Divide and Conquer: A Faster Approach to CircuitSAT

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Author: Denis Avetisyan

A new parallel algorithm efficiently decomposes complex CircuitSAT problems into manageable parts, accelerating solutions for critical applications like logical equivalence checking and cryptographic analysis.

This paper details a parallel algorithm for decomposing hard CircuitSAT instances, utilizing adaptive interval partitioning and a time-limited SAT solver to improve performance.

Despite advances in SAT solving, decomposing exceptionally hard CircuitSAT instances remains a significant challenge. This paper, ‘Efficient Parallel Algorithm for Decomposing Hard CircuitSAT Instances’, introduces a novel parallel algorithm that adaptively partitions such instances into weaker, more manageable subproblems. By leveraging parallel hardness estimations and a time-limited SAT solver, our approach efficiently identifies high-quality decompositions, demonstrating improved performance on challenging problems like Logical Equivalence Checking and cryptographic preimage attacks. Could this technique unlock further scalability for solving increasingly complex problems in formal verification and cryptography?

The CircuitSAT Imperative: A Foundation for Reliable Computation

The CircuitSAT problem, which centers on verifying whether a complex network of logic gates – a Boolean circuit – can achieve a specific output, is fundamentally important to both formal verification and modern hardware design. Essentially, it asks if there exists a set of inputs that will make the circuit ‘true’. This isn’t merely an abstract mathematical exercise; ensuring the correctness of digital circuits – from microprocessors to embedded systems – relies heavily on efficiently solving CircuitSAT instances. A failure to do so can result in flawed designs, leading to system malfunctions, security vulnerabilities, and costly redesigns. Consequently, advancements in tackling this problem directly translate into more reliable and secure technology, impacting a vast range of applications.

Despite decades of refinement, conventional SAT solvers encounter significant limitations when applied to contemporary circuit designs. These solvers, which systematically explore possible truth assignments to determine if a circuit’s output can satisfy given conditions, face an exponential growth in computational demands as circuit size and complexity increase. While effective for smaller problems, the sheer number of logic gates and interconnections in modern circuits quickly overwhelms these algorithms, leading to computation times that render formal verification impractical. This scalability issue isn’t simply a matter of needing more powerful computers; it represents a fundamental challenge in how these solvers explore the vast search space inherent in complex Boolean formulas, hindering progress in crucial areas like chip design and software verification.

The escalating complexity of modern Boolean circuits presents a significant computational challenge for formal verification and hardware design, driving innovation in solver technology. Current serial approaches to the CircuitSAT problem increasingly falter as circuit sizes grow, necessitating a shift towards parallel algorithms capable of distributing the computational load across multiple processors. Researchers are actively exploring techniques such as circuit partitioning – dividing a large circuit into smaller, more manageable sub-circuits – coupled with sophisticated parallel search strategies. These efforts aim to not only accelerate the solving process but also to overcome the inherent limitations of traditional solvers, enabling verification of increasingly intricate digital designs and ultimately reducing time-to-market for new hardware.

Deconstructing Complexity: The Adaptive Parallel Algorithm

The Adaptive Parallel Algorithm tackles the CircuitSAT problem through a process of intelligent partitioning, dividing the overall circuit into numerous smaller, logically independent subproblems. This decomposition is not arbitrary; the algorithm analyzes the circuit’s structure to identify sections that can be evaluated without dependency on others, maximizing the potential for concurrent processing. By breaking down the complex CircuitSAT instance into these independent units, the algorithm facilitates parallel execution, allowing each subproblem to be assigned to a separate processing core or thread. This approach reduces the overall solution time by distributing the computational workload and eliminating sequential bottlenecks inherent in solving the entire circuit as a single unit.

Interval Partitioning, employed within the Adaptive Parallel Algorithm, decomposes a Boolean circuit into independent subproblems by identifying intervals of logic gates with minimal interdependencies. This is achieved by topologically sorting the gates and assigning each gate to an interval based on its position and the fan-in/fan-out relationships with other gates. The algorithm aims to maximize the number of gates within each interval while maintaining independence, thus creating parallelizable subproblems. Specifically, a gate can be assigned to an interval if all its inputs and outputs are contained within that same interval, or if the interval boundary contains a complete fan-in or fan-out cut. The size and number of intervals are dynamically determined by the circuit’s structure, allowing the algorithm to adapt to varying circuit complexities and maximize parallel execution.

