent to the input CNF formulas. Our toolchain for generating and verifying POGs scales to all but the largest graphs produced by D4 for formulas from a recent model counting competition. Additionally, we have developed a formally verified CPOG checker and model counter for POGs in the Lean 4 proof assistant. In doing so, we proved the soundness of our proof framework. These programs comprise the first formally verified toolchain for weighted and unweighted model counting. 

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