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Clausal proofs, particularly those based on the deletion resolution asymmetric tautology (DRAT) proof system, are widely used by Boolean satisfiability solvers for expressing proofs of unsatisfiability. Their success stems from their simplicity and scalability. When solvers go beyond pure propositional reasoning, however, generating clausal proofs becomes more difficult. Solvers that employ pseudo-Boolean reasoning, including cutting-planes operations, can express proofs in the VeriPB proof system, but its adoption is not widespread. We introduce PBIP (Pseudo-Boolean Implication Proof), a framework that provides an intermediate representation between VeriPB and clausal proofs. We also introduce a toolchain comprising 1) a VeriPB-to-PBIP translator that performs proof trimming and optimization, and 2) a PBIP-to-LRAT translator that makes use of proof-generating operations on ordered binary decision diagrams (BDDs) to generate clausal proofs in LRAT format, a variant of the DRAT that allows efficient checking. We demonstrate the viability of our approach, the effectiveness of our trimming, and the performance of our clausal proof generator on a set of native PB benchmarks and compare our approach to direct checking of VeriPB proofs. 

Abstract Program verification languages such as Dafny and F$$ ^\star $$ ${}^{\star }$often rely heavily on Satisfiability Modulo Theories (SMT) solvers for proof automation. However, SMT-based verification suffers from instability, where semantically irrelevant changes in the source program can cause spurious proof failures. While existing mitigation techniques emphasize preemptive measures, we propose a complementary approach that focuses on diagnosing and repairing specific instances of instability-induced failures. Our key technique is a novel differential analysis to pinpoint problematic quantified formulas in an unstable query. We implement this technique in Cazamariposas, a tool that automatically identifies such quantified formulas and suggests fixes. We evaluate Cazamariposas on multiple large-scale systems verification projects written in three different program verification languages. Our results demonstrate Cazamariposas ’ effectiveness as an instability debugger. In the majority of cases, Cazamariposas successfully isolates the issue to a single problematic quantifier, while providing a stabilizing fix. 

Satisfiability (SAT) solvers have been using the same input format for decades: a formula in conjunctive normal form. Cardinality constraints appear frequently in problem descriptions: over 64% of the SAT Competition formulas contain at least one cardinality constraint, while over 17% contain many large cardinality constraints. Allowing general cardinality constraints as input would simplify encodings and enable the solver to handle constraints natively or to encode them using different (and possibly dynamically changing) clausal forms. We modify the modern SAT solver CaDiCaL to handle cardinality constraints natively. Unlike the stronger cardinality reasoning in pseudo-Boolean (PB) or other systems, our incremental approach with cardinality-based propagation requires only moderate changes to a SAT solver, preserves the ability to run important inprocessing techniques, and is easily combined with existing proof-producing and validation tools. Our experimental evaluation on SAT Competition formulas shows our solver configurations with cardinality support consistently outperform other SAT and PB solvers. 

A recent breakthrough in computer-assisted mathematics showed that every set of 30 points in the plane in general position (i.e., no three points on a common line) contains an empty convex hexagon. Heule and Scheucher solved this problem with a combination of geometric insights and automated reasoning techniques by constructing CNF formulas ϕ_n, with O(n⁴) clauses, such that if ϕ_n is unsatisfiable then every set of n points in general position must contain an empty convex hexagon. An unsatisfiability proof for n = 30 was then found with a SAT solver using 17 300 CPU hours of parallel computation. In this paper, we formalize and verify this result in the Lean theorem prover. Our formalization covers ideas in discrete computational geometry and SAT encoding techniques by introducing a framework that connects geometric objects to propositional assignments. We see this as a key step towards the formal verification of other SAT-based results in geometry, since the abstractions we use have been successfully applied to similar problems. Overall, we hope that our work sets a new standard for the verification of geometry problems relying on extensive computation, and that it increases the trust the mathematical community places in computer-assisted proofs. 

Quantum Computing (QC) is a new computational paradigm that promises significant speedup over classical computing in various domains. However, near-term QC faces numerous challenges, including limited qubit connectivity and noisy quantum operations. To address the qubit connectivity constraint, circuit mapping is required for executing quantum circuits on quantum computers. This process involves performing initial qubit placement and using the quantum SWAP operations to relocate non-adjacent qubits for nearest-neighbor interaction. Reducing the SWAP count in circuit mapping is essential for improving the success rate of quantum circuit execution as SWAPs are costly and error-prone. In this work, we introduce a novel circuit mapping method by combining incremental and parallel solving for Boolean Satisfiability (SAT). We present an innovative SAT encoding for circuit mapping problems, which significantly improves solver-based mapping methods and provides a smooth trade-off between compilation quality and compilation time. Through comprehensive benchmarking of 78 instances covering 3 quantum algorithms on 2 distinct quantum computer topologies, we demonstrate that our method is 26× faster than state-of-the-art solver-based methods, reducing the compilation time from hours to minutes for important quantum applications. Our method also surpasses the existing heuristics algorithm by 26% in SWAP count. 

Satisfiability solving has been used to tackle a range of long-standing open math problems in recent years. We add another success by solving a geometry problem that originated a century ago. In the 1930s, Esther Klein’s exploration of unavoidable shapes in planar point sets in general position showed that every set of five points includes four points in convex position. For a long time, it was open if an empty hexagon, i.e., six points in convex position without a point inside, can be avoided. In 2006, Gerken and Nicolás independently proved that the answer is no. We establish the exact bound: Every 30-point set in the plane in general position contains an empty hexagon. Our key contributions include an effective, compact encoding and a search-space partitioning strategy enabling linear-time speedups even when using thousands of cores. 

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. 

Satisfiability (SAT) solvers are versatile tools that can solve a wide array of problems, and the models and proofs of unsatisfiability emitted by SAT solvers can be checked by verified software. In this way, the SAT toolchain is trustworthy. However, many applications are not expressed natively in SAT and must instead be encoded into SAT. These encodings are often subtle, and implementations are error-prone. Formal correctness proofs are needed to ensure that implementations are bug-free. In this paper, we present a library for formally verifying SAT encodings, written using the Lean interactive theorem prover. Our library currently contains verified encodings for the parity, at-most-one, and at-most-k constraints. It also contains methods of generating fresh variable names and combining sub-encodings to form more complex ones, such as one for encoding a valid Sudoku board. The proofs in our library are general, and so this library serves as a basis for future encoding efforts