Parallel execution of decomposed subproblems is central to the Adaptive Parallel Algorithm’s efficiency. Modern multi-core processors provide multiple computational units capable of simultaneously executing independent tasks. By distributing the independent subproblems generated by Interval Partitioning across these cores, the algorithm achieves a substantial reduction in overall computation time. The speedup is not necessarily linear due to overhead associated with task distribution and synchronization, but the algorithm is designed to minimize these costs and maximize parallel throughput. The degree of achievable speedup is directly related to the number of available cores and the granularity of the subproblems; finer-grained decomposition allows for greater parallelism but introduces increased overhead.

Empirical Validation: Benchmarking Performance

The algorithm was implemented utilizing the Kissat SAT solver, a contemporary, high-performance solver recognized for its efficiency and scalability. Experimental validation was conducted on a computing cluster provisioned with Intel Xeon E5-2695 v4 processors, providing a standardized and repeatable testing environment. This hardware configuration facilitated rigorous performance analysis and allowed for benchmarking against existing methodologies under controlled conditions. The use of Kissat and a dedicated cluster ensured both the accuracy and reliability of the obtained results.

Rigorous testing of the algorithm utilized a diverse set of benchmark circuits, specifically focusing on Large Encoding Challenge (LEC) instances. Results demonstrate the algorithm’s capability to solve LEC instances previously considered intractable for sequential solvers, meaning those unsolvable within a 24-hour timeframe on a single core processor. This indicates a significant performance improvement over existing methods when applied to this class of problem, offering solutions where prior approaches failed to achieve completion within practical time constraints.

The algorithm demonstrated the capability to solve Logic Equivalence Checking (LEC) instances that previously exceeded a 24-hour time limit when processed on a single core processor. This indicates a significant improvement in performance for computationally intensive LEC problems. These instances, intractable for sequential solvers within the specified timeframe, were successfully resolved using the implemented approach and the Kissat SAT solver on a cluster equipped with Intel Xeon E5-2695 v4 processors. This functionality broadens the scope of solvable LEC problems, enabling analysis of more complex digital circuits.

Regarding the MD4-43 preimage attack, our algorithm required 264,039 seconds to reach a solution. This performance surpasses that of previously published methods for this specific attack vector. Critically, this solution time is equivalent to the performance achieved by the Cube-and-Conquer approach, representing a significant result given the established efficiency of that algorithm in this domain.

Testing on the MD4 hash function demonstrated a performance advantage for the implemented algorithm. Specifically, the MD4-40 preimage attack was successfully solved, while the Cube-and-Conquer method failed to construct solution cubes within a 100,000-second time limit. This indicates a capacity to efficiently address cryptographic challenges where other methods encounter computational bottlenecks, demonstrating improved performance for this particular hash length.

Testing on MD4 hash preimage attacks demonstrated the algorithm’s ability to solve MD4-41 and MD4-42 within a reasonable timeframe, whereas the Cube-and-Conquer method failed to find solutions for these instances after exceeding 300,000 seconds of computation. This performance difference highlights the efficiency of the implemented approach when tackling these specific cryptographic challenges, indicating a significant improvement over existing methods for these problem sizes.

Synergistic Optimization: The Power of Cube-and-Conquer

The synergistic combination of Cube-and-Conquer with a parallel algorithm significantly boosts problem-solving efficiency through a complementary approach. While parallel algorithms excel at broadly exploring solution spaces, they can be hampered by the intricacies of complex problems. Cube-and-Conquer addresses this by strategically identifying and isolating critical regions within the problem instance – effectively reducing its complexity before applying the parallel solver. This pre-processing step allows the parallel algorithm to focus its resources on the most impactful areas, drastically reducing computation time and enabling solutions for instances previously considered intractable. The result is not simply an additive improvement, but a multiplicative effect where each technique amplifies the strengths of the other, pushing the boundaries of what’s solvable.

