Curtis Bright

Piyush Jha, Zhengyu Li, Zhengyang Lu et al.

This paper introduces AlphaMapleSAT, a novel Monte Carlo Tree Search (MCTS) based Cube-and-Conquer (CnC) SAT solving method aimed at efficiently solving challenging combinatorial problems. Despite the tremendous success of CnC solvers in solving a variety of hard combinatorial problems, the lookahead cubing techniques at the heart of CnC have not evolved much for many years. Part of the reason is the sheer difficulty of coming up with new cubing techniques that are both low-cost and effective in partitioning input formulas into sub-formulas, such that the overall runtime is minimized. Lookahead cubing techniques used by current state-of-the-art CnC solvers, such as March, keep their cubing costs low by constraining the search for the optimal splitting variables. By contrast, our key innovation is a deductively-driven MCTS-based lookahead cubing technique, that performs a deeper heuristic search to find effective cubes, while keeping the cubing cost low. We perform an extensive comparison of AlphaMapleSAT against the March CnC solver on challenging combinatorial problems such as the minimum Kochen-Specker and Ramsey problems. We also perform ablation studies to verify the efficacy of the MCTS heuristic search for the cubing problem. Results show up to 2.3x speedup in parallel (and up to 27x in sequential) elapsed real time.

Noah Rubin, Curtis Bright, Kevin K. H. Cheung et al.

In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking. Using state-of-the-art solvers as black boxes we were able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to ten. Moreover, we improve the effectiveness of the solvers by formulating an extended symmetry breaking method as well as an improvement to the straightforward CP encoding. We also analyze the effectiveness of using CP and IP solvers to search for triples of MOLS, compare our timings to those which have been previously published, and estimate the running time of using this approach to resolve the longstanding open problem of determining the existence of a triple of MOLS of order ten.

Curtis Bright, Kevin K. H. Cheung, Brett Stevens et al.

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry$\unicode{x2014}$the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches$\unicode{x2014}$highlighting the difficulty of relying on special-purpose search code for nonexistence results.

Curtis Bright, Ilias Kotsireas, Vijay Ganesh

Over the last few decades, many distinct lines of research aimed at automating mathematics have been developed, including computer algebra systems (CASs) for mathematical modelling, automated theorem provers for first-order logic, SAT/SMT solvers aimed at program verification, and higher-order proof assistants for checking mathematical proofs. More recently, some of these lines of research have started to converge in complementary ways. One success story is the combination of SAT solvers and CASs (SAT+CAS) aimed at resolving mathematical conjectures. Many conjectures in pure and applied mathematics are not amenable to traditional proof methods. Instead, they are best addressed via computational methods that involve very large combinatorial search spaces. SAT solvers are powerful methods to search through such large combinatorial spaces---consequently, many problems from a variety of mathematical domains have been reduced to SAT in an attempt to resolve them. However, solvers traditionally lack deep repositories of mathematical domain knowledge that can be crucial to pruning such large search spaces. By contrast, CASs are deep repositories of mathematical knowledge but lack efficient general search capabilities. By combining the search power of SAT with the deep mathematical knowledge in CASs we can solve many problems in mathematics that no other known methods seem capable of solving. We demonstrate the success of the SAT+CAS paradigm by highlighting many conjectures that have been disproven, verified, or partially verified using our tool MathCheck. These successes indicate that the paradigm is positioned to become a standard method for solving problems requiring both a significant amount of search and deep mathematical reasoning. For example, the SAT+CAS paradigm has recently been used by Heule, Kauers, and Seidl to find many new algorithms for $3\times3$ matrix multiplication.