仅用23次乘法实现3x3矩阵乘法的新通用方法
A New General-Purpose Method to Multiply 3x3 Matrices Using Only 23 Multiplications
AI总结：
通过将Brent方程转化为SAT问题并利用约500个SAT求解器，找到了Laderman问题的新解，证明解空间比预期更大，为22次乘法的存在提供了可能性。
摘要：
计算机代数中最著名的猜想之一是矩阵乘法可能可以在接近二次时间内实现。已知的最佳指数是2.376，归功于Coppersmith和Winograd。文献中许多解决该问题的尝试通过解决固定大小的问题，然后递归应用解决方案。这导致了具有固定大小的纯组合优化问题。这些问题不太可能在多项式时间内可解。1976年，Laderman发表了一种仅用23次乘法实现两个3x3矩阵相乘的方法。该结果是非交换的，因此可以递归应用于更小的子矩阵。35年来，没有人能够做得更好，并且是否可以用22次乘法实现仍然是一个开放问题。我们通过求解所谓的Brent方程[7]来推进。我们实现了一种方法，将这个非常困难的问题转化为SAT问题，并尝试用我们约500个SAT求解器的组合来解决。通过这种新方法，我们能够为Laderman问题生成新的解。我们提出了一个具有23次乘法的全新通用非交换解，并证明该解是新的，并且不是Laderman原始解的等价变体。这一结果表明Laderman问题的解空间比预期更大，因此现在更有可能存在22次乘法的解。如果存在，我们可能很快就能通过更长时间运行我们的算法，或者由于SAT求解器算法的进一步改进而找到它。
英文摘要：
One of the most famous conjectures in computer algebra is that matrix multiplication might be feasible in not much more than quadratic time. The best known exponent is 2.376, due to Coppersmith and Winograd. Many attempts to solve this problems in the literature work by solving, fixed-size problems and then apply the solution recursively. This leads to pure combinatorial optimisation problems with fixed size. These problems are unlikely to be solvable in polynomial time. In 1976 Laderman published a method to multiply two 3x3 matrices using only 23 multiplications. This result is non-commutative, and therefore can be applied recursively to smaller sub-matrices. In 35 years nobody was able to do better and it remains an open problem if this can be done with 22 multiplications. We proceed by solving the so called Brent equations [7]. We have implemented a method to converting this very hard problem to a SAT problem, and we have attempted to solve it, with our portfolio of some 500 SAT solvers. With this new method we were able to produce new solutions to the Laderman's problem. We present a new fully general non-commutative solution with 23 multiplications and show that this solution is new and is NOT an equivalent variant of the Laderman's original solution. This result demonstrates that the space of solutions to Laderman's problem is larger than expected, and therefore it becomes now more plausible that a solution with 22 multiplications exists. If it exists, we might be able to find it soon just by running our algorithms longer, or due to further improvements in the SAT solver algorithms.
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