The integration of Cube-and-Conquer centers on a strategic decomposition of the complex circuit into manageable regions. This approach doesn’t attempt a blanket solution across the entire problem, but instead employs a method to pinpoint areas-or ‘cubes’-that are most influential in determining the circuit’s overall behavior. By isolating these critical regions, the solving process becomes dramatically more targeted and efficient. Rather than exhaustively searching all possible configurations, computational resources are concentrated on these key areas, accelerating the path to a solution and enabling the handling of significantly larger and more complex CircuitSAT instances. This focused approach represents a substantial improvement over traditional methods, offering a more intelligent and streamlined path to circuit verification and optimization.

The efficiency of the Cube-and-Conquer method hinges on the rapid generation of suitable cubes, and the March_cu111 tool directly addresses this need. This specialized software automates the process of identifying and creating these crucial cubes, significantly reducing the computational burden previously required for manual or less sophisticated cube generation techniques. By streamlining this initial step, March_cu111 allows researchers and engineers to focus on the core logic of the CircuitSAT problem, accelerating the overall solving process and enabling exploration of more complex circuit designs. The tool’s automated approach not only enhances speed but also minimizes the potential for human error, contributing to more reliable and consistent results in circuit verification and optimization.

The combined application of parallel algorithms and the Cube-and-Conquer strategy represents a significant advancement in tackling complex CircuitSAT instances. This synergy arises because each method compensates for the other’s limitations; parallel algorithms rapidly explore the solution space, while Cube-and-Conquer intelligently focuses computational resources on the most critical circuit regions. By first identifying and isolating these crucial areas – often representing the core challenges within the circuit – the Cube-and-Conquer technique dramatically reduces the scope of the problem requiring intensive parallel processing. This focused approach not only accelerates the solving process but also improves the overall efficiency, allowing for the resolution of instances previously considered intractable. The resulting performance gains demonstrate that the integrated methodology transcends the sum of its parts, providing a robust and highly effective solution for complex circuit validation and analysis.

The pursuit of efficient solutions to complex problems, as demonstrated by this work on CircuitSAT decomposition, echoes a fundamental tenet of mathematical thought. The algorithm’s adaptive partitioning and parallel processing strive for a logically complete breakdown of intractable instances, mirroring the desire for provable correctness. As Carl Friedrich Gauss once stated, “If others would think as hard as I do, they would not have so little to think about.” This sentiment aptly describes the rigorous approach taken in dissecting these hard SAT problems; the method isn’t simply about finding a solution, but about establishing logical equivalence through a demonstrably correct and scalable process, akin to seeking mathematical certainty.

Further Directions

The demonstrated efficacy of adaptive partitioning, while promising, merely shifts the fundamental bottleneck. The core limitation remains the inherent exponential complexity of SAT solving itself. While parallelization offers a constant-factor improvement, it does not address the asymptotic behavior. Future work must, therefore, prioritize algorithms that demonstrably reduce the search space – not simply explore it faster. The reliance on a time-limited solver, though pragmatic, introduces an element of statistical uncertainty. A truly elegant solution would guarantee a result – or definitively prove unsatisfiability – within a bounded computational framework.

Furthermore, the observed performance gains are currently tied to instances arising from Logical Equivalence Checking and cryptographic attacks. Generalizing this approach to arbitrary SAT problems necessitates a deeper understanding of instance structure and the development of partitioning heuristics that transcend domain-specific characteristics. Any heuristic, however, introduces a potential abstraction leak – a compromise of mathematical purity for the sake of practical expediency.

Ultimately, the field requires a re-evaluation of its fundamental assumptions. The current paradigm favors increasingly complex solvers operating on increasingly large instances. A more fruitful path might lie in the development of provably correct, minimal algorithms capable of tackling inherently difficult problems with mathematical rigor – even if it means accepting a degree of computational cost. The pursuit of elegance, after all, is rarely cheap.

Original article: https://arxiv.org/pdf/2602.17130.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-20 15